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This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. Für die Wellengleichung
\[
\sum ^{2n+1}_{\nu = 1} \frac {\partial ^2u}{\partial x^2_{\nu }} = \frac {1}{c^2} \frac {\partial ^2u}{\partial t^2}
\]
im \((2n+1)\)-dimensionalen Raume \(R_{2n+1}\) läßt sich die Lösung des \textit{Cauchy}schen Problems bekanntlich durch eine der \textit{Poisson}schen analoge Formel darstellen. In der vorliegenden Note wird diese Lösung mit elementaren Mitteln in neuer Form gewonnen, die analog ist der bekannten \textit{Kirchhoff}schen Formel. Die benutzten Überlegungen gestatten überdies, die Lösung des \textit{Cauchy}schen Problems für den (allgemeineren) Fall der Wellengleichung in heterogenen Medien zu gewinnen, wobei \(c^2\) eine Funktion des Ortes im \(R_{2n+1}\) ist. (Für \(n=1\) vgl. Publications de l'Institut séismologique de l'Académie des Sciences de l'URSS Nr. 6 (1930), p. 1.)
| 1 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifolds onto almost paracontact metric manifolds. We discuss the transference of structures on total manifolds and base manifolds and provide some examples. We also obtain the integrability condition of horizontal distribution and investigate curvature properties under such submersions.
| 0 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. This volume contains the reflection of scientific ideas, approaches and methods of S.L. Sobolev's work. These works laid the foundations for an intensive development of the modern theory of partial differential equations and equations of mathematical Physics. The book is divided into two parts.
Part I is under the heading ``Equations of Mathematical Physics''. In this part there are eleven articles viz. i. Application of the theory of plane waves to the Lamb problem, ii. On a New Method in the plane problem on Elastic Vibrations, iii. On Application of a New Method to study Elastic Vibrations in a space with Axial symmetry, iv. On Variations of a Half plane and a Layer with Arbitrary Initial conditions, v. On a New Method of solving problems about propagation of vibrations, vi. Functionally Invariant solutions of the Wave Equation, vii. General theory of Diffraction of waves on Riemann Surfaces, viii. The problem of propagations of a plastic state, ix. On a New Problem of mathematical physics, x. On Motion of a symmetric Top with a cavity filled with Fluid, xi. On a class of problems of Mathematical physics.
Part 2 is under the heading ``computational Mathematics and cubature Formulas''. In this section there are twenty nine articles. They are: i. Schwarz's Algorithm in Elastic theory, ii. On Solution uniqueness of difference equations of Elliptic type, iii. On one difference equation, iv. Certain comments on the Numeric solutions of Integral equations, v. Certain modern questions of computational mathematics, vi. Functional analysis and computational mathematics, vii. Formulas of mechanical cubature in \(n\)-dimensional space, viii. On interpolation of functions of n- variables, ix. Various types of convergence of cubature and quadrature formulas, x. Cubature formulas on the sphere invariant under finite groups of rotations, xi. The number of Nodes in cubature formulas on the sphere, xii. Certain questions of the theory of cubature formulas, xiii. A method for calculating the co-efficient in mechanical cubature formulas, xiv. On the rate of convergence of cubature formulas, xv. Theory of cubature formulas, xvi. Convergence of approximate integration formulas for functions from \(L(m)2\), xvii. Evolution of integrals of infinitely differentiable functions xviii. Cubature formulas with regular boundary layer, xix. A difference Analogue of the polyharmonic equation, xx. Optimal mechanical cubature formulas with nodes on a regular Lattice, xxi. Constructing cubature formulas with regular boundary layer, xxii. Convergence of cubature formulas on infinitely differentiable functions xxiii. Convergence of cubature formulas on the elements of \(L(m)2\), xxiv. The co-efficients of optimal quadrature formulas, xxv. On the roots of Euler polynomials xxvi. On the end roots of the Euler polynomials, xxvii. On the asymptotes of the roots of the Euler polynomials xxviii. More on the zeros of Euler polynomials, xxix. On the algebraic order of exactness of formulas of approximate in Integration.
This volume is well edited and useful to those who are working in the field of history of modern mathematics and Mathematical physics.
| 1 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. The formation of Primordial Black Holes (PBHs) through the collapse of large fluctuations in the early universe is a rare event. This manifests itself, for instance, through the non-Gaussian tail of the formation probability. To compute such probability and the abundance of PBHs, the curvature perturbation is frequently adopted. In this note we emphasize that its use does not provide the correct PBH formation probability. Through a path-integral approach we show that the exact calculation of the PBH abundance demands the knowledge of multivariate joint probabilities of the curvature perturbation or, equivalently, of all the corresponding connected correlators.
| 0 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. See JFM 62.0266.02
| 1 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. This paper presents the formal verification of a compiler front-end that translates a subset of the C language into the Cminor intermediate language. The semantics of the source and target languages as well as the translation between them have been written in the specification language of the Coq proof assistant. The proof of observational semantic equivalence between the source and generated code has been machine-checked using Coq. An executable compiler was obtained by automatic extraction of executable Caml code from the Coq specification of the translator, combined with a certified compiler back-end generating PowerPC assembly code from Cminor, described in previous work.
| 0 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. Verf. hat bereits in einer früheren Note dieselbe quasilineare hyperbolische Differentialgleichung betrachtet und gibt hier in russischer Sprache die ausführliche Theorie des dort angegebenen Verhaltens (C. R. Acad. Sci. URSS (2) 20 (1938), 79-83; F. d. M. \(64_{\text I}\), 490).
| 1 |
This is the second volume of the ``Selected works'' of Sergeĭ L'vovich Sobolev (1908--1989), who invented, among other things, the notion of the generalized derivative of a function, Sobolev spaces, and Sobolev imbedding theorems. Information about the first volume is available in [Zbl 1229.01147] (for its English translation, see [Zbl 1117.01022]).
The second volume contains the papers of S. L. Sobolev on functional analysis and partial differential equations. For this reason, the volume is divided into two parts. Within each part, the papers are arranged in chronological order. In this volume, all the papers are available in Russian, i.e., the papers originally published by S. L. Sobolev in French are translated into Russian.
Part I is under the heading ``Functional analysis''. In this part, there are 13 articles: {\parindent=6mm \begin{itemize}\item[1.] The Cauchy problem in a space of functionals [Dokl. Akad. Nauk SSSR 1935, No. 3, 291--294 (1935; Zbl 0012.40603; JFM 61.1200.02)]. \item[2.] On some estimates related to families of functions which have square integrable derivatives [Dokl. Akad. Nauk SSSR 1936, No. 1, 279--282 (1936; Zbl 0014.05702; JFM 62.0266.01)]. \item[3.] On a class of integro-differential equations for several independent variables. I [Izv. Akad. Nauk SSSR, Ser. Mat. 1937, 515--549 (1937; JFM 63.1020.02)]. \item[4.] Idem. II [Izv. Akad. Nauk SSSR, Ser. Mat. 1938, No. 1, 61--90 (1938; Zbl 0018.36401; JFM 64.1121.02)]. \item[5.] On a theorem from functional analysis [Mat. Sb. 4, 471--497 (1938; Zbl 0022.14803; JFM 64.1100.02)]. \item[6.] On estimations of certain sums for functions defined on a net [Izv. Akad. Nauk SSSR, Ser. Mat. 4, 5--16 (1940; Zbl 0025.39802; JFM 66.0389.02)]. \item[7.] On some transformation groups of \(n\)-dimensional space [Dokl. Akad. Nauk SSSR 32, 380--382 (1941; Zbl 0061.21505)]. \item[8.] The extensions of abstract function spaces connected with the theory of the integral [Dokl. Akad. Nauk SSSR 114, 1170--1173 (1957; Zbl 0083.34003)]. \item[9.] Imbedding theorems for abstract functions of sets [Dokl. Akad. Nauk SSSR 115, 57--59 (1957; Zbl 0083.34101)]. \item[10.] Some generalizations of imbedding theorems [Fundam. Math. 47, 277--324 (1959; Zbl 0100.31802)]. \item[11.] The density of compactly supported functions in the space \(L^{(m)}_p(E_n)\) [Sib. Mat. Zh. 4, 673--682 (1963; Zbl 0204.43802)]. \item[12.] Representation of periodic analytic functions by a sum of squares [Dokl. Akad. Nauk SSSR 165, 40--43 (1965; Zbl 0146.12801)]. \item[13.] The density of finite functions in \(l^{(l)}_p\) [Dokl. Akad. Nauk SSSR 167, 516--518 (1966; Zbl 0185.20505)].
\end{itemize}} Part II is under the heading ``Partial differential equations''. In this part, there are 18 articles: {\parindent=6mm \begin{itemize}\item[1.] The wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 6, 57 p. (1930)]. \item[2.] On a generalization of Kirchhoff's formula [Dokl. Akad. Nauk SSSR 1933, No. 6, 256--262 (1933; Zbl 0008.20805; JFM 59.0492.03)]. \item[3.] On a question about analytic solutions of a system of partial differential equations with two independent variables [Tr. Fiz.-Mat. Inst. Steklova 5, 265--282 (1934; Zbl 0009.21001; JFM 61.1247.03)]. \item[4.] On a question about the integration of the wave equation for a non-homogeneous medium [Tr. Inst. Seismolog. No. 42, 26 p. (1934)]. \item[5.] A new method for solving the Cauchy problem for linear normal hyperbolic equations [Mat. Sb. 1, No. 1, 39--71 (1936; Zbl 0014.05902; JFM 62.0568.01)]. \item[6.] On a boundary value problem for polyharmonic equations [Mat. Sb. 2, No. 3, 465--499 (1937; Zbl 0018.02603; JFM 63.0465.01). \item[7.] On the theory of nonlinear hyperbolic partial differential equations [Mat. Sb. 5, No. 1, 71--99 (1939; Zbl 0021.22702; JFM 65.0441.02)]. \item[8.] Some new problems of the theory of partial differential equations of hyperbolic type [Mat. Sb. 11, No. 3, 155--203 (1942; Zbl 0061.21509)]. \item[9.] On almost periodicity of solutions to the wave equation. I [Dokl. Akad. Nauk SSSR 48, No. 8, 542--545 (1945; Zbl 0061.21702)]. \item[10.] Idem. II [Dokl. Akad. Nauk SSSR 48, No. 9, 618--620 (1945; Zbl 0061.21702)]. \item[11.] Idem. III [Dokl. Akad. Nauk SSSR 49, No. 1, 12--15 (1945; Zbl 0061.21702)]. \item[12.] An example of a correct boundary value problem for the equation of string vibration with the conditions given all over the boundary [Dokl. Akad. Nauk SSSR 109, No. 1, 707--709 (1956; Zbl 0074.07801)]. \item[13.] General formulation of certain boundary value problems for elliptic partial differential equations [Dokl. Akad. Nauk SSSR 111, No. 3, 521--523 (1956; Zbl 0074.08301)]. (Joint work with M. I. Vishik) \item[14.] Some soviet articles on application of functional analysis to differential equations [Czech. Math. J. 6, No. 3, 289--310 (1956; Zbl 0075.08404)]. \item[15.] Functional methods in the theory of partial differential equations [Bull. Soc. Math. Phys. Serbie 9, No. 3--4, 215--244 (1957; Zbl 0138.34503)]. \item[16.] A note on Petrovsky's test for the uniform correctness of the Cauchy problem in the case of partial differential equations [Dokl. Akad. Nauk SSSR 121, No. 4, 598--601 (1958; Zbl 0113.08201)]. \item[17.] The fundamental solution of the Cauchy problem for the equation \(\frac{\partial^3u}{\partial x\partial y\partial z}- \frac{1}{4}\frac{\partial u}{\partial t} = F(x,y,z,t)\) [Dokl. Akad. Nauk SSSR 129, No. 6, 1246--1249 (1959; Zbl 0092.09602)]. \item[18.] On mixed problems for partial differential equations with two independent variables [Calcutta Math. Soc., Golden Jubilee Commemorat. Vol. (1958--1959), Part 2, 447--484 (1959; Zbl 0119.30603)].
\end{itemize}} This volume is intended for researchers and university students in mathematics and mechanics. The author analyses the Conjugate Gradient Normal Residual algorithm (CGNR) and proves that it is both residual minimizing and error reducing. Then, by using a polynomial preconditioning he constructs an adaptive version of the CGNR algorithm. Numerical tests and comparisons with GMRES and classical CGNR algorithms are also reported.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. The author gives a brief history of \(q\)-series indicating a few of prominent theorems in the subject. Also, he gives some of the primary tools that are used to prove theorems about \(q\)-series concentrating on the use of combinatorial methods. The theorems of basic hypergeometric series are often q-analogues of the results about ordinary or generalized hypergeometric series. Finally some authors have been mentioned because their work connects the basic hypergeometric series not only with other areas in mathematics but also in physics.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. In this paper, we consider a Timoshenko system subject to viscoelastic damping acting on a part of the boundary. We use a general assumption on the relaxation function and establish explicit energy decay result from which we can recover the optimal exponential and polynomial rates. Our result generalizes the earlier related results in the literature.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. We introduce natural q-analogs of hypergeometric series well-poised in SU(n), the related hypergeometric series in U(n), and invariant G-functions. We prove that both the SU(n) multiple q-sereis and the invariant G-functions satisfy general q-difference equations. Both the SU(N) and U(n) q-series are new multivariable generalizations of classical basic hypergeometric series of one variable. We prove an identity which expresses our U(n) multiple q-series as a finite sum of finite products of classical basic hypergeometric series. These U(n) q- sereis also satisfy an elegant reduction formula which is analogous to the ''inclusion lemma'' for classical invariant G-functions.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. It is well known that the analysis of a horizontally layered elastic material can be considerably simplified by the introduction of a Fourier or Hankel transform and the application of the finite layer approach. The conventional finite layer (and finite element) stiffness approach breaks down when applied to incompressible materials. In this paper these difficulties are overcome by the introduction of an exact finite layer flexibility matrix. This flexibility matrix can be assembled in much the same way as the stiffness matrix and does not suffer from the disadvantage of becoming infinite for an incompressible material. The method is illustrated by a series of examples drawn from the geotechnical area, where it is observed that many natural and man-made deposits are horizontally layered and where it is necessary to consider incompressible behaviour for undrained conditions.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. The third order mock theta functions, introduced and named by S. Ramanujan, are a class of functions similar to but distinct from theta functions and yet whose behaviour under the fundamental transformations of the modular group can be described. The fifth and seventh order mock theta functions, also introduced by Ramanujan, probably have analyzable behaviour under the fundamental transformations. As a first step in this direction, the author expands the known fifth and seventh order functions into Hecke type series. For the fifth order functions, the expansion is achieved very slickly using the technique of Bailey chains [see the author, Pac. J. Math. 114, 267--283 (1984; Zbl 0547.10012)].
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Determining the local orientation of crystals in engineering and geological materials has become routine with the advent of modern crystallographic mapping techniques. These techniques enable many thousands of orientation measurements to be made, directing attention towards how such orientation data are best studied. Here, we provide a guide to the visualization of misorientation data in three-dimensional vector spaces, reduced by crystal symmetry, to reveal crystallographic orientation relationships. Domains for all point group symmetries are presented and an analysis methodology is developed and applied to identify crystallographic relationships, indicated by clusters in the misorientation space, in examples from materials science and geology. This analysis aids the determination of active deformation mechanisms and evaluation of cluster centres and spread enables more accurate description of transformation processes supporting arguments regarding provenance.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. From the authors' superb and very rewarding introduction, (Chapter 1): ``Number theory and classical analysis are in the spotlight in the present book, which is the fourth of five projected volumes, wherein the authors plan to discuss all the claims made by \textit{S. Ramanujan} in [(*) ``The Lost Notebook and other unpublished papers.'' Berlin: Springer-Verlag (1988; Zbl 0639.01023)]. As in our previous volumes, and in the sequel, we liberally interpret lost notebook not only to include the original lost notebook found by the first author in the library at Trinity College, Cambridge, in March 1976, but also to include all of the material published in (*). This includes letters that Ramanujan wrote to G. H. Hardy from nursing homes, several partial manuscripts, and miscellaneous papers. Some of these manuscripts are located at Oxford University, are in the handwriting of G. N. Watson, and are ``copied from loose papers.'' However, it should be emphasized that the original manuscripts in Ramanujan's handwriting can be found at Trinity College Library; Cambridge.''
``Chapter 2 is devoted to two intriguing identities involving double series of Bessel functions found on page 335 of (*). One is connected with the classical circle problem, while the other is conjoined to the equally famous Dirichlet divisor problem. The double series converge very slowly, and the identities were extremely difficult to prove. Initially, the second author and his collaborators, \textit{S. Kim} and \textit{A. Zaharescu}, were not able to prove the identities with the order of summation as prescribed by Ramanujan, i.e., the identities were proved with the order of summation reversed [Adv. Math. 229, No. 3, 2055--2097 (2012; Zbl 1236.33010); Math. Ann. 335, No. 2, 249--283 (2006; Zbl 1100.33001)]. It is possible that Ramanujan intended that the summation indices should tend to infinity ``together''. The three authors therefore also proved the two identities with the product of the summation indices tending to \(\infty\) (*). Finally, these authors proved Ramanujan's first identity with the order of summation as prescribed by Ramanujan [Adv. Math. 236, 24--59 (2013; Zbl 1326.11059)].
In Chap. 2, we provide proofs of the two identities with the order of summation indicated by Ramanujan in the first identity and with the order of summation reversed in the second identity. We also establish the identities when the product of the two indices of summation tends to infinity.''
``It came as a huge surprise to us while examining pages in ``The Lost Notebook and other unpublished papers'' when we espied famous formulas of N. S. Koshliakov and A. P. Guinand, although Ramanujan wrote them in slightly disguised forms. Moreover, we discovered that Ramanujan had found some consequences of these formulas that had not theretofore been found by any other authors. We are grateful to Y. Lee and J. Sohn for their collaboration on these formulas, which are the focus of Chap.3.
Chapter 4, on the classical gamma function, features two sets of claims. We begin the chapter with some integrals involving the gamma function in the integrands. Secondly, We examine a claim that reverts to a problem (*) that Ramanujan submitted to the Journal of the Indian Mathematical Society, which was never completely solved. On page 339 in (*), Ramanujan offers a refinement of this problem, which was proved by the combined efforts of \textit{E. Karatsuba} [J. Comput. Appl. Math. 135, No. 2, 225--240 (2001; Zbl 0988.33001)] and \textit{H. Alzer} [Bull. Lond. Math. Soc. 35, No. 5, 601--607 (2003; Zbl 1027.33002)].
Hypergeometric functions are featured in Chap. 5. This chapter contains two particularly interesting results. The first is an explicit representation for a quotient of two particular bilateral hypergeometric series.''
``Ramanujan's formula is so unexpected that no one but Ramanujan could have discovered it! The second is a beautiful continued fraction, for which \textit{S. Y. Kang, S. G. Lim}, and \textit{J. Sohn} [J. Math. Anal. Appl. 307, No. 1, 153--166 (2005; Zbl 1068.33018)] found two entirely different proofs, each providing a different understanding of the entry.
Chapter 6 contains accounts of two incomplete manuscripts on Euler's constant \(\gamma\).
\textit{S. Kim} kindly collaborated with the second author on Chap. 7, on an unusual problem examined in a rough manuscript by Ramanujan on Diophantine approximation [Ramanujan J. 31, No. 1-2, 83--95 (2013; Zbl 1329.11073)]. This manuscript was another huge surprise to us, for it had never been noticed by anyone, to the best of our knowledge, that Ramanujan had derived the best possible Diophantine approximation to \(e^{2/a}\), which was first proved in print approximately 60 years after Ramanujan had found his proof.''
``At the beginning of Chap. 8, in Sect. 8.1, we relate that Ramanujan had anticipated the famous work of \textit{L. G. Sathe} [J. Indian Math. Soc., N. Ser. 17, 63--82 (1953; Zbl 0050.27102); J. Indian Math. Soc., N. Ser. 17, 83--141 (1953; Zbl 0051.28008)] and \textit{A. Selberg} [J. Indian Math. Soc., N. Ser. 18, 83--87 (1954; Zbl 0057.28502)] on the distribution of primes, although Ramanujan did not state any specific theorems. In prime number theory, Dickman's function is a famous and useful function, but in Sect. 8.2., we see that Ramanujan had discovered Dickman's function at least 10 years before \textit{K. Dickman} did in [Ark. Mat. A 22, No. 10, 14 p. (1930; JFM 56.0178.04)]. We then turn to a formula for \(\zeta(\tfrac 12)\), first given in Sect. 8 of Chap. 15 in Ramanujan's second notebook. In (*), Ramanujan offers an elegant reinterpretation of this formula, which renders an already intriguing result even more fascinating. Next, we examine a fragment on sums of powers that was very difficult to interpret. One of the most interesting results in the chapter yields an unusual algorithm for generating solutions to Euler's Diophantine equation \(a^3+ b^3= c^3+ d^3\).
Chapter 9 is devoted to discarded fragments of manuscripts and partial manuscripts concerning the divisor functions \(\sigma_k(n)\) and \(d(n)\), respectively, the sum of the \(k\)th powers of the divisors of \(n\), and the number of divisors of \(n\).
In the next chapter, Chap. 10, we prove all of the results on page 196 of (*). Two of the results evaluating certain Dirichlet series are especially interesting.
Chapter 11 contains some unusual old and new results on primes arranged in two rough, partial manuscripts. Ramanujan's manuscripts contain several errors, and we conjecture that this work predates his departure for England in 1914. Harold Diamond helped us enormously in both interpreting and correcting the claims made by Ramanujan in the two partial manuscripts examined in Chap. 11.
In Chap. 12, we discuss a manuscript that was either intended to be a paper by itself or, more probably, was slated to be the concluding portion of Ramanujan's paper [Messenger 45, 81--84 (1915; JFM 45.1250.01)]. The results in this paper hark back to Ramanujan's early preoccupation with infinite series identities. Our account here includes a closer examination of two of Ramanujan's series by Johann Thiel, to whom we are very grateful for his contributions.
Perhaps the most fascinating formula found in the three manuscripts on Fourier analysis in the handwriting of Watson is a transformation formula involving the Riemann \(\Xi\)-function and the logarithmic derivative of the gamma function in Chap. 13.
The second of the aforementioned manuscripts features integrals that possess transformation formulas like those satisfied by theta functions. Two of the integrals were examined by Ramanujan in two papers, where he considered the integrals to be analogues of Gauss' sums, a view that we corroborate in Chap. 14. One of the integrals, to which page 198 of (*) is devoted, was not examined earlier by Ramanujan.
In the third manuscript, on Fourier analysis, which we discuss in Chap. 15, Ramanuian considers some problems on Mellin transforms.
The next three chapters pertain to some of Ramanujan's earlier published papers. We then consider miscellaneous collections of results in classical analysis and elementary mathematics in the next two chapters.
Chapter 21 is devoted to some strange, partially incorrect claims of Ramanujan that likely originate from an early part of his career''.
This is the ninth volume in the Ramanujan notebooks series. Through nearly 20 years the care and attention to all manner of scholarly detail has been extraordinarily and consistently of the highest quality. Of course Ramanujan's life was tragically short. Yet in having Berndt, and later Andrews and Berndt as editors he has surely been fortunate indeed.
The smallest fragments are lovingly handled here. The reviewer took particular delight in Chapter 6 on \(\gamma\), Euler's constant, where a rapidly convergent series is found; also, in \S7.3 we find the best possible Diophantine approximation of \(e^{2/a}\), where a \(a\neq 0\) is an integer. As this chapter remarks, Ramanujan found this 60 years before it appeared in the literature. -- One stands in awe of this series of books.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Given a set \(A\) of natural numbers, i.e., nonnegative integers, the representation function of \(A\) is the function which to every natural number \(n\) associates the number \(r_A(n)\) of ordered pairs \((a, b)\) of elements \(a, b\) of \(A\) such that \(a+b = n\). We characterize intrinsically the representation function, answering an open problem.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. As it well known, Ramanujan defined several mock theta functions in his last letter to Hardy (January, 1920). Despite the passage of 88 years, the properties of these functions have remained somewhat mysterious up to the present day. In this important paper the authors prove a conjecture made by George Andrews more than 40 years ago concerning the Fourier expansion of
\[
f(q)= 1+\sum^\infty_{n=1} {q^{(n^2)}\over (1+ q)^2\cdots (1+ q^n)^2},\tag{1}
\]
one of Ramanujan's ``third-order'' mock theta functions.
The transformation properties under the inversion \(z\to -1/2\) were first treated by \textit{G. N. Watson} in [J. Lond. Math. Soc. 11, 55--80 (1936; Zbl 0013.11502)]. In [Trans. Am. Math. Soc. 72, 474--500 (1952; Zbl 0047.27902)], an article based upon her doctoral dissertaion, \textit{L. A. Dragonette} applied Watson's results to obtain approximate transformation formulae for \(f(q)\) under all elements of \(\text{SL}(2,\mathbb{Z})\), and she used these, together with the circle method, to determine the first \(\sqrt{n}\) terms in the asymptotic expansion of \(\alpha(n)\), the \(n\)th Fourier coefficient of \(f(q)\), with a remainder term of the order of growth \(\sqrt{n}\log n\), as \(n\to\infty\). George Andrews continued Dragonette's work in his Ph.D. thesis (mid-1960's; like Dragonette's, directed by Rademacher), reducing the estimate of the order of growth of the remainder to \(O(n^\varepsilon)\), arbitrary \(\varepsilon> 0\). Andrews achieved this striking improvement by deriving more precise transformation formulae of \(f(q)\) under \(\text{SL}(2,\mathbb{Z})\) through application of Poisson summation. The results of Dragonette and Andrews bring to mind the growth of \(p(n)\), the partition function of number theory. Indeed, the order of growth in \(n\) of \(p(n)\) is \(\exp(\pi\sqrt{2n/3})\), while that of \(\alpha(n)\) is the square root of the latter expression, that is, \(\exp(\pi\sqrt{n/6})\).
The paper under review completes the work of Dragonette and Andrews, establishing an exact convergent series formula for \(\alpha(n)\) and thereby proving the conjecture of Andrews, to which the authors have, quite reasonably, appended Dragonette's name. The series in question is:
\[
\alpha(n)= \pi(24n-1)^{-1/4} \sum^\infty_{k=1} (-1)^{[{k+1\over 2}]}A_{2k}(n- \rho(k))\times I_{1/2}(\pi\sqrt{24n-1}/12k),\tag{2}
\]
where \(I_{1/2}\) is the modified Bessel function of the first kind, \(A_k(n)\) is the usual Kloosterman sum occurring in Rademacher's convergent series representation of \(p(n)\) (i.e., the summands of \(A_k(n)\) include the multiplier system of the Dedekind eta-function as factors), and \(p(k)= k/2\) for even \(k\); \(0\) for odd \(k\). The expression (2) is strikingly similar to Rademacher's series for \(p(n)\), in which \(I_{3/2}(\pi\sqrt{24n-1}/6k)\) occurs (in place of \(I_{1/2}(\pi\sqrt{24n-1}/12k))\) and \(A_k(n)\) replaces the more complicated \(A_{2k}(n- \rho(k))\) of (2).
Having proved (2), the authors likewise obtain the long-sought exact formulas for \(N_e(n)\) and \(N_0(n)\), where these arithmetic functions are, respectively, the number of partitions of \(n\) of even and odd rank. (The rank of a partition is its largest part minus the number of its parts.) This follows from the identities \(p(n)= N_e(n)+ N_0(n)\), \(\alpha(n)= N_e(n)- N_0(n)\), the first one obvious and the second well known.
The proof that Bringmann and Ono adduce to prove (2) is remarkable for its depth and ingenuity, not surprisingly in light of the widely-recognized difficulty of the problem they have solved. It entails work of \textit{S. P. Zwegers} [Contemp. Math. 291, 269--277 (2001; Zbl 1044.11029)], which recasts Watson's modular transformation properties (of \(f(q)\)) as the transformation formulae of a real analytic three-dimensional vector-valued modular form (weight \({1\over 2}\)). The proof involves as well Maass forms and the Serre-Stark basis theorem for holomorphic modular terms of weight \({1\over 2}\). Especially interesting is the application (\S5) of a weight-changing differential operator which transforms weight \(k\) Maass forms to weight \(2-k\) holomorphic modular forms, and therefore can be understood as an antilinear analogue of the Bol differential operator.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Not reviewed
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. The principal objects of study in this paper are the two \(q\)-series
\[
R_2(q) := \prod_{k \geq 1}\frac{1}{(1-q^k)}\sum_{n \in \mathbb{Z} \backslash \{0\}} \frac{(-1)^nq^{n(3n+1)/2}}{(1-q^n)^2}
\]
and
\[
R_2^o(q) := \prod_{k \geq 1}\frac{1}{(1-q^{2k})}\sum_{n \in \mathbb{Z}} \frac{(-1)^nq^{n(3n+5)+2}}{(1-q^{2n+1})^3}.
\]
These series are generating functions for certain rank moments and Durfee symbols which play an important role in recent work of \textit{G. E. Andrews} [Invent. Math. 169, No. 1, 37--73 (2007; Zbl 1214.11116)].
The author shows that up to a linear combination of quasimodular forms, \(q^{-1}R_2(q^{24})\) is the holomorphic part of a weight \(3/2\) weak Maass form. For this reason, she calls \(R_2(q)\) a ``quasimock theta function'' (ordinary mock theta functions being essentially holomorphic parts of weak Maass forms). As applications, an asymptotic formula for the coefficients of \(R_2(q)\) as well as congruences for these coefficients (and related combinatorial functions) are derived.
As for \(R_2^o(q)\), the author computes the transformation properties and derives an asymptotic formula for the coefficients, but states that the ``transformation law for \(R_2^o(q)\) \dots is more complicated than the transformation law for \(R_2(q)\) arising from higher order poles of the generating function. For this reason no usual weak Maass forms arise.''
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. This paper describes an algorithm for discovery of convoys in database with proximity log. Traditionally, discovery of convoys covers trajectories databases. This paper presents a model for context-aware browsing application based on the network proximity. Our model uses mobile phone as proximity sensor and proximity data replaces location information. As per our concept, any existing or even especially created wireless network node could be used as presence sensor that can discover access to some dynamic or usergenerated content. Content revelation in this model depends on rules based on the proximity. Discovery of convoys in historical user's logs provides a new class of rules for delivering local content to mobile subscribers.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. The authors prove 11 identities from Ramanujan's lost notebook. These involve seven functions defined by \(q\)-series, which are called mock theta functions. Let
\[
\begin{aligned} (x)_ n &=(x;q)_ n =\begin{cases} \prod_{i=0}^{n-1}(1-q^ ix), &n\geq 0,\\ \prod_{i=1}^{-n}(1- q^{-i}x)^{-1}, &n<0,\;x\neq q,q^ 2,\dots\hbox{ or } q^{-n}, \end{cases} \\ (x)_ \infty &=(x;q)_ \infty=\prod_{i\geq 0}(1-q^ i x), \hbox{ where } | q| <1, \\ \phi(q) &=\sum_{n\geq 0} {(- 1)^ n q^{n^ 2}(q;q^ 2)_ n \over (-q)_{2n}}, \\ \psi(q) &=\sum_{n\geq 0} {(-1^ n)q^{(n+1)^ 2}(q;q^ 2)_ n \over (- q)_{2n+1}}, \\ j(x,q)&=\sum_ n(-1)^ nq^{\left({x \atop 2}\right)}x^ n, \quad | q|<1. \\ \end{aligned}
\]
Two of the identities are
\[
\begin{aligned} \phi(q^ 9)-\psi(q)-q^{-3}\psi(q^ 9) &={{j(- q^ 3,q^{12})(q^ 6;q^ 6)^ 2_ \infty} \over {j(-q,q^ 4)j(- q^ 9,q^{36})}}, \\ {\psi(\omega q)-\psi(\omega^ 2 q)\over (\omega- \omega^ 2)q} &= {{j(-q,q^ 4)j(-q^ 9,q^{36})(q^ 3;q^ 6)_ \infty} \over {j(-q^ 3,q^{12})}} \\ \end{aligned}
\]
where \(\omega\) is a primitive cubic root of 1. Results concerning \(\Theta\) functions are proved first. Bailey pair method and constant term method are described next. These are used to derive Hecke type identities for the sixth order mock theta functions. The asymptotics of the functions for \(q\) near a root of unity are also discussed.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. We suggest an augmented symmetry approach to reduce the computational complexity of the DFT electronic structure calculations based on the Wien2k package and to extend its applicability to the studies of heterogeneous ring-shape molecular nanomagnets. The approach is tested for the reference chromium-based rings \({\mathrm{Cr}_8}\), \({\mathrm{Cr}_7\mathrm{Cd}}\) and \({\mathrm{Cr}_7\mathrm{Ni}}\), and a good agreement with the results of the previous standard studies is reached with the substantial gain in the computing time.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. This revised and expanded new edition will surely continue to meet the needs for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or \(q\)-series. Simplicity, clarity, deductive proofs, thoughtfully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, orthogonal polynomials in several variables, and generating functions.
Chapters 9-11 are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base \(q\)-series.
Some sections and exercises have been added to reflect recent developments, and the Bibliography has been revised to maintain its comprehensiveness.
For the review of the first edition see Zbl 0695.33001.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. When count data from event sequences display a non-Poissonian dispersion, instability effects usually arise in the classical estimation especially when the binomial distribution is used. They are a bar to making inferences about the dispersion behaviour mostly up to and around the point at which it changes from underdispersion to overdispersion. To this end, the hyperbinomial process (HBP) of process-index \(\alpha (t)\) comes out as a suitable analysis model given that it involves both the dispersion types. In practice, however, it also presents some similar drawbacks. Here a bootstrap-type procedure allowing one to ``robustize'' the usual classical estimators with respect to the most common instability causes is therefore described. A comparison with some pseudo-jackknife procedures is made by using accident count data. The example shows the bootstrap-type procedure used as being very useful for making inferences about the HBP-functions.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to \(4n^2\) or \(4n(n+1)\) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turán determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the eta-function identities of Macdonald. Moreover, the powers \(4n(n+1)\), \(2n^2+n\), \(2n^2-n\) that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac-Wakimoto conjectured identities involving representing a positive integer by sums of \(4n^2\) or \(4n(n+1)\) triangular numbers, respectively.
The article has also been published in monograph form [Developments in Mathematics 5. Boston MA: Kluwer Academic Publishers (2002; Zbl 1125.11316)]. An announcement appeared in Proc. Natl. Acad. Sci. USA 93, No. 26, 15004--15008 (1996; Zbl 1125.11346).
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Inventory models are widely used in a variety of real-world applications. In particular, inventory systems with perishable items have received a significant amount of attention. We consider an \((s,S)\) continuous inventory model with perishable items, impatient customers, and random lead times. Two characteristic behaviors of impatient customers are balking and reneging. Balking is when a customer departs the system if the item they desire is unavailable. Reneging occurs when a waiting customer leaves the system if their demand is not met within a set period of time. The proposed system is modeled as a two-dimensional Markov process with level-dependent G/M/\(1\)-type structure. We also consider independent and identically distributed replenishment lead times that follow a phase-type distribution. We find an efficient approximation method for the joint stationary distribution of the number of items in the system, and provide formulas for several performance measures. Moreover, to minimize system costs, we find the optimal values of \(s\) and \(S\) numerically and perform a sensitivity analysis on key parameters.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Im Anschluß an die klassische Identität
\[
(1) \quad \prod_{\lambda=1}^{\infty}(1+x^{\lambda})(1-x^{2\lambda-1})=1 \;(| x| < 1)
\]
und deren bekannte zahlentheoretische Interpretation wird der folgende Satz bewiesen: Man bezeichne mit \(P_n(x)\) und \(Q_n(x)\) die Näherungszähler bzw. -nenner des Kettenbruches
\[
(2) \quad K(x)= +\frac{x\mid }{\mid 1}+\frac{x^2\mid}{\mid 1}+\frac{x^3\mid}{\mid 1}+\cdots.
\]
Dann ist für \(| x| < 1\)
\[
(3')\quad \lim_{n=\infty}P_n(x) = \frac{1}{\prod_{\lambda=1}^{\infty}(1-x^{5\lambda-4})(1- x^{5\lambda-1}}
\]
und
\[
(3'')\quad \lim_{n=\infty}Q_n(x) = \frac{1}{\prod_{\lambda=1}^{\infty}(1-x^{5\lambda-3})(1- x^{5\lambda-2}}.
\]
Um auf obige Gleichungen zu kommen, gibt der Verf. zwei Wege: einen zahlentheoretischen und einen analytischen. Der erste führt ganz analog zum Ziele, wie bei \textit{Franklin} (C. R. 92, 448, 1881). Der zweite Beweis stützt sich hingegen auf die \textit{Gauß}schen Ausdrücke
\[
\left[ \begin{matrix} k \\ l \end{matrix} \right] = \frac{(1-x^k)(1-x^{k-1})\cdots(1-x^{k-l+1})}{(1-x)(1-x^2)\cdots (1-x^l)},
\]
mit deren Hilfe eine explizite Darstellung der Polynome \(P_n(x), Q_n(x)\) gegeben wird.
Die zahlentheoretische Interpretation der Sätze (3'), (3'') lautet nun wie folgt: Die Anzahl \(Z_1(n)\) der Zerlegungen einer positiven ganzen Zahl \(n\) in voneinander verschiedene Summanden von der Eigenschaft
\[
(4)\quad n=b_1+b_2+\cdots, b_{\lambda-1}>b_{\lambda}+1, b_{\lambda}\geqq 1,
\]
ist gleich der Anzahl \(F_1(n)\) der Zerlegungen von \(n\) in gleiche oder verschiedene Summanden von der Form \(5v\pm 1\). Weiter: Die Anzahl derjenigen Zerlegungen, (4), wo \(b_{\lambda}\geqq 2\) ist, ist gleich der Anzahl \(F_2(n)\) von \(n\) in gleiche oder verschiedene Summanden von der Form \(5v\pm 2\).
Endlich beschäftigt sich der Verf. mit dem Verhalten von (2) für \(| x| =1\) und gewinnt, wenn \(x\) eine primitive \(m\)- Tetraeders Einheitswurzel ist, das Ergebnis, daß \(K(x)\) divergiert bzw. konvergiert, je nachdem \(m\) durch 5 teilbar ist oder nicht. (IV 2.)
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. No review copy delivered.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. For a review see Zbl 0371.10001. (A Russian translation (Nauka, Moscow) was published in 1982, see Zbl 0499.10001.)
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Not reviewed
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Zu dem Referat F. d. M. d. Bd. S. 201 sei bemerkt, daß\ die analytischen Formeln sich bereits bei Rogers (Lond. M. S. Proc. 25, 329, 1894) finden.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. This article is a review of the dissipation processes in nanoelectromechanical systems (NEMS). As NEMS technology becomes more and more prevalent in research and engineering applications, it is of great importance to understand the dissipative mechanisms that in part define the dynamic response of such devices. The purpose of this work is to understand, sort, and categorize dominant dissipation sources and to determine their significance with respect to physics processes and engineering considerations.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. This reviewer can do no better than reproduce portions of the authors' Introduction, which is absolutely superb!
From the introduction: ``The centerpiece of this volume is the partition function \(p(n)\). Featured in this book are congruences for \(p(n)\), ranks and cranks of partitions, the Ramanujan \(\tau\)-function, the Rogers-Ramanujan functions, and the unpublished portion of Ramanujan's paper on highly composite numbers [Proc. Lond. Math. Soc. (2) 14, 347--409 (1915; JFM 45.1248.01, JFM 45.0286.02)].
``The first three chapters are devoted to ranks and cranks of partitions. In 1944, \textit{F. J. Dyson} [Eureka 8, 10--15 (1944)] defined the rank of a partition to be the largest part minus the number of parts. If \(N(m,t,n)\) denotes the number of partitions of \(n\) with rank congruent to \(m\) modulo \(t\), then Dyson conjectured that
\[
N(k,5,5n+4)=\frac{p(5n+4)}{5},\quad 0\leq k\leq 4,\tag{1.0.1}
\]
and
\[
N(k,7,7n+5)=\frac{p(7n+5)}{7},\quad 0\leq k\leq 6\tag{1.0.2}
\]
Thus, if (1.0.1) and (1.0.2) were true, the partitions counted by \(p(5n+4)\) and \(p(7n+5)\) would fall into five and seven equinumerous classes, respectively, thus providing combinatorial explanations and proofs for Ramanujan's famous congruences \(p(5n + 4)\equiv 0 \pmod {5}\) and \(p(7n+5)\equiv 0\pmod {7}\). Dyson's conjectures were first proved in 1954 by \textit{A. O. L. Atkin} and \textit{H. P. F. Swinnerton-Dyer} [Proc. Lond. Math. Soc. (3) 4, 84--106 (1954; Zbl 0055.03805)].
``Dyson observed that the corresponding analogue to (1.0.1) and (1.0.2) does not hold for the third famous Ramanujan congruence \(p(11n+6)\equiv 0 \pmod {11}\), and so he conjectured the existence of a statistic that he called the crank that would combinatorially explain this congruence. In his doctoral dissertation [Generalizations of Dyson's rank, Ph.D. Thesis, Pennsylvania State University, University Park, PA (1986)], \textit{F.~G. Garvan} defined a crank for vector partitions, which became the forerunner of the true crank, which was discovered by \textit{G. E. Andrews} and \textit{F.~G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167--171 (1988; Zbl 0646.10008)]
[\dots]
``Although Ramanujan did not record any written text about ranks and cranks in his lost notebook [The Lost Notebook and other unpublished papers. Berlin: Springer-Verlag; New Delhi: Narosa Publishing House (1988; Zbl 0639.01023)], he did record theorems about their generating functions. Chapter 2 is devoted to the five and seven-dissections of each of these two generating functions. Cranks are the exclusive topic of Chapter 3, where dissections for the generating function for cranks are studied, but now in the context of congruences. A particular formula found in the lost notebook and proved in Chapter 4 is employed in our proofs in Chapter 3. [\dots]
``\textit{G. H. Hardy} (ed.) [Math. Z. 9, 147--153 (1921; JFM 48.0150.02)] extracted a portion of Ramanujan's manuscript and added several details in giving proofs of his aforementioned famous congruences for the partition function, namely,
\[
p(5n+4)\equiv 0 \pmod {5}, \;p(7n+5)\equiv 0 \pmod {7},\;p(11n+6)\equiv 0\pmod {11}.\tag{1.0.3}
\]
[\dots]
``These congruences (1.0.3) are the first cases of the infinite families of congruences
\[
p(5^kn+\delta_{5,k})\equiv 0 \pmod {5^k},\tag{1.0.4}
\]
\[
p(7^kn+\delta_{7,k})\equiv 0 \pmod {7^{[k/2]+1}},\tag{1.0.5}
\]
\[
p(11^kn+\delta_{11,k})\equiv 0 \pmod {11^k},
\]
where \(\delta_{p,k}:\equiv 1/24 \pmod{p^k}\). In Ramanujan's manuscript, he actually gives a complete proof of (1.0.4), but many of the details are omitted. These details were supplied by \textit{G. N. Watson} [J. Reine Angew. Math. 179, 97--128 (1938; Zbl 0019.15302, JFM 64.0122.02)]
[\dots]
``Since proofs of (1.0.4) and (1.0.5) can now be found in several sources (which we relate in Chapter 5), there is no need to give proofs here.
[\dots]
It was surprising for us to learn that Ramanujan had also found congruences for \(p(n)\) for the moduli 13, 17, 19, and 23 and had formulated a general conjecture about congruences for any prime modulus. However, unlike (1.0.3), these congruences do not give divisibility of \(p(n)\) in any arithmetic progressions. In his doctoral dissertation, \textit{J. M. Rushforth} [Congruence properties of the partition function and associated functions, Doctoral Thesis, University of Birmingham (1950), Proc. Camb. Philos. Soc. 48, 402--413 (1952; Zbl 0047.04302)] supplied all of the missing details for Ramanujan's congruences modulo 13, 17, 19, and 23. Since Rushforth's work has never been published and since his proofs are motivated by those found by Ramanujan, we have decided to publish them here for the first time. In fact, almost all of Rushforth's thesis is devoted to Ramanujan's unpublished manuscript on \(p(n)\) and \(\tau(n)\), and so we have extracted from it further proofs of results claimed by Ramanujan in this famous manuscript.
``Ramanujan's general conjecture on congruences for prime moduli was independently corrected, proved, and generalized in two distinct directions by H. H. Chan and J.-P. Serre and by \textit{S. Ahlgren} and \textit{M. Boylan} [Invent. Math. 153, No. 3, 487--502 (2003; Zbl 1038.11067)]. The proof by Chan and Serre is given here for the first time.
[\dots]
``Chapter 6 is devoted to six entries on page 189 of the lost notebook [loc. cit.], all of which are related to the content of Chapter 5, and to entries on page 182, which are related to Ramanujan's paper on congruences for \(p(n)\) [Proc. Camb. Philos. Soc. 19, 207--210 (1919; JFM 47.0885.01)] and of
course also to Chapter 5. In particular, we give proofs of two of Ramanujan's most famous identities, immediately yielding the first two congruences in (1.0.3). On page 182, we also see that Ramanujan briefly examined congruences for \(p_r(n)\), where \(p_r(n)\) is defined by
\[
(q;q)^r_\infty=\sum_{n=0}^\infty p_r(n)q^n,\quad |q|<1.
\]
Apparently, page 182 is page 5 from a manuscript, but unfortunately all of the remaining pages of this manuscript are likely lost forever. We have decided also to discuss in Chapter 6 various scattered, miscellaneous entries on \(p(n)\). Most of this mélange can be found in Ramanujan's famous paper with Hardy establishing their asymptotic series for \(p(n)\) [Proc. Lond. Math. Soc. (2) 17, 75--115 (1917; JFM 46.0198.04)].
``In Chapter 7, we examine nine congruences that make up page 178 in the lost notebook. These congruences are on generalized tau functions and are in the spirit of Ramanujan's famous congruences for \(\tau(n)\) discussed in Chapter 5.
``The Rogers-Ramanujan functions are the focus of Chapter 8, wherein Ramanujan's 40 famous identities for these functions are examined. Having been sent some, or possibly all, of the 40 identities in a letter from Ramanujan,
\textit{L. J. Rogers} [Proc. Lond. Math. Soc. (2) 19, 387--397 (1921; JFM 48.0151.02)] proved eight of them, with \textit{G. N. Watson} [J. Indian Math. Soc. 20, 57--69 (1934; Zbl 0009.33707, JFM 60.0184.01)] later providing proofs for six further identities as well as giving different proofs of two of the identities proved by Rogers. For several years after Ramanujan's death, the list
of 40 identities was in the hands of Watson, who made a handwritten copy for himself, and it is this copy that is published in [The Lost Notebook and Other Unpublished Papers (loc. cit.)]. Fortunately, he did not discard the list in Ramanujan's handwriting, which now resides in the library at Trinity College, Cambridge. Approximately ten years after Watson's death, \textit{B. J. Birch} [A look back at Ramanujan's notebooks. Math. Proc. Camb. Philos. Soc. 78, 73--79 (1975; Zbl 0305.10002)] found Watson's copy in the library at Oxford University and published it in 1975, thus bringing it to the mathematical public for the first time. \textit{D. Bressoud} [Proof and generalization of certain identities conjectured by Ramanujan, Ph.D. Thesis, Temple University (1977)] and \textit{A. J. F. Biagioli} [Glasg. Math. J. 31, No. 3, 271--295 (1989; Zbl 0679.10018)] subsequently proved several further identities from the list.
``Our account of the 40 identities in Chapter 8 is primarily taken from a Memoir [Mem. Am. Math. Soc. 880, 96 p. (2007; Zbl 1118.11044) by \textit{B. C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B. P. Yeap, A. J. Yee, H. Yesilyurt} and \textit{J. Yi}. The goal of these authors was to provide proofs for as many of these identities as possible that were in the spirit of Ramanujan's mathematics. In doing so, they borrowed some proofs from Rogers, Watson, and Bressoud, while supplying many new proofs as well. After the publication of this Memoir in which proofs of 35 of the 40 identities were given in the spirit of Ramanujan, \textit{H. Yesilyurt} [J. Number Theory 129, No. 6, 1256--1271 (2009; Zbl 1219.11063), J. Math. Anal. Appl. 388, No. 1, 420--434 (2012; Zbl 1272.11037)] devised ingenious and difficult proofs of the remaining five identities, and so these papers are the second primary source on which Chapter 8 is constructed.
``Chapter 9 is devoted to one general theorem on certain sums of positive integral powers of theta functions, and five examples in illustration.
[\dots]
``In 1915, the London Mathematical Society published Ramanujan's paper on highly composite numbers [loc. cit.], [Collected Papers, 78--128]. However, this is only part of the paper that Ramanujan submitted. The London Mathematical Society was in poor financial condition at that time, and to diminish expenses, they did not publish all of Ramanujan's paper. Fortunately, the remainder of the paper has not been lost and resides in the library at Trinity College, Cambridge. In its original handwritten form, it was photocopied along with Ramanujan's lost notebook in 1988 [loc. cit.]. \textit{J.-L. Nicolas} and \textit{G. Robin} prepared an annotated version of the paper for the first volume of the Ramanujan Journal in 1997 [Ramanujan J. 1, No.~2, 119--153 (1997; Zbl 0917.11043)]. In particular, they inserted text where gaps occurred, and at the end of the paper, they provided extensive commentary on research in the field of highly composite numbers accomplished since the publication of Ramanujan's original paper. Chapter 10 contains this previously unpublished manuscript of Ramanujan on highly composite numbers, as completed by Nicolas and Robin, and a moderately revised and extended version of the commentary originally written by Nicolas and Robin.''
Reviewer's remark: The Introduction includes wonderful historical context for Ramanujan's partition work; in the same vein, the first section of Chapter 5 is not to be missed!
For the two previous volumes see Zbl 1075.11001 and Zbl 1180.11001.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. This paper provides a guide and tutorial to type 2 fuzzy sets. Type 2 fuzzy sets allow for linguistic grades of membership thus assisting in knowledge representation. They also offer improvement on inferencing with type 1 sets. The various approaches to knowledge representation and inferencing are discussed, with worked examples, and some of the applications of type 2 sets are reported.
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This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. In this talk we present the discoveries made in the theory of Rogers-Ramanujan identities in the last five years which have been made because of the interchange of ideas between mathematics and physics. We find that not only does every minimal representation \(M(p,p')\) of the Virasoro algebra lead to a Rogers-Ramanujan identity but that different coset constructions lead to different identities. These coset constructions are related to the different integrable perturbations of the conformal field theory. We focus here in particular on the Rogers-Ramanujan identities of the \(M(p,p')\) models for the perturbations \(\phi_{1,3},\sim\phi_{2,1},\sim\phi_{1,2}\) and \(\phi_{1,5}.\)
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. [For the entire collection see Zbl 0706.00007.]
This paper deals with topological partition problems. The expression \(X\rightarrow(Y)^ n_ \omega\) has the established meaning with the proviso that \(X\) and \(Y\) be topological spaces and that the desired homogeneous set be homeomorphic with \(Y\).
The principal theorem shows that one may not expect interesting homogeneous sets when one partitions the set of quadruples of a Hausdorff space: under the assumption of the GCH and \(\square_ \kappa\) for every singular \(\kappa\) one may find for every Hausdorff space \(X\) of cardinality at most the first weakly compact cardinal a partition \(F: [X]^ 4\rightarrow\omega\) with only discrete homogeneous sets.
The author quotes several results that show that the theorem is the best one can get, except for a ZFC result.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. [For part III of this paper see the author in ibid. 41, 186--208 (1981; Zbl 0477.33009).]
While the author was examining the papers of the late G. N. Watson in the library of Trinity College, Cambridge in the year 1976, an additional book of unpublished notes by the Indian mathematician was discovered. The presence of this co-called ``lost'' notebook had for some time been suspected on circumstantial grounds, but it had not hitherto come to light. See \textit{G. E. Andrews}, Am. Math. Mon. 86, 89--108 (1979; Zbl 0401.01003). In his notebooks, Ramanujan gave a large number of fascinating identities, mostly without proof. These results are closely associated with \(q\)-hypergeometric function theory and the theory of partitions.
In this paper, the author starts from the generating function of the number of partitions of \(n\) in which the parts (arranged in ascending order) alternate in parity starting with the smallest part odd. This generating function is associated with a number of identities stated but not proved by Ramanujan and which provide substantial combinatorial information on the above mentioned partition. The author then proves these identities by some extremely interesting but rather lengthy manipulations. This paper is included by relating Ramanujan's identities to the combinatorial theory of stacks. The formulae given are too lengthy for inclusion in a brief review of this nature.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. We updated our Peggy tool, previously presented in [6], to perform translation validation for the LLVM compiler using a technique called Equality Saturation. We present the tool, and illustrate its effectiveness at doing translation validation on SPEC 2006 benchmarks.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. In his last letter to G. H. Hardy, S. Ramanujan spoke of a new class of functions which behaved asymptotically like \(\theta\)-functions but which were not. He called them mock \(\theta\)-functions. Specifically, these are q-series, f(q), convergent for \(| q| <1\) and satisfying: (0) For every root of unity, \(\zeta\), there is a \(\theta\)-function \(\theta_{\zeta}(q)\) such that \(f(q)-\theta_{\zeta}(q)\) is bounded as \(q\to \zeta\) radially. (1) No single \(\theta\)-function works for all \(\zeta\).
As Andrews and Hickerson have pointed out, no one has proved that mock \(\theta\)-functions exist. In his letter, Ramanujan listed 17 functions which he proposed as likely candidates. Four of these were described as 3rd order functions, ten as 5th order, and three as 7th order. No one knows precisely what he meant by the order of a mock theta function.
\textit{G. N. Watson} [J. Lond. Math. Soc. 11, 55-80 (1936; Zbl 0013.11502)] proved that Ramanujan's 3rd order functions satisfy (0) and are not \(\theta\)-functions, and in [Proc. Lond. Math. Soc., II. Ser. 42, 274-304 (1936; Zbl 0015.30402)] showed that the 5th order functions satisfy (0). The situation rested pretty much at that point until Andrews' discovery of Ramanujan's ``Lost Notebook'' [see Am. Math. Mon. 86, 89-108 (1979; Zbl 0401.01003)] which brought to light ten identities which imply that the 5th order functions are not \(\theta\)-functions. These identities have recently been proven by \textit{D. Hickerson} [Invent. Math. 94, No. 3, 639-660 (1988; Zbl 0661.10059)].
This article is an investigation of those ten identities, proving that they separate into two sets of equivalent identities and investigating their combinatorial implications.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. State commands refer to states, not actions. They have a temporal dimension explicitly or implicitly. They indirectly change what we are permitted, forbidden or obligated to do. This paper presents \({\mathsf {DTNL}}\), a deontic logic meant to handle state commands based on the branching-time temporal logic \(\mathsf {PCTL}^*\). The models of \({\mathsf {DTNL}}\) are trees with bad states, which are identified by a propositional constant \(\mathfrak {b}\) introduced in the language. To model state commands, a dynamic operator that adds states to the extension of \(\mathfrak {b}\) is introduced.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. This thesis is the result of my research on the following two questions, both posed by Don Zagier:
1. How do the mock \(\theta\)-functions fit in the theory of modular forms?
2. Is there a theory of indefinite theta functions?
Since most of the mock \(\theta\)-functions had been related to sums like the one in (1) \(f(q)=\frac {2}{(q)_\infty} \sum_{n\in\mathbb Z} \frac{(-1)^nq^{\frac32 n^2+\frac12 n}}{1+q^n}\) for the third order mock \(\theta\)-function (Watson), I first considered this type of sum. The result of this research is Chapter 1. In it we consider the series
\[
\sum_{n\in\mathbb Z}\frac{(-1)^n e^{\pi i(n^2+n)\tau+2\pi inv}}{1-e^{2\pi in\tau+2\pi iu}}\qquad (\tau\in\mathcal H, v\in\mathbb C, u \in\mathbb C\setminus (\mathbb Z\tau+ \mathbb Z)).
\]
This function was also studied by \textit{M. Lerch} in [Rozpravy České Akad. Císaré I 24, 465--480 (1892)] (see JFM 24.0442.02 for an abstract). Therefore we call this a Lerch sum. This sum is of the same type as the sum in (1). The function does not transform like a Jacobi form. However, we find that on addition of a (relatively easy) correction term the function does transform like a Jacobi form. This correction term is real-analytic.
In Chapter 2 we consider certain indefinite \(\theta\)-functions, in an attempt to give a partial answer to the second question. These indefinite \(\theta\)-functions are modified versions of the sums considered by \textit{L. Göttsche} and \textit{D. Zagier} in [Sel. Math., New Ser. 4, No. 1, 69--115 (1998; Zbl 0924.57025)]. We find elliptic and modular transformation properties for these functions. Because of the modifications
the indefinite \(\theta\)-functions are no longer holomorphic (in general). Although the results in this chapter are more general than the results in [Göttsche and Zagier, loc. cit.], the second question is far from being solved. This is because we only consider indefinite quadratic forms of type \((r-1,1)\). It remains a problem of considerable interest to develop a theory of theta-series for quadratic forms of arbitrary type.
In [Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 47--56 (1988; Zbl 0646.10018)] \textit{G. E. Andrews} gives most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients.
In Chapter 4 we use the results from Chapter 2, together with identities for the fifth and seventh order function and similar identities for other mock \(\theta\)-functions, to get the modular transformation properties of the seventh-order mock \(\theta\)-functions and most of the fifth-order functions. The final result is that we can write each of these mock \(\theta\)-functions as the sum of two functions \(H\) and \(G\), where:
-- \(H\) is a real-analytic modular form of weight \(1/2\) and is an eigenfunction of the appropriate Casimir operator with eigenvalue \(3/16\) (this is also the eigenvalue of holomorphic modular forms of this weight; for the theory of real-analytic modular forms see for example [\textit{H. Maaß}, Lectures on modular functions of one complex variable. Lectures on Mathematics and Physics. Mathematics. 29. Bombay: Tata Institute of Fundamental Research (1964; Zbl 0254.10018), Ch. IV]); and
-- \(G\) is a theta series associated to a negative definite unary quadratic form, i.e. has the form
\(\sum\text{sgn}(f)\beta(2f^2y)e^{-\pi if^2\tau-2\pi ifb}\),
where \(f\) ranges over a certain arithmetic progression \(a\mathbb Z+b\) \((a, b\in\mathbb Q)\), \(\tau=x+iy\in\mathcal H\) and \(\beta(x)=\int_x^\infty u^{-1/2} e^{-\pi u}\,du\). Moreover \(G\) is bounded as \(\tau\) tends vertically to any rational limit.
This decomposition is thus similar to the one found in [\textit{D. Zagier}, C. R. Acad. Sci., Paris, Sér. A 281, 883--886 (1975; Zbl 0323.10021)] for an Eisenstein series of weight \(3/2\), the holomorphic part of the series, in that case, having class numbers as Fourier coefficients.
Many of the results of Chapter 4 could also be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2, i.e. we have actually given 3 approaches to proving modularity properties of the mock \(\theta\)-functions.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Strong lattice reduction is the key element for most attacks against lattice-based cryptosystems. Between the strongest but impractical HKZ reduction and the weak but fast LLL reduction, there have been several attempts to find efficient trade-offs. Among them, the BKZ algorithm introduced by \textit{C. P. Schnorr} and \textit{M. Euchner} [Math. Program. 66, No. 2(A), 181--199 (1994; Zbl 0829.90099); FCT 1991, Lect. Notes Comput. Sci. 529, 68--85 (1991; Zbl 0925.11049)] seems to achieve the best time/quality compromise in practice. However, no reasonable complexity upper bound is known for BKZ, and \textit{N. Gama} and \textit{P. Q. Nguyen} [Eurocrypt 2008, Lect. Notes Comput. Sci. 4965, 31--51 (2008; Zbl 1149.94314)] observed experimentally that its practical runtime seems to grow exponentially with the lattice dimension. In this work, we show that BKZ can be terminated long before its completion, while still providing bases of excellent quality. More precisely, we show that if given as inputs a basis \((b _{i })_{i \leq n } \in \mathbb Q^{n \times n }\) of a lattice \(L\) and a block-size \(\beta \), and if terminated after
\[
\Omega\left(\frac{n^3}{\beta^2}(\log n + \log \log \max_i \|{b}_i\|)\right)
\]
calls to a \(\beta \)-dimensional HKZ-reduction (or SVP) subroutine, then BKZ returns a basis whose first vector has norm
\[
\leq 2 \nu _{\beta}^{\frac{n-1}{2(\beta-1)}+\frac{3}{2}} \cdot (\det L )^{\frac{1}{n}},
\]
where \(\nu _{\beta } \leq \beta \) is the maximum of Hermite's constants in dimensions \(\leq \beta \). To obtain this result, we develop a completely new elementary technique based on discrete-time affine dynamical systems, which could lead to the design of improved lattice reduction algorithms.
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. In Ramanujan's ``lost'' notebook, it is claimed that
\[
\sum_{n=0}^{\infty} a^nq^{n^2} = \prod_{n=1}^{\infty} \left(1 + aq^{2n-1}(1 + y_1(n) + y_2(n) + \cdots) \right),
\]
where
\[
\begin{aligned} y_1(n) &= \frac{\sum_{j \geq n} (-1)^jq^{j^2+j}}{\sum_{j \geq 0} (-1)^j(2j+1)q^{j^2+j}}\\ \text{and} y_2(n) &= \frac{\left(\sum_{j \geq n} (j+1)(-1)^jq^{j^2+j}\right) \left(\sum_{j \geq n} (-1)^jq^{j^2+j} \right) } {\left(\sum_{j\geq 0} (-1)^j(2j+1)q^{j^2+j} \right)^2}. \end{aligned}
\]
This is reminiscent of the identity treated Part VIII: The entire Rogers-Ramanujan function [Adv. Math. 191, No. 2, 393--407 (2005; Zbl 1067.11062)]. Indeed, Andrews uses more or less the same method to prove the above assertion. One notable difference is that the key polynomials were orthogonal in that paper but are not orthogonal in the present case.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. A new model for sporadic task systems is introduced. This model -- the generalized multiframe task model -- further generalizes both the conventional sporadic-task model, and the more recent multiframe model of Mok and Chen. A framework for determining feasibility for a wide variety of task systems is established; this framework is applied to this task model to obtain a feasibility testing algorithm that runs in time pseudo-polynomial in the size of the input for all systems of such tasks whose densities are bounded by a constant less than one. (Provider: Leibiger)
| 0 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Let \(\operatorname{spt}(n)\) denote the total number of appearances of the smallest parts in all the partitions of \(n\). In 1988 [Trans. Am. Math. Soc. 305, No. 1, 47--77 (1988; Zbl 0641.10009); Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 29-45 (1988; Zbl 0652.10010)], the second author gave new combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions.
In 2008 [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)], the first author found Ramanujan-type congruences for the spt-function mod 5, 7 and 13. We give new combinatorial interpretations of the spt-congruences mod 5 and 7. These are in terms of the same crank but for a restricted set of vector partitions. The proof depends on relating the spt-crank with the crank of vector partitions and the Dyson rank of ordinary partitions. We derive a number of identities for spt-crank modulo 5 and 7. We prove the surprising result that all the spt-crank coefficients are nonnegative.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Quarantine is often proposed and sometimes used to control the spread of infectious diseases through a human population. Yet there is usually little or no information on the effectiveness of attempting to quarantine humans that is not of an anecdotal or conjectural nature. This paper describes how a compartmental model for the geographic spread of infectious diseases can be used to address the potential effectiveness of human quarantine. The model is applied to data from the historical record in Central Canada around the time of the 1918--19 influenza epidemic. Information on the daily mobility patterns of individuals engaged in the fur trade throughout the region prior to, during, and immediately after the epidemic are used to determine whether rates of travel were affected by informal quarantine policies imposed by community leaders. The model is then used to assess the impact of observed differences in travel on the spread of the epidemic. Results show that when mobility rates are very low, as in this region, quarantine practices must be highly effective before they alter disease patterns significantly. Simulation results suggest, though, that effectiveness varies depending on when the limitation on travel between communities is implemented and how long it lasts, and that a policy of introducing quarantine at the earliest possible time may not always lead to the greatest reduction in cases of a disease.
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This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. Every once in a while a book is published that is so beautiful that one rejoices to be around to read it. The present book is one such. The subject is basic hypergeometric series, or \(q\)-series. These are series on the form \(\sum a_ n\) with \(a_{n+1}/a_ n\) a rational function of \(n\), and some multidimensional extensions. This seems an unpromising topic, yet there are beautiful applications in combinatorics, number theory, statistical mechanics, orthogonal polynomials, Lie and Kac-Moody algebras, group theory, and a marvelous illustration of the power of symbolic algebra on a computer, in this case SCRATCHPAD.
\(q\)-series start with Euler, but the modern period really starts in England, first with Sylvester's work on partitions, and second with a very important series of papers by L. J. Rogers. The work of Rogers was completely ignored until one of his papers was read by Ramanujan twenty years after it was written. The main series of three papers of Rogers have been read seriously by a few people, but the paper that led to this trilogy probably had not been read carefully before Andrews read it, and redid it in Chapter 2. Andrews than goes along the same path Rogers took, but uses SCRATCHPAD to obtain enough data to explain to the reader how this material could have been discovered. After Ramanujan's rediscovery of the Rogers-Ramanujan identities, and his reading of Rogers's paper on them, new proofs of these identities were found by Rogers and Ramanujan. Bailey had read one of the later papers of Rogers carefully, and found a simple method of forming new series from old. This is explained, and illustrated by obtaining Rogers-Ramanujan identities for mod \((2k+1)\), \(k=2,3,....\)
Integrals and series are clearly analogous to each other. There are times when there is much more than an analogy. Using the orthogonality of \(e^{in\theta}\) it is possible to change many integrals into sums, and vice versa. One very important instance of this is the set of integrals, or constant term identities, that start with Selberg's multidimensional beta integral, includes the Dyson-Gunson-Wilson identity, the extension of it proved by Zeilberger and Bressoud, and then up to the Macdonald conjectures. Part of this is treated, including the first inclusion of Selberg's integral with a proof in a book. Since Selberg's paper is not readily obtainable, this is a service to the many who can use this integral.
The other start of a modern treatment of \(q\)-series is Sylvester's work on partitions, with the aim of providing bijective proofs of identities, or showing that two sets are equinumerous by providing an explicit bijection. Some beautiful bijections of Bressoud are given, as is a proof of the very important involution principle of Garsia and Milne. Next, generalized Frobenius partitions are introduced, and are shown to be a very powerful tool to handle certain questions.
The application of most interest outside of pure mathematics is Baxter's solution of the hard hexagon model in two dimensional statistical mechanics. This is a fascinating story, and the end is not in sight. Andrews, Baxter, Forrester and SCRATCHPAD have done some marvelous work, adding many more exactly solved models to those previously solved by Onsager, Baxter and a few others. A small part of this work is outlined here. A most tantalizing set of functions was introduced by Ramanujan in his last letter to Hardy. He called these mock theta functions, and gave examples of some of orders 3, 5 and 7. Watson explained some of the theory of the third order ones in two papers in the mid 1930's, and wrote a bit about the fifth order ones. There is still a lot we do not know about these functions, but Andrews with SCRATCHPAD has made substantial progress. The book ends with a description of how to use SCRATCHPAD to attack problems in this area. First, Andrews shows how the Lusztig, Macdonald, Wall conjecture could have been solved using SCRATCHPAD. This is much easier than the original proof by Andrews. Bailey's work along with the desire to understand the mock theta functions led Andrews to infinitely many identities, with the mock theta functions on the second level. On the first, or easiest, level, he found the remarkable identity;
\[
(\sum^{\infty}_{n=0}q^{n(n+1)/2})^ 3=\sum^{\infty}_{n=0}\sum^{2n}_{j=0}q^{2n^ 2+2n- j(j+1)/2}(1+q^{2n+1})(1-q^{2n+1})^{-1}.
\]
This gives a new proof of Gauß's theorem that every positive integer is the sum of three triangular numbers. If you have not found something of beauty that you like by the end of this book, you have my sympathies. I found a lot.
| 1 |
This book contains a reprinting of 58 papers of George Andrews with commentary. These papers were selected in consultation with Andrews from 292 papers covering the period until 2011. As of May 2015, Andrews has written a total of 309 papers. The papers collected in this volume are divided into 8 chapters:
1. The geometry of numbers.
2. \(q\)-series.
3. Partition identities.
4. Plane partitions.
5. Combinatorics, Fibonacci numbers, and computers.
6. Number theory.
7. Surveys.
8. Education, history, etc.
Each chapter is divided into subsections. A commentary is given at the beginning of each chapter or at the beginning of each subsection. The commentary made by the author gives an overview of the papers and shows how they are related to other papers by the author not included in this volume. A complete list of the 292 papers is given in the front matter. The front matter also contains the author's curriculum vitae, biographical notes, a list of his Ph.D. students and some photos.
The first chapter contains one paper on boundary lattice points of strictly convex bodies in \(\mathbb{R}^n\) written while Andrews was a Fullbright scholar at the University of Cambridge.
The second chapter contains 10 subsections, 374 pages and covers 19 papers. George Andrews is one the world's leading experts on \(q\)-series. Basically, a \(q\)-series is a series that involves the \(q\)-Pochhammer symbol
\[
(a)_n= (a;q)_n= (1- a)(1- aq)\cdots(1- aq^{n-1}).
\]
See \textit{B. C. Berndt}'s article [in: Ramanujan rediscovered. Proceedings of a conference on elliptic functions, partitions, and \(q\)-series in memory of K. Venkatachaliengar, Bangalore, India, June 1--5, 2009. Mysore: Ramanujan Mathematical Society. 31--51 (2010; Zbl 1245.33013)] for a discussion on what a \(q\)-series is. The chapter begins with a paper from the author's Ph.D. thesis on transformation formulas and asymptotic estimates for Ramanujan's third-order mock theta functions. This improved a result of Dragonette. It took over 40 years for \textit{K. Bringmann} and \textit{K. Ono} [Invent. Math. 165, No. 2, 243--266 (2006; Zbl 1135.11057)] to improve on Andrews' results. Often \(q\)-series are also called basic hypergeometric series and these are \(q\)-analogs of hypergeometric series. Basic hypergeometric series were first systematically studied by Heine. The next two papers are on Heine's method of using the \(q\)-binomial theorem to obtain further identities. The now standard reference for basic hypergeometric series is \textit{G. Gasper} and \textit{M. Rahman}'s book [Basic hypergeometric series. 2nd ed. Cambridge: Cambridge University Press (2004; Zbl 1129.33005)]. The next paper is a survey of new results on \(q\)-orthogonal polynomials and is a sample of the work that grew out of the author's sabbatical with Richard Askey during 1975--1976. The quintessential \(q\)-series identity is the First Rogers-Ramanujan Identity
\[
1+ \sum^\infty_{n=1} {q^{n^2}\over(q)_n}= \prod^\infty_{n=0} {1\over (1- q^{5n+1})(1- q^{5n+4})}.\tag{1}
\]
See \textit{G. E. Andrews} [Encyclopedia of Mathematics and its Applications, Vol. 2. Section: Number Theory. Reading, Massachusetts, etc.: Addison-Wesley Publishing Company (1976). Cambridge etc.: Cambridge University Press (1984; Zbl 0655.10001)] for some early history of this identity. It was originally discovered and proved by Rogers in 1894. It was made famous by Ramanujan who rediscovered it without proof sometime before 1913. The author reminds us that only three people independently discovered proofs of the Rogers-Ramanujan identities: \textit{L. J. Rogers} [Proc. Lond. Math. Soc. 24, 337--352 and ibid. 25, 318--343 (1893/1894; JFM 25.0432.01)], \textit{I. Schur} [Berl. Ber. 1917, 302--321 (1917; JFM 46.0201.01; JFM 46.1448.02)], and Baxter. \textit{G. N. Watson} [J. Lond. Math. Soc. 4, 4--9 (1929; JFM 55.0219.09)] showed how (1) is a limiting case of a certain very-well poised hypergeometric series. The next paper illustrates how \(q\)-difference techniques applied to series of this type lead to Rogers-Ramanujan-type identities. The next six papers illustrate Andrews' very powerful method of Bailey pairs and the Bailey chain to discover and prove deep \(q\)-series identities including multiple-series analogs of Rogers-Ramanujan-type identities. This method originated with \textit{W. N. Bailey} [Proc. Lond. Math. Soc., II. Ser. 50, 1--10 (1948; Zbl 0031.39203)] and involved what is now called Bailey's Lemma. A pair of sequences \(\{\alpha_n\}\), \(\{\beta_n\}\) is called a Bailey Pair if
\[
\beta_n= \sum^n_{r=0} {\alpha_r\over (q)_{n-r} (aq)_{n+r}}.
\]
We illustrate with an example from \textit{G. E. Andrews'} paper [Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018)] on the fifth- and seventh-order mock theta functions. If we let \(\beta_n= {1\over (-q;q)_n}\) and \(a=1\) we find using a computer algebra system that
\[
\begin{aligned} \alpha_0 &= 1,\\ \alpha_1 &= q(-3+q),\\ \alpha_2 &= (q^4- 2q^3- q^2+ 2q+2) q^3,\\ \alpha_3 &= (q^9- 2q^8- q^6+ 4q^5- 2q^2- 2) q^6,\\ \alpha_4 &= (q^{16}- 2q^{15}+ q^{12}+ 2q^{11}- 2q^8- 2q^7+ 2q^3+ 2) q^{10}.\end{aligned}
\]
This leads to the conjecture
\[
\alpha_n= q^{n(3n+1)/2} \sum^n_{j=-n} (-1)^j q^{-j^2}- q^{n(3n-1)/2} \sum^{n-1}_{j=-n+1} (-1)^j q^{-j^2}.
\]
Applying a limiting form of Bailey's Lemma [\textit{G. E. Andrews}, Trans. Am. Math. Soc. 293, 113--134 (1986; Zbl 0593.10018), Eq. (3.1)] gives the result that
\[
f_0(q)={1\over (q)_\infty} \sum^\infty_{j=-\infty}\, \sum_{n\geq|j|} (-1)^j q^{n(5n+1)/2-j^2} (1-q^{4n+2}),\tag{2}
\]
where \(f_0(q)\) is Ramanujan's fifth-order mock theta function
\[
f_0(q)= \sum^\infty_{n=0} {q^{n^2}\over (-q;q)_n}.
\]
Andrews' identity (2) was crucial in \textit{D. Hickerson}'s [Invent. Math. 94, No. 3, 639--660 (1988; Zbl 0661.10059)] proof of the mock theta conjectures and crucial in the work of \textit{S. P. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht, Faculteit Wiskunde en Informatica (Diss.) (2002; Zbl 1194.11058)] on transformation formulae for \(f_0(q)\) and other mock theta functions. The next section is on \(q\)-trinomials and the Borwein conjecture. In 1990, Peter Borwein conjectured that the polynomials \(A_n(q)\), \(B_n(q)\), and \(C_n(q)\) each have nonnegative coefficients, where
\[
\prod^n_{j=1} (1-q^{3j-2})(1- q^{3j-1})= A_n(q^3)- qB_n(q^3)- q^2C_n(q^3).
\]
This problem remains unsolved. The next paper is Andrews' introduction to Ramanujan's ``lost'' notebook, and includes a statement of the mock theta conjecture for fifth-order functions mentioned above. Andrews has written 20 papers on the ``lost'' notebook. \textit{G. E. Andrews} and \textit{B. C. Berndt} [Ramanujan's lost notebook. Part I. New York, NY: Springer (2005; Zbl 1075.11001); Part II. New York, NY: Springer (2009; Zbl 1180.11001); Part III. London: Springer (2012; Zbl 1248.11003); Part IV. New York, NY: Springer (2013; Zbl 1288.11002)] have completed the first four of five volumes on the ``lost'' notebook. The events leading Andrews to the discovery of the ``lost'' notebook are chronicled in [\textit{G. E. Andrews}, in: The legacy of Srinivasa Ramanujan. Proceedings of the international conference in celebration of the 125th anniversary of Ramanujan's birth, University of Delhi, Delhi, India, December 17--22, 2012. Mysore: Ramanujan Mathematical Society, 77--88 (2013; Zbl 1308.01007)]. \textit{G. E. Andrews} also wrote a series of nine papers on results from the ``lost'' notebook in [Adv. Math. 41, 137--172 (1981; Zbl 0477.33001); ibid. 41, 173--185 (1981; Zbl 0477.33002); ibid. 41, 186--208 (1981; Zbl 0477.33009); ibid. 53, 55--74 (1984; Zbl 0539.33002); ibid. 61, 156--164 (1986; Zbl 0601.10007)]; with the reviewer, ibid. 73, No. 2, 242--255 (1989; Zbl 0677.10010); with \textit{D. Hickerson}, ibid. 89, No. 1, 60--105 (1991; Zbl 0739.11042); ibid. 191, No. 2, 393--407 (2005; Zbl 1067.11062); ibid. 191, No. 2, 408--422 (2005; Zbl 1067.11063)]. On p. 268 of the volume under review ref. [Book3] should be [Book4]. See [\textit{G. E. Andrews}, Reg. Conf. Ser. Math. 66, xii, 130 p. (1986; Zbl 0594.33001), Section 1.6] on how Andrews' collaboration with Baxter began. \textit{A. Berkovich} and \textit{B. M. McCoy} [Doc. Math., J. DMV Extra Vol. ICM Berlin 1998, Vol. III, 163--172 (1998; Zbl 0973.11090)] outline how the theory of Rogers-Ramanujan identities relate to lattice models of statistical mechanics and conformal field theory. This development began with the Andrews-Baxter collaboration. Two of these papers are included. The last two papers in Chapter 2 are on Pfaff's method and MacMahon's ``Partition analysis''.
The third chapter covers 18 papers. In 1948, \textit{H. L. Alder} [Bull. Am. Math. Soc. 54, 712--722 (1948; Zbl 0035.31201)] proved the nonexistence of certain partition identities of Rogers-Ramanujan-type. This could have proved the death knell of the subject. Luckily, Andrews and Gordon developed the right setting and conditions for these type of partition identities. We mention in particular \textit{G. A. Andrews}' memoir [Mem. Am. Math. Soc. 152, ii, 86 pp. (1974; Zbl 0296.10010)]. We quote Andrews from p. 401: ``The sheer implausibility of partition identities existing at all, combined with cornucopia of results I was finding, provided an almost intoxicating initial decade in my career.'' This discovery of deep partition identities has continued culminating the relatively recent amazing four paramater generalization of the Göllnitz partition identity by \textit{K. Alladi} et al. [in: Unusual applications of number theory. Proceedings of the DIMACS workshop held at Rutgers University, Piscataway, NJ, USA, January 10--14, 2000. Providence, RI: American Mathematical Society (AMS), 1--7 (2004; Zbl 1079.11054); Invent. Math. 153, No. 2, 231--260 (2003; Zbl 1097.05006)]. We let \(\text{spt}(n)\) denote the number of smallest parts in the partitions of \(n\). For example, \(\text{spt}(4)=10\) since the partitions of 4 are
\[
\overline 4,\quad 3+\overline 1,\quad \overline 2+\overline 2,\quad 2+\overline 1+\overline 1,\quad\overline 1+\overline 1+\overline 1+\overline 1,
\]
with the smallest parts overlined. \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)] discovered and proved the amazing congruences
\[
\begin{aligned} \text{spt}(5n+ 4) &\equiv 0\pmod 5,\tag{3}\\ \text{spt}(7n+ 5) &\equiv 0\pmod 7,\tag{4}\\ \text{spt}(13n+6) &\equiv 0\pmod{13}.\tag{5}\end{aligned}
\]
The spt-generating function is related to the rank and crank of partitions, so the congruences \(\text{mod}\,5\) and \(7\) are not too surprising. However, the mod \(13\) congruence (5) is a total surprise. We have found a crank type result [\textit{G. E. Andrews} et al., Ramanujan J. 29, No. 1--3, 321--338 (2012; Zbl 1256.05012)] that explains the mod \(5\) and \(7\) congruences. It is an open problem to find a rank type function that combinatorially explains the mod \(13\) congruence. \textit{K. Bringmann} [Duke Math. J. 144, No. 2, 195--233 (2008; Zbl 1154.11034)] has shown how the spt-generating function is related to a weak harmonic Maass form of weight \(\frac{3}{2}\). \textit{C. Jennings-Shaffer} [with the reviewer, ``Exotic Bailey-Slater spt-functions. II: Hecke-Rogers-type double sums and Bailey pairs from groups \(A\), \(C\), \(E\)'', Preprint, \url{arXiv:1501.06843}; ``Exotic Bailey-Slater spt-functions. I: Group \(A\)'', Preprint, \url{arXiv:1408.5395}; ``Exotic Bailey-Slater spt-functions. III: Bailey pairs from groups \(B\), \(F\), \(G\), and \(J\)'', Preprint, \url{arXiv:1504.05120}; ``Two partition functions with congruences modulo 3, 5, 7, and 13'', Preprint, \url{arXiv:1505.06467}] has shown how to generalize the spt-function using Bailey pairs.
Chapter 4 contains four papers on plane partitions: ``MacMahon's conjecture on symmetric plane partitions'', ``Plane partitions. III: The weak Macdonald conjecture'', ``Plane partitions. IV: A conjecture of Mills-Robbins-Rumsey'', and ``Determinant identities'' (joint with W. H. Burge). The proofs involve incredible binomial and \(q\)-binomial determinant evaluations using the theory of generalized hypergeometric and basic hypergeometric series. We state MacMahon's Conjecture: Let \(M(j,s;n)\) denote the number of symmetric plane partitions with at most \(s\) rows and each part at most \(j\). Then
\[
\sum^\infty_{n=0} M(j,s;n)q^n= \prod^s_{i=1} {(1- q^{j+2i-1})\over (1-q^{2i-1})}\,\prod^s_{h=i+1} {(1- q^{2(j+i+h-1)})\over (1-q^{2(i+h-1)})}.
\]
Chapter 5 contains three papers: ``Catalan numbers, \(q\)-Catalan numbers and hypergeometric series'', ``Fibonacci numbers and the Rogers-Ramanujan identities'', and ``SCRATCHPAD explorations for elliptic theta functions'' (joint with R. J. Baxter). In the last paper, the EIPRA (Euler Product Representation Algorithm) is described. This is used to recognize whether a given \(q\)-series can be written as a nice infinite product. The reviewer has written a MAPLE implementation [Sémin. Lothar. Comb. 42, B42d, 27 p. (1999; Zbl 1010.11072)]. This tool is essential for research in \(q\)-series. Andrews was one of the pioneers for using computer algebra systems to do research in \(q\)-series.
Chapter 6 contains six papers: ``A note on the Bombieri-Selberg formula for algebraic curves'', ``MacMahon's prime numbers of measurement'', ``E\(\Upsilon\)PHKA! \(\text{num}= \Delta+\Delta+\Delta\)'', ``Ramanujan's method in \(q\)-series congruences'' (with R. Roy), ``\(q\)-analogs of the binomial coefficient congruences of Babbage, Wolstenhome, and Glaisher'', and ``On the number distinct multinomial coefficients'' (joint with A. Knopfmacher and B. Zimmermann). My favorite is the third paper in which Andrews proves Gauss' theorem that every positive integer is the sum of three triangular numbers by proving that
\[
\Biggl(\sum^\infty_{n=0} q^{n(n+1)/2}\Biggr)^3= \sum^\infty_{n=0} q^n \sum^{2n}_{j=0} {q^{j(4n+1-j)/2}(1+ q^{2n+1})\over (1- q^{2n+1})},
\]
by his method of Bailey pairs.
Chapter 7 consists of six surveys: ``Partition identities'', ``A general theory of identities of the Rogers-Ramanujan type'', ``Partitions: at the interface of \(q\)-series and modular forms'', ``Applications of basic hypergeometric functions'', ``Problems and prospects for basic hypergeometric functions'', and ``The well-poised thread: an organized chronicle of some amazing summations and their implications''. The fourth paper contains a proof of Bailey's \({_6\Psi_6}\) summation theorem and how it can be used to prove Jacobi's classical sums of squares identity. This was in part \textit{S. C. Milne}'s inspiration of his monograph on sums of squares [Ramanujan J. 6, No. 1, 7--149 (2002; Zbl 1125.11315)]. The fifth paper contains the \(q\)-Dyson conjecture
\[
\text{constant term of} \quad \prod_{1\leq i<j\leq n} \Biggl({x_i\over x_j}\Biggr)_{a_i} \Biggl(q{x_j\over x_i}\Biggr)_{a_i}= {(q)_{a_1+a_2+\cdots+ a_n}\over (q)_{a_1}(q)_{a_2}\cdots(q)_{a_n}},
\]
which was subsequently proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201--224 (1985; Zbl 0565.33001)]. Also, Problems 2, 3, 4 of the fifth paper were in part inspirational to \textit{S. C. Milne} to develop multiple-well-poised series for \(\mathrm{SU}(n)\) [Adv. Math. 36, 169--211 (1980; Zbl 0451.33010); ibid. 58, 1--60 (1985; Zbl 0586.33014)].
The final chapter contains the two papers: ``L. J. Rogers and the Rogers-Ramanujan identities'', and ``Mathematical education: a case for balance''. A flexible job-shop-scheduling problem is an extension of classical job-shop problems that permit an operation of each job to be processed by more than one machine. The research methodology is to assign operations to machines (assignment) and determine the processing order of jobs on machines (sequencing) such that the system objectives can be optimized. This problem can explore very well the common nature of many real manufacturing environments under resource constraints. A genetic algorithm-based approach is developed to solve the problem. Using the proposed approach, a resource-constrained operations-machines assignment problem and flexible job-shop scheduling problem can be solved iteratively. In this connection, the flexibility embedded in the flexible shop floor, which is important to today's manufacturers, can be quantified under different levels of resource availability.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. Gravitational perturbations of the de Sitter spacetime are investigated using the Regge-Wheeler formalism. The set of perturbation equations is reduced to a single second order differential equation of the Heun-type for both electric and magnetic multipoles. The solution so obtained is used to study the deviation from an initially radial geodesic due to the perturbation. The spectral properties of the perturbed metric are also analyzed. Finally, gauge- and tetrad-invariant first-order massless perturbations of any spin are explored following the approach of Teukolsky. The existence of closed-form, i.e., Liouvillian, solutions to the radial part of the Teukolsky master equation is discussed.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. Natural tags based on DNA fingerprints or natural features of animals are now becoming very widely used in wildlife population biology. However, classic capture-recapture models do not allow for misidentification of animals which is a potentially very serious problem with natural tags. Statistical analysis of misidentification processes is extremely difficult using traditional likelihood methods but is easily handled using Bayesian methods. We present a general framework for Bayesian analysis of categorical data arising from a latent multinomial distribution. Although our work is motivated by a specific model for misidentification in closed population capture-recapture analyses, with crucial assumptions which may not always be appropriate, the methods we develop extend naturally to a variety of other models with similar structure.
Suppose that the observed frequencies \(f\) are a known linear transformation \(f=A'x\) of a latent multinomial variable \(x\) with cell probability vector \(\pi=\pi(\theta)\). Given that full conditional distributions \([\theta\,|\,x]\) can be sampled, implementation of Gibbs sampling requires only that we can sample from the full conditional distribution \([x\,|\,f,\theta ]\), which is made possible by knowledge of the null space of \(A'\). We illustrate the approach using two data sets with individual misidentification, one simulated, the other summarizing recapture data for salamanders based on natural marks.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. In this book the author carries on the study on the interaction between mathematics and physics that he began with the book entitled: ``Topology, Geometry and Gauge Fields: Foundations'' [Texts in applied mathematics 25, New York, Springer (1997; Zbl 0876.53002)]. Hereafter this book will be denoted by [N]. In the present book, the subtitle is ``Interactions'', which means the study of the geometry and topology of the interplay between gauge fields and matter fields.
The first chapter (Geometrical Background) provides the geometrical framework (contained in Chapters 3 and 4 of [N]) that is necessary in Chapter 2 (Physical Motivation). This chapter contains the main feature of the book: the study of some events in the world around us can lead to abstract mathematical notions. The examples considered are electromagnetic fields (Coulomb field and monopole field), spin zero electrodynamics (Klein-Gordon equations), spin one-half electrodynamics and SU(2)-Yang-Mills-Higgs theory on \(\mathbb{R}^n\). These examples of classical gauge theories are heuristic and informally discussed and the mathematical notions that arise from its description are principal bundles, characteristic classes, and spinor structures.
Hereafter the contents of the book adapts itself to the structure of the standard texts in differential geometry: Frame Bundles and Spacetimes (Chapter 3), Differential Forms and Integration (Chapter 4), de Rham Cohomology (Chapter 5) and Characteristic Classes (Chapter 6). The book concludes with an Appendix that contains a brief description of the Seiberg-Witten monopoles on \(\mathbb{R}^4\). As in [N], there are many exercises (228) that essentially constitute fragments of proofs of theorems and propositions.
The reviewer considers that the present book (jointly with [N]) gives an excellent pattern to follow in the study of the interaction between mathematics and physics from a mathematical viewpoint, i.e., by introducing heuristically and informally physical problems that motivate the basic mathematical ingredients required for its description and then develop in detail the necessary results of the mathematical theory that arise to obtain the highest information about the initial problems. On the other hand, the book gives a very good motivation for the study of characteristic classes (Chern classes and Stiefel-Whitney classes) and spinor structures (one proves that several spacetimes admit spinor structures).
The proofs are well structured and the parts left to the reader have as goal, according to the author, that the reader participates actively in their development. However, some of them can be slightly simplified. For example, the author constructs a partition of unity subordinated to an open covering of a smooth manifold with the same index set (Corollary 3.1.5, where there is an erratum: \(\sum^\infty_{k=1} \varphi_k (x)=1\) ought to be \(\sum_{\alpha \in A}\varphi_\alpha (x)=1)\) and this is not used in many proofs, with the consequent simplification (Theorem 3.1.6 (existence of a Riemannian metric), Theorem 3.1.7 (existence of a connection) and some others).
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This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane-layer geometry, this can be seen as classical Rayleigh-Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appear as the depth -- and therefore the effective Rayleigh number -- of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.
| 0 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. Dieser Artikel erschien in dem im Zbl 0213.00103 angezeigten Sammelwerk.]
Aus der Einleitung des Verf.: ``The \(\ell\)-adic étale cohomology of algebraic varieties is much richer than the classical cohomology, in that Galois groups operate on it. This opens up a new field of inquiry, even in the classical case. Although theorems seem scarce, the soil is fertile for conjectures\(\ldots\). The main idea is, roughly speaking, that a cohomology class which is fixed under the Galois group should be algebraic when the ground field is finitely generated over the prime field.''
In den ersten beiden Paragraphen (\S1. The \(\ell\)-adic cohomology; \S2. Cohomology classes of algebraic cycles) wird diese Idee näher ausgeführt. Sei \(k\) ein Körper, \(\bar k\) eine algebraisch abgeschlossene Erweiterung von \(k\), \(G(\bar k,k)\) die Automorphismengruppe von \(\bar k,k\). Sei \(V\) ein geometrisch irreduzibles und glattes projektives Schema über \(k\), sei \(\bar V=V\times\bar k\). Auf den \(\ell\)-adischen Kohomologiegruppen \(H_{\ell}^i(\bar V)\) operiert \(G(\bar k,k)\); um die ,,richtigen \(\ell\)-adischen Orientierungen'' zu erhalten, werden ,,tordierte'' Kohomologiegruppen \(H_{\ell}^i(\bar V)(m)\) eingeführt. Einem irreduziblen Unterschema \(X\) von \(\bar V\) der Kodimension \(i\) wird eine Kohomologieklasse \(c(X)\in H_{\ell}^{2i}(\bar V)(i)\) zugeordnet. \(X\) ist ,,definiert'' über einer endlichen Erweiterung von \(k\) und wird daher durch eine offene Untergruppe \(U\) von \(G(\bar k,k)\) festgelassen. Dann wird auch \(c(X)\) durch \(U\) festgelassen. Die erste Vermutung des Verf. besteht darin, daß die Umkehrung dieser Tatsache richtig ist (falls \(k\) über dem Primkörper endlich erzeugt wird). Aus dieser Vermutung ergibt sich u.a. fur Abelsche Mannigfaltigkeiten eine Konsequenz (Formel (8)), die inzwischen vom Verf. selbst bewiesen wurde (falls \(k\) endlicher Körper ist; s. die Arbeit in Invent. Math. 2, 134--144 (1966; Zbl 0147.20303). Gegenwärtig gibt es auch weitere Information über die in \S1 betrachteten(von Serre eingeführten) Lieschen Algebren. Siehe \textit{J.-P. Serre} [Abelian \(\ell\)-adic representations and elliptic curves. New York etc.: Benjamin (1968; Zbl 0186.25701)].
In \S3 diskutiert Verf. den Zusammenhang seiner Vermutungen mit den Zetafunktionen von Varietäten über endlichen Körpern. Im Mittelpunkt steht die Vermutung, daß die Ordnung des Pols der Zetafunktion \(\zeta(V,s)\) an der Stelle \(s=i\) gleich dem Rang der durch die algebraischen Zyklen der Kodimension \(i\) erzeugten Gruppe ist (Formel (12)).
In \S4 wird nur vorausgesetzt, daß \(k\) über seinem Primkorper endlich erzeugt ist. Verf. kombiniert seine Vermutungen mit den bekannten Hypothesen von Birch und Swinnerton-Dyer uber die Zetafunktion von \(V/k\). Ein angekündigtes Ergebnis von \textit{H. Pohlmann} ist inzwischen publiziert [Ann. Math. (2) 88, 161--180 (1968; Zbl 0201.23201)].
Weitere ergänzende bibliographische Angaben:
\textit{A. Grothendieck} [``Standard conjectures on algebraic cycles.'' Algebr. Geom., Bombay Colloq. 1968, 193--199 (1969; Zbl 0201.23301)];
\textit{P. Swinnerton-Dyer} [``The conjectures of Birch and Swinnerton-Dyer, and of Tate.'' Proc. Conf. local Fields, NUFFIC Summer School Driebergen 1966, 132--157 (1967; Zbl 0197.47101)];
the author [``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog.'' Sém. Bourbaki 1965/66, Exp. No. 306, 415--440 (1966; Zbl 0199.55604)];
\textit{J.-P. Serre} [Sém. Delange-Pisot-Poitou 11 (1969/70), Exp. No. 19 (1970; Zbl 0214.48403)].
| 1 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. In this paper we study the existence and uniqueness of weak solutions for a nonlinear wave equation associated with a nonlinear boundary condition of many-point type.
| 0 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. The \(l\)-primary part of \(K_2F\), \(F\) an algebraic number field, can be described in terms of the Galois cohomology of \(F\) provided one knows the \(\mathbb Z_l\)-rank of a particular cohomology group. In this letter the author shows that this rank can be determined by using Iwasawa's theory of \(\mathbb Z_l\)-extensions.
For the entire collection see Zbl 0265.00008.
| 1 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. Let \((E,\|\cdot\|_E)\) be a separable real Banach space, \(\Theta\subset\mathbb{R}^d\) be a nonempty subset. Let \(X_t\), \(t\in \Theta\), be an \(E\)-valued random field satisfying \(E\| X_t \|^p_E< \infty\), \(t\in\Theta\), and \(E\| X_t- X_s \|^p_E \leq c| t-s|^{d+\beta}\), \(t,s\in \Theta\), for some constants \(c> 0\), \(\beta> 0\) and \(p\geq 1\). Furthermore, suppose that one of the following two conditions hold: (i) \((d+\beta)/p\in (0,1]\); (ii) \(\Theta\) is a domain in \(\mathbb{R}^d\). Then there exists a continuous modification \(\widetilde X\) of \(X\) that is locally Hölder continuous with exponent \(\gamma\in (0, \beta/ p]\).
| 0 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. Review of [Zbl 1407.01030; Zbl 1386.11002].
| 1 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. Dieser Band umfasst den Briefwechsel der Jahre 1660 und 1661. Obgleich der Zeitraum kurz ist, beläuft sich dennoch die Anzahl der hier mitgeteilten Briefe auf 246 nebst 24 im Supplement. Die bedeutendsten Correspondenten sind: Boulliau, Fr. van Schooten, Hevelius, Stevin, Wallis, Leopold de Medicis, Heynsius, die Brüder Constantijn und Lodewijk Huygens, Moray, Oldenburg, Thevenot und Fermat. Es werden in diesem Briefwechsel die verschiedensten Gegenstände auf mathematischen und physikalischem Gebiet abgehandelt, u. a. die wichtigsten Entdeckungen, bei denen Ch. Huygens das Seinige gethan hat. So liegt hier auch der vollständige Briefwechsel über die Erfindung der Pendeluhr vor, wobei Huygen's Berechtigung Galilei gegenüber deutlich hervortritt. Von nicht geringerer Wichtigkeit ist der Briefwechsel über die Erfindung des Fernrohres, über die Versuche mit der eben erfundenen Luftpumpe und über den Ring des Saturn. Ein alphabetisches Verzeichnis der Correspondenten und der in den Briefen genannten Personen und Sachen ist wieder beigegeben. Dann folgen eine kurze Angabe aller behandelten Gegenstände und ein Verzeichnis der in diesem und in den beiden vorigen Bänden anzubringenden Verbesserungen. Im Texte findet man Abbildungen der von Galilei entworfenen Pendeluhren, eines von Hevelius wahrgenommenen Halo, des Planeten Saturn, wie Huygens denselben zeichnete, und eines seiner langen Fernröhre. Am Ende stehen Briefe in Facsimile von Fr. van Schooten und Bl. Pascal. Die ganze Ausstattung und Ausführung ist wieder mustergültig.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. We construct the graviton propagator in the \(n\)-dimensional anti-de Sitter spacetime in the most general covariant gauge. We then study the behaviour of this propagator for different values of the gauge parameters. We will show that in any gauge, apart from the Landau gauge, the graviton propagator in the AdS spacetime contains a complicated term involving the derivative of a hypergeometric function which cannot be expressed in terms of elementary functions. We do our calculations in the Euclidean approach.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. Recent research on mathematics reforms in the United States indicates that the reforms are not yet widely implemented. Generally, this claim results from looking at the extent to which teachers use curricular materials or engage in particular classroom practices. This article moves beyond disparate questions of use and practice to examine interactions between teachers and curricula as evidenced by their enactments of whole-number lessons from a Standards-based curriculum. Specifically, we analyze videorecorded 1st- and 2nd-grade classroom lessons in terms of students' opportunities to reason and communicate about mathematics. This analysis indicates that the level of fidelity to the written curriculum differs from the level of fidelity to the authors' intended curriculum during lesson enactments. Drawing on this analysis, this article explores how curricula support and hinder teachers as they engage students in opportunities to learn mathematics and how teachers' instructional moves and choices impact the enactment of curricula.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. This monograph is a good introduction to the mathematical foundations of special relativity, distinguishing strictly between doing mathematics and appealing to physical arguments to motivate or interpret mathematics. It consists of three chapters and two appendices. The first chapter is concerned with the basic geometrical aspects of a 4-dimensional real vector space endowed with a nondegenerate, symmetric, bilinear form of index one and its associated group of orthogonal transformations. In the second chapter the reader can enjoy the beautiful interactions between the mathematics of skew-symmetric linear transformations and the physics of electromagnetic fields. The final chapter gives a readable introduction to the algebraic theory of spinors. The first appendix deals with the topology devised by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174-181 (1976; Zbl 0319.54005)]. The second appendix elaborates upon the essential 2-valuedness of spinors and its physical significance. Familiarity with linear algebra and general topology will do to read this rewarding monograph.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by \(\sqrt{n}\)) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. The book is intended as a text for a second course in linear algebra. The first six chapters cover most of the material contained in the author's Introduction to Linear Algebra (First edition 1970; Zbl 0216.06001, second edition 1986; Zbl 0577.15001). The presentation is more abstract, however, and proceeds appreciably faster. The author emphasizes that this is to make the book selfcontained but the text is not intended as a substitute for the introductory one. Linear spaces are considered over an arbitrary subfield of the field of complex numbers. As opposed to the introductory text there is a complete theory of determinants, a broader discussion of Hermitian and unitary operators, more about eigenvalue problems. Two chapters are devoted to polynomial matrices and the Jordan normal form. A brief treatment of convexity culminating in the Krein- Milman theorem concludes the book.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. A methodology is presented that determines the staff requirements for a baggage handling operation at an international airport facing high demand. It consists of two sequentially solved integer linear programming models. The first model determines the handler requirement using historical demand data and incorporates the operation's spatial constraints. The second model, taking as input the results of the first model, defines each handler's shift and task assignments. This formulation can be considered as the solution of a set covering problem using feasible shift structures pre-defined by an algorithm. The proposed approach is validated in an application to a real-world case, demonstrating significant improvements over the baggage handling operator's own staff planning decisions.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. This is the second part of the paper and begins with the section 6. The first part was published in the same journal [J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010)].
The author discusses the equivariant localization in order to describe the finite dimensional analog of Witten's partition function. He presents the generalized Duistermaat-Heckman theorem, and the equivariant localization theorem. Next he is concerned with duality and Seiberg-Witten. The notion of duality symmetries has its roots in classical electromagnetic theory and the symmetry might interchange strong and weak coupling. This notion is studied by using the Clifford algebra and the spin structures on a manifold. The coupled Dirac operator acts on the sections in certain spinor bundles \(\mathcal{S(L)}\) and the Seiberg-Witten equations fulfilled by this operator are the Dirac equation and the curvature equation. To any solution \((A,\psi)\) of the Seiberg-Witten map one can associate the fundamental elliptic complex and its cohomology groups have nice interpretations. The author presents an outline of a construction with no details and scarcely a word of explanation for the Witten conjecture asserting that, for certain four-manifolds, the zero-dimensional Seiberg-Witten invariants contain all the information available in all the Donaldson invariants.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. The infinite-dimensional symmetry groups of the (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation is found by the classical Lie group method. The symmetry groups are used to perform symmetry reduction and together with tanh function method to obtain soliton-like solutions. Moreover, a characterization of the corresponding Lie algebra is given.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. See the review of the 2nd edition (1979) in Zbl 0439.53005. Reviews of the 1st edition appeared in Zbl 0202.52201 (Vol. I, II; 1970) and Zbl 0306.53003 (Vol. III-V, 1975).
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. The eighth-century Latin manuscript Milan, Veneranda Biblioteca Ambrosiana, L 99 Sup. contains fifteen palimpsest leaves previously used for three Greek scientific texts: a text of unknown authorship on mathematical mechanics and catoptrics, known as the \textit{Fragmentum Mathematicum Bobiense} (three leaves), Ptolemy's \textit{Analemma} (six leaves), and an astronomical text that has hitherto remained unidentified and almost entirely unread (six leaves). We report here on the current state of our research on this last text, based on multispectral images. The text, incompletely preserved, is a treatise on the construction and uses of a nine-ringed armillary instrument, identifiable as the ``meteoroscope'' invented by Ptolemy and known to us from passages in Ptolemy's \textit{Geography} and in writings of Pappus and Proclus. We further argue that the author of our text was Ptolemy himself.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. Through the gauge field theory, we obtain the solution of the Dirac-Born-Infeld-Anti-de Sitter (DBI-AdS) black hole. In the meantime, according to the relations between the action and the grand partition function, we obtain the grand partition function in the DBI-AdS black hole. The temperature and the potential of the DBI-AdS black hole are gained from differential of the grand partition function. With the thermodynamic relations, other thermodynamics are also obtained. The solution and the thermodynamics of the DBI-AdS black hole are turned out that they can reduce to the case of a charged black hole in four-dimensional space-times.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. See the review in Zbl 0572.30023.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. This is a book on topology and geometry, and like any book on subjects as vast as these, it has a point of view that guided the selection of topics. The author's point of view is that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The goal is to weave together rudimentary notions from the classical gauge theories of physics and the topological and geometrical concepts that became the mathematical models of these notions. The reader is assumed to have a minimal understanding of what an electromagnetic field is, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, and a solid background in real analysis and linear algebra with some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual \(SU(2)\)-connections on \(S^4\) with instanton number \(-1\). This second edition of the book includes a new chapter on singular homology theory and a new appendix outlining Donaldson's beautiful application of gauge theory to the topology of compact, simply connected, smooth 4-manifolds with definite intersection form.
See also the review of the first edition [New York: Springer (1997; Zbl 0876.53002)].
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. The reproducing kernels \(\widetilde{K}_k(x, x')\) and \(K_k(x, x')\) of two subspaces over the space of all polynomials in \(x = (x_1,\dots, x_n)\in\mathbb{R}^n\) with respect to a weight scalar product are considered. The representation of \(\widetilde{K}_k(x, x')\) and \(K_k(x, x')\) in terms of the Gegenbauer polynomials is proved in the case when \(x\) or \(x^\prime\) lie on the ball's surface \(|x|= 1\).
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Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. See the review of the entire collection in Zbl 0153.07403.
| 1 |
Even someone only vaguely familiar with the work of John Tate will be able to guess that his collected works begin with his ``Fourier analysis in number fields and Hecke's zeta-functions'', Tate's thesis written in 1950 and first published in the Brighton proceedings [in: J. W. S. Cassels (ed.) and A. Fröhlich (ed.), Algebraic number theory. London etc.: Academic Press. 305--347 (1967)], where Tate worked out Emil Artin's suggestion to derive the functional equation of Hecke's zeta functions using the newly developed tool of ideles.
Later, Tate worked on the Galois cohomology of number fields (where he formulated a generalization of Artin's reciprocity law as an isomorphism of Tate cohomology groups), function fields, elliptic curves and abelian varieties; the keywords here are Tate cohomology groups, Poitou-Tate duality, and Tate-Shafarevich groups. The Galois-cohomological approach to global class field theory is summarized in his survey [in: Algebraic number theory. London etc.: Academic Press. 162--203 (1967; Zbl 1179.11041)] in the Brighton proceedings.
In between fundamental work on Lubin-Tate formal groups, \(p\)-divisible groups, group schemes and rigid analytic spaces, Tate published on some concrete problems, such as symbols in arithmetic [in: Actes Congr. Intern. Math. 1970, 1, 201--211 (1971; Zbl 0229.12013)] and Milnor groups [with \textit{H. Bass}, Lect. Notes Math. 342, 349--446 (1973; Zbl 0299.12013)], the non-existence of elliptic curves defined over the rationals with rational torsion points of order \(13\) [with \textit{B. Mazur}, Invent. Math. 22, 41--49 (1973; Zbl 0268.14009)] or his beautiful survey on the arithmetic of elliptic curves [Invent. Math. 23, 179--206 (1974; Zbl 0296.14018)]. The last 70 pages of this first volume of Tate's collected works present letters of Tate to Dwork, Serre, Springer, Birch and Atkin. A detailed review of Tate's collected works was published by \textit{J. S. Milne} [Bull. Am. Math. Soc., New Ser. 54, No. 4, 551--558 (2017; Zbl 1369.00040)]; see also [\textit{J. S. Milne}, in: The Abel Prize 2008--2012. Heidelberg: Springer. 259--340 (2014; Zbl 1317.01011)].
The individual articles are: ``Fourier analysis in number fields and Hecke's zeta-functions'', ``A note on finite ring extensions'' (with \textit{E. Artin}) [Zbl 0043.26701], ``On the relation between extremal points of convex sets and homomorphisms of algebras'' [Zbl 0043.11403], ``Genus change in inseparable extensions of function fields'' [Zbl 0047.03901], ``On Chevalley's proof of Lüroth's theorem'' (with \textit{S. Lang}) [Zbl 0047.03802], ``The higher dimensional cohomology groups of class field theory'' [Zbl 0047.03703], ``The cohomology groups in algebraic number fields'' [in: Proceedings of the international congress of mathematicians 1954. Amsterdam, September 2--9. Vol. II. Short lectures. Groningen: Erven P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. 66--67 (1954)], ``On the Galois cohomology of unramified extensions of function fields in one variable'' (with \textit{Y. Kawada}) [Zbl 0068.03402], ``On the characters of finite groups'' (with \textit{R. Brauer}) [Zbl 0065.01401], ``Homology of Noetherian rings and local rings'' [Zbl 0079.05501], ``WC-groups over \(p\)-adic fields'' [Zbl 0091.33701], ``On the inequality of Castelnuovo-Severi'' (with \textit{A. Mattuck}) [Zbl 0081.37604], ``On the inequality of Castelnuovo-Severi, and Hodge's theorem'' [unpublished], ``Principal homogeneous spaces over abelian varieties'' (with \textit{S. Lang}) [Zbl 0097.36203], ``Principal homogeneous spaces for abelian varieties'' [Zbl 0116.38201], ``A different with an odd class'' (with \textit{A. Fröhlich} and \textit{J.-P. Serre}) [Zbl 0105.02903], ``Nilpotent quotient groups'' [Zbl 0125.01503], ``Duality theorems in Galois cohomology over number fields'' [Zbl 0126.07002], ``Ramification groups of local fields'' (with \textit{S. Sen}) [Zbl 0136.02702], ``Formal complex multiplication in local fields'' (with \textit{J. Lubin}) [Zbl 0128.26501], ``Algebraic cycles and poles of zeta functions'' [Zbl 0213.22804], ``Elliptic curves and formal groups'' (with \textit{J. Lubin} and \textit{J.-P. Serre}) [unpublished], ``On the conjectures of Birch and Swinnerton-Dyer and a geometric analog'' [Zbl 0199.55604], ``Formal moduli for one-parameter formal Lie groups'' (with \textit{J. Lubin}) [Zbl 0156.04105], ``The cohomology groups of tori in finite Galois extensions of number fields'' [Zbl 0146.06501], ``Global class field theory'' [Zbl 1179.11041], ``Endomorphisms of Abelian varieties over finite fields'' [Zbl 0147.20303], ``The rank of elliptic curves'' (with \textit{I. R. Shafarevich}) [Zbl 0168.42201], ``Residues of differentials on curves'' [Zbl 0159.22702], ``\(p\)-divisible groups'' [Zbl 0157.27601], ``The work of David Mumford'' [Zbl 0333.01015], ``Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda)'' [Zbl 0212.25702], ``Good reduction of abelian varieties'' [Zbl 0172.46101], ``Group schemes of prime order'' (with \textit{F. Oort}) [Zbl 0195.50801], ``Symbols in arithmetic'' [Zbl 0229.12013], ``Rigid analytic spaces'' [Zbl 0212.25601], ``The Milnor ring of a global field'' [Zbl 0299.12013], ``Appendix to The Milnor ring of a global field'' [unpublished], ``Letter from Tate to Iwasawa on a relation between \(K_2\) and Galois cohomology'' [Zbl 0284.12004], ``Points of order \(13\) on elliptic curves'' (with \textit{B. Mazur}) [Zbl 0268.14009], ``The arithmetic of elliptic curves'' [Zbl 0296.14018], ``The 1974 Fields Medals. I: An algebraic geometer'' [Zbl 1225.01087], ``Algorithm for determining the type of a singular fiber in an elliptic pencil'' [Zbl 1214.14020]. The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle.
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. The author investigates a mechanism in the early universe, or at the Planck length, which could have controlled space-time topological modifications and led to the onset of a spinor structure. This mechanism is thought to provide a gravitational analog of the Aharonov-Susskind Gedankenexperiment to detect the relative rotation of the universe, spinor behavior, and the two homotopy classes of the Lorentz Lie group. Contents include: an introduction, preliminary remarks, magnetic mass and characteristic classes, conformal null boundary and Aharonov-Susskind apparatus, and concluding remarks.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. While intra-atomic f-d hybridization is expected, experimental confirmation of f-d hybridization in the photoemission final state leading to 4f band structure has been limited to 5f systems and compound systems with very shallow 4f levels. We demonstrate that core 4f states can contribute to the valence band structure in a wide band gap dielectric, in this case HfO\(_{2}\) in the photoemission final state. In spite of the complications of sample charging, we find evidence of symmetry in the shallow 4f levels and wave vector dependent band dispersion, the latter consistent with the crystal structure of HfO\(_{2}\).
| 0 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. The paper surveys some background for the Witten conjecture that relates Donaldson invariants with Seiberg-Witten invariants. Starting with an elementary discussion of connections, curvature, etc. and their physical interpretations, the paper reviews the definition of Donaldson invariants. In the case of zero dimensional anti-self-dual moduli spaces, Donaldson invariant is viewed as the Euler number of an infinite rank vector bundle in a familiar way. Following \textit{M. F. Atiyah} and \textit{L. Jeffrey} [J. Geom. Phys. 7, 119--136 (1990; Zbl 0721.58056)], the Euler number in turn is interpreted as a path integral using the Mathai-Quillen formalism. The path integral has been derived by Witten via a physical approach, and it is the starting point for the Witten conjecture, the detail of which will be given in the second part of the series.
| 1 |
This is the second edition of the masterpiece by \textit{G. L. Naber} [The geometry of Minkowski spacetime. An introduction to the mathematics of the special theory of relativity. Applied Mathematical Sciences 92. New York etc.: Springer-Verlag (1992; Zbl 0757.53046)] which received the 1993 CHOICE award for Outstanding Academic Title with the following citation: ``Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.'' Enthusiastic reviews of the first edition came also from the American Mathematical Society (1993) that defined the book ``\dots a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics'' and from the Dutch Mathematical Society (1993) which praised the author by writing ``\dots his talent in choosing the most significant results and ordering them within the book can't be denied. The reading of the book is, really, a pleasure''.
According to the author's intention this monograph provides an introduction to the Special Theory of Relativity emerging from the interaction between \textit{A. Einstein} and \textit{H. Minkowski} [The principle of relativity. Original papers by A. Einstein and H. Minkowski, translated into English by M. N. Saba and S. N. Bose. Calcutta: University Press (1921; JFM 48.1059.09)] that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. As correctly reported by the editorial review on its back cover, in addition to the usual menu of topics one is accustomed to finding in introductions to Special Relativity, this book treats a wide variety of results of more contemporary origin; anyway the editorial suggestion about a treatment presuming only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and some elementary point-set topology in the two appendices, seems undervaluing the threshold of full comprehension that is not inferior to a graduate level in Mathematics or Physics.
In the \textit{Introduction} the author examines the reasons why Minkowski spacetime is generally regarded as the appropriate arena within which to formulate the laws of Physics not referring to gravitational phenomena. After having described the Relativity Principle in the terms suggested by \textit{H. A. Lorentz, A. Einstein, H. Minkowski} and \textit{H. Weyl} [The principle of relativity: A collection of original memoirs on the special and general theory of relativity. Reprint of the 1923 edition. New York: Dover Publications (1952; Zbl 0047.20601)], the author motivates our underlying model of the ``event world'' through the characterization of the causal automorphisms (a mapping composed of translations, positive scalar multiples and linear transformations) proved by \textit{E. C. Zeeman} [J. Math. Phys. 5, 490--493 (1964; Zbl 0133.23205)]. It is abundantly clarified how such a model consists of a 4-dimensional real vector space on which a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) is defined, and its associated group of orthogonal transformations (the Lorentz group) is presented.
The basic geometrical information about this model is supplied in \textit{Chapter 1} with preliminary material on indefinite inner product spaces, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parameterization of timelike curves, the reversed Schwarz and triangle inequalities and the theorem on measuring proper spatial separation with clocks proved by \textit{A. A. Robb} [Geometry of time and space. Cambridge: Univ. Press (1936; Zbl 0013.23303)]. In this chapter the reader can also find a kinematic discussion of time dilation, the relativity of simultaneity, length contraction, the composition law for velocities, the hyperbolic motion, the construction of 2-dimensional Minkowski diagrams and a variety of related paradoxes. \textit{Chapter 1} also contains the definitions of the causal and chronological precedence relations and a detailed proof of Zeeman's theorem on the causal automorphism based on some results by \textit{G. Salmon} [A treatise on the analytic geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition, in 2 vols. Vol. I. London: Longmans, Green \(\and\) Co. (1911; JFM 42.0587.01)] and by \textit{N. H. Kuiper} [Linear algebra and geometry. Amsterdam: North-Holland Publishing Company (1962; Zbl 0101.37802)]. Built upon the one-to-one correspondence between vectors in Minkowski spacetime and \(2\times2\) complex Hermitian matrices the intermediate part of \textit{Chapter 1} shows that the fractional linear transformation of the ``celestial sphere'' has the same effect on past null directions as the Lorentz transformation under the spinor map with the stereographic projection by \textit{L. V. Ahlfors} [Complex analysis. An introduction to the theory of analytic functions of one complex variable. 3rd ed. Düsseldorf etc.: McGraw-Hill Book Company (1979; Zbl 0395.30001)]. Immediate consequences are: Penrose's theorem on the apparent shape of a relativistically moving sphere, the existence of invariant null directions for an arbitrary Lorentz transformation and the conclusion that a general Lorentz transformation is completely determined by its effects on any three distinct past null directions. \textit{Chapter 1} ends with the world momentum of material particles and photons and its conservation in contact interactions, from which it is possible to obtain most of the relativistic particle mechanics included the transverse Doppler effect (TDE) experimentally verified by \textit{Herbert E. Ives} and \textit{G. R. Stilwell} in 1938.
\textit{Chapter 2} describes an electromagnetic field at a point in Minkowski spacetime as a linear transformation skew-symmetric with respect to the Lorentz inner product whose algebraic structure is analyzed also using the Cayley-Hamilton Theorem reported by \textit{I. N. Herstein} [Topics in algebra. New York-Toronto-London: Blaisdell Publishing Company, a division of Ginn and Company (1964; Zbl 0122.01301)] and a result from \textit{S. Lang} [Linear algebra. 3rd ed. New York etc.: Springer-Verlag (1987; Zbl 0618.15001)]. The energy-momentum transformation is introduced for an arbitrary skew-symmetric linear transformation and then the Lorentz World Force is solved for charged particles moving in constant electromagnetic fields. \textit{Chapter 2} culminates with variable fields and introduces the Maxwell's (source free) equations from the skew-symmetric bilinear form (bivector associated with the linear transformation representing the field) and its dual. In spite of his elegant illustration of the Electromagnetic Theory the author admits the existence of logical and calculational difficulties which could be better deepened by reading \textit{S. Parrott} [Relativistic electrodynamics and differential geometry. New York etc.: Springer-Verlag (1987; Zbl 0609.53045)].
\textit{Chapter 3} is a detailed exposition of the algebraic Theory of Spinors devised by \textit{E. Cartan} [The theory of spinors. Rev. ed. Paris: Hermann \(\and\) Cie. (1966; Zbl 0147.40101)], applied to Maxwell equations by \textit{O. Laporte} and \textit{G. E. Uhlenbeck} [Phys. Rev., II. Ser. 37, 1380--1397 (1931; Zbl 0002.09001)], developed by \textit{O. Veblen} [Science, New York 80, 415--419 (1934; Zbl 0010.13301); C. R. Congr. Int. Math. 1, 111--127 (1937; Zbl 0018.32604)] and by \textit{W. T. Payne} [Am. J. Phys. 20, 253--262 (1952; Zbl 0046.43705)] and by \textit{W. L. Bade} and \textit{H. Jehle} [Rev. Mod. Phys. 25, 714--728 (1953; Zbl 0051.20705)] and further refined by \textit{E. D. Bolker} [Am. Math. Mon. 80, 977--984 (1973; Zbl 0287.55001)]. Some results from \textit{I. M. Gel'fand, R. A. Minlos} and \textit{Z. Ya. Shapiro} [Representations of the rotation and Lorentz groups and their applications. Oxford-London-New York-Paris: Pergamon Press (1963); Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. (1958; Zbl 0108.22005)] play an essential role in applying spinors in Minkowski spacetime, especially when representing \(\mathrm{SL}(2,{\mathbb C})\). \textit{Chapter 3} offers also a ``Petrov-type'' classification of electromagnetic fields (in both tensor and spinor form) and a spinor equivalent of the energy-momentum transformation used to give a proof of the Dominant Energy Condition.
Recent astronomical observations suggest that the expansion of our own Universe is accelerating, rather than slowing down, according to the studies by \textit{W. de Sitter} [Bull. Astron. Inst. Netherlands 7, 97--105 (1933; Zbl 0007.33103); Bull. Astron. Inst. Netherlands 7, 205--216 (1934; Zbl 0009.33403); Proc. Akad. Wet. Amsterdam 37, 597--601 (1934; Zbl 0010.28301)]. Therefore, beyond indicating how to adapt Special Relativity to the presence of not negligible gravitational fields, the new \textit{Chapter 4} explores some features of the ``de Sitter Universe'', a model markedly different from Minkowski spacetime that is leading to an unexpected flourishing literature from many researchers like \textit{S. Akcay} and \textit{R. A. Matzner} [Classical Quantum Gravity 28, No. 8, Article ID 085012, 26 p. (2011; Zbl 1216.83033)], \textit{Ion I. Cot{\b a}escu} and \textit{C. Crucean} [Prog. Theor. Phys. 124, No. 6, 1051-1066 (2010; Zbl 1213.83141)], \textit{M. Faizal} [Classical Quantum Gravity 29, No. 3, Article ID 035007, 10 p. (2012; Zbl 1235.83045)], \textit{D. Bini, G. Esposito} and \textit{A. Geralico} [Gen. Relativ. Gravitation 44, No. 2, 467--490 (2012; Zbl 1235.83039)], \textit{D.-Y. Jia, R.-H. Yue} and \textit{S.-M. Huang} [Commun. Theor. Phys. 55, No. 1, 75--79 (2011; Zbl 1223.83030)].
\textit{Appendix A} investigates the ``path topology'' for \(M\) by \textit{S. W. Hawking, A. R. King} and \textit{P. J. McCarthy} [J. Math. Phys. 17, 174--181 (1976; Zbl 0319.54005)] who based their work on the homeomorphism group of the ``fine topology'' by \textit{E. C. Zeeman} [Topology 6, 161--170 (1967; Zbl 0149.41204)], not neglecting to underline how, in many topological ways, \(R^ 4\) is unique among the Euclidean spaces \(R^ n\) as shown by \textit{M. H. Freedman} and \textit{F. Luo} [Selected applications of geometry to low-dimensional topology. Providence, RI (USA): American Mathematical Society (AMS) (1989; Zbl 0691.57001)]. Although the author has massively contributed to topology [\textit{G. L. Naber}, Topology, geometry, and gauge fields. Interactions. New York, NY: Springer (2000; Zbl 0979.53001); 2nd ed. New York, NY: Springer (2011; Zbl 1233.53004); Topology, geometry and gauge fields: Foundations. New York, NY: Springer (1997; Zbl 0876.53002); 2nd ed. Berlin: Springer (2011; Zbl 1231.53002); J. Geom. Symmetry Phys. 2, 27--123 (2004; Zbl 1079.58010); J. Geom. Symmetry Phys. 3, 1--83 (2005; Zbl 1080.58014)], in this book he has decided, with commendable humility, to adopt \textit{S. Willard} [General topology. Reading, Mass. etc.: Addison-Wesley Publishing Company (1970; Zbl 0205.26601)] as canonical reference.
In \textit{Appendix B} the author elaborates upon the essential 2-valuedness of spinors and its physical significance and he also discusses Dirac's famous ``Scissors Problem'' and its relation to the notion of a two-valued representation of the Lorentz group. The best source for most of the necessary material is \textit{M. J. Greenberg} [Lectures on algebraic topology. New York-Amsterdam: W.A. Benjamin, Inc. (1967; Zbl 0169.54403)]; the remaining part can be derived from \textit{R. P. Feynman, R. B. Leighton} and \textit{M. Sands} [The Feynman lectures on physics. I: Mainly mechanics, radiation, and heat. II: Mainly electromagnetism and matter. 2nd printing. III: Quantum mechanics. Reading, Mass.-Palo Alto-London: Addison Wesley Publishing Company, Inc. (1965; Zbl 0131.38703)] and from \textit{A. M. R. Magnon} [J. Math. Phys. 28, 1364--1369 (1987; Zbl 0638.53067)] and further from \textit{M. Spivak} [A comprehensive introduction to differential geometry. Vol. 1--5. 3rd ed. with corrections. Houston, TX: Publish or Perish (1999; Zbl 1213.53001)].
The large amount of exercises disseminated in the book, each one fundamental for the development, is a peculiarity of the author in order to encourage an active participation on the part of the reader. Event-driven programming has found pervasive acceptance, from high-performance servers to embedded systems, as an efficient method for interacting with a complex world. The fastest research Web servers are event-driven, as is the most common operating system for sensor nodes.
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Die Verff. definieren als \(k\)-Graph eine Menge \(S\), deren Elemente Punkte heißen, zusammen mit einer Menge von \(k\)-elementigen Teilmengen von \(S\), die \(k\)-Kanten von \(G_k\) heißen. Man färbe die Punkte von \(G_k\) irgendwie mit \(l\) Farben. Man bestimme diejenigen \(k\)-Kanten von \(G_k\), die wenigstens einen Punkt jeder Farbe enthalten, und bilde die Quotienten der Zahl dieser Kanten und der Zahl aller \(k\)-Kanten von \(G_k\). Die größtmögliche so erhaltene Zahl (also bei allen \(l\)-Färbungen von \(G_k\)) wird mit \(p(G_k,l)\) bezeichnet. Weiter bezeichne \(m(n,k,l)\) den Minimalwert von \(p(G_k,l)\), wo \(G_k\) alle \(k\)-Graphen mit \(n\) Punkten durchläuft, und \(m(k,l)\) den Minimalwert aller \(p(G_k,l)\) mit \(G_k\) endlich. Es wird gezeigt, daß \(p(G_k,l) = m(n,k,l)\) gilt, wenn \(G_k\) der vollständige \(k\)-Graph mit \(n\) Punkten ist. Ferner wird die Abschätzung \(m(n,k,l) > S_2(k,l)\cdot l! \cdot l^{-k}\) bewiesen (wo \(S_2(k,l)\) eine Stirling-Zahl zweiter Art ist). Im Spezialfall, daß \(n\) durch \(l\) teilbar ist, wird \(m(n,k,l)\) exakt angegeben. Als weitere Folgerung ergibt sich zum Beispiel, daß \(\lim_{k \to \infty} m(k,l)=1\) (für jedes \(l\)) ist; für hinreichend großes \(k\) gibt es also für jeden \(k\)-Graphen Färbungen mit \(l\) Farben, so daß die meisten \(k\)-Kanten alle Farben enthalten. Weitere möglichen Verallgemeinerungen werden diskutiert.
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Capillary waves excited by the vertical oscillation of a thin elongated plate below an air-water interface are analyzed using time-resolved measurements of the surface topography. A parametric instability is observed above a well defined acceleration threshold, resulting in a so-called cross-wave, a staggered wave pattern localized near the wavemaker and oscillating at half the forcing frequency. This cross-wave, which is stationary along the wavemaker but propagative away from it, is described as the superposition of two almost anti-parallel propagating parametric waves making a small angle of the order of 20^{o} with the wavemaker edge. This contrasts with the classical Faraday parametric waves, which are exactly stationary because of the homogeneity of the forcing. Our observations suggest that the selection of the cross-wave angle results from a resonant mechanism between the two parametric waves and a characteristic length of the surface deformation above the wavemaker.{
\copyright 2012 American Institute of Physics}
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. This report examines algebraic and combinatorial properties of codes from the point of view of association schemes in which the relations are not necessarily symmetric. For an association scheme with \(n\) classes on the set \(X\), the matrices \(D_i\), \(i=0,\dots,n\), denote the adjacency matrices of the \(n\) relations together with \(D_0=I\). If \(P_{ik}\) is the \(i\)th eigenvalue of \(D_k\) then the eigenmatrices of the scheme are defined as \(P=(P_{ik})\) and \(Q=|X|P^{-1}\). These two matrices turn out to be of great importance to the theory.
Two schemes in particular play prominent roles, namely, the Hamming and Johnson schemes. Let \(F\) be a finite set, \(|F|=q\geq 2\) and \(X=F^n\) its \(n\)-fold Cartesian product. Two points \(x\) and \(y\) are said to be \(i\)th related if the Hamming distance between them, \(d_H(x,y)\), is \(i\). This defines the Hamming scheme \(H(n,q)\) of length \(n\) over \(F\). The Johnson scheme \(J(n,v)\) is defined as follows. Let \(1\leq n\leq v/2\), \(F=\{0,1\}\) and let \(X\) be the set of binary \(v\)-tuples of Hamming weight \(n\). Two points \(x\) and \(y\) of \(X\) are \(i\)th associates in \(J(n,v)\) if \(d_H(x,y)/2=i\). \(J(n,v)\) is a symmetric scheme with \(n\) classes. Concepts of duality and extensions of association schemes are also considered.
Distance distributions of subsets of the set \(X\) on which an association scheme is defined are studied, and these considerations lead to interesting linear programming problems involving the eigenmatrices \(P\) and \(Q\). The notion of a \(T\)-design on a subset \(Y\) of \(X\) is defined and, for sets of the form \(T=\langle 1,2,\dots,\tau\rangle\) and a Hamming scheme defined on \(X\), these are simply orthogonal arrays of strength \(\tau\). For Johnson schemes they are the usual \(t\)-designs. For both Hamming and Johnson schemes, explicit forms for \(P\) and \(Q\) are obtained and, in the case of Hamming schemes, the simple connection between \(P\) and the MacWilliams weight enumeration formula is given. Some classical coding inequalities follow directly from the linear programming bound applied to these two schemes.
In Chapter five, polynomial schemes are introduced and an important analog to Lloyd's theorem on perfect codes in \(P\)-polynomial schemes is proven. The duality between \(P\)-polynomial and \(Q\)-polynomial schemes is considered. Inequalities of Rao (for orthogonal arrays in \(H(n,q)\)) and Wilson and Ray-Chaudhuri (for \(t\)-designs in \(J(n,v)\)) are extended to \(t\)-designs in any \(Q\)-polynomial scheme. The final chapter treats additive codes that are subgroups of \(F^n\), the \(n\)-fold direct product of the abelian group \(F\). The duality among subgroups is studied using group characters. A criterion for deciding when an additive code is a subscheme of a Hamming scheme is given and the result is applied to the Golay code.
This report, which forms the author's thesis at the Université Catholique de Louvain, demonstrates the utility of association schemes in coding and is one of the most important and innovative works in this area in recent years.
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. The author considers general Randers spaces and general Ingarden spaces, which are important classes of Finsler spaces. These spaces are Finsler spaces with the metric function defined by
\[
F(x,y)=\alpha(x,y)+ \beta(x,y),
\]
where \(\alpha (x,y)\) is a pseudo-Riemannian metric which represents the gravitational part of \(F(x,y)\) and the 1-form \(\beta^2\) is the electromagnetic part of \(F(x,y)\).
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Let \(n \geq k \geq l \geq 2\) be integers, and let \(\mathcal{F}\) be a family of \(k\)-element subsets of an \(n\)-element set. Suppose that \(l\) divides the size of the intersection of any two (not necessarily distinct) members in \(\mathcal{F}\). We prove that the size of \(\mathcal{F}\) is at most \(\binom{\lfloor n / l \rfloor}{k / l}\) provided \(n\) is sufficiently large for fixed \(k\) and \(l\).
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Die Sätze über die rationalen und elliptischen Regelflächen, welche Hr. Segre in seinen Arbeiten aufgestellt hat, welche in den Bänden XIX u. XXI der Atti di Torino erschienen sind (vgl. F. d. M. XVI. 1884. 604, JFM 16.0604.01; JFM 16.0604.02; XVIII. 1886. 448, 617, JFM 18.0448.01; JFM 18.0617.02), bilden besondere Fälle allgemeinerer Sätze betreffs der Regelflächen beliebigen Geschlechtes, deren Beweis der Verfasser in einem umfangreichen Aufsatze im XXX. Bande der Math. Ann. unternommen hat. In der Mitteilung, über die hier zu berichten ist und welche jener Abhandlung voraufgeht, hat der Verf. manche seiner Ergebnisse ausgesprochen. Wir beschränken uns auf die Wiedergabe der folgenden:
Eine Regelfläche von dem Geschlechte \(p\) und der Ordnung \(n \geqq 4p\), welche einem \(R_{n-p-i-1}\) (linearen Raume von \(n-p-i+1\) Dimensionen) angehört, wo \(0 < i < p\), besitzt immer eine Leitcurve, welche einem \(R_h\) angehört, wo \(h \leqq i\), und deren Ordnung (wenn man auf ihre Vielfachbeit Rücksicht nimmt) \(\overset{=}< i+h\) ist. Im allgemeinsten Falle, d. h. wenn \(i+1\) willkürliche der Erzeugenden der Regelfläche unabhängig sind, ist \(h = i\), die Oberfläche ist hyperelliptisch und besitzt als doppelte Leitlinie eine rationale Normalcurve \(i^{\mathrm ter}\) Ordnung, welche einem Raume \(R_i\) angehört. Eine Ausnahme findet nur für den Fall \(i = p-1\) statt; denn dann hat die allgemeinste Regelfläche eine einfache einem \(R_{p-1}\) angehörige Leitlinie vom Geschlechte \(p\) und von der Ordnung \(2p-2\). Die Regelflächen von \(R_{n-2p+1}\) ergeben durch ihre Projectionen alle Regelflächen \(n^{\mathrm ter}\) Ordnung und \(p^{\mathrm ten}\) Geschlechts der niederen Räume. Jede Regelfläche ist mit zwei Leitcurven von einer Ordnung \(\overset{=}<\frac12(n+p)\) versehen. Ist \(n+p\) eine gerade Zahl, so besitzt die Oberfläche \(\infty^1\) Minimalcurven von der Ordnung \(\frac 12(n+p)\); ist dagegen \(n+p\) eine ungerade Zahl, so hat sie eine endliche Anzahl von Minimalcurven von der Ordnung \(\frac 12(n+p-1)\). Auf die Betrachtung der Minimalcurven kann man eine neue Einteilung der Regelflächen gründen.
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Let v, k, \(\ell_ 1,...,\ell_ s\) be positive integers with \(v>k>\ell_ s>...>\ell_ 1\geq 0.\) A family \({\mathcal F}\) of k-subsets of a v-set X is called a (v,L,k)-system if \(| F\cup F'|\in L=\{\ell_ 1,...,\ell_ s\}\) for all F, F' in \({\mathcal F}\). The fundamental theorem of \textit{M. Deza, P. Erdős} and \textit{P. Frankl} [Proc. Lond. Math. Soc., III. Ser. 36, 369-384 (1978; Zbl 0407.05006)] is that \(| {\mathcal F}|\leq \prod_{1\leq i\leq s}(v-\ell_ i)/(k-\ell_ i)\) for \(v\geq v_ 0(L,k).\) This upper bound can be regarded as of order \(cv^ s\). So a family of order \(c'v^{s'}\) is large if s-s' is small. Here, families of sections of quadrics and Hermitian varieties in PG(n,q) for sufficiently large n are investigated. All families with \(s-s'\leq 5\) are tabulated. the only ones with \(s-s'=0\) are those with X a conic or a Hermitian curve in PG(2,q) or X an elliptic quadric in PG(3,q).
| 1 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Authenticated key exchange (AKE) is a cryptographic scheme with the aim to establish a high-entropy and secret session key over a insecure communications network. Password-authenticated key exchange (PAKE) assumes that the parties in play share a simple password, which is cheap and human-memorable and is used to achieve the authentication. PAKEs are practically relevant as these features are extremely appealing in an age where most people access sensitive personal data remotely from more-and-more pervasive hand-held devices. Theoretically, PAKEs allow the secure computation and authentication of a high-entropy piece of data using a low-entropy string as a starting point. In this paper, we apply the recently proposed technique introduced in [\textit{J. Ding} et al., ``A simple provably secure key exchange scheme based on the learning with errors problem'', Cryptology ePrint Archive, Report 2012/688 (2012)] to construct two lattice-based PAKE protocols enjoying a very simple and elegant design that is an parallel extension of the class of Random Oracle Model (ROM)-based protocols PAK and PPK [\textit{V. Boyko} et al., Eurocrypt 2000, Lect. Notes Comput. Sci. 1807, 156--171 (2000; Zbl 1082.94535), \textit{P. MacKenzie}, ``The PAK suite: protocols for password-authenticated key exchange'', DIMACS technical report 2002-46 (2002)], but in the lattice-based setting. The new protocol resembling PAK is three-pass, and provides mutual explicit authentication, while the protocol following the structure of PPK is two-pass, and provides implicit authentication. Our protocols rely on the ring-learning-with-errors (RLWE) assumption, and exploit the additive structure of the underlying ring. They have a comparable level of efficiency to PAK and PPK, which makes them highly attractive. We present a preliminary implementation of our protocols to demonstrate that they are both efficient and practical. We believe they are suitable quantum safe replacements for PAK and PPK.
| 0 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. A new and very short proof of Sperner's theorem is given. This is done counting the number of maximal chains containing a given set \(A\) in the lattice of a collection of subsets of a set \(S\).
| 1 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. The weak well-posedness of the weak solution to a three-component Camassa-Holm system with peakons is studied. After using the technique of flow to get an integral system equivalent to the original one in weak sense under the Lagrange coordinates, the main result is accomplished.
| 0 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. The ultimate extension of the Erdös-Ko-Rado theorem is proved. Concluding the work of Frankl, Wilson, and others the largest size of a \(t\)-intersecting \(k\)-family is determined as well as the extreme systems. Specifically, the \(4m\)-conjecture is proved.
| 1 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. A Heron triangle is a triangle whose side lengths and area are integers. Two Heron triangles are amicable provided the area of one is equal to the perimeter of the other (and vice versa).
This paper is devoted to proving that there only exists one pair of amicable Heron triangles, namely the one given by the triangles of side lengths \((3,25,26)\) and \((9,12,15)\). To do so, the authors solely rely on elementary techniques based just on divisibility properties.
| 0 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Two families \(\mathcal{A}\) and \(\mathcal{B}\), of \(k\)-subsets of an \(n\)-set, are cross \(t\)-intersecting if for every choice of subsets \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\) we have \(| A \cap B | \geq t\). We address the following conjectured cross \(t\)-intersecting version of the Erdős-Ko-Rado theorem: For all \(n \geq(t + 1)(k - t + 1)\) the maximum value of \(| \mathcal{A} | | \mathcal{B} |\) for two cross \(t\)-intersecting families \(\mathcal{A}, \mathcal{B} \subset {{[n]} \choose k }\) is \({{n - t} \choose {k - t}}^2\). We verify this for all \(t \geq 14\) except finitely many \(n\) and \(k\) for each fixed \(t\). Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a \(p\)-weight version of the problem, which comes from the product measure on the power set of an \(n\)-set.
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. The author considers the Cauchy problem for the perturbed Liouville equation \(\partial^2_t u-\partial^2_xu+8\exp u=\varepsilon (\partial_x F_1(\partial_xu, \partial_tu)+\partial_tF_2(\partial_xu, \partial_tu))\) supplemented with smooth rapidly decaying initial conditions. An asymptotic solution is constructed with respect to the perturbation parameter \(\varepsilon\) by the two-scale expansion method.
| 0 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. We present a semidefinite programming approach to bound the measures of cross-independent pairs in a bipartite graph. This can be viewed as a far-reaching extension of Hoffman's ratio bound on the independence number of a graph. As an application, we solve a problem on the maximum measures of cross-intersecting families of subsets with two different product measures, which is a generalized measure version of the Erdős-Ko-Rado theorem for cross-intersecting families with different uniformities.
| 1 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Transdermal alcohol biosensors that do not require active participation of the subject and yield near continuous measurements have the potential to significantly enhance the data collection abilities of alcohol researchers and clinicians who currently rely exclusively on breathalyzers and drinking diaries. Making these devices accessible and practical requires that transdermal alcohol concentration (TAC) be accurately and consistently transformable into the well-accepted measures of intoxication, blood alcohol concentration/breath alcohol concentration (BrAC). A novel approach to estimating BrAC from TAC based on covariate-dependent physics-informed hidden Markov models (HMMs) with two emissions is developed. The hidden Markov chain serves as a forward full-body alcohol model with BrAC and TAC, the two emissions, assumed to be described by a bivariate normal which depends on the hidden Markovian states and person-level and session-level covariates via built-in regression models. An innovative extension of hidden Markov modeling is developed wherein the HMM framework is regularized by a first-principles PDE model to yield a hybrid that combines prior knowledge of the physics of transdermal ethanol transport with data-based learning. Training, or inverse filtering, is effected via the Baum-Welch algorithm and 256 sets of BrAC and TAC signals and covariate measurements collected in the laboratory. Forward filtering of TAC to obtain estimated BrAC is achieved via a new physics-informed regularized Viterbi algorithm which determines the most likely path through the hidden Markov chain using TAC alone. The Markovian states are decoded and used to yield estimates of BrAC and to quantify the uncertainty in the estimates. Numerical studies are presented and discussed. Overall good agreement between BrAC data and estimates was observed with a median relative peak error of 22\% and a median relative area under the curve error of 25\% on the test set. We also demonstrate that the physics-informed Viterbi algorithm eliminates non-physical artifacts in the BrAC estimates.
| 0 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. The author proves a generalization of an old theorem by \textit{K. N. Majumdar} [Ann. Math. Stat. 24, 377--389 (1953; Zbl 0051.10802)]. In fact, this paper closes a long process of improvements involving also \textit{P. Frankl} and \textit{Z. Füredi} [Families of finite sets with missing intersections, Finite and infinite sets, 6th Hung. Combin. Colloq., Eger/Hung. 1981, Coll. Math. Soc. János Bolyai 37, 305--318 (1984; Zbl 0575.05001)] and \textit{L. Pyber} [Discrete Math. 52, 253--268 (1984; Zbl 0557.05004)]. If \(\mathcal {L}\) is a set of \(k\) positive integers and \(\{A_1, \dots, A_m \}\) is a family of subsets of an \(n\)-element set such that \(| A_i \cap A_j| \in {\mathcal L}\) for all \(i \not = j\), then \(m \leq \sum_{i=0}^k {n-1 \choose i}\).
| 1 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. This small book represents an almost popular deduction of special relativity theory together with some historical notes accompanied by citations from Einstein and others.
What is different from the usual approach is the use of the functions sinh and cosh in the calulation of eigentime of motion, whereas the velocity is then related to the tanh of the same argument.
| 0 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Let \((r - 1)n \geq rk\) and let \({\mathcal{F}}_1,\dots,{\mathcal{F}}_r \subset\binom{[n]}{k}\). Suppose that \(F_1 \cap \dots \cap F_r \not= \emptyset\) holds for all \(F_i \in {\mathcal{F}}_i\), \(1 \leq i \leq r\). Then we show that \(\prod_{i=1}^r |{\mathcal{F}}_i| \leq {\binom{n-1}{k-1}}^r\).
| 1 |
This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. This article deals with some modification of the well-known convergence conditions of the projective-iterate methods to the equation \(x=f+Tx+\lambda Fx\) with operators \(T,F:\mathbb{X}\to\mathbb{X}\) in a Banach space \(\mathbb{X}=U+V\) that is a direct sum of some of its subspaces \(U\) and \(V\). The corresponding approximations are defined by means of the formulas \(x_ k=f+Tz_ k+\lambda F_{k-1}\) where \(z_ k=x_{k-1}+w_ k\), \(w_ k\in U\), \(x_ k-z_ k\in V\).
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Suppose we are given a bipartite graph with vertex set X, Y, \(| X| =n\), \(| Y| =N\), each point in X(Y) has degree D(d) fixed, respectively, moreover, each pair of points x,x'\(\in X\) has at most D/(log n)\({}^ 3\) (common) neighbours. Let t(X,Y) denote the minimum number of vertices of Y needed to cover all vertices of X. We prove (Theorem 1.1) that t(X,Y)d/n tends to 1 as n tends to infinity.
This result has many applications.
Theorem [the second author, Eur. J. Comb. 6, 69-78 (1985; Zbl 0565.05016)]. Suppose \(k>r>1\) are fixed, \(n\to \infty\). Then there exists a collection of \((1+o(1))\times \left( \begin{matrix} n\\ r\end{matrix} \right)/\left( \begin{matrix} k\\ r\end{matrix} \right)\) k-subsets of an n-set so that each r-subset is contained in at least one member of the collection.
Analogues and strengthenings of this result are deduced. E.g. for vector spaces, orthogonal or simplectic geometries, random collections of k-sets with constant probabilities, etc.
Theorem 3.3. Suppose \({\mathcal G}\) is a graph on v vertices and e edges and \({\mathcal R}\) is a random graph on n vertices and edge probability \(e/\left(\begin{matrix} v\\ 2\end{matrix} \right)\). Then there exists a collection of \((1+o(1))\left( \begin{matrix} n\\ 2\end{matrix} \right)/\left( \begin{matrix} v\\ 2\end{matrix} \right)\) induced subgraphs of \({\mathcal R}\) on v vertices, isomorphic to \({\mathcal G}\) and such that each edge (non-edge) of \({\mathcal R}\) is covered by an edge (non-edge) of a graph in the collection.
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This book is authored by two of the leading researchers in the subject. It presents one of the great appeals of extremal set theory as a subject in which the statements are easily accessible without a lot of mathematical background, yet the ideas and proofs have applications in a wide range of fields including combinatorics, number theory, and probability theory. This book is aimed at mathematically mature undergraduates and highlights the elegance and power of the field of study. The first half of the book provides classic results with some new proofs including a complete proof of the Ahlswede-Khachatrian theorem as well as some recent progress on the Erdős matching conjecture. The second half of the book presents some combinatorial structural results and linear algebra methods including the Deza-Erdős-Frankl theorem, an application of Rödl's packing theorem, an application of semidefinite programming, and very recent progress (obtained in 2016) on the Erdős-Szemeredi sunflower conjecture and capset problem. This book is divided into 32 chapters, each chapter is presented in systematic and understandable ways. The book concludes with a collection of challenging open problems.
\par Chapter 1 is devoted to the introduction of combinatorial, mostly extremal problems concerning systems of subsets of a given finite set. In Chapter 2, the authors discuss natural and useful operations on sets and families of subsets, which are also known as sets systems. Chapter 3 is devoted to some almost trivial proofs of classical results. One important result, which was proved independently by three sets of authors namely Perles and Shelah, Sauer, and Vapnik and Chervonensky is also described. In Chapter 4, the Erdős-Ko-Rado theorem via shifiting is presented. Kindly note that this is one of the oldest results in extremal set theory, a theorem that still influences a good part of the ongoing research in this field. In Chapter 5, the basic ideas and properties of Katona's circle are explained. An alternative proof of the Erdős-Ko-Rado theorem using Katona's circle is presented. The Lubell-Yamamoto-Meshalkin (LYM) inequality [\textit{D. Lubell}, J. Comb. Theory 1, 299 (1966; Zbl 0151.01503); \textit{K. Yamamoto}, J. Math. Soc. Japan 6, 343--353 (1954; Zbl 0056.26301); \textit{L. D. Meshalkin}, Theory Probab. Appl. 8, 203--204 (1963; Zbl 0123.36303); translation from Teor. Veroyatn. Primen. 8, 219--220 (1963)] is presented.
\par Chapter 6 is devoted to the Kruskal-Katona theorem which has many applications outside extremal set theory. In Chapter 7, a short proof of an important classic result of \textit{D. J. Kleitman} [J. Comb. Theory 5, 157--163 (1968; Zbl 0245.05003)] is presented. Chapter 8 is devoted to the Hilton-Milner theorem [\textit{A. J. W. Hilton} and \textit{E. C. Milner}, Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)], and also discusses its latest proof given by Frankl. Chapter 9 is devoted to the Erdős matching conjecture [\textit{P. Erdős}, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8, 93--96 (1965; Zbl 0136.21302)], its proof is discussed in brief. Chapter 10--11 are devoted to the Ahlswede-Khachatrian theorem and its two proofs, one based on the method of generating sets and the other using the pushing-pulling method [\textit{R. Ahlswede} and \textit{L. H. Khachatrian}, Eur. J. Comb. 18, No. 2, 125--136 (1997; Zbl 0869.05066); Combinatorica 19, No. 1, 1--15 (1999; Zbl 0980.05049)], both approaches are purely combinatorial and may be considered ``dual'' to each other in some sense. Those readers who are interested to know more about the Ahlswede and Khachatrian theorem may consult the survey written by \textit{C. Bey} and \textit{K. Engel} [in: Numbers, information and complexity. Dedicated to Rudolf Ahlswede on the occasion of his 60th birthday. Dordrecht: Kluwer Academic Publishers. 45--74 (2000; Zbl 1122.05316)].
\par Chapter 12 is devoted to uniform measure versus product measure, which are closely related to each other. The Erdős-Ko-Rado theorem (EKR) is described and proved on the lines of uniform measure and product measure by \textit{R. Ahlswede} and \textit{G. O. H. Katona} [Discrete Math. 17, 1--22 (1977; Zbl 0368.05001)] and \textit{Y. Filmus} [``Erdős-Ko-Rado for \(\mu_p\) using Katona's circle method'', Preprint, \url{http://www.cs.toronto.edu/~yuvalf/pEKR.pdf}]. Several related results are also discussed. Chapter 13 is devoted to \textit{D. J. Kleitman}'s correlation inequality [J. Comb. Theory 1, 209--214 (1966; Zbl 0148.01105)], and recent results are presented. In Chapter 14, the authors discuss about $r$-cross union families and the extension of the EKR theorem given by \textit{P. Frankl} and \textit{N. Tokushige} [Comb. Probab. Comput. 20, No. 5, 749--752 (2011; Zbl 1283.05259)]. In Chapter 15, some simple applications of the random walk method in extremal problems for finite sets are described. The first serious application of the random walk method is due to \textit{P. Frankl} [in: Combinatorics, Keszthely 1976, Colloq. Math. Soc. Janos Bolyai 18, 365--375 (1978; Zbl 0401.05001)], and it is applicable to cross $t$-interesting families, $r$-wise $t$-interesting families etc. [\textit{P. Frankl} et al., J. Comb. Theory, Ser. A 128, 207--249 (2014; Zbl 1301.05316), \textit{S. J. Lee} et al., Discrete Appl. Math. 216, Part 3, 627--645 (2017; Zbl 1358.05298)].
\par Chapter 16--17 are devoted to $L$-systems or $(K,L)$ systems, the authors introduce their concept and rank, and present fundamental results. P. Frankl showed that, for every rational number $q\ge 1$ there are infinitely many choices of $K$ and $L$ such that $\alpha(K, L)= q$ [J. Comb. Theory, Ser. A 42, 200--206 (1986; Zbl 0603.05001)]. The main idea of this theorem is described in Chapter 17. The general upper bound for $m(n,k,L)$ due to \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 36, 369--384 (1978; Zbl 0407.05006)] is also discussed and the Füredi structure theorem [\textit{Z. Füredi}, Discrete Math. 47, 129--132 (1983; Zbl 0531.05002)] is shown. Lastly, the authors deal with the exponent of a $(10,\{0,1,3,6\})$-system, its properties and related proofs. Chapter 18 is devoted to the DEF theorem. The analogue results to DEF and its related properties and subsequent proofs are presented. Chapter 19 describes the proof of the Füredi structure theorem using properties of hypergraphs, and also related results given by \textit{P. Erdős} and \textit{D. J. Kleitman} [J. Comb. Theory 5, 164--169 (1968; Zbl 0167.22302)]. In Chapter 20, the authors consider sufficient conditions for a hypergraph to have an almost perfect matching and suggest two applications such as Rödl's packing theorem and the construction of a larger $L$-system whose intersection structure is a Steiner system. The detailed description of \textit{V. Rödl}'s packing theorem [Eur. J. Comb. 6, 69--78 (1985; Zbl 0565.05016)] and the extended technique to prove this theorem given by \textit{P. Frankl} and \textit{V. Rödl} [Eur. J. Comb. 6, 317--326 (1985; Zbl 0624.05055)] are shown. The Rödl-Tengan construction of an $L$-system on a Steiner system [\textit{V. Rödl} and \textit{E. Tengan},J. Comb. Theory, Ser. A 113, No. 6, 1214--1218 (2006; Zbl 1098.05078)] and related properties are described in detail.
\par Chapter 21 is devoted to upper bounds using multilinear polynomials. The authors describe \textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}'s theorem [Osaka J. Math. 12, 737--744 (1975; Zbl 0342.05018)], \textit{P. Frankl} and \textit{R. M. Wilson}'s theorem [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)], \textit{M. Deza} et al.'s theorem [Combinatorica 3, 331--339 (1983; Zbl 0528.05012)], and \textit{H. S. Snevily}'s theorem [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)]. Chapter 22 is devoted to applications to discrete geometry. In some problems of discrete geometry, the geometric constraints can be expressed in terms of intersections in hypergraphs, then such problems may be solved by applying the corresponding results on $L$-system. The findings by \textit{R. A. Fisher} [Ann. Eugenics 10, 52--75 (1940; Zbl 0063.01383)], \textit{X. Chen} and \textit{V. Chvátal} [Discrete Appl. Math. 156, No. 11, 2101--2108 (2008; Zbl 1157.05019)], and \textit{P. Frankl} and \textit{R. M. Wilson} [Combinatorica 1, 357--368 (1981; Zbl 0498.05048)] are also discussed. Chapter 23 is devoted to upper bounds using inclusion matrics. The authors present recent findings due to Frankl and Wilson [loc. cit.], and an extension of these results to non-uniform families by \textit{L. Babai} and \textit{P. Frankl} [Linear algebra methods in combinatorics. II. University of Chicago, Department of Computer Science (1992)]. Further, \textit{G. Katona}'s intersecting shadow theorem [Acta Math. Acad. Sci. Hung. 15, 329--337 (1964; Zbl 0134.25101)] and its related properties are discussed. Chapter 24 is devoted to some algebraic constructions for $L$-systems. Please note that designs, codes, and finite geometrics are useful sources for the construction for large $L$-systems. The basic definitions, examples and properties of algebraic constructions are explained. The valuable findings by \textit{M. Deza} et al. [Proc. Lond. Math. Soc. (3) 50, 405--425 (1985; Zbl 0549.05006)] and \textit{N. Tokushige} [J. Comb. Theory, Ser. A 113, No. 3, 420--434 (2006; Zbl 1088.05073)] are presented. Chapter 25 is devoted to oddtown and eventown problems. The eventown/oddtown theorems were proved independently by \textit{E. R. Berlekamp} [Can. Math. Bull. 12, 471--474 (1969; Zbl 0272.05004)] and \textit{J. E. Graver} [Linear Algebra Appl. 10, 111--128 (1975; Zbl 0301.05012)]. Also, \textit{P. Frankl} and \textit{N. Tokushige} [Eur. J. Comb. 51, 280--286 (2016; Zbl 1321.05015)] have discussed uniform eventown problems.
\par Chapter 26 is devoted to the tensor product method. The authors present a literature review based on the works of prior researchers and present basic terminology such as multilinear map, tensor product, symmetric tensors, alternating tensors, symmetric algebra, exterior algebra etc. The applications of symmetric products, alternating products are due to \textit{B. Bollobás} [Acta Math. Acad. Sci. Hung. 16, 447--452 (1965; Zbl 0138.19404)], \textit{L. Lovácz} [Graph theory and related topics, Proc. Conf. Honour W. T. Tutte, Waterloo/Ont. 1977, 1--14 (1979; Zbl 0462.05037)], \textit{P. Frankl} [Eur. J. Comb. 3, 125--127 (1982; Zbl 0488.05004)], and \textit{Z. Füredi} [Eur. J. Comb. 5, 133--136 (1984; Zbl 0546.05048)]. Chapter 27 is devoted to the ratio bound. The Erdős-Ko-Rado theorem is proved using this concept. The ratio bound gives an upper bound for graph invariants using eigenvalues of an adjacency matrix. The independence number is one of the most important graph invariants. The bounding of the independence number by a positive semidefinite matrix is presented. The ratio bound theorem was proved independently by \textit{A. J. Hoffman} [Graph Theory Appl., Proc. advanced Sem. Wisconsin, Madison 1969, 79--91 (1970; Zbl 0221.05061)] and \textit{P. Delsarte} [An algebraic approach to the association schemes of coding theory. Ann Arbor, MI: Historical Jrl. (1973; Zbl 1075.05606)]. The applications of the ratio bound are describe based on the findings of \textit{R. M. Wilson} [Combinatorica 4, 247--257 (1984; Zbl 0556.05039)]. Chapter 28 is devoted to measures of cross independent sets, where the ratio bound (as given in Chapter 27) is extended into two ways by considering, (i) the measure of an independent set instead of an independence number, and (ii) cross independent sets in a bipartite graph, which are a natural expansion of independent sets in a graph. The idea of bounding measures using singular values and singular decomposition, the extension of the ratio bound and its applications are presented. Chapter 29 is devoted to the application of semidefinite programming (an extension of linear programming), which is a strong tool for some combinatorial problems. The semidefinite programming is useful for measure of cross independent sets. The findings by \textit{S. Suda} and \textit{H. Tanaka} [Bull. Lond. Math. Soc. 46, No. 2, 342--348 (2014; Zbl 1285.05181)] and \textit{S. Suda} et al. [Math. Program. 166, No. 1--2 (A), 113--130 (2017; Zbl 1375.05261)] are presented. Chapter 30 is devoted to the cross intersection problem with measures, which is actually a continuation of Chapter 29. The findings by \textit{P. C. Fishburn} et al. [SIAM J. Algebraic Discrete Methods 7, 73--79 (1986; Zbl 0582.60014)], and its extension to two families by Suda, Tanaka and Tokushige [loc. cit.] are presented.
\par Chapter 31 is devoted to capsets and sunflowers. The recent developments of a polynomial method (which is also known as slice rank method), and its two applications are discussed. A result on slice rank given by \textit{T. Tao} [``A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound'', \url{http://terrytao.wordpress.com/2016/05/18/a}] is presented. The authors also describe the proofs for results obtained by \textit{J. S. Ellenberg} and \textit{D. Gijswijt} [Ann. Math. (2) 185, No. 1, 339--343 (2017; Zbl 1425.11020)] and the conjecture introduced by \textit{P. Erdős} and \textit{E. Szemerédi} [J. Comb. Theory, Ser. A 24, 308--313 (1978; Zbl 0383.05002)]. Chapter 32 includes several challenging open problems such as the Chvátal conjecture, Frankl's union-closed conjecture, maximal families without $s+1$ pairwise disjoint sets, the Erdős matching conjecture, the Kleitman matching problem, and a diversity of interesting hypergraphs and missing intersections. Purpose - The purpose of this paper is to examine the calculation of magnetic field distribution in the modular amorphous transformers under short-circuit state including the flux by the voltage supplying. The magnetically asymmetrical transformer (amorphous asymmetrical transformer - AAT) has been compared also with the symmetrical one (amorphous symmetrical transformer - AST).
Design/methodology/approach - 3D field problems were analyzed with total \(\psi\) and reduced \(\phi\) potentials within the finite element method (FEM). The calculated fluxes have been verified experimentally.
Findings - The field method which includes voltage excitation is helpful for flux density (B) calculation and winding reactances determination, as well. Calculations and tests yield similar flux distributions in both AST and AAT constructions. One should emphasize that AAT is better for manufacturing and repairing.
Research limitations/implications - Owing to very thin (\(80\mu\)m) amorphous ribbon, the solid core has been assumed for computer simulations.
Originality/value - Employment of a field method for calculation of the innovative three-phase amorphous modular transformers. New construction of amorphous transformer, i.e. AAT, has been manufactured at Opole University of Technology.
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