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Let \((R,m)\) be a regular local ring of dimension \(d\), \(I\subset R\) an ideal and \(f:X\rightarrow \mathrm{Spec} R\) a map that factors as a finite sequence of blowups with smooth centers and is such that \(I{\mathcal O}_X\) is invertible. Let \(E\) be the closed fiber \(f^{-1}\{m\}\). The vanishing conjecture of the author [Math. Res. Lett. 1, No. 6, 739--755 (1994; Zbl 0844.13015)] says that \(H^i_E(I, (I{\mathcal O}_X)^{-1})=0\) for all \(i\not =d\). \textit{S. D. Cutkosky} [Math. Res. Lett. 1, No. 6, 752--755 (1994; Zbl 0844.13015)] proved it when \(d=2\) for \(R\) essentially of finite type over a field of characteristic \(0\). Here it shows that vanishing holds for those ideals that are finitely supported, that is those for which the centers of the blowups are closed points. Let \(I\) be an ideal in a regular local ring \(R\). The author associates with \(I\) an integrally closed ideal \(\widetilde I\). Several Briançon-Skoda type theorems are proved and used to improve previous results. For example, if the ideal \(I\) is generated by \(\ell\) elements, then \(\widetilde {I^{n + \ell}} \subseteq I^n\) for all \(n \geq 0\). Several conjectures are formulated, motivated and proved in some special cases. One of these conjectures is the vanishing conjecture (related to Grauert-Riemenschneider vanishing), which asserts that certain cohomology groups are zero.
This conjecture is obtained in the appendix for rings essentially of finite type over a field of characteristic zero, as a consequence of a more general vanishing theorem. The paper also elaborates on the two-dimensional case.
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Let \((R,m)\) be a regular local ring of dimension \(d\), \(I\subset R\) an ideal and \(f:X\rightarrow \mathrm{Spec} R\) a map that factors as a finite sequence of blowups with smooth centers and is such that \(I{\mathcal O}_X\) is invertible. Let \(E\) be the closed fiber \(f^{-1}\{m\}\). The vanishing conjecture of the author [Math. Res. Lett. 1, No. 6, 739--755 (1994; Zbl 0844.13015)] says that \(H^i_E(I, (I{\mathcal O}_X)^{-1})=0\) for all \(i\not =d\). \textit{S. D. Cutkosky} [Math. Res. Lett. 1, No. 6, 752--755 (1994; Zbl 0844.13015)] proved it when \(d=2\) for \(R\) essentially of finite type over a field of characteristic \(0\). Here it shows that vanishing holds for those ideals that are finitely supported, that is those for which the centers of the blowups are closed points. Advances in networking and database technology have made global information sharing a reality. Multidatabase systems (MDBSs) represent a promising approach to addressing the challenges of achieving interoperability among multiple pre-existing databases that are highly autonomous and possibly heterogeneous. The performance of an MDBS is greatly dependent on effectiveness of multidatabase query optimization (MQO). However, the unavailability of and uncertainty in the statistics essential to query optimization have made multidatabase query optimization (MQO) significantly more challenging than distributed query optimization.
This research undertook to develop a fuzzy statistics-based MQO approach to addressing statistics estimation and uncertainty problems in an MDBS environment. We analyzed the statistics needed in an MDBS environment and classified them into three categories: point-based, distribution-function-based and dependency-based. Fuzzy numbers were adopted to represent point-based statistics, and a fuzzy polynomial regression method was developed for estimating distribution function-based statistics (i.e., attribute or join selectivity) from a set of subquery results. For dependency-based statistics, a fuzzy regression method was employed for estimating logical-parameter-based local cost functions. Furthermore, methods for ranking the fuzzy numbers that are fundamental to fuzzy-statistics-based MQO were also discussed. The proposed fuzzy statistics estimation methods were illustrated using examples to demonstrate its applicability in supporting MQO.
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Eisenstein's reciprocity law for \(p\)-th powers states that \((\alpha/a) = (a/\alpha)\) for the \(p\)-th power residue symbol, where \(a\) and \(\alpha\) are coprime to \(p\) and \(\alpha\) is semi-primary, i.e., congruent to an integer modulo \(p\). This reciprocity law was a necessary tool for proving higher reciprocity laws in the work of Kummer, Furtwängler and Takagi; Artin first managed to prove the general reciprocity law without using Eisenstein reciprocity. Immediately after Artin had proved his reciprocity law, \textit{H. Hasse} [Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie. I a: Beweise zu Teil I. II: Reziprozitätsgesetz. Leipzig: B. G. Teubner (1930; JFM 56.0165.01)] showed how to derive Eisenstein's law from Artin's.
In the present article, the author gives a simple proof of Eisenstein's reciprocity law for \(p\)-th powers using the reciprocity law for Hilbert symbols. Die Teile I und Ia sind in [Jahresber. Dtsch. Math.-Ver. 35, 1--55 (1926; JFM 52.0150.19) und 36, 233--311 (1927; JFM 53.0143.01)] erschienen und nun als Sonderdruck neu herausgegeben worden. Teil II bildet den Ergänzungsband VI zum Jahresbericht der D. M. V.
Teil II bringt zuerst das allgemeine Artinsche Reziprozitätsgesetz, anschließend die darauf aufbauende Normenresttheorie des Verf., sodann das aus beiden folgende Reziprozitätsgesetz der Potenzreste mit Ergänzungssätzen, die Produktformel und das Reziprozitätsgesetz der Hilbertschen Normenrestsymbole. Es folgen dann als Hauptanwendung des Artinschen Rezipro\-zitätsgesetzes der Furtwänglersche Hauptidealsatz in Artins gruppentheoretischer Deutung, die Verlagerung der Idealklassen bei Übergang zu Oberkörpern, die im Zusammenhang mit dem Artinschen Gesetz stehenden Dichtigkeitssätze von Frobenius und Tschebotaröw (Chebotarev), die Artinschen \(L\)-Reihen und noch eine Übersicht über die mit dem expliziten Reziprozitätsgesetz operierende Fermat-Literatur.
Besprechungen: L. Weisner, Am. Math. Mon. 39, 296 (1932); L. Zányi, Acta Szeged 5, 254--255 (1932); T. Rella, Monatsh. Math. 39, 10--11 (1932); D., Nieuw Arch. (2) 17, II, 192 (1932); Th. Skolem, Norsk Mat. Tidsskr. 14, 97--99, 99--100 (1932).
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Eisenstein's reciprocity law for \(p\)-th powers states that \((\alpha/a) = (a/\alpha)\) for the \(p\)-th power residue symbol, where \(a\) and \(\alpha\) are coprime to \(p\) and \(\alpha\) is semi-primary, i.e., congruent to an integer modulo \(p\). This reciprocity law was a necessary tool for proving higher reciprocity laws in the work of Kummer, Furtwängler and Takagi; Artin first managed to prove the general reciprocity law without using Eisenstein reciprocity. Immediately after Artin had proved his reciprocity law, \textit{H. Hasse} [Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie. I a: Beweise zu Teil I. II: Reziprozitätsgesetz. Leipzig: B. G. Teubner (1930; JFM 56.0165.01)] showed how to derive Eisenstein's law from Artin's.
In the present article, the author gives a simple proof of Eisenstein's reciprocity law for \(p\)-th powers using the reciprocity law for Hilbert symbols. The purpose of this letter is to describe the stability properties of the inverse problem of determining a straight-line crack in a specimen, from measurements of voltages and currents on its boundary. We discuss a stability theorem and its implications for practical applications. The stability behaviour is further studied in several numerical calculations.
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Let \(k\) be a field. The author considers Krull-Schmidt \(k\)-categories \(\mathcal C\) with \(\text{ind}\,\mathcal C\) finite such that \(\hom_{\mathcal C}(I,J)\) is at most one-dimensional for \(I,J\in\text{ind}\,\mathcal C\) and the composition of two non-isomorphisms in \(\text{ind}\,\mathcal C\) is zero. Thus \(\mathcal C\) is essentially given by a quiver \(Q\) with vertex set \(\text{Ob}\,\mathcal C\). The main result states that \(\mathcal C\) is preabelian if and only if \(Q\) has no path of length \(> 2\). Moreover, it is shown that \(\mathcal C\) is abelian if and only if, in addition, every edge of \(Q\) belongs to a path of length two. The author shows that such categories \(\mathcal C\) arise as categories of representations of certain clans without special loops in the sense of \textit{W. W. Crawley-Boevey} [J. Lond. Math. Soc., II. Ser. 40, No. 1, 9--30 (1989; Zbl 0725.16012)]. The author introduces a class of matrix problems, which he calls `clans', as a convenient means of expressing a problem posed by \textit{I. M. Gelfand} [Actes Congr. Int. Math., Nice 1970, 1, 95--111 (1971; Zbl 0239.58004)], as well as problems considered by \textit{L. A. Nazarova} and \textit{A. V. Roĭter} [Funkts. Anal. Prilozh. 7, No. 4, 54--69 (1973; Zbl 0375.15007)], \textit{V. M. Bondarenko} and \textit{Yu. A. Drozd} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 71, 24--41 (1977; Zbl 0429.16026)], and \textit{S. M. Khoroshkin} [Funkts. Anal. Prilozh. 15, No. 2, 50--60 (1981; Zbl 0463.22008)]. Representations of clans are related to representations of quivers and of partially ordered sets.
The author states and proves a classification of the indecomposable representations for a certain class of clans which includes the aforementioned problems. He uses the methods developed in part I [J. Lond. Math. Soc., II. Ser. 38, 385--402 (1988; Zbl 0692.15008)] which are based on the functorial filtration technique of \textit{I. M. Gelfand} and \textit{V. A. Ponomarev} [Usp. Mat. Nauk 23, No. 2, 3--60 (1968; Zbl 0236.22012)] and \textit{C. M. Ringel} [Math. Ann. 214, 19--34 (1975; Zbl 0299.20005)].
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Let \(k\) be a field. The author considers Krull-Schmidt \(k\)-categories \(\mathcal C\) with \(\text{ind}\,\mathcal C\) finite such that \(\hom_{\mathcal C}(I,J)\) is at most one-dimensional for \(I,J\in\text{ind}\,\mathcal C\) and the composition of two non-isomorphisms in \(\text{ind}\,\mathcal C\) is zero. Thus \(\mathcal C\) is essentially given by a quiver \(Q\) with vertex set \(\text{Ob}\,\mathcal C\). The main result states that \(\mathcal C\) is preabelian if and only if \(Q\) has no path of length \(> 2\). Moreover, it is shown that \(\mathcal C\) is abelian if and only if, in addition, every edge of \(Q\) belongs to a path of length two. The author shows that such categories \(\mathcal C\) arise as categories of representations of certain clans without special loops in the sense of \textit{W. W. Crawley-Boevey} [J. Lond. Math. Soc., II. Ser. 40, No. 1, 9--30 (1989; Zbl 0725.16012)]. We show that the ordered Matsumoto \(K_0\)--group of a sustitutive system is derivable from the ordered Cech cohomology of a simple augmentation of the tiling space associated with the substitution. Besides supplying a topological interpretation of the Matsumoto \(K_0\)--group, this leads to finer invariants of the flow equivalence class of substitutive system.
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Let \(k\) be a ring with an identity element and \(k[\![x]\!]\) the ring of formal power series in \(x\) over \(k\). \(\mathcal J(k)\) is defined to be the group of normalised formal power series in \(k[\![x]\!]\) under substitution. That is, the elements of \(\mathcal J(k)\) are formal power series of the form \(f(x)=x+\alpha_1 x^2+\alpha_2x^3+\cdots\). When \(p\) is a prime and \(k=\mathbb{F}_p\) is the finite field of \(p\) elements, \(\mathcal J(\mathbb{F}_p)\) is well studied and known as the `Nottingham group' [see \textit{R. Camina} in New horizons in pro-\(p\) groups, Prog. Math. 184, 205-221 (2000; Zbl 0977.20020) for a survey of results].
In this paper the authors focus on the case when \(k=\mathbb{Z}\), the integers. The authors begin by proving that as a topological group \(\mathcal J(\mathbb{Z})\) is 4-generator. These generators are not defined unambiguously, and the authors go on to show that for almost every choice of generators they generate a free subgroup of rank 4. The term `almost every' is made precise by using definitions borrowed from the theory of entire functions regarding densities of subsets.
The authors go on to compute the real cohomology of \(\mathcal J(\mathbb{Z})\) with uniformly constant support and show that this is naturally isomorphic to the cohomology of the nilpotent part of the Witt algebra. In order to compute the cohomology the authors study the coset space \(\mathcal J(\mathbb{R})/\mathcal J(\mathbb{Z})\), and prove various topological and geometric results about this space. The paper under review provides an extremely useful survey of the current knowledge about the so-called Nottingham group \(\mathcal N\), and of the techniques that have been used to investigate it. This group has been studied originally by group-theorists as a group of power series under substitution; see the papers by \textit{S.~A.~Jennings} [Can. J. Math. 6, 325-340 (1954; Zbl 0058.02201)], \textit{D.~L.~Johnson} [J. Aust. Math. Soc., Ser. A 45, No. 3, 296-302 (1988; Zbl 0666.20016)] and his Ph.D.\ student \textit{I.~O.~York} [Proc. Edinb. Math. Soc., II. Ser. 33, No. 3, 483-490 (1990; Zbl 0723.20010)]. \(\mathcal N\) can be also regarded, number-theoretically, as the group of wild automorphisms of the local field \(\mathbb{F}_q( (t))\); this approach has led among others to the striking result by the author, that every countably-based pro-\(p\) group can be embedded in \(\mathcal N\) [J. Algebra 196, No. 1, 101-113 (1997; Zbl 0883.20015)].
As the author notes, the Nottingham group on the one hand lends itself to detailed and computational investigations; on the other hand, it exhibits many interesting properties, thus providing many examples and counterexamples. This paper is a very handy reference for such an interesting object.
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Let \(k\) be a ring with an identity element and \(k[\![x]\!]\) the ring of formal power series in \(x\) over \(k\). \(\mathcal J(k)\) is defined to be the group of normalised formal power series in \(k[\![x]\!]\) under substitution. That is, the elements of \(\mathcal J(k)\) are formal power series of the form \(f(x)=x+\alpha_1 x^2+\alpha_2x^3+\cdots\). When \(p\) is a prime and \(k=\mathbb{F}_p\) is the finite field of \(p\) elements, \(\mathcal J(\mathbb{F}_p)\) is well studied and known as the `Nottingham group' [see \textit{R. Camina} in New horizons in pro-\(p\) groups, Prog. Math. 184, 205-221 (2000; Zbl 0977.20020) for a survey of results].
In this paper the authors focus on the case when \(k=\mathbb{Z}\), the integers. The authors begin by proving that as a topological group \(\mathcal J(\mathbb{Z})\) is 4-generator. These generators are not defined unambiguously, and the authors go on to show that for almost every choice of generators they generate a free subgroup of rank 4. The term `almost every' is made precise by using definitions borrowed from the theory of entire functions regarding densities of subsets.
The authors go on to compute the real cohomology of \(\mathcal J(\mathbb{Z})\) with uniformly constant support and show that this is naturally isomorphic to the cohomology of the nilpotent part of the Witt algebra. In order to compute the cohomology the authors study the coset space \(\mathcal J(\mathbb{R})/\mathcal J(\mathbb{Z})\), and prove various topological and geometric results about this space. A two degrees of freedom model of two coupled suspension systems characterised by piecewise linear stiffness has been studied. The system, representing a pantograph current collector head, is shown to be sensitive to changes in excitation and system parameters, possessing chaotic, periodic and quasiperiodic behaviour. The coupled system has a more irregular behaviour with larger motions than the uncoupled suspension system, indicating that the response from the uncoupled suspension system cannot be used as a worst case measure. Since small changes in system parameters and excitation affect the results drastically then wear and mounting as well as actual operating conditions are crucial factors for the system behaviour.
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Chaotic behavior in nonlinear autonomous ODEs is often associated with homoclinic orbits. Such homoclinic structures, in turn, are characterized by the celebrated Smale-Shil'nikov-Birkhoff Theorem. In this setting, the corresponding stable and unstable manifolds have the entire homoclinic orbit in common. Contrary to this autonomous situation, the stable and unstable integral manifolds (the authors speak of fiber bundles) of nonautonomous ODEs typically intersect transversally in isolated points.
In this interesting paper, the authors consider homoclinic orbits of nonautonomous ODEs and study their behavior under one-step discretization. The first part provides a corresponding persistence result and related error estimates. Moreover, an algorithm due to \textit{J. P. England} et al. [SIAM J. Appl. Dyn. Syst. 3, No. 2, 161--190 (2004; Zbl 1059.37067)] is generalized to nonautonomous systems. Three examples convincingly illustrate the obtained result: An artificial one serves to confirm the error estimates, a periodic one reveals the influence of an underlying autonomous system and the final realistic example comes from mathematical biology. We present an algorithm to compute the one-dimensional stable manifold of a saddle point for a planar map. In contrast to current standard techniques, here it is not necessary to know the inverse or approximate it, for example, by using Newton's method. Rather than using the inverse, the manifold is grown starting from the linear eigenspace near the saddle point by adding a point that maps back onto an earlier segment of the stable manifold. The performance of the algorithm is compared to other methods using an example in which the inverse map is known explicitly. The strength of our method is illustrated with examples of noninvertible maps, where the stable set may consist of many different pieces, and with a piecewise-smooth model of an interrupted cutting process. The algorithm has been implemented for use in the DsTool environment and is available for download with this paper.
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Chaotic behavior in nonlinear autonomous ODEs is often associated with homoclinic orbits. Such homoclinic structures, in turn, are characterized by the celebrated Smale-Shil'nikov-Birkhoff Theorem. In this setting, the corresponding stable and unstable manifolds have the entire homoclinic orbit in common. Contrary to this autonomous situation, the stable and unstable integral manifolds (the authors speak of fiber bundles) of nonautonomous ODEs typically intersect transversally in isolated points.
In this interesting paper, the authors consider homoclinic orbits of nonautonomous ODEs and study their behavior under one-step discretization. The first part provides a corresponding persistence result and related error estimates. Moreover, an algorithm due to \textit{J. P. England} et al. [SIAM J. Appl. Dyn. Syst. 3, No. 2, 161--190 (2004; Zbl 1059.37067)] is generalized to nonautonomous systems. Three examples convincingly illustrate the obtained result: An artificial one serves to confirm the error estimates, a periodic one reveals the influence of an underlying autonomous system and the final realistic example comes from mathematical biology. We study the fixed point theorems for nonspreading mappings, defined by Kohsaka and Takahashi, in Banach spaces but using the sense of norm instead of using the function \(\phi\). Furthermore, we prove a weak convergence theorem for finding a common fixed point of two quasi-nonexpansive mappings having demiclosed property in a uniformly convex Banach space. Consequently, such theorem can be deduced to the case of the nonspreading type mappings and some generalized nonexpansive mappings.
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The authors compare two different Yang-Mills theories -- one introduced by \textit{A. Connes} and \textit{M. A. Rieffel} [Contemp. Math. 62, 237--266 (1987; Zbl 0633.46069)], and the second coming from the notion of a spectral triple. It is shown that for generic quantum Heisenberg manifolds the two theories coincide. A draft of the paper is available at \url{arxiv:1304.7617}. [For the entire collection see Zbl 0602.00004.]
The authors extend the well known Yang-Mills theory to the finitely generated projective modules over a \(C^*\)-algebra with the natural apparatus of connections, curvature, Hermitian structure, YM-functional etc.... For the noncommutative (ir)rational rotation algebra \(A_{\theta}\) with smooth structure \(A^{\infty}_{\theta}\) the authors prove ``that if \(\Xi\) is a finite projective module over \(A^{\infty}_{\theta}\) which is not a multiple of any other projective module, then the moduli space for connections on \(\Xi^ d\) which minimize the Yang-Mills functional is homeomorphic to \(({\mathbb T}^ 2)^ d/\Sigma_ d\), where \({\mathbb T}^ 2\) is the ordering 2-torus and \(\Xi_ d\) is the group of permutation of \(d\) objects acting by permuting the components of \(({\mathbb T}^ 2)^ d\). Any finite projective module over \(A^{\infty}_{\theta}\) is of the above form \(\Xi^ d.''\)
Reviewer's remark: The corresponding variational (anti-) self-dual equation is \(\theta_{\nabla}(X,Y)^*=\pm \theta_{\nabla}(X,Y)\), the solutions of which are thus described (Theorem 2.1). In a geometric version of Hilbert vector bundles the reviewer studied the quantization procedure equation \(\theta_{\nabla}(X,Y)=-(i/\hslash)\omega (X,Y)Id\), where \(\omega\) is the symplectic structure on \(K\)--orbits [C. R. Acad. Sci., Paris, Sér. A 291, 295--298 (1980; Zbl 0459.58006); ibid. Sér. I 295, 345--348 (1982; Zbl 0501.58024); Acta Math. Vietnam 8, No.2, 35--131 (1983; Zbl 0567.43001)].
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The authors compare two different Yang-Mills theories -- one introduced by \textit{A. Connes} and \textit{M. A. Rieffel} [Contemp. Math. 62, 237--266 (1987; Zbl 0633.46069)], and the second coming from the notion of a spectral triple. It is shown that for generic quantum Heisenberg manifolds the two theories coincide. A draft of the paper is available at \url{arxiv:1304.7617}. We prove that if the Bishop-Phelps theorem is correct for a uniform dual algebra \(R\) of operators in a Hilbert space, then the algebra \(R\) is selfadjoint.
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The author studies the geometry of the family of simply connected homogeneous 3-manifolds \((M,g_{\kappa,\tau})\). First, basing on [\textit{W. P. Thurston}, Three dimensional geometry and topology. Vol. 1. Princeton Mathematical Series. 35. Princeton, NJ: Princeton University Press (1997; Zbl 0873.57001)] the author proves the following:
Theorem. Let \((M,g)\) be a simply connected homogeneous 3-manifold with at least 4-dimensional isometry group. Then \((M,g)\) either has constant sectional curvature or it fibres as an 1-dimensional principal fiber bundle over an homogeneous 2-manifold of constant curvature \(\kappa\). The horizontal distribution of this fibration is the kernel of a connection form with constant curvature \(\tau\).
If \((M,g)\) is a principal bundle as in the above theorem then it is denoted by \((M,g_{\kappa,\tau})\).
Next the author investigates the normal Jacobi fields, the conjugate radius and the closed geodesics on \((M,g_{\kappa,\tau})\) and determines the cut locus and the injectivity radius for all \((M,g_{\kappa,\tau})\). Finally, the author investigates the trigonometry on \((M,g_{\kappa,\tau})\). He proves the so called law of sine. In the end he derives the laws of trigonometry for the projected triangle and the law of holonomy, which together give a complete set of relations for determine the trigonometry on \((M,g_{\kappa,\tau})\). The book originates from notes of a graduate course Thurston gave at Princeton in the period 1978-1980. These notes have been widely distributed and have become the most important and influent text in low-dimensional topology and hyperbolic geometry of the last two decades. In fact they founded the new field of hyperbolic 3-manifolds, with connections to various other subjects as Kleinian groups, reflection groups, Teichmüller theory, differential geometry etc. The notes constitute the background and the basis for the important hyperbolization theorem for Haken 3-manifolds. Written in a very intuitive way, many of the beautiful ideas have been extended and formalized by various authors, sometimes using also different approaches and methods. After 1990, the notes have been thoroughly revised and extended, and the present book presents the first four chapters of this revised version.
The first chapter (``What is a manifold'') discusses polygons and surfaces, hyperbolic surfaces and some 3-manifolds as the 3-torus, the 3-sphere and elliptic 3-space, the Poincaré and the Seifert-Weber dodecahedral spaces, lens spaces and the complement of the figure-eight knot. In the second chapter (``Hyperbolic geometry and its friends'') various models of hyperbolic spaces are discussed, their isometries, trigonometric formulas, volume formulas etc. The third chapter is on ``Geometric manifolds'', that is on geometric structures on manifolds, the developing map and completeness and the eight model geometries in dimension three, containing also sections on discrete groups, bundles and connections, contact structures, PL-manifolds and smoothings. The last and fourth chapter is entitled ``The structure of discrete groups'', with sections on groups generated by small elements, Euclidean manifolds and crystallographic groups, Euclidean and spherical (elliptic) 3-manifolds, the `thick-thin'' decomposition of hyperbolic manifolds, Teichmüller space and 3-manifolds modeled on fibered geometries. (The next chapter of the revised version continues with the theory of orbifolds, in particular the classification of 2-orbifolds and geometric 3-orbifolds.)
Even more than the original notes, the book is full of ideas, examples, pictures, exercises and problems, side-remarks, hints to further developments and connections with other fields. It is written very densely, and often in a more intuitive and ``experimental'' than formal way (but much more complete and detailed than the original notes). This makes it both fascinating and challenging to read and one of the most beautiful and original texts in topology and geometry. Hopefully also a revised version of the other parts of the original notes will appear in the near future.
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The author studies the geometry of the family of simply connected homogeneous 3-manifolds \((M,g_{\kappa,\tau})\). First, basing on [\textit{W. P. Thurston}, Three dimensional geometry and topology. Vol. 1. Princeton Mathematical Series. 35. Princeton, NJ: Princeton University Press (1997; Zbl 0873.57001)] the author proves the following:
Theorem. Let \((M,g)\) be a simply connected homogeneous 3-manifold with at least 4-dimensional isometry group. Then \((M,g)\) either has constant sectional curvature or it fibres as an 1-dimensional principal fiber bundle over an homogeneous 2-manifold of constant curvature \(\kappa\). The horizontal distribution of this fibration is the kernel of a connection form with constant curvature \(\tau\).
If \((M,g)\) is a principal bundle as in the above theorem then it is denoted by \((M,g_{\kappa,\tau})\).
Next the author investigates the normal Jacobi fields, the conjugate radius and the closed geodesics on \((M,g_{\kappa,\tau})\) and determines the cut locus and the injectivity radius for all \((M,g_{\kappa,\tau})\). Finally, the author investigates the trigonometry on \((M,g_{\kappa,\tau})\). He proves the so called law of sine. In the end he derives the laws of trigonometry for the projected triangle and the law of holonomy, which together give a complete set of relations for determine the trigonometry on \((M,g_{\kappa,\tau})\). For any semisimple \(f\)-ring \(A\) with bounded inversion, we show that the frame of radical ideals of \(A\) and the frame of \(z\)-ideals of \(A\) have isomorphic subfit coreflections. If we assume the Axiom of Choice, then the two coreflections are actually identical. If the \(f\)-ring has the property that the sum of two \(z\)-ideals is a \(z\)-ideal (as in the case of rings of continuous functions), then the epicompletion of the frame of \(z\)-ideals is shown to be a dense quotient of the epicompletion of the frame of radical ideals. Baer rings, exchange rings, and normal rings that lie in the class of semisimple \(f\)-rings with bounded inversion are shown to have characterizations in terms of frames of \(z\)-ideal which are similar to characterizations in terms of frames of radical ideals.
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Let G be a connected, simply connected, almost simple algebraic group defined over \(F_ p\), and T a maximal torus of G contained in a Borel subgroup of G. Let X(T), W be the character group and the Weyl group of T, respectively. Let \(G_ 1\) (resp. \(B_ 1)\) be the kernel of the Frobenius endomorphism of G (resp. B). The authors prove that certain \(G_ 1T\)-modules are rigid, i.e. their socle series and radical series coincide, assuming Lusztig's conjecture. They first show that the induced \(G_ 1T\)-modules \(Ind^{G_ 1T}_{w_{B_ 1T}}(\lambda)\), where \(w\in W\) and \(\lambda\) is a p-regular character of T (i.e. \(\lambda\) lies in the interior of an alcove in \(X(T)\otimes_ ZR)\), are rigid. Furthermore, the composition factor multiplicities in the Loewy layers of the induced modules are coefficients in the Lusztig Q-polynomials. Then they show that the injective \(G_ 1T\)-modules \(\hat Q_ 1(\lambda)\) are rigid. If \(p\geq 3h-3\) (where h is the Coxeter number) then there is an injective G-module \(Q_ 1(\lambda)\) which restricts to \(\hat Q_ 1(\lambda)\). It is shown that \(Q_ 1(\lambda)\) is rigid, and a formula is obtained for its Loewy series. Some of the arguments are analogous to those of \textit{R. S. Irving} [in Trans. Am. Math. Soc. 291, 733-754 (1985; Zbl 0594.17005) and Ann. Sci. Ec. Norm. Supér., IV. Sér. 21, 47-65 (1988)]. Let \({\mathfrak p}_S\) be a parabolic subalgebra of a complex semisimple Lie algebra \({\mathfrak g}\), and let \({\mathcal O}_S\) be the subcategory of \({\mathcal O}\) consisting of \({\mathfrak p}_S\)-finite modules. The first of these papers determines which objects of \({\mathcal O}_S\) are both projective and injective: they are the projective covers of the irreducible modules of largest Gelfand-Kirillov dimension. (The proof is due in part to Devra Garfinkle.) Several interesting equivalent formulations of the problem are presented as well.
The second paper studies the Loewy length of generalized Verma modules and their projective covers. (The Loewy length is the length of the shortest filtration with semisimple subquotients.) For Verma modules with regular infinitesimal character, the Loewy length is computed exactly, using the Kazhdan-Lusztig conjecture; in fact the result is shown to be essentially equivalent to the Kazhdan-Lusztig conjecture. Various more precise results are also proved, including some for singular infinitesimal character. There are some results for \({\mathcal O}_S\), but these are mostly confined to special cases.
Some of the results appeared first in \textit{D. Barbasch}'s paper [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, No.3, 489--494 (1984; Zbl 0581.17002)].
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Let G be a connected, simply connected, almost simple algebraic group defined over \(F_ p\), and T a maximal torus of G contained in a Borel subgroup of G. Let X(T), W be the character group and the Weyl group of T, respectively. Let \(G_ 1\) (resp. \(B_ 1)\) be the kernel of the Frobenius endomorphism of G (resp. B). The authors prove that certain \(G_ 1T\)-modules are rigid, i.e. their socle series and radical series coincide, assuming Lusztig's conjecture. They first show that the induced \(G_ 1T\)-modules \(Ind^{G_ 1T}_{w_{B_ 1T}}(\lambda)\), where \(w\in W\) and \(\lambda\) is a p-regular character of T (i.e. \(\lambda\) lies in the interior of an alcove in \(X(T)\otimes_ ZR)\), are rigid. Furthermore, the composition factor multiplicities in the Loewy layers of the induced modules are coefficients in the Lusztig Q-polynomials. Then they show that the injective \(G_ 1T\)-modules \(\hat Q_ 1(\lambda)\) are rigid. If \(p\geq 3h-3\) (where h is the Coxeter number) then there is an injective G-module \(Q_ 1(\lambda)\) which restricts to \(\hat Q_ 1(\lambda)\). It is shown that \(Q_ 1(\lambda)\) is rigid, and a formula is obtained for its Loewy series. Some of the arguments are analogous to those of \textit{R. S. Irving} [in Trans. Am. Math. Soc. 291, 733-754 (1985; Zbl 0594.17005) and Ann. Sci. Ec. Norm. Supér., IV. Sér. 21, 47-65 (1988)]. We discuss the problem of finding acceptable models for (propositional) logic programs with negation and constraints. It is well known that although the well-founded semantics is defined for every general program, not every program with constraints has a well-founded model. We argue that, in case the program is consistent but has no well-founded model, we should look for an expansion of the current program having a well-founded model. ''We discuss some properties these expansions and the methods generating them should have. In particular we show that there are tractable and complete expansion methods, i.e., efficient methods returning an expansion having a well-founded model whenever the original program is consistent. Furthermore, we investigate the complexity of expansion minimization problems, showing that in general they are NP-hard. If, however, we restrict these problems to local minimization problems, they can be solved efficiently. This work can be viewed as a logical reconstruction of ideas presented (procedurally) in truth-maintenance (e.g., dependency-directed backtracking), autoepistemic logic and abductive reasoning.
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This paper considers generalized entropies of dynamical systems introduced by Rényi, using the one-parameter family of generalized entropy functions defined by:
\[
H_q(p_1,\dots,p_n)=-\frac1{q-1}\log(\sum_{i=1}^n p_i^1),\qquad q\neq 1.
\]
The usual Shannon entropy function is obtained as a limit of these functions: \(\lim_{q\rightarrow 1}H_q(p_1,\dots,p_n)\). This paper is one in a series which attempts to define new metric invariants for dynamical systems by introducing a generalized entropy of a dynamical system using the Rényi entropy functions. In an earlier paper [Nonlinearity 11, 771--782 (1998; Zbl 0943.37004)], the authors showed that simply replacing \(H_1\) with \(H_q\) yields no new results for ergodic transformations of positive entropy. For \(q<1\) one simply obtains \(+\infty\) and for \(q\geq 1\) one recovers the usual measure theoretic entropy.
In this paper, they extend this result to nonergodic transformations where the situation is slightly different. It turns out that for \(q>1\) the generalized Rényi entropy can, potentially, detect ergodicity. Namely, if the hypothesis of ergodicity is weakened to aperiodicity, then for \(q<1\) one obtains \(+\infty\), for \(q=1\) one recovers the usual measure theoretic entropy, but for \(q>1\) the generalized entropy yields the essential infimum of the usual entropies of the ergodic components of \(T\).
Since the usual measure-theoretic entropy is the average value of the entropies of the ergodic components, this result allows for the existence of examples of nonergodic transformations \(T\) where the generalized entropy is strictly smaller than the measure theoretic entropy of \(T\). Unfortunately, the authors do not have an example for which this observation is (in their words) useful.
The paper is well written, the techniques are interesting, and the result is surprising and interesting in the sense that it suggests the existence of an entropy-based invariant which can detect ergodicity. The authors study the generalized Rényi entropies which were introduced in the physics literature. The proper defintion of these entropies needs, in their opinion, further clarification. They show that the Rényi entropies do not contain any new information. On the other hand, they introduce the notion of the correlation entropies, which are not invariants of the dynamical systems in the measure-theoretic sense, but they are invariant under transformations of the state space with bounded distortion. With these correlation entropies they provide a formal definition which can serve as a basis for the results reported in the physics literature.
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This paper considers generalized entropies of dynamical systems introduced by Rényi, using the one-parameter family of generalized entropy functions defined by:
\[
H_q(p_1,\dots,p_n)=-\frac1{q-1}\log(\sum_{i=1}^n p_i^1),\qquad q\neq 1.
\]
The usual Shannon entropy function is obtained as a limit of these functions: \(\lim_{q\rightarrow 1}H_q(p_1,\dots,p_n)\). This paper is one in a series which attempts to define new metric invariants for dynamical systems by introducing a generalized entropy of a dynamical system using the Rényi entropy functions. In an earlier paper [Nonlinearity 11, 771--782 (1998; Zbl 0943.37004)], the authors showed that simply replacing \(H_1\) with \(H_q\) yields no new results for ergodic transformations of positive entropy. For \(q<1\) one simply obtains \(+\infty\) and for \(q\geq 1\) one recovers the usual measure theoretic entropy.
In this paper, they extend this result to nonergodic transformations where the situation is slightly different. It turns out that for \(q>1\) the generalized Rényi entropy can, potentially, detect ergodicity. Namely, if the hypothesis of ergodicity is weakened to aperiodicity, then for \(q<1\) one obtains \(+\infty\), for \(q=1\) one recovers the usual measure theoretic entropy, but for \(q>1\) the generalized entropy yields the essential infimum of the usual entropies of the ergodic components of \(T\).
Since the usual measure-theoretic entropy is the average value of the entropies of the ergodic components, this result allows for the existence of examples of nonergodic transformations \(T\) where the generalized entropy is strictly smaller than the measure theoretic entropy of \(T\). Unfortunately, the authors do not have an example for which this observation is (in their words) useful.
The paper is well written, the techniques are interesting, and the result is surprising and interesting in the sense that it suggests the existence of an entropy-based invariant which can detect ergodicity. In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta function with respect to its first argument that does not need recurrences. As a by-product, we correct some formulas that have appeared in the literature.
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This paper is concerned with the zeros of differential polynomials, that is, polynomials of the form \(P(f,f',\dots,f^{(q)})\) where \(f\) and the coefficients of \(P\) are meromorphic in a domain \(D\). Lower bounds for the number of zeros of \(P\) in a closed subset \(E\) of \(D\) are obtained under the assumption that there exists a solution \(w\) of the corresponding differential equation \(P(w,w',\dots,w^{(q)})\) which is meromorphic in \(D\), and certain other hypotheses. The case that \(D\) is the complex plane and that \(E\) is a closed disk had been considered by the author in an earlier paper [Acta Sci. Nat. Univ. Pekin. 26, No. 5, 513-529 (1990; Zbl 0746.30023)]. Lower bounds for the common zeros of sets of differential polynomials (to a fixed function \(f)\) are also obtained. Let \(f\) be a transcendental meromorphic function in \(\mathbb{C}\) and \(a_ 1,\ldots,a_ p\) meromorphic in \(\mathbb{C}\) and linearly independent, with the property that \(a_ j\), \(j=1,\ldots,p\), is small with respect to \(f\). Define for \(s\geq 1\)
\[
M_ s:=\{a_ 1^{s_ 1}a_ 2^{s_ 2}\cdots a_ p^{s_ p}\mid s_ j\in\mathbb{N}_ 0\text{ and } s_ 1+\cdots+s_ p=s\}
\]
and assume that \(u_ 1,\ldots,u_ n\) is a basis of \(M_ s\) and \(v_ 1,\ldots,v_ k\) a basis of \(M_{s+1}\). Then we define
\[
P[f]:=W(v_ 1,\ldots,v_ k,\;u_ 1f,\ldots,u_ nf)
\]
where \(W\) means the Wronskian. Denote by \(A_ j:=\sum^ p_{m=1}C_{j,m}a_ m\), \(j=1,2,\ldots,q\) any finite number of distinct linear combinations of the \(a_ j\)'s with constant coefficients. Let \(a\) be a pole of the meromorphic function \(g\) with multiplicity \(m_ a\), then we denote by
\[
n_ \lambda(r,g):=\sum_{| a|\leq r,\;a \text{ pole of }g}\min(m_ a,\lambda),
\]
where \(\lambda\) is a nonnegative number, and
\[
N_ \lambda(r,g):=\int^ r_ 0{n_ \lambda(t,g)-n_ \lambda(0,g)\over t}dt+n_ \lambda(0,g)\log r.
\]
As the main result the author proves
\[
{1\over n}N\left(r,{1\over P[f]}\right)\geq\sum^ q_{j=1}\left(N\left(r,{1\over f-A_ j}\right)-N_ \lambda\left(r,{1\over f-A_ j}\right)\right)+o(T(r,f))
\]
where \(\lambda=k+{n-1\over 2}\).
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This paper is concerned with the zeros of differential polynomials, that is, polynomials of the form \(P(f,f',\dots,f^{(q)})\) where \(f\) and the coefficients of \(P\) are meromorphic in a domain \(D\). Lower bounds for the number of zeros of \(P\) in a closed subset \(E\) of \(D\) are obtained under the assumption that there exists a solution \(w\) of the corresponding differential equation \(P(w,w',\dots,w^{(q)})\) which is meromorphic in \(D\), and certain other hypotheses. The case that \(D\) is the complex plane and that \(E\) is a closed disk had been considered by the author in an earlier paper [Acta Sci. Nat. Univ. Pekin. 26, No. 5, 513-529 (1990; Zbl 0746.30023)]. Lower bounds for the common zeros of sets of differential polynomials (to a fixed function \(f)\) are also obtained. We study two smooth 2-surfaces in Euclidean 4-space: a translation surface and its evolute.
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The first edition from 1994, see the review Zbl 0809.62064, provided an extension of generalized linear models (GLM) for multivariate and multicategorical models, longitudinal data analysis, random effects and nonparametric predictors. The aim of the new edition is to reflect the major new developments over the past years. The book is clearly written, with emphasis on basic ideas. The authors illustrate concepts with numerous examples, using real data from biological sciences, economics and social sciences.
The organization parallels that of the first edition. Ch. 8 is now called ``State space models and hidden Markov models'', and the other titles remain the same. In this second edition, Bayesian concepts are considered more comprehensively. In Ch. 3, marginal models are treated, and marginal means of correlated binary and categorical data are estimated. Ch. 5, on nonparametric and semiparametric generalized regression, has been totally rewritten, it contains new sections on local smoothing and Bayesian inference. Ch. 6 now covers both time series and longitudinal data.
Ch. 7 contains a new subsection on a nonparametric approach based on finite mixtures, and a new section on a fully Bayesian approach, where all parameters are regarded as random. MCMC techniques are described in greater detail in Ch. 8 and in the appendix. Ch. 8 also extends the main ideas from state space models to models with spatial and spatio-temporal data.
In summary, this book gives a thorough exposition of recent developments in categorical data based on GLMs. In the last decade there are various extensions of generalized linear models (GLM) in various directions such as multivariate and multicategorical models, longitudinal data analysis, random effects and nonparametric predictors. The aim of this book is to bring together and to review a large part of the recent advances in this area. Although the continuous case is sketched sometimes, throughout the book the focus is on categorical data.
The book is aimed at applied statisticians, graduate students of statistics and researchers with a strong interest in statistics and data analysis from areas like econometrics, biometrics and social sciences. It is written on an intermediate level with emphasis on basic ideas. For rigorous proofs the reader is referred to the literature.
The book contains 9 chapters the titles of which are as follows: Ch. 1: Introduction. Ch. 2: Modelling and analysis of cross-sectional data: a review of univariate generalized linear models. Ch. 3: Models for multicategorical responses: multivariate extensions of generalized linear models. Ch. 4: Selecting and checking models. Ch. 5: Semi- and nonparametric approaches to regression analysis. Ch. 6: Fixed parameter models for time series and longitudinal data. Ch. 7: Random effects models. Ch. 8: State space models. Ch. 9: Survival models. The book also contains 63 examples, 45 figures and 48 tables.
There are two appendices. Appendix A provides some material which is necessary to study this book, e.g., exponential families, basic ideas for asymptotics, EM-algorithm and Monte Carlo methods. In Appendix B, some software is described that can be used for fitting GLM and extensions.
In summary, this book gives a fairly helpful picture of recent advances in categorical data based generalized linear models.
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The first edition from 1994, see the review Zbl 0809.62064, provided an extension of generalized linear models (GLM) for multivariate and multicategorical models, longitudinal data analysis, random effects and nonparametric predictors. The aim of the new edition is to reflect the major new developments over the past years. The book is clearly written, with emphasis on basic ideas. The authors illustrate concepts with numerous examples, using real data from biological sciences, economics and social sciences.
The organization parallels that of the first edition. Ch. 8 is now called ``State space models and hidden Markov models'', and the other titles remain the same. In this second edition, Bayesian concepts are considered more comprehensively. In Ch. 3, marginal models are treated, and marginal means of correlated binary and categorical data are estimated. Ch. 5, on nonparametric and semiparametric generalized regression, has been totally rewritten, it contains new sections on local smoothing and Bayesian inference. Ch. 6 now covers both time series and longitudinal data.
Ch. 7 contains a new subsection on a nonparametric approach based on finite mixtures, and a new section on a fully Bayesian approach, where all parameters are regarded as random. MCMC techniques are described in greater detail in Ch. 8 and in the appendix. Ch. 8 also extends the main ideas from state space models to models with spatial and spatio-temporal data.
In summary, this book gives a thorough exposition of recent developments in categorical data based on GLMs. Efficient grooming of traffic can greatly reduce the cost of the network. To deal with the changing traffic in SONET/WDM tree networks, two heuristic algorithms are developed by combining genetic algorithm with traffic-splitting heuristics. To evaluate the algorithms, the bound is also derived. Computer simulations show that the proposed algorithms can achieve good results in reducing the blocking rate of new traffic.
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The authors consider the convex mathematical programming problem with inclusion constraints: \(\min\{f(x): x \in C\), \(0 \in F(x)\}\). They show that the Lagrangian of this problem is constant on its solution set. Basing on this they derive various Lagrange-multiplier based characterizations of its solution set. Consider the following cone-constrained convex programming problem. (P) \(\min ~f(x),\) s.t. \(x\in C,\) \(-g(x)\in K,\) where \(X\) is a Banach space, \(Y\) is a locally convex (Hausdorff) space, \(C\) is a closed convex subset of \(X\), \(K\) is a closed convex cone in \(Y\), \( f:X\rightarrow \mathbb{R}\) is a continuous convex function, and \(g:X\rightarrow Y\) is a continuous \(K\)-convex mapping. The purpose of this paper is to present characterizations of the solution set of problem (P), which in particular covers semidefinite convex programs and programs involving explicit convex inequality constraints.
First, the authors establish that the Lagrangian function of (P) is constant on the solution set of (P). Then, they present various simple Lagrange multiplier-based characterizations of the solution set of (P). It is shown that, for a finite-dimensional convex program with inequality constraints, the characterizations illustrate the property that the active constraints with positive Lagrange multipliers at an optimal solution remain active at all optimal solutions of (P). Finally, they present applications of these results to derive corresponding Lagrange multiplier characterizations of the solution sets of semidefinite programs and fractional programs. In particular, they characterize the solution set of a semidefinite linear program in terms of a complementary slackness condition with a fixed Lagrange multiplier. Specific examples are given to illustrate the significance of the results.
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The authors consider the convex mathematical programming problem with inclusion constraints: \(\min\{f(x): x \in C\), \(0 \in F(x)\}\). They show that the Lagrangian of this problem is constant on its solution set. Basing on this they derive various Lagrange-multiplier based characterizations of its solution set. The authors classify all harmonic maps with finite uniton number from a Riemann surface into a compact simple Lie group \(G\), in terms of certain pieces of the Bruhat decomposition of the group of algebraic loops in \(G\) and corresponding canonical elements. They estimate the minimal uniton number of the corresponding harmonic maps with respect to different representations and they make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, they give some explicit descriptions of harmonic spheres in low dimensional spin groups making use of spinor representations.
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Let \(R\) be a ring. A right \(R\)-module \(M\) is called small when the functor \(\text{Hom}_R(M,-)\) commutes with direct sums in the category Mod-\(R\) of right \(R\)-modules. It is well known that every finitely generated right \(R\)-module is small but the converse is not true in general; the rings for which both classes of modules coincide are called right steady.
It was known by results of \textit{R. Colpi} and \textit{J. Trlifaj} [Commun. Algebra 22, No. 10, 3985-3995 (1994; Zbl 0818.16003)] that a simple ring containing an infinite set of orthogonal idempotents is neither right nor left steady but the problem of whether this is true for not necessarily simple rings remained open. The main result of the paper under review provides a negative answer to this question by showing that, for each cardinal \(\kappa\), there exists a left and right steady ring containing a set of orthogonal central idempotents of cardinality \(\kappa\). In the last part of the paper, further constructions of non-steady rings are presented and it is shown that a direct product of rings is (right) steady if and only if so is each factor and the number of factors is finite. The object of this paper is to study the class \({\mathcal{STAR}}\) of all \(*\)- modules, which had been introduced by \textit{C. Menini} and \textit{A. Orsatti} [Rend. Semin. Mat. Univ. Padova 82, 203-231 (1989; Zbl 0701.16007)] in order to characterize certain equivalences between subcategories of module categories. The main tool used in this investigation are the classes \({\mathcal S}_ \lambda\) of \(*_ \lambda\)- modules, where \(\lambda\) is a nonzero cardinal. A module \(P\) belongs to \({\mathcal S}_ \lambda\) whenever \(P\) is finitely generated and satisfies, for every \(\kappa< \lambda\), the following condition \(C(\kappa)\): ``For each submodule \(M\) of \(P^{( \kappa)}\), the condition \(M\in \text{Gen} (P_ R)\) is equivalent to the injectivity of the canonical group homomorphism \(\text{Ext}_ R (P,M)\to \text{Ext}_ R (P, P^{( \kappa)})\)''. Since the \(*\)-modules are precisely those modules \(P\) which satisfy \(C(\kappa)\) for all cardinals \(\kappa\), the classes \({\mathcal S}_ \lambda\), starting with the class \({\mathcal S}_ 1\) of all finitely generated modules and including the class \({\mathcal{ASTAR}}= {\mathcal S}_{\aleph_ 0}\) whose elements are called almost \(*\)-modules, form a decreasing chain that approaches \({\mathcal{STAR}}\) ``from above'' (the intersection of all the \({\mathcal S}_ \lambda\) is precisely \({\mathcal{STAR}}\)). The authors show that \(*_ \lambda\)-modules behave in the expected way and induce equivalences between certain natural subcategories of \(\text{Gen} (P_ R)\) and \(\text{Cog} (P^*_ S)\) depending on \(\lambda\) (where \(S\) denotes the endomorphism ring of \(P_ R\)). Finally, they answer in the negative a question of C. Menini by showing that, in general, the inclusion of \({\mathcal{STAR}}\) in \({\mathcal{ASTAR}}\) is proper. In fact, using the Cohn-Schofield solution of Artin's problem for skew-field extensions, they show that for each infinite cardinal \(\lambda\) there exists a 2-generated \(*_ \lambda\)- module \(P\), over a suitable ring \(R\), that does not satisfy \(C(\lambda)\).
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Let \(R\) be a ring. A right \(R\)-module \(M\) is called small when the functor \(\text{Hom}_R(M,-)\) commutes with direct sums in the category Mod-\(R\) of right \(R\)-modules. It is well known that every finitely generated right \(R\)-module is small but the converse is not true in general; the rings for which both classes of modules coincide are called right steady.
It was known by results of \textit{R. Colpi} and \textit{J. Trlifaj} [Commun. Algebra 22, No. 10, 3985-3995 (1994; Zbl 0818.16003)] that a simple ring containing an infinite set of orthogonal idempotents is neither right nor left steady but the problem of whether this is true for not necessarily simple rings remained open. The main result of the paper under review provides a negative answer to this question by showing that, for each cardinal \(\kappa\), there exists a left and right steady ring containing a set of orthogonal central idempotents of cardinality \(\kappa\). In the last part of the paper, further constructions of non-steady rings are presented and it is shown that a direct product of rings is (right) steady if and only if so is each factor and the number of factors is finite. In a Hilbert space we construct a regularized continuous analog of the Newton method for nonlinear equation with a Fréchet differentiable and monotone operator. We obtain sufficient conditions of its strong convergence to the normal solution of the given equation under approximate assignment of the operator and the right-hand of the equation.
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Let \(R\) and \(R'\) denote noncompact Riemann surfaces and assume there is some homeomorphism \(h\) of \(R\) onto \(R'\). Then, in general, \(h\) will not respect the conformal structure of the surfaces, i.e. it is nowhere expected to be holomorphic. The author asks for conditions which allow to call such a mapping \(h\) to be ``good'' in the sense of function theory; this will be the case if \(h\) is homotopic to some holomorphic injection \(f:R\to R'\) (of course it would be too much restrictive to require \(f\) as a biholomorphic function). For hyperbolic surfaces \(R\), \(R'\) we can consider the hyperbolic length of a closed curve \(\gamma\) on \(R\) and also the infimum \(l_R(c)\) of these numbers where the curves run in the free homotopy class \(c\) of \(\gamma\) (the curvature of the hyperbolic metric should be normalized). Now, if \(h\) is a ``good' homeomorphism, then we obtain the Schwarz-Pick-type inequality \(l_{R'} (h^*(c))\leq l_R(c)\), where \(h^*(c)\) is the image of \(c\) under the induced map \(h^*\) of the homotopy classes. One main result of the article is that the converse is not true in general: a homeomorphisms which fulfills this arclength-inequality is not necessarily a ``good'' one. But, if we replace the hyperbolic length in the definition above by the length, then at least in special cases (if \(R\), \(R'\) are one-punctured tori) the inequality is sufficient for \(h\) to be a ``good'' mapping. The techniques of the proofs are essentially based on results of \textit{M. Shiba} [Holomorphic functions and moduli I, Publ. Math. Sci. Res. Inst. Berkeley 10, 237-246 (1986; Zbl 0653.30028)]. An interesting walk along the border between geometry and function theory. [For the entire collection see Zbl 0646.00004.]
Let R be an open Riemann surface of genus \(g>0\) together with a canonical homology basis \(\chi\). If a closed Riemann surface R' of the same genus g contains R, \(\chi\) gives rise to a canonical homology basis \(\chi\) ' of R'. The pair (R',\(\chi\) ') defines a point in the Torelli space of genus g. Denote by \(C^*(R,\chi)\) the set of all such Torelli surfaces (R',\(\chi\) '). The normalized period matrices \(\tau^*(R',\chi ')\) of surfaces \((R',\chi ')\in C^*(R,\chi)\) and their diagonal \(\Delta^*(R',\chi ')\) are studied in this paper. One of the results is the following: There exists a closed polydisc P in \({\mathbb{C}}^ g\) such that (i) \(\Delta^*(R',\chi ')\in P\) for all \((R',\chi ')\in C^*(R,\chi)\) and (ii) \(\Delta^*(R',\chi ')\in \partial P\) if and only if (R',\(\chi\) ') is a canonical hydrodynamic continuation of (R,\(\chi)\). The results generalize preceding work of the author [Trans. Am. Math. Soc. 301, 299-311 (1987; Zbl 0626.30046)].
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Let \(R\) and \(R'\) denote noncompact Riemann surfaces and assume there is some homeomorphism \(h\) of \(R\) onto \(R'\). Then, in general, \(h\) will not respect the conformal structure of the surfaces, i.e. it is nowhere expected to be holomorphic. The author asks for conditions which allow to call such a mapping \(h\) to be ``good'' in the sense of function theory; this will be the case if \(h\) is homotopic to some holomorphic injection \(f:R\to R'\) (of course it would be too much restrictive to require \(f\) as a biholomorphic function). For hyperbolic surfaces \(R\), \(R'\) we can consider the hyperbolic length of a closed curve \(\gamma\) on \(R\) and also the infimum \(l_R(c)\) of these numbers where the curves run in the free homotopy class \(c\) of \(\gamma\) (the curvature of the hyperbolic metric should be normalized). Now, if \(h\) is a ``good' homeomorphism, then we obtain the Schwarz-Pick-type inequality \(l_{R'} (h^*(c))\leq l_R(c)\), where \(h^*(c)\) is the image of \(c\) under the induced map \(h^*\) of the homotopy classes. One main result of the article is that the converse is not true in general: a homeomorphisms which fulfills this arclength-inequality is not necessarily a ``good'' one. But, if we replace the hyperbolic length in the definition above by the length, then at least in special cases (if \(R\), \(R'\) are one-punctured tori) the inequality is sufficient for \(h\) to be a ``good'' mapping. The techniques of the proofs are essentially based on results of \textit{M. Shiba} [Holomorphic functions and moduli I, Publ. Math. Sci. Res. Inst. Berkeley 10, 237-246 (1986; Zbl 0653.30028)]. An interesting walk along the border between geometry and function theory. The main objective of this paper is to propose an approach to solve the multiple-category attribute reduct problem. The \((\alpha ,\beta )\) lower approximate and \((\alpha ,\beta )\) upper approximate distribution reduct are introduced into decision-theoretic rough set model. On the basis of this, the judgement theorems and discernibility matrices associated with the above two types of distribution reduct are examined as well, from which we can obtain attribute reducts. Finally, an example is used to illustrate the main ideas of the proposed approaches.
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This is a continuation of the author's previous paper (Part I) [Inf. Comput. 209, No. 10, 1312--1354 (2011; Zbl 1243.03075)] and the unpublished manuscript (Part II) [``Introduction to clarithmetic. II'', Preprint, \url{arxiv:1004.3236}], familiarity with which is assumed for studying the present paper. The newly introduced arithmetical theories (based on computability logic) are called CLA8, CLA9 and CLA10 with the following properties:
-- CLA8 is sound and complete with respect to PA-provably recursive-time computability (in the same sense as CLA7, studied in Part II, is sound and complete with respect to primitive recursive- time computability.) It augments CLA7 by the rule (1) \(\{F(x) \sqcup \neg F(x), \;\exists x F(x)\} / \{\sqcup x F(x)\}\) for elementary \(F(x)\).
``The story does not end with provably recursive-time computability though. Not all computable problems have recursive (let alone provably) time complexity bounds. In other words, not all computable problems are recursive-time computable. An example is \((\ast) \,\sqcap\!x\big(\exists y\,p(x,y)\rightarrow\sqcup y\,p(x,y)\big)\), where \(p(x,y)\) is a decidable binary predicate such that the unary predicate \(\exists y\, p(x,y)\) is undecidable (for instance, \(p(x,y)\) means `Turing machine \(x\) halts within \(y\) steps'). Problem \((\ast)\) is solved by the following effective strategy: Wait till Environment chooses a value \(m\) for \(x\). After that, for \(n=0,1,2,\cdots\), figure out whether \(p(m,n)\) is true. If and when you find an \(n\) such that \(p(m,n)\) is true, choose \(n\) for \(y\) in the consequent and retire. On the other hand, if there was a recursive bound \(\tau\) for the time complexity of a solution \({\mathcal M}\) of \((\ast)\), then the following would be a decision procedure for (the undecidable) \(\exists y\, p(x,y)\): Given an input \(m\) (in the role of \(y\)), run \({\mathcal M}\) for \(\tau(|m|+1)\) steps in the scenario where Environment chooses \(m\) for \(x\) at the very beginning of the play of \((\ast)\), and does not make any further moves. If, during this time, \({\mathcal M}\) chooses a number \(n\) for \(y\) in the consequent of \((\ast)\) such that \(p(m,n)\) is true, accept; otherwise reject.''
``A next natural step on the road of constructing incrementally strong clarithmetical theories for incrementally weak concepts of computability is to go beyond PA-provably recursive-time computability and consider the weaker concept of constructively PA-provable computability of (the problem represented by) a sentence \(X\). The latter means existence of a machine \({\mathcal M}\) such that PA proves that \({\mathcal M}\) computes \(X\), even if the running time of \({\mathcal M}\) is not bounded by any recursive function.''
-- CLA9 is sound and complete with respect to constructively PA-provable computability. ``That is, a sentence \(X\) is provable in CLA9 if and only if it is constructively PA-provably computable. Deductively, CLA9 only differs from CLA8 in that, instead of (1), it has the following, stronger, rule:'' \(\{F(x)\sqcup\neg F(x)\} / \{\exists x F(x)\rightarrow\sqcup x F(x)\}\) for elementary \(F(x)\).
-- CLA10 is sound and complete with respect to PA-provable computability. It augments CLA9 by the rule \(\{\exists x F(x)\} / \{\sqcup x F(x)\}\) where \(\exists x F(x)\) is an elementary sentence. In this paper the author introduces an(other) arithmetical theory based on his computability logic (CoL), which is an analogue of Peano arithmetic (based on classical logic). CoL, as the author puts it, is ``a semantical, mathematical and philosophical platform, and a long-term program, for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth which logic has more traditionally been.'' The author also notes in the introduction of the paper that ``the main value of CoL \dots will eventually be determined by whether and how it relates to the outside, extra-logical world.''
``Unlike the mathematical or philosophical constructivism, however, and even unlike the early-day theory of computation, modern computer science has long understood that, what really matters, is not just \textit{computability}, but rather \textit{efficient computability}.''
In his earlier paper [``Towards applied theories based on computability logic'', J. Symb. Log. 75, No. 2, 565--601 (2010; Zbl 1201.03055)] the author introduced the CoL-based arithmetic \textbf{CLA1}, in which ``every formula expresses a number-theoretic computational \textit{problem} (rather than just a true/false \textit{fact}), every theorem expresses a problem that has an algorithmic solution, and every proof encodes such a solution.'' That system ``proves formulas expressing computable but often intractable arithmetical problems.''
``A purpose of the present paper is to construct a CoL-based system for arithmetic which, unlike \textbf{CLA1}, proves only efficiently -- specifically, polynomial time -- computable problems. The new applied formal theory \textbf{CLA4} presented in Section 11 achieves this purpose.''
The author proves the soundness of \textbf{CLA4} and also what he calls \textit{extensional completeness} of the theory, in the sense that ``every number-theoretic computational problem that has a polynomial time solution is expressed by some theorem of \textbf{CLA4}.'' It is noted that this is weaker than \textit{intensional completeness}, which amounts to ``any formula representing an (efficiently) computable problem is provable.'' (Intentional completeness of \textbf{CLA4} has not been proven in the paper -- it might not hold.)
``Syntactically, our \textbf{CLA4} is an extension of \textbf{PA}, and the semantics of the former is a conservative generalization of the semantics of the latter.'' In the final section, the author notes that ``Both classical-logic-based and intuitionistic-logic-based systems of bounded arithmetic happen to be \textit{inherently weak} theories, as opposed to our CoL-based version, which is as strong as Gödel's incompleteness phenomenon permits, and which can be indefinitely strengthened without losing computational soundness.''
The paper is rather long and requires familiarity with the earlier works of the author where CoL was introduced and developed.
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This is a continuation of the author's previous paper (Part I) [Inf. Comput. 209, No. 10, 1312--1354 (2011; Zbl 1243.03075)] and the unpublished manuscript (Part II) [``Introduction to clarithmetic. II'', Preprint, \url{arxiv:1004.3236}], familiarity with which is assumed for studying the present paper. The newly introduced arithmetical theories (based on computability logic) are called CLA8, CLA9 and CLA10 with the following properties:
-- CLA8 is sound and complete with respect to PA-provably recursive-time computability (in the same sense as CLA7, studied in Part II, is sound and complete with respect to primitive recursive- time computability.) It augments CLA7 by the rule (1) \(\{F(x) \sqcup \neg F(x), \;\exists x F(x)\} / \{\sqcup x F(x)\}\) for elementary \(F(x)\).
``The story does not end with provably recursive-time computability though. Not all computable problems have recursive (let alone provably) time complexity bounds. In other words, not all computable problems are recursive-time computable. An example is \((\ast) \,\sqcap\!x\big(\exists y\,p(x,y)\rightarrow\sqcup y\,p(x,y)\big)\), where \(p(x,y)\) is a decidable binary predicate such that the unary predicate \(\exists y\, p(x,y)\) is undecidable (for instance, \(p(x,y)\) means `Turing machine \(x\) halts within \(y\) steps'). Problem \((\ast)\) is solved by the following effective strategy: Wait till Environment chooses a value \(m\) for \(x\). After that, for \(n=0,1,2,\cdots\), figure out whether \(p(m,n)\) is true. If and when you find an \(n\) such that \(p(m,n)\) is true, choose \(n\) for \(y\) in the consequent and retire. On the other hand, if there was a recursive bound \(\tau\) for the time complexity of a solution \({\mathcal M}\) of \((\ast)\), then the following would be a decision procedure for (the undecidable) \(\exists y\, p(x,y)\): Given an input \(m\) (in the role of \(y\)), run \({\mathcal M}\) for \(\tau(|m|+1)\) steps in the scenario where Environment chooses \(m\) for \(x\) at the very beginning of the play of \((\ast)\), and does not make any further moves. If, during this time, \({\mathcal M}\) chooses a number \(n\) for \(y\) in the consequent of \((\ast)\) such that \(p(m,n)\) is true, accept; otherwise reject.''
``A next natural step on the road of constructing incrementally strong clarithmetical theories for incrementally weak concepts of computability is to go beyond PA-provably recursive-time computability and consider the weaker concept of constructively PA-provable computability of (the problem represented by) a sentence \(X\). The latter means existence of a machine \({\mathcal M}\) such that PA proves that \({\mathcal M}\) computes \(X\), even if the running time of \({\mathcal M}\) is not bounded by any recursive function.''
-- CLA9 is sound and complete with respect to constructively PA-provable computability. ``That is, a sentence \(X\) is provable in CLA9 if and only if it is constructively PA-provably computable. Deductively, CLA9 only differs from CLA8 in that, instead of (1), it has the following, stronger, rule:'' \(\{F(x)\sqcup\neg F(x)\} / \{\exists x F(x)\rightarrow\sqcup x F(x)\}\) for elementary \(F(x)\).
-- CLA10 is sound and complete with respect to PA-provable computability. It augments CLA9 by the rule \(\{\exists x F(x)\} / \{\sqcup x F(x)\}\) where \(\exists x F(x)\) is an elementary sentence. The Fast Aerosol Spectrometer (FASP) is a device for spectral aerosol measurements. Its purpose is to safely monitor the atmosphere inside a reactor containment. First we describe the FASP and explain its basic physical laws. Then we introduce our reconstruction methods for aerosol particle size distributions designed for the FASP. We extend known existence results for constrained Tikhonov regularization by uniqueness criteria and use those to generate reasonable models for the size distributions. We apply a Bayesian model-selection framework on these pre-generated models. We compare our algorithm with classical inversion methods using simulated measurements. We then extend our reconstruction algorithm for two-component aerosols, so that we can simultaneously retrieve their particle-size distributions and unknown volume fractions of their two components. Finally we present the results of a numerical study for the extended algorithm.
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In [Eur. J. Comb. 28, No. 5, 1493--1529 (2007; Zbl 1117.51016)], the author characterized shadows of apartments of residues in the point-line truncations of \(J\)-Grassmannians of buildings. In the paper under review, she more generally considers shadows of residues in the point-line truncations of \(J\)-Grassmann geometries \(\Sigma\) of buildings and presents two characterizations by local properties when full subgeometries of \(\Sigma\) are such shadows. The author then applies her results to certain buildings with simply laced diagrams. The author gives a beautiful characterization of the shadows of apartments of residues in the point-line truncations of \(J\)-Grassmannians of buildings. In fact, these are characterized by two conditions on a given connected induced subgraph of the point graph of the point-line truncation (a local one and one involving circuits and planes of the \(J\)-Grassmannian); roughly the former controls the thickness (or rather the thinness) and the latter controls the diameter and girth of the rank 2 residues of the subgraph.
A number of other results is presented, such as the theorem that the shadow of any residue in the point-line truncation of a J-Grassmannian is a convex subspace. The usefulness of the main theorem is illustrated with a number of examples.
The reviewer likes to stress that the results proved in this paper hold for any \(J\) (of course, \(J\) must meet every connected component of the diagram). As such, this paper is a marvellous contribution to the theory of building Grassmannians.
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In [Eur. J. Comb. 28, No. 5, 1493--1529 (2007; Zbl 1117.51016)], the author characterized shadows of apartments of residues in the point-line truncations of \(J\)-Grassmannians of buildings. In the paper under review, she more generally considers shadows of residues in the point-line truncations of \(J\)-Grassmann geometries \(\Sigma\) of buildings and presents two characterizations by local properties when full subgeometries of \(\Sigma\) are such shadows. The author then applies her results to certain buildings with simply laced diagrams. This paper presents the experimental study conducted over the INEX 2007 Document Mining Challenge corpus employing a frequent subtree-based incremental clustering approach. Using the structural information of the XML documents, the closed frequent subtrees are generated. A matrix is then developed representing the closed frequent subtree distribution in documents. This matrix is used to progressively cluster the XML documents. In spite of the large number of documents in INEX 2007 Wikipedia dataset, the proposed frequent subtree-based incremental clustering approach was successful in clustering the documents.
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In this short note the authors prove that an externally saturated class \(\Sigma\) of morphisms in a category \(C\) is an internally saturated class (in the sense of P. J. Freyd and G. M. Kelly) iff it is externally saturated and admits a calculus of left fractions. They show, using a suitable shape functor, that every internal saturated class is also externally saturated. In a previous paper [Glas. Mat., III. Ser. 42, No. 2, 309--318 (2007; Zbl 1152.18001)], the second author proved that every internally saturated class has a calculus of left fractions, and in this paper the authors prove that the converse holds true provided that the category \(C\) has finite colimits and a terminal object. For a pair of categories \(({\mathcal C},{\mathcal K})\) the shape equivalences form the class \(\Sigma\) of morphisms of \({\mathcal C}\) that are orthogonal to the class of objects of \({\mathcal K}\). The author addresses the question of determing possible existence of other objects in \({\mathcal C}\), out of \({\mathcal K}\), which every shape equivalence for \(({\mathcal C},{\mathcal K})\) is orthogonal to. This is done when \({\mathcal K}\) is reflective or proreflective in \({\mathcal C}\). The general case is also discussed. Conditions for an internally saturated class \({\mathcal K} \subset\text{Ob}({\mathcal C})\) to be reflective or proreflective are also considered.
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In this short note the authors prove that an externally saturated class \(\Sigma\) of morphisms in a category \(C\) is an internally saturated class (in the sense of P. J. Freyd and G. M. Kelly) iff it is externally saturated and admits a calculus of left fractions. They show, using a suitable shape functor, that every internal saturated class is also externally saturated. In a previous paper [Glas. Mat., III. Ser. 42, No. 2, 309--318 (2007; Zbl 1152.18001)], the second author proved that every internally saturated class has a calculus of left fractions, and in this paper the authors prove that the converse holds true provided that the category \(C\) has finite colimits and a terminal object. We derive a general theory for elastic phase transitions in solids subject to diffusion under possibly large deformations. After stating the physical model, we derive an existence result for measure-valued solutions that relies on a new approximation result for cylinder functions in infinite settings.
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This is a problem book in Galois theory. It contains 122 completely solved exercises. These exercises, following the author's introduction, were taken either from the exam subjects at the University of Alger, or from certain research papers. The exercises deal mainly with effective calculations of situations, e.g., as describing the lattice of all subfields of a given field. Some of the exercises solved in the book are well-known, some being very difficult. The selection of the exercises does not cover, however, some basic sections from Galois theory. The book is intended, following the author's introduction, to address the students expecting to pass various exams in France (Licence-Maîtrise de Mathématiques, or Agrégation).
Further advanced problems are presented in the Niveau II problem book reviewed below. This is a new problem book following the author's former one (see the preceding review). It contains 115 completely solved exercises, and, compared to his former problem book, they are more difficult (see the review of Niveau I problem book). Following the author's introduction, the book is addressed to the students expecting to pass various exams in France (Maîtrise, DEA, Agrégation).
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This is a problem book in Galois theory. It contains 122 completely solved exercises. These exercises, following the author's introduction, were taken either from the exam subjects at the University of Alger, or from certain research papers. The exercises deal mainly with effective calculations of situations, e.g., as describing the lattice of all subfields of a given field. Some of the exercises solved in the book are well-known, some being very difficult. The selection of the exercises does not cover, however, some basic sections from Galois theory. The book is intended, following the author's introduction, to address the students expecting to pass various exams in France (Licence-Maîtrise de Mathématiques, or Agrégation).
Further advanced problems are presented in the Niveau II problem book reviewed below. Consider the system \((1)\quad \dot x_ i=\sum^{n}_{j=1}a_{ij}(t)g_ j(x_ j),\quad 1\leq i\leq n,\) where \(A(t)=(a_{ij}(t))\) is an almost periodic matrix function of t with \(a_{ij}(t)\geq 0\) for \(i\neq j\); \(\sum^{n}_{i=1}a_{ij}(t)=0\), for \(1\leq j\leq n\); and \(g_ j(u)\in C^ 1({\mathbb{R}}^+,{\mathbb{R}})\), \(g_ j(0)=0\), \(\dot g_ j(u)>0\) on (0,\(\infty)\), for \(1\leq j\leq n\). The author proves that system (1) possesses an almost periodic invariant manifold which is a ``cone'' in \(D=\{x\in {\mathbb{R}}^ n| x\geq 0\}\) with its vertex at the origin. An analogous result is also established for \(\dot x=A(t)g(t),\) where the \(n\times n\) matrix \(A(t)=(a_{ij}(t))\) is almost periodic and symmetric; \(a_{ij}(t)\geq 0\) for \(i\neq j\), \(\sum^{n}_{i=1}a_{ij}(t)=0\) for \(1\leq j\leq n\); \(g(x)=(g_ 1(x),...,g_ n(x))\in C^ 1(D)\), \((\partial /\partial x_ j)(g_ k(x)-g_ i(x))\geq 0\), \((\partial /\partial x_ j)(g_ j(x)-g_ i(x))>0\) in D for \(i\neq j\), and \(1\leq i,j,k\leq n\); g(x)\(\geq 0\) and \(g_ i(x_ 1,...,x_{i-1},0,x_{i+1},...,x_ n)=0\) in D for \(1\leq i\leq n\), \(\partial g_ i(x)/\partial x_ i>0\), \(\forall x>0\).
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\textit{J. Geronimus} [Ann. Math., II. Ser. 31, 681-686 (1930)] introduced polynomials \(K_ n(x)\) defined by
\[
K_ 0(x)=a,\quad K_ 1(x)=bx+d,\quad K_{n+1}(x)+K_{n-1}(x)=2xK_ n(x),\quad n\geq 1.
\]
When \(d=0\), he showed they are orthogonal on [-1,1] with respect to \((1- x^ 2)^{1/2}(1-\mu x^ 2)^{-1}\) when \(\mu =(2ab-b^ 2)/a^ 2\) and \(\mu\leq 1\). He also found a second order linear differential equation for \(K_ n(x)\) when \(d=0\). The general case when \(d\neq 0\) is considered here. A second order differential equation is discovered for \(K_ n(x)\). The orthogonality when the measure is absolutely continuous can be obtained from old work of Szegö. The details are worked out here. \textit{J. Wilson} and the reviewer have a general set of classical type orthogonal polynomials that depends on five parameters. When the right three are set equal to zero, these polynomials reduce to the Geronimus polynomials. The orthogonality for the polynomials with five free parameters is given in Mem. Am. Math. Soc. 319, 55 p. (1985; Zbl 0572.33012). This memoir is a detailed exposition of a new family of orthogonal polynomials which has five free parameters and a continuous weight distribution. Indeed, there are cases where this distribution has a finite discrete part in addition. The orthogonality relation is based on a new contour integral. To state the most important results some notation is needed: (throughout \(q\in {\mathbb{C}}\) and \(| q| <1)\)
\[
(a;q)_ n:=\prod^{n-1}_{j=0}(1-aq^ j),\quad n=0,1,2,...;e_ q(a):=\prod^{\infty}_{j=0}(1-aq^ j),
\]
\[
p_ n(x;a,b,c,d| q):=a^{-n}(ab;q)_ n(ac;q)_ n(ad;q)_ n 4\phi_ 3\left[ \begin{matrix} q^{-n},q^{n-1}abcd,ae^{i\theta},ae^{-i\theta};\\ ab,ac,ad\quad q,q\end{matrix} \right]
\]
where \(x=\cos \theta\), \(n=0,1,2,..\). (also denoted by \(p_ n(x))\), a terminating basic hypergeometric series.
The underlying integral depends on five parameters q, \(a_ j\), (1\(\leq j\leq 4)\) such that \(| q| <1\) and \(a_ ja_ k\neq q^{\ell}\) for \(\ell =0,1,2,...,(1\leq j,k\leq 4):\)
\[
(1/2\pi i)\int_{C}e_ q(z^ 2)e_ q(z^{-2})\prod^{4}_{j=1}(e_ q(a_ jz)e_ q(a_ j/z))^{-1}(dz/z)
\]
\[
=\frac{2e_ q(abcd)}{e_ q(q)}\prod_{1\leq j<k\leq 4}e_ q(a_ ja_ k)^{-1},
\]
where C is a closed positively oriented contour consisting of the unit circle deformed so as to separate the sequences of poles converging to zero from the sequences of poles converging to infinity.
It is shown that \(\{p_ n(x)\}\) is a family of polynomials in x, with degree \((p_ n)=n\), and satisfying a three-term recurrence. Further, each \(p_ n\) is symmetric in a,b,c,d. The purely continuous weight distribution occurs for \(-1<q<1\), \(\max (| a|,| b|,| c|,| d|)<1\), and a,b,c,d all real or appearing in conjugate pairs: then
\[
\int^{1}_{-1}p_ n(x)p_ m(x)w(x)(1-x^ 2)^{- 1/2}dx=0\quad for\quad n\neq m,
\]
where
\[
w(x)=e_ q(e^{2i\theta})e_ q(e^{-2i\theta})\prod^{4}_{j=1}(e_ q(a_ je^{i\theta})e_ q(a_ je^{-i\theta}))^{-1},\quad where\quad x=\cos \theta,
\]
and \(a_ j\) (1\(\leq j\leq 4)\) takes the values a,b,c,d. The integral
\[
\int^{1}_{-1}p_ n(x)^ 2w(x)(1-x^ 2)^{-1/2}dx
\]
is explicitly found. When any of the parameters a,b,c,d exceed 1, a finite discrete part (explicitly known) is added to the weight distribution.
The power of these results stems from the large number of free parameters. Special choices lead to previously studied families. For example, the case \(c=-a\), \(b=aq^{1/2}=-d\) gives the Rogers continuous q-ultraspherical polynomials (\textit{R. Askey} and \textit{M. E.-H. Ismail}, Studies in pure mathematics, Mem. of P. Turán, 55-78 (1983; Zbl 0532.33006).
Another example comes from the choice \(a=q^{\alpha /2+1/4}\), \(c=q^{1/2}\), \(b=-q^{\beta /2+1/4}\), \(d=bq^{1/2}\); this is the family of little q-Jacobi polynomials studied by G. Andrews and R. Askey.
Other special cases are also discussed in the paper, including the example \(q=0\), and a q-analogue of Meixner-Pollaczek polynomials. There is a Rodrigues type formula for \(p_ n\) which depends on a divided difference operator. Also, the connection coefficients between two different families \(\{p_ n(x;a,b,c,d| q)\}\) and \(\{p_ n(x;a',b',c',d'| q)\}\) are given as \({}_ 5\phi_ 4\)-series, and some tractable special cases are discussed. This is a paper of fundamental importance in the theory of orthogonal polynomials in one variable of hypergeometric type.
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\textit{J. Geronimus} [Ann. Math., II. Ser. 31, 681-686 (1930)] introduced polynomials \(K_ n(x)\) defined by
\[
K_ 0(x)=a,\quad K_ 1(x)=bx+d,\quad K_{n+1}(x)+K_{n-1}(x)=2xK_ n(x),\quad n\geq 1.
\]
When \(d=0\), he showed they are orthogonal on [-1,1] with respect to \((1- x^ 2)^{1/2}(1-\mu x^ 2)^{-1}\) when \(\mu =(2ab-b^ 2)/a^ 2\) and \(\mu\leq 1\). He also found a second order linear differential equation for \(K_ n(x)\) when \(d=0\). The general case when \(d\neq 0\) is considered here. A second order differential equation is discovered for \(K_ n(x)\). The orthogonality when the measure is absolutely continuous can be obtained from old work of Szegö. The details are worked out here. \textit{J. Wilson} and the reviewer have a general set of classical type orthogonal polynomials that depends on five parameters. When the right three are set equal to zero, these polynomials reduce to the Geronimus polynomials. The orthogonality for the polynomials with five free parameters is given in Mem. Am. Math. Soc. 319, 55 p. (1985; Zbl 0572.33012). The author deals with a parabolic two-phase system with memory occupying a bounded and smooth domain \(\Omega\subset \mathbb{R}^N\) \((N\geq 1)\), whose state is described by the pair \((\theta,\chi)\). Here \(\theta\) is the relative temperature \((\theta=0\) being the equilibrium temperature at which the two phases, for instance solid and liquid, can coexist) and \(\chi\) is the concentration of the more energetic phase (that is water in water-ice system). Assuming on the phase variable either a relaxation dynamics or a Stefan condition, the author proves existence and uniqueness results for feedback control problems corresponding to two different types of thermostat: the relay switch and the Preisach operator. Moreover, the author shows that, when the data enjoy suitable regularity properties, one can obtain a stronger regularity for the solution.
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This paper is concerned with Linnik's constant \(L\) such that the least prime \(p\) with \(p\equiv k \pmod q\) is \(\ll q^ L\) for \((k,q)=1\), and the estimate \(L\leq 8\) is proved. Quite recently, \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc., III. Ser. 64, 265--338 (1992; Zbl 0739.11033)] established the somewhat sharper bound \(L\leq 5.5\), which seems to be the present record in this problem. The two main purposes of this paper are to give new zero-free regions for Dirichlet \(L\)-functions, and to give a new estimate for Linnik's constant. If \(q\) is sufficiently large it is shown that the \(L\)-functions to modulus \(q\) have at most a Siegel zero in the region \(\sigma \geq 1-0.348/\log q\), \(| t| \leq 1\). The previous best result [see \textit{J.-R. Chen}, Sci. Sin., Ser. A 26, 714--731 (1983; Zbl 0513.10045)] had a constant \(0.103\). Moreover it is shown that there is at most one character \(\chi_ 1\), together with its conjugate, for which the \(L\)-function \(L(s,\chi\)) can have a zero in the region \(\sigma \geq 1-0.702/\log q\), \(| t| \leq 1\); and there are at most two characters \(\chi_ 1,\chi_ 2\), together with their conjugates, which can have a zero in the region \(\sigma \geq 1-0.857/\log q\), \(| t| \leq 1.\)
Using these results it is shown that if \((a,q)=1\) then there is a prime \(p\equiv a\pmod q\) satisfying \(p\ll q^{11/2}\). The previous best bound for Linnik's constant [\textit{J.-R. Chen} and \textit{J.-M. Liu}, Sci. China, Ser. A 32, 654--673, 792--807 (1989; Zbl 0684.10040)] had \(27/2\) in place of \(11/2\).
The proofs of these results depend on a number of new ideas, of which we mention three. Firstly it is shown how estimates for zero-free regions can be made to depend on Burgess's bounds for \(L\)-functions. In particular, his recent work [\textit{D. A. Burgess}, J. Lond. Math. Soc., II. Ser. 33, 219--226 (1986; Zbl 0593.10033)] on the ``\(r=3\)'' case is used. Secondly, the partial fraction decomposition for \(\frac{L'}{L}(s,\chi)\) which has previously been employed in obtaining zero-free regions, is replaced by an ``explicit formula'' for a sum \(\sum \Lambda (n)\chi (n)n^{-s}w(n)\) with a general non-negative weight \(w(n)\). Thirdly, the zero-density estimate used by other authors is augmented by a new kind of density estimate, the origins of which are related to the apparatus for proving zero-free regions.
It should also be mentioned that the paper requires a considerable amount of numerical work -- much more than in previous works on the subject. The paper concludes by describing a number of ways in which small improvements to the results may be obtained.
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This paper is concerned with Linnik's constant \(L\) such that the least prime \(p\) with \(p\equiv k \pmod q\) is \(\ll q^ L\) for \((k,q)=1\), and the estimate \(L\leq 8\) is proved. Quite recently, \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc., III. Ser. 64, 265--338 (1992; Zbl 0739.11033)] established the somewhat sharper bound \(L\leq 5.5\), which seems to be the present record in this problem. The articles of this volume will be reviewed individually. The preceding summer school has been reviewed (see Zbl 1014.00008).
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For \(k\in \mathbb N\), the closed subalgebra \(W_k\) of \(\ell^\infty (\mathbb Z)\) generated by functions \(n\mapsto \lambda^{n^i}\), with \(\lambda\in\mathbb T, i=0,\cdots,k,\) is a closed subalgebra of the so called Weyl algebra. This paper finds the topological center \(\Lambda\) of the enveloping semigroup \(\Sigma\) of the dynamical system on \(W_k\) given by the shift operator, that is the subset of \(\Sigma\) for which the composition is left and right continuous. The author identifies \(\Lambda\) with the group \(\mathbb Z\times \Hom(\mathbb T/H,\mathbb T) \), \(H\) being the torsion subgroup of \(\mathbb T\), and shows that, up to isomorphism, the group structure of the former is independent of \(k\). The same result for the Weyl algebra was already proved by the author and \textit{I. Namioka} in [Milan J. Math. 78, No. 2, 503--522 (2010; Zbl 1238.37002)]. The authors study the Ellis group and its topological center of the dynamical system \((X_f,U)\), where \(U\) is the shift operator on \(l^{\infty}(\mathbb{Z})\), \(f(n)=\lambda^{n^k}\) and \(\lambda\) is an irrational number of the unit circle. It is proved that for each natural number \(k\), the shift-orbit closure \(X_f\) of the function \(f\) is homeomorphic to the \(k\)-torus. Further it is shown that the topological center of the spectrum of the Weyl algebra is the image of \(\mathbb{Z}\) in the spectrum.
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For \(k\in \mathbb N\), the closed subalgebra \(W_k\) of \(\ell^\infty (\mathbb Z)\) generated by functions \(n\mapsto \lambda^{n^i}\), with \(\lambda\in\mathbb T, i=0,\cdots,k,\) is a closed subalgebra of the so called Weyl algebra. This paper finds the topological center \(\Lambda\) of the enveloping semigroup \(\Sigma\) of the dynamical system on \(W_k\) given by the shift operator, that is the subset of \(\Sigma\) for which the composition is left and right continuous. The author identifies \(\Lambda\) with the group \(\mathbb Z\times \Hom(\mathbb T/H,\mathbb T) \), \(H\) being the torsion subgroup of \(\mathbb T\), and shows that, up to isomorphism, the group structure of the former is independent of \(k\). The same result for the Weyl algebra was already proved by the author and \textit{I. Namioka} in [Milan J. Math. 78, No. 2, 503--522 (2010; Zbl 1238.37002)]. Using stereo vision in the field of mapping and localization is an intuitive idea, as demonstrated by the number of animals that have developed the ability. Though it seems logical to use vision, the problem is a very difficult one to solve. It requires the ability to identify objects in the field of view, and classify their relationship to the observer. A procedure for extracting and matching object data using a stereo vision system is introduced, and initial results are provided to demonstrate the potential of this system.
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The authors consider the parabolic-parabolic chemotaxis model with logistic source
\[
\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla b) + u(a - bu), \\
\tau v_t = \Delta v - \lambda v + \mu v \end{cases}
\]
in \(\mathbb R^N\) with \(\tau = 1\) and positive parameters \(\chi, a, b, \lambda, \mu\), and prove that that the spreading speed for solutions to nontrivial initial data with finite support is \(2\sqrt a\), provided \(\chi < \frac{4b}{N\mu}\). Since the spreading speed for the Fisher-KPP equation, i.e.\ for the first equation in the system above with \(\chi = 0\), is also \(2\sqrt a\), this shows that (at least when \(\chi\) is small or when \(b\) is large) chemotaxis has no effect on the spreading speed.
Thereby, the authors transfer results such as [\textit{R. B. Salako} et al., J. Math. Biol. 79, No. 4, 1455--1490 (2019; Zbl 1479.35116)] for the parabolic-elliptic version, i.e\ the system above with \(\tau = 0\), to the fully parabolic setting. Upper bounds on the spreading speed are obtained by applying the comparison principle to \(w = u + \frac{\chi}{2\mu} |\nabla v|^2\) which fulfills \(w_t \le \Delta w + a w\). This technique makes essential use of the assumption \(\tau=1\). In this work the authors consider the spatial spreading speed and minimal wave speed of a Keller-Segel chemoattraction system. They prove upper and lower bounds of the spreading interval \([c^*_-,c^*_+]\). Moreover, they show that the chemotaxis does not slow down the spreading speed of the solutions with nonempty compactly supported initials, and that when under some conditions the chemotaxis does not speed up the spreading speed of the solutions with nonempty compactly supported initials. They also discuss the spreading properties of solutions with initial functions satisfying some exponential decay property at infinity. Finally, they prove the existence of travelling wave solutions.
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The authors consider the parabolic-parabolic chemotaxis model with logistic source
\[
\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla b) + u(a - bu), \\
\tau v_t = \Delta v - \lambda v + \mu v \end{cases}
\]
in \(\mathbb R^N\) with \(\tau = 1\) and positive parameters \(\chi, a, b, \lambda, \mu\), and prove that that the spreading speed for solutions to nontrivial initial data with finite support is \(2\sqrt a\), provided \(\chi < \frac{4b}{N\mu}\). Since the spreading speed for the Fisher-KPP equation, i.e.\ for the first equation in the system above with \(\chi = 0\), is also \(2\sqrt a\), this shows that (at least when \(\chi\) is small or when \(b\) is large) chemotaxis has no effect on the spreading speed.
Thereby, the authors transfer results such as [\textit{R. B. Salako} et al., J. Math. Biol. 79, No. 4, 1455--1490 (2019; Zbl 1479.35116)] for the parabolic-elliptic version, i.e\ the system above with \(\tau = 0\), to the fully parabolic setting. Upper bounds on the spreading speed are obtained by applying the comparison principle to \(w = u + \frac{\chi}{2\mu} |\nabla v|^2\) which fulfills \(w_t \le \Delta w + a w\). This technique makes essential use of the assumption \(\tau=1\). We introduce an extension of the classical space of distributions which allows us to define the correct operations of multiplication of distributions by discontinuous functions and differentiation of distributions, as well as to pose correctly the Cauchy problem for ordinary linear differential equation with distributions in coefficients, which was the subject of research of many authors. We define this extension using a modification of the Perron--Stieltjes integral, whose properties are also studied in the present paper.
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The author gives a decomposition formula of the Bartholdi zeta function of a group covering of a digraph \(D\) (Theorem 5). He also defines an \(L\)-function of \(D\) and gives a determinant expression of it (Theorem 6). The proof uses Amitsur's identity (Theorem 4) and a variant of \textit{H. Bass}' method [Int. J. Math. 3, No. 6, 717--797 (1992; Zbl 0767.11025)]. Theorems 5 and 6 imply a decomposition formula for the Bartholdi zeta function in terms of \(L\)-functions (Corollary 1). This paper deals with a study of Ihara-Selberg zeta function of a finite graph. But his approach is slightly different from the one that has been usually done. He first defines the noncommutative determinant ``\(\det_{P/A}(\alpha)\)''. Here \(A\) is a \(k\)-algebra, \(k\) a commutative ring containing the rational field \(\mathbb Q\), and \(\alpha\) is, say, a power series with constant term \(I\) with coefficients in \(\text{End}_ A(P)\). After developing general properties of determinants, the author proceeds to study the general theory of zeta functions of a uniform lattice on a tree. The zeta function is given as an ``Euler product''
\[
Z(u)=\prod_{\varepsilon\in\mathcal P} \det_{\mathbb C[\Gamma]/\mathbb C[\Gamma]}(1-\sigma_ \varepsilon u^{l_ \varepsilon})^{-1}.
\]
For a finite-dimensional representation \(\rho: \Gamma\to \text{GL}(V_ \rho)\), the \(L\)-function is defined by \(L(u,\rho)=\prod_{\varepsilon\in{\mathcal P}}\text{det}(I-\rho (\sigma_ \varepsilon) u^{l_ \varepsilon})^{-1}\). The author gives some properties of \(L(u,\rho)\) and the non-trivial zeros of \(L(u,\rho)\).
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The author gives a decomposition formula of the Bartholdi zeta function of a group covering of a digraph \(D\) (Theorem 5). He also defines an \(L\)-function of \(D\) and gives a determinant expression of it (Theorem 6). The proof uses Amitsur's identity (Theorem 4) and a variant of \textit{H. Bass}' method [Int. J. Math. 3, No. 6, 717--797 (1992; Zbl 0767.11025)]. Theorems 5 and 6 imply a decomposition formula for the Bartholdi zeta function in terms of \(L\)-functions (Corollary 1). The investigation object of the paper is a dynamic economic model of pure exchange \((X^i, P^i, \omega^i\), \(T^i: i\in N)\), where \(N\) is a countable infinite set of consumers and, for each consumer \(i\), \(X^i\) is his consumption set, \(P^i\) his preference over \(X^i\), \(\omega^i\) his endowment and \(T^i\) is the collection of his lifetime periods. For each period \(t\) the commodity-price duality is represented by a symmetric Riesz dual system \(\langle E_t, E_t'\rangle\), where \(E_t\) and \(E_t'\) are respectively the commodity space and price space. The spaces may be infinite-dimensional. The preferences \(P^i\) are not assumed to be transitive or complete. Each agent can consume only a finite set of periods and at each period only a finite set of agents are alive. The overlapping generations model is a particular case of this model. Main equilibrium existence results are obtained under the assumption that there exists a finite set of non- negligible endowment owners and under relaxation of the assumption to the infinite-lived assets owners.
To get advanced results the authors construct a price space \(P\) and a commodity space \(\Lambda(P)\) of the entire economy which are subsets of the space products \(\prod^\infty_{t= 1} E_t'\) and \(\prod^\infty_{t= 1} E_t\), respectively. The spaces compose a commodity- price dual system \(\langle\Lambda(P), P\rangle\). Notions of equilibrium, quasi-equilibrium and weak quasi-equilibrium are introduced in relation with the duality. Several equilibrium existence theorems in corresponding terms are obtained under different sets of assumptions on the preferences and endowments of agents.
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Let \(f\) be a convex function on an interval~\(J=[m,M]\). The authors prove that
\[
f\big(M+m-\frac{x+y}{2}\big)\leq f(M)+f(m)-f\big(\frac{x+y}{2}\big)
\]
for all \(x,y\in J\). Using an idea of \textit{O. E. Tikhonov} [Linear Algebra Appl. 416, No. 2--3, 773--775 (2006; Zbl 1100.15011)], they generalize this inequality to operators on a finite-dimensional Hilbert space \(\mathcal H\) as follows: Let \(A_1,\dots,A_n\) be positive linear operators on \(\mathcal H\). If \(A_1+\dots+A_n=I\) and \(0\in J\), then
\[
f\Big(M+m-\sum_{i=1}^nx_iA_i\Big)\leq f(M)+f(m)-\sum_{i=1}^nf(x_i)A_i
\]
for all \(x_1,\dots,x_n\in J\). They also present several related inequalities. The author obtains the following
Theorem: For a function \(f: S\rightarrow \mathbb R\), (\(S\) is any interval in \(\mathbb R\)) the following conditions are equivalent:
(i) \(f\) is a matrix convex function, i.e., \(f(\alpha X+(1-\alpha )Y)\leq \alpha f(X)+(1-\alpha )f(Y)\) for all self-adjoint operators \(X,~Y\) in a finite-dimensional Hilbert space and every \(\alpha \in [0,1]\) provided that the spectra of \(X\) and \(Y\) lie in \(S\);
(ii) for any natural \(k\), for all families of positive operators \(\{A_i\}_{i=1}^k\) in a finite-dimensional Hilbert space \(H\), such that \(\sum_{i=1}^k A_i=1_H\), and arbitrary numbers \(x_i \in S\), the inequality
\[
f\left( \sum_{i=1}^k x_i A_i \right) \leq \sum_{i=1}^k f(x_i)A_i
\]
holds true.
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Let \(f\) be a convex function on an interval~\(J=[m,M]\). The authors prove that
\[
f\big(M+m-\frac{x+y}{2}\big)\leq f(M)+f(m)-f\big(\frac{x+y}{2}\big)
\]
for all \(x,y\in J\). Using an idea of \textit{O. E. Tikhonov} [Linear Algebra Appl. 416, No. 2--3, 773--775 (2006; Zbl 1100.15011)], they generalize this inequality to operators on a finite-dimensional Hilbert space \(\mathcal H\) as follows: Let \(A_1,\dots,A_n\) be positive linear operators on \(\mathcal H\). If \(A_1+\dots+A_n=I\) and \(0\in J\), then
\[
f\Big(M+m-\sum_{i=1}^nx_iA_i\Big)\leq f(M)+f(m)-\sum_{i=1}^nf(x_i)A_i
\]
for all \(x_1,\dots,x_n\in J\). They also present several related inequalities. [For the entire collection see Zbl 0564.00027.]
The string matching problem, in its simplest form, is the following: given two strings, the pattern and the text, find all occurrences of the pattern in the text. In this paper the author surveys several open problems concerning the algorithmic aspects of combinatorics of strings. Many of these are variants, extensions or variations of the basic string matching problem. The problems are divided into four groups: string matching, generalizations of string matching, index construction and miscellaneous. The survey is useful and up-to-date.
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Harvey and Lawson's notion of a calibration on a Riemannian manifold is extended to the semi-Riemannian category. Unlike the Riemannian case, where the submanifolds calibrated by a calibration are homologically volume minimizing, the submanifolds calibrated by a calibration on a semi-Riemannian manifold are homologically volume maximizing among a restricted class of submanifolds. Various examples of such homologically maximizing submanifolds can be found by dualizing examples of \textit{R. Harvey} and \textit{H. B. Lawson jun.} [Acta Math. 148, 47-157 (1982; Zbl 0584.53021)] to the signature case. This paper is perhaps best characterized as a foundational essay on the geometries of minimal varieties associated to closed forms. The fundamental observation here is the following: Let \(X\) be a Riemannian manifold, and suppose \(\phi\) is a closed exterior p-form with the property that
\[
(1)\quad \phi | \xi \leq vol_{\xi}
\]
for all oriented tangent \(p\)-planes \(\xi\) on \(X\). Then any compact oriented \(p\)- dimensional submanifold \(M\) of \(X\) with the property that
\[
(2)\quad \phi |_ M=vol_ M
\]
is homologically volume minimizing in \(X\), i.e. vol(M)\(\leq vol(M')\) for any \(M'\) such that \(\partial M=\partial M'\) and \([M-M']=0\) in \(H_ p(X;R)\). To see this, one simply notes that \(vol(M)=\int_ M\phi =\int_{M'}\phi \leq vol(M')\). Condition (2) enables us to associate to an exterior \(p\)-form \(\phi\) a family of oriented \(p\)-dimensional submanifolds in \(X\) which we call \(\phi\)-submanifolds. If \(\phi\) is closed and is normalized to satisfy condition (1), then each \(\phi\)-submanifold is homologically mass minimizing in \(X\).
A closed exterior \(p\)-form \(\phi\) satisfying (1) will be called a calibration and the Riemannian manifold \(X\) together with this form will be called a calibrated manifold. As an example, let \(X\) be a complex Hermitian \(n\)-manifold with Kähler form \(\omega\), and consider \(\phi =(1/p!)\omega^ p\) for some \(p\), \(1\leq p\leq n\). Then the \(\phi\)- submanifolds are just the canonically oriented complex submanifolds of dimension \(p\) in \(X\). If \(d\phi =0\), i.e., if \(X\) is a Kähler manifold, then the complex submanifolds are homologically mass minimizing. This is the classical observation of \textit{H. Federer} [Geometric measure theory. Berlin: Springer (1969; Zbl 0176.00801)]. One of the main points of this paper is to exhibit and study some beautiful geometries of minimal subvarieties which are really not visible from this first viewpoint. We shall concentrate primarily on geometries in \(\mathbb R^ n\) associated to forms with constant coefficients. A significant part of the work will be to derive a tractable system of partial differential equations whose solutions represent subvarieties in the given geometry. These systems are in a specific sense generalizations of the Cauchy-Riemann equations.
The first geometry to be studied in depth is associated to the form
\[
\phi =\Re\{dz_ 1\bigwedge...\bigwedge\, dz_ n\}
\]
in \(\mathbb C^ n\). It consists of Lagrangian submanifolds of ''constant phase'', and is therefore called special Lagrangian geometry. In fact the only Lagrangian submanifolds which are stationary are special Lagrangian.
Up to \(SU_ n\)-coordinate changes, special Lagrangian submanifolds are locally graphs of the form \(\{y=(\nabla F)(x)\}\) where \(F\) is a scalar potential function satisfying a nonlinear elliptic equation. When \(n=3\), this equation has the following beautiful form:
\[
(3)\quad \Delta F=\det (\text{Hess }F).
\]
We conclude that the graph of the gradient of any solution to (3) is an absolutely volume-minimizing three-fold in \(\mathbb R^ 6\). In particular, any \(C^ 2\) solution of (3) is real analytic. The equation (3) bears an intimate relation to the work of \textit{H. Lewy} on harmonic gradient maps [Ann. Math. (2) 88, 518--529 (1968; Zbl 0164.13803)] and explains the mysterious appearance there of the minimal surface equation. This is discussed in Chapter III.
The geometry of special Lagrangian submanifolds in richly endowed (see Sections III.3 and 4), and constitutes a large new class of minimizing currents in \(\mathbb R^ n\). In particular, we are able to explicitly construct simple minimizing cones which are not real analytic (see Section III.3.C).
Chapter IV is devoted to the study of three exceptional geometries. There is a geometry of three-folds (and a dual geometry of four-folds) in \(\mathbb R^ 7\), which is invariant under the standard representation of \(G_ 2\). This geometry is associated to the three-form \(\phi (x,y,z)=(x,yz)\) where \(x,y,z\in \mathbb R^ 7\) are considered as imaginary Cayley numbers. A three- manifold \(M\subset \mathbb R^ 7=\text{Im }O\) belongs to this geometry if each of its tangent planes is a (canonically oriented) imaginary part of a quaternion subalgebra of the Cayley numbers \(O\). The local system of differential equations for this geometry is essentially deduced from the vanishing of the associator \([x,y,z]=(xy)z-x(yz)\), and thus the geometry is called associative. The most fascinating and complex geometry discussed here is the geometry of Cayley four-folds in \(\mathbb R^ 8\cong O\). This is the family of subvarieties associated to the four-form \(\psi (x,y,z,w)=\langle x(\bar yz)- z(\bar yx),w\rangle.\) It is invariant under the eight-dimensional representation of \(\text{Spin}_ 7\) and contains the coassociative geometry (the dual geometry of four-folds in \({\mathbb R}^ 7)\). It also contains both the (negatively oriented) complex and the special Lagrangian geometries for a seven-dimensional family of complex structures on \({\mathbb{R}}^ 8\). In fact for any of these structures, the form \(\psi\) can be expressed as
\[
\psi =-1/2\omega^ 2+\Re\{dz\}
\]
where \(\omega\) is the Kähler form and \(dz=dz_ 1\bigwedge...\bigwedge\, dz_ 4\) as above.
Chapter V contains a number of comments concerning generalizations of the main ideas and results of the paper. These comments include the observation that every Cayley four-fold naturally carries a twentyone- dimensional family of anti-self-dual \(SU_ 2\) Yang-Mills fields.
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Harvey and Lawson's notion of a calibration on a Riemannian manifold is extended to the semi-Riemannian category. Unlike the Riemannian case, where the submanifolds calibrated by a calibration are homologically volume minimizing, the submanifolds calibrated by a calibration on a semi-Riemannian manifold are homologically volume maximizing among a restricted class of submanifolds. Various examples of such homologically maximizing submanifolds can be found by dualizing examples of \textit{R. Harvey} and \textit{H. B. Lawson jun.} [Acta Math. 148, 47-157 (1982; Zbl 0584.53021)] to the signature case. A boundary value problem is considered for a singularly perturbed parabolic equation that
involves a small parameter multiplying all derivatives and degenerates into a finite equation when the parameter is equal to zero. An existence theorem is proved for a periodic solution with an internal layer, i.e., for a steplike contrast structure. A theorem on passage to the limit when the small parameter vanishes is proved as well.
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The author uses \(\lambda\)-determinants to count domino-tilings of a \(2n\times 2n\) square. The \(\lambda\)-determinant is a generalization of the determinant of a matrix and was introduced by \textit{D. P. Robbins} and \textit{H. Rumsey jun.} [Adv. Math. 62, 169--184 (1986; Zbl 0611.15008)]. Consider the \(2n\times 2n\) matrix \(M=(m_{ij})_{i,j=1}^{2n}\) with \(m_{ij}=1\) if \(| 2i-2n-1| + | 2j-2n-1| \leq 2n\) and \(m_{ij}=0\) for all other \(i,j\) (thus \(M\) consists of a central diamond of \(1\)'s surrounded by \(0\)'s). When the \(0\)'s in \(M\) are replaced by a parameter \(t\), the \(\lambda\)-determinant of the resulting matrix is well-defined and is a polynomial in \(\lambda\) and \(t\). The limit of this polynomial as \(t \rightarrow 0\) is a polynomial in \(\lambda\) whose value at \(\lambda =1\) is the number of domino-tilings of a \(2n\times 2n\) square. Es sei M eine \(n\times n\)-Matrix. Für \(k\leq n\) ist dann ein zusammenhängender k-Minor von M ein aus k aufeinander folgenden Zeilen und Spalten gebildeter Minor. Die Determinante von M läßt sich als rationale Funktion von zusammenhängenden k-Minoren ausdrücken. Für die Fälle \(k=2\) und \(k=3\) werden von den Verfassern die entsprechenden rationalen Funktionen explizit angegeben. Dabei werden als Beschreibungshilfsmittel alternierende Vorzeichen-Matrizen benutzt, die in dem erforderlichen Umfang eingehend diskutiert werden. Derartige Matrizen sind quadratische Matrizen mit Elementen aus \(\{\)-1,0,1\(\}\), deren Zeilen- und Spaltensummen alle den Wert Eins besitzen und bei denen in jeder Zeile und jeder Spalte die von Null verschiedenen Elemente alternierende Vorzeichen besitzen.
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The author uses \(\lambda\)-determinants to count domino-tilings of a \(2n\times 2n\) square. The \(\lambda\)-determinant is a generalization of the determinant of a matrix and was introduced by \textit{D. P. Robbins} and \textit{H. Rumsey jun.} [Adv. Math. 62, 169--184 (1986; Zbl 0611.15008)]. Consider the \(2n\times 2n\) matrix \(M=(m_{ij})_{i,j=1}^{2n}\) with \(m_{ij}=1\) if \(| 2i-2n-1| + | 2j-2n-1| \leq 2n\) and \(m_{ij}=0\) for all other \(i,j\) (thus \(M\) consists of a central diamond of \(1\)'s surrounded by \(0\)'s). When the \(0\)'s in \(M\) are replaced by a parameter \(t\), the \(\lambda\)-determinant of the resulting matrix is well-defined and is a polynomial in \(\lambda\) and \(t\). The limit of this polynomial as \(t \rightarrow 0\) is a polynomial in \(\lambda\) whose value at \(\lambda =1\) is the number of domino-tilings of a \(2n\times 2n\) square. We prove that in an arbitrary Delone set \(X\) in the three-dimensional space, the subset \(X_6\) of all points from \(X\) at which the local group has no rotation axis of order larger than 6 is also a Delone set. Here, under the local group at point \(x \in X\) we mean the symmetry group \(S_x(2R)\) of the cluster \(C_x(2R)\) of \(x\) with radius \(2R\), where \(R\) is the radius of the largest ball free of points of \(X\) (according to Delone's empty spheretheory). The main result seems to be the first rigorously proved statement for absolutely generic Delone sets which implies substantial statements for Delone sets with strong crystallographic restrictions. For instance, an important observation of Shtogrin on the boundedness of local groups in Delone sets with equivalent \(2R\)-clusters immediately follows from the main result. Further, we propose a crystalline kernel conjecture and its two weaker versions. According to the crystalline kernel conjecture, in an arbitrary Delone set, points with locally crystallographic axes only (i.e., of order 1, 2, 3, 4, or 6) inevitably constitute the essential part of the set. These conjectures significantly generalize the famous crystallography statement on the impossibility of a (global) fivefold symmetry in a three-dimensional lattice.
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Recently, \textit{A.~Knutson} and \textit{P.~Zinn-Justin} introduced a non-standard multiplication in \(M_n(\mathbb C)\), the space of all \(n\) by \(n\) matrices, denoted \(\bullet\) [see Adv. Math. 214, No. 1, 40--77 (2007; Zbl 1193.14068)]. By definition, the Brauer loop scheme is \(E:=\{M\in M_n({\mathbb C})\mid M\bullet M=0\}\). Knutson and Zinn-Justin have classified all the top dimensional irreducible components of \(E\). The main result of the article under review is that \(E\) is equidimensional, i.e., there are no other irreducible components. This was already conjectured in [loc. cit]. We introduce the Brauer loop scheme
\[
E:=\{M\in M_ N({\mathbb C}):\;M\bullet M=0\},
\]
where \(\bullet\) is a certain degeneration of the ordinary matrix product. Its components of top dimension, \(\lfloor N^2/2\rfloor\), correspond to involutions \(\pi\in S_ N\) having one or no fixed points. In the case \(N\) even, this scheme contains the upper--upper scheme from [\textit{A. Knutson}, J. Algebr. Geom. 14, No. 2, 283--294 (2005; Zbl 1074.14044), see also \url{arXiv:math.AG/0306275}] as a union of \((N/2)!\) of its components. One of those is a degeneration of the commuting variety of pairs of commuting matrices.
The Brauer loop model is an integrable stochastic process studied in [\textit{J. de Gier} and \textit{B. Nienhuis}, J. Stat. Mech. Theory Exp. 2005, No. 1, Paper P01006, 10 p., electronic only (2005; Zbl 1072.82585), see also math.AG/0410392], based on earlier related work in [\textit{M. J. Martins, B. Nienhuis} and \textit{R. Rietman}, An intersecting loop model as a solvable super spin chain, Phys. Rev. Lett. 81, No. 3, 504--507 (1998; Zbl 0944.82006), see also cond-mat/9709051], and some of the entries of its Perron--Frobenius eigenvector were observed (conjecturally) to equal the degrees of the components of the upper--upper scheme.
Our proof of this equality follows the program outlined in [\textit{P. Di Francesco} and \textit{P. Zinn-Justin}, Commun. Math. Phys. 262, No. 2, 459--487 (2006; Zbl 1113.82026), see also math-ph/0412031]. In that paper, the entries of the Perron--Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on \(E\). As a consequence, we obtain a formula for the degree of the commuting variety, previously calculated up to \(4\times 4\) matrices.
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Recently, \textit{A.~Knutson} and \textit{P.~Zinn-Justin} introduced a non-standard multiplication in \(M_n(\mathbb C)\), the space of all \(n\) by \(n\) matrices, denoted \(\bullet\) [see Adv. Math. 214, No. 1, 40--77 (2007; Zbl 1193.14068)]. By definition, the Brauer loop scheme is \(E:=\{M\in M_n({\mathbb C})\mid M\bullet M=0\}\). Knutson and Zinn-Justin have classified all the top dimensional irreducible components of \(E\). The main result of the article under review is that \(E\) is equidimensional, i.e., there are no other irreducible components. This was already conjectured in [loc. cit]. Let \(\gamma _{pr }(G)\) denote the paired domination number of graph \(G\). A graph \(G\) with no isolated vertex is paired domination vertex critical if for any vertex \(v\) of \(G\) that is not adjacent to a vertex of degree one, \(\gamma_{pr} (G - v) < \gamma_{pr}(G)\). We call these graphs \(\gamma_{pr}\)-critical. In this paper, we present a method of constructing \(\gamma_{pr}\)-critical graphs from smaller ones. Moreover, we show that the diameter of a \(\gamma_{pr}\)-critical graph is at most \(\frac{3}{2}(\gamma_{pr} (G)-2)\) and the upper bound is sharp, which answers a question proposed by \textit{M.A. Henning} and \textit{C.M. Mynhardt} [''The diameter of paired-domination vertex critical graphs,'' Czech. Math. J. 58, No.\,4, 887--897 (2008; Zbl 1174.05093)].
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Let \(T\) be a piecewise isometry, this means \(X\) is a subset of~\({\mathbb R}^{d}\), there exists a finite collection~\({\mathcal Z}\) of pairwise disjoint open polytopes such that the closure of their union equals~\(X\), and \(T:X\to X\) satisfies that \(T|_{Z}\) is an isometry for every \(Z\in{\mathcal Z}\). The map~\(T\) need not be invertible. In this paper it is proved that the topological entropy of~\(T\) equals zero. This result holds also if \(T\) is a contracting piecewise affine map. However, the author provides a counterexample to this result, if \(T\) is contracting, but not piecewise affine.
Although the result seems to be intuitively clear, it has been only conjectured. Only much weaker results were proved before (for example the special cases \(d=1\) and \(d=2\)).
The proof relies on an idea developped in \textit{M.~Tsujii} [Invent. Math. 143, 349-373 (2001; Zbl 0969.37012)]. Besides of solving a problem, which has been open for long time, this paper is also written very well. Suppose that \(U\) is a (not necessarily bounded or connected) polyhedron in \({\mathbb{R}}^{d}\), and let \({\mathcal P}\) be a finite family of polyhedra whose interiors are pairwise disjoint. A map \({\mathcal T}:U\to U\) is called piecewise linear, if for each \(P\in{\mathcal P}\) the restriction of \({\mathcal T}\) to the interior of \(P\) is an affine map \(A_{P}x+v_{P}\). If for every \(P\in{\mathcal P}\) all eigenvalues of \(A_{P}\) have modulus strictly larger than \(1\), then \({\mathcal T}\) is called expanding.
The author proves in this paper that every expanding piecewise linear map \({\mathcal T}\) on a bounded polyhedron has an invariant probability measure, which is absolutely continuous with respect to the Lebesgue measure. Moreover, there exist only finitely many ergodic absolutely continuous invariant probability measures \(\mu_{1},\mu_{2},\dots ,\mu_{p}\). Furthermore there exist open sets \(U_{1},U_{2},\dots ,U_{p}\) with \(\bigcup_{k=1}^{p}U_{k}\) has full Lebesgue measure in \(U\), such that \(\frac{1}{n}\sum_{j=1}^{n-1} \delta_{{\mathcal T}^{j}(x)}\) converges weakly to \(\mu_{k}\) for every \(k\) and each \(x\in U_{k}\). As in \textit{F. Hofbauer} and \textit{G. Keller} [Math. Z. 180, 119-140 (1982; Zbl 0485.28016)] one can derive strong ergodic properties of the \(\mu_{k}\) (weak Bernoullicity, exponential decay of correlation, central limit theorems). It is also proved that the associated Perron-Frobenius operator on the space of functions of bounded variation is quasi compact and its essential spectral radius is bounded by a certain expansion constant of \({\mathcal T}\).
Similar results are known for other types of dynamical systems. These are generalizations of the classical result obtained by \textit{A. Lasota} and \textit{J. Yorke} [Trans. Am. Math. Soc. 186, 481-488 (1973; Zbl 0298.28015)] in the one-dimensional case for a class of maps much larger than the family of piecewise linear maps.
In the proofs a quantity called ``weighted multiplicity'' is used. Using delicate estimations the author shows a result on the spectrum of the associated Perron-Frobenius operator on the space of functions of bounded variation. From this result the existence of absolutely continuous invariant probability measures and their ergodic properties can be derived in the classical way.
Besides of presenting interesting new results this paper is also written very well. It provides an important step towards the understanding of absolutely continuous invariant measures in the more-dimensional case.
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Let \(T\) be a piecewise isometry, this means \(X\) is a subset of~\({\mathbb R}^{d}\), there exists a finite collection~\({\mathcal Z}\) of pairwise disjoint open polytopes such that the closure of their union equals~\(X\), and \(T:X\to X\) satisfies that \(T|_{Z}\) is an isometry for every \(Z\in{\mathcal Z}\). The map~\(T\) need not be invertible. In this paper it is proved that the topological entropy of~\(T\) equals zero. This result holds also if \(T\) is a contracting piecewise affine map. However, the author provides a counterexample to this result, if \(T\) is contracting, but not piecewise affine.
Although the result seems to be intuitively clear, it has been only conjectured. Only much weaker results were proved before (for example the special cases \(d=1\) and \(d=2\)).
The proof relies on an idea developped in \textit{M.~Tsujii} [Invent. Math. 143, 349-373 (2001; Zbl 0969.37012)]. Besides of solving a problem, which has been open for long time, this paper is also written very well. Ancient Yi is one of six kinds of ancient word in the world and it recorded human development history in thousands of years. The recognition technology of ancient Yi character will enable us to transform lots of precious ancient Yi literature into electronic documents which is convenient for storage and spread. Due to imbalance of historical development and territorial limitation, the research on recognition of ancient Yi is rare. In this article, we apply novel deep learning to ancient character recognition. Our framework consists of five models which are extended based on four-layer convolutional neural network (CNN). Then we take Alpha-Beta divergence as penalty term to implement the coding for output neurons of five models. Next, two fully connected layers finish characteristics compression. Finally, we use softmax layer to reevaluate characteristics of ancient Yi character and get the probability distribution. The character owning the highest probability is identified as the target character. Experiments show that our method has higher precision compared with the traditional CNN model for handwriting recognition of the ancient Yi.
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Some results of the first author and \textit{T. Struppeck} given in [Mathematika 40, 215-225 (1993; Zbl 0796.11026)] about partial quotients of \(U\)-numbers are extended here to the field of formal Laurent series. The paper deals with the statistical behavior of the partial quotients of the certain set of real \(U_ 2\)-numbers. The existence of the subset of \(U_ 2\)-numbers with the property that if it is translated by any nonnegative integer and then squared, the result is a Liouville number, is demonstrated, too. Proofs use the diophantine approximations by quadratic irrationals.
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Some results of the first author and \textit{T. Struppeck} given in [Mathematika 40, 215-225 (1993; Zbl 0796.11026)] about partial quotients of \(U\)-numbers are extended here to the field of formal Laurent series. Let \(\mathcal{F}(\mathbb{R}^2)=\{f\in\mathbf{L}_{\infty}(\mathbb{R}^2)\cap\mathbf{L}_1(\mathbb{R}^2):f\geq 0\}\). Suppose \(s\in\mathcal{F}(\mathbb{R}^2)\) and \(\gamma:\mathbb{R}\rightarrow[0,\infty)\). Suppose \(\gamma\) is zero at zero, positive away from zero, and convex. For \(f\in\mathcal{F}(\mathbb{R}^2)\) let \(F(f)=\int_{\mathbb{R}^2}\gamma(f(x)-s(x))\,d\mathcal{L}^2x\); \(\mathcal L^2\) here is Lebesgue measure on \(\mathbb{R}^2\). In the denoising literature \(F\) would be called a \(fidelity\) in that it measures how much \(f\) differs from \(s\), which could be a noisy grayscale image. Suppose \(0<\varepsilon<\infty\), and let \(\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\) be the set of those \(f\in\mathcal{F}(\mathbb{R}^2)\) such that \(\mathbf{TV}(f)<\infty\) and \(\varepsilon\mathbf{TV}(f)+F(f)\leq\varepsilon\mathbf{TV}(g)+F(g)\) for \(g\in\mathbf{k}(f)\); here \(\mathbf{TV}(f)\) is the total variation of \(f\), and \(\mathbf{k}(f)\) is the set of \(g\in\mathcal{F}(\mathbb{R}^2)\) such that \(g=f\) off some compact subset of \(\mathbb{R}^2\). A member of \(\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\) is called a total variation regularization of \(s\) (with smoothing parameter \(\varepsilon\)). \textit{L. I. Rudin, S. Osher} and \textit{E. Fatemi} in [Physica D 60, No.~1--4, 259--268 (1992; Zbl 0780.49028)] and \textit{T. F. Chan} and \textit{S. Esedoglu} in [SIAM J. Appl. Math. 65, No.~5, 1817--1837 (2005; Zbl 1096.94004)] have studied total variation regularizations of \(F\) where \(\gamma(y)=y^2\) and \(\gamma(y)=y, y\in\mathbb{R}\), respectively.
Our purpose in this paper is to determine \(\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\) when \(s\) is the indicator function of a compact convex subset of \(\mathbb{R}^2\). It will turn out that if \(f\in\mathbf{m}^{\text{loc}}_{\varepsilon}(F)\), then, for \(0<y<1\), the set \(\{f>y\}\) is essentially empty or is essentially the union of the family of closed balls of a certain radius depending in a simple way on \(\gamma, \varepsilon\), and \(y\). While taking \(s=1_S, S\) compact and convex, is certainly not representative of the functions \(s\) which occur in image denoising, we hope this result sheds some light on the nature of total variation regularizations. In addition, one can test computational schemes for total variation regularization against these examples. Examples where \(S\) is \(not\) convex will appear in a later paper [cf. Part III, SIAM J. Imaging Sci. 2, No. 2, 532--568, electronic only (2009; Zbl 1175.49038)].
For Part I, see the author, SIAM J. Math. Anal. 39, No.~4, 1150--1190 (2007; Zbl 1185.49047).
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The paper is closely related to the paper of (*) \textit{V. K. Dzyadyk} [Ukr. Mat. Zh. 35, No.3, 297-302 (1983; Zbl 0531.30032)] and completes that paper. In (*) some biorthogonal polynomials are investigated. Under some additional assumptions the author deduces for these polynomials recurrence formulas consisting of three terms. The generalized moment problem is to find a measure \(\mu\) (x) and two sequences \(\{a_ j(x)\}_ 0^{\infty}\) and \(\{b_ k(x)\}_ 0^{\infty}\) in \(L^ 2(X,d\mu(x))\) so that
\[
s_{j+k}=\int_{X}a_ j(x)b_ k(x)d\mu(x)\quad for\quad some\quad set\quad X\subset {\mathbb{R}}.
\]
Assuming the existence of a solution the author constructs a system of biorthogonal polynomials from \(\{a_ j(x)\}\) and \(\{b_ k(x)\}\) and relates them to the Padé approximants of the series \(f(z)=\sum^{\infty}_{0}s_ jz^ j\) assumed analytic in \(\{z:\quad | z|<1\}\) and such that the Hankel determinants of the \(\{s_ j\}_ 0^{\infty}\) are different from zero.
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The paper is closely related to the paper of (*) \textit{V. K. Dzyadyk} [Ukr. Mat. Zh. 35, No.3, 297-302 (1983; Zbl 0531.30032)] and completes that paper. In (*) some biorthogonal polynomials are investigated. Under some additional assumptions the author deduces for these polynomials recurrence formulas consisting of three terms. For the Edwards-Anderson model we find an integral representation for some surface terms on the Nishimori line. Among the results are expressions for the surface pressure for free and periodic boundary conditions and the adjacency pressure, i.e., the difference between the pressure of a box and the sum of the pressures of adjacent sub-boxes in which the box can been decomposed. We show that all those terms indeed behave proportionally to the surface size and prove the existence in the thermodynamic limit of the adjacency pressure.
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Let \(S\subset X\) be an isolated invariant set of a discrete dynamical system \(f:X\to X\) on a locally compact metric space \(X\). In [Trans. Am. Math. Soc. 352, No. 7, 3305--3322 (2000; Zbl 0956.37010)] \textit{J. Franks} and \textit{D. Richeson} defined the Conley index of \(S\) to be the shift equivalence class of the induced map \(f_P:N/L\to N/L\) for a filtration pair \(P=(N,L)\).
In the present paper the author defines the (pointed) mapping torus index of \(S\) as the (pointed) homotopy type of the mapping torus of \(f_P\). It is proved that this is well defined, and that typical properties of the Conley index hold. The notion is weaker than the discrete Conley index of \textit{J. Franks} and \textit{D. Richeson} [Trans. Am. Math. Soc. 352, No. 7, 3305--3322 (2000; Zbl 0956.37010)]
but more accessible to computations. In the case of a continuous flow the pointed mapping torus index is equivalent to the Conley index of the suspension flow.
The paper also contains an idea for a numerical approach for computing the mapping torus index. We introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.
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Let \(S\subset X\) be an isolated invariant set of a discrete dynamical system \(f:X\to X\) on a locally compact metric space \(X\). In [Trans. Am. Math. Soc. 352, No. 7, 3305--3322 (2000; Zbl 0956.37010)] \textit{J. Franks} and \textit{D. Richeson} defined the Conley index of \(S\) to be the shift equivalence class of the induced map \(f_P:N/L\to N/L\) for a filtration pair \(P=(N,L)\).
In the present paper the author defines the (pointed) mapping torus index of \(S\) as the (pointed) homotopy type of the mapping torus of \(f_P\). It is proved that this is well defined, and that typical properties of the Conley index hold. The notion is weaker than the discrete Conley index of \textit{J. Franks} and \textit{D. Richeson} [Trans. Am. Math. Soc. 352, No. 7, 3305--3322 (2000; Zbl 0956.37010)]
but more accessible to computations. In the case of a continuous flow the pointed mapping torus index is equivalent to the Conley index of the suspension flow.
The paper also contains an idea for a numerical approach for computing the mapping torus index. This paper describes the network architecture of an Asia-wide satellite Internet that considers the situations in developing regions. The design considerations for the architecture are costs, effective use of satellite bandwidth, scalability, and routing strategy when combined with terrestrial links. The architecture includes using one-way shared satellite links to reduce costs, IP multicast to leverage the broadcast nature of satellite links, QoS, audio-video application gateway to adapt to the limited bandwidth of satellite links. This architecture is implemented in an operational network testbed connecting 13 countries and supporting a distance learning project.
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In order to treat problems in which one has to handle sheaves associated with, for instance, holomorphic functions with growth conditions, the authors introduced in [Astérisque, 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)] the category of ind-sheaves. In the present paper, they give the definition and the elementary properties of the micro-support \(SS(\cdot)\) of ind-sheaves as well as the notion of regularity. In particular, they prove that \(SS(\cdot)\) and the regular micro-support \(SS_{\text{reg}}(\cdot)\) of ind-sheaves behave naturally with respect to distinguished triangles, and that they are invariant through the action of quantized contact transformations. They then apply these results to the ind-sheaves of temperate holomorphic solutions of \({\mathcal D}\)-modules, by proving that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding \({\mathcal D}\)-module and that the ind-sheaf is regular if the \({\mathcal D}\)-module is regular holonomic. They finally compute an example of the ind-sheaf of temperate solutions of an irregular \({\mathcal D}\)-module in dimension one. Sheaf theory works well in situations where compatible local information can be glued together to produce global information. Since the 1960s, thanks to Grothendieck and his school [see, for example, ``Sémin. Géom. algébrique 1963-64'' (SGA4), \textit{M. Artin, A. Grothendieck}, and \textit{J.-L. Verdier}, Lect. Notes Math. 269 (1972; Zbl 0234.00007)], more and more mathematicians have viewed the category of sheaves on a space as a valuable generalization of that space.
The authors of this monograph are interested in analysis on manifolds, and using sheaf theory in that study. But on an analytic manifold there are problems with tempered functions and distributions, for example. Their approach to meeting some of these problems is to replace the ``space'' of sheaves on the manifold with the category Mod\(^{c}(k_{X})\) of sheaves of \(k\)-modules on \(X\) with compact support, and then to work in the category of ind-objects (systems of objects defined on a filtered category, again an idea that goes back at least to Grothendieck; see the above reference) in that category of sheaves -- hence the title for the monograph. This serves as a kind of completion for the generalized manifold, and in it they find much of what they are looking for. Passing then to the derived category of this completion [see, for example, ``Catégories derivées (Etat O)'', \textit{J.-L. Verdier}, Lect. Notes Math. 569, 262-311 (1977; Zbl 0407.18008)], which is something like the homotopy category of the generalized space, they produce the familiar external and internal operations (\(\Hom\), \(R\Hom,\) \(Rf_{\ast},\) composition, etc.) and a few new ones useful for their purposes. As they say:\ ``\dots ind-sheaves allow us to treat functions with growth conditions in the formalism of sheaves. On a complex manifold \(X\), we can define the ind-sheaf of ``tempered holomorphic functions'' \(\mathcal{O}_{X}^{t}\), or the ind-sheaf of ``Whitney holomorphic functions'' \(\mathcal{O}_{X}^{w}\), and obtain for example the sheaves of distributions or of \(C^{\infty}\)-functions using Sato's construction of hyperfunctions, simply replacing \(\mathcal{O}_{X}\) with \(\mathcal{O}_{X}^{t}\) or \(\mathcal{O}_{X}^{w}\). We also prove an adjunction formula for integral transforms in this framework''.
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In order to treat problems in which one has to handle sheaves associated with, for instance, holomorphic functions with growth conditions, the authors introduced in [Astérisque, 271. Paris: Société Mathématique de France (2001; Zbl 0993.32009)] the category of ind-sheaves. In the present paper, they give the definition and the elementary properties of the micro-support \(SS(\cdot)\) of ind-sheaves as well as the notion of regularity. In particular, they prove that \(SS(\cdot)\) and the regular micro-support \(SS_{\text{reg}}(\cdot)\) of ind-sheaves behave naturally with respect to distinguished triangles, and that they are invariant through the action of quantized contact transformations. They then apply these results to the ind-sheaves of temperate holomorphic solutions of \({\mathcal D}\)-modules, by proving that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding \({\mathcal D}\)-module and that the ind-sheaf is regular if the \({\mathcal D}\)-module is regular holonomic. They finally compute an example of the ind-sheaf of temperate solutions of an irregular \({\mathcal D}\)-module in dimension one. By properly accounting for the invariance of a Calabi-Yau sigma-model under shifts of the \(B\)-field by integral amounts (analogous to the \(\theta\)-angle in QCD), we show that the moduli spaces of such sigma-models can often be enlarged to include ``large radius limit'' points. In the simplest cases, there are holomorphic coordinates on the enlarged moduli space which vanish at the limit point, and which appear as multipliers in front of instanton contributions to Yukawa couplings. (Those instanton contributions are therefore suppressed at the limit point.) In more complicated cases, the instanton contributions are still suppressed but the enlarged space is singular at the limit point. This singularity may have interesting effects on the effective four-dimensional theory, when the Calabi-Yau is used to compactify the heterotic string.
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Quantum Chern-Simons theory leads to a number of interesting ``quantum'' invariants in low dimensional topology, including the Jones and HOMFLYPT polynomials of knots and representations of mapping class groups of surfaces. Some of these objects admit interesting deformations, a.k.a. ``refinements''. The most basic example of such a refinement is replacing a Schur polynomial -- a character of \(\mathfrak{sl}_N\) -- by the corresponding Macdonald polynomial. Based on this Schur-to-Macdonald replacement, \textit{M. Aganagic} and \textit{S. Shakirov} [Commun. Math. Phys. 333, No. 1, 187--228 (2015; Zbl 1322.81069)] constructed deformations of the representations of genus-one MCGs, and from this found deformations of the HOMFLYPT polynomials of torus knots. (When the torus knot is coloured by a totally symmetric or totally antisymmetric representation, the deformations agree with the Poincaré polynomials of triply graded knot homology, but this agreement breaks down for the multi-component torus links and for other representations.)
Whether a Schur-to-Macdonald refinement extends to all of quantum Chern-Simons theory continues to be debated. A dogmatist about locality properties of quantum field theories will quickly answer that Chern-Simons TQFT cannot be deformed. A less dogmatic person will ask not ``whether'' Chern-Simons theory is deformable, but ``how much'' of it is deformable, perhaps in a not-fully-local way. For example, are the higher-genus MCG representations deformable?
This paper contributes significant evidence supporting the existence of refined higher-genus MCG representations. Specifically, the paper proposes refinements of the genus-2 representations. The refinements owe their existence to two curious facts about quantum groups and Chern-Simons TQFT. First, the quantum \(6j\)-symbols do not admit any reasonable refinements, but their squares do; these refined squared \(6j\)-symbols are the main algebraic contribution of this paper. Second, the quantum \(6j\)-symbol is the essential local information in Chern-Simons TQFT, but it never appears by itself in final MCG representations (at least in genus \(\leq 2\)): only its square does.
Combining these facts, the authors propose refined representations of genus-\(2\) mapping class groups, and compute the corresponding refined Jones and HOMFLYPT invariants of a number of genus-\(2\) knots. They do not prove that their proposals are truly topological invariants, but they do provide a number of explicit nontrivial calculations confirming topological invariance in many examples. The Jones and HOMFLY polynomials, invariants of knots in \(S^3\) defined using two-dimensional projections and invariance under Reidemeister moves, can be heuristically re-defined by physicists using the Chern-Simons path integral over the space of connections in the trivial principal \(\mathrm{SU}(n)\) bundle. The integrality of their coefficients follows from Khovanov's construction of a bigraded knot cohomology theory associated to each knot; the Euler characteristic of Khovanov's cohomology with respect to one grading yields the Jones polynomial, hence proving its integrality. In this paper, the authors define a \(\mathrm{SU}(n)\) Chern-Simons theory on every \(3\)-manifold with a semi-free circle action. They obtain new topological invariants of Seifert fibered \(3\)-manifolds and of torus knots inside such manifolds. Their results are viewed as evidence for a conjecture of \textit{S. Gukov} et al. [Lett. Math. Phys. 74, No. 1, 53--74 (2005; Zbl 1105.57011)] about \(\mathrm{SL}_n\) knot homology.
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Quantum Chern-Simons theory leads to a number of interesting ``quantum'' invariants in low dimensional topology, including the Jones and HOMFLYPT polynomials of knots and representations of mapping class groups of surfaces. Some of these objects admit interesting deformations, a.k.a. ``refinements''. The most basic example of such a refinement is replacing a Schur polynomial -- a character of \(\mathfrak{sl}_N\) -- by the corresponding Macdonald polynomial. Based on this Schur-to-Macdonald replacement, \textit{M. Aganagic} and \textit{S. Shakirov} [Commun. Math. Phys. 333, No. 1, 187--228 (2015; Zbl 1322.81069)] constructed deformations of the representations of genus-one MCGs, and from this found deformations of the HOMFLYPT polynomials of torus knots. (When the torus knot is coloured by a totally symmetric or totally antisymmetric representation, the deformations agree with the Poincaré polynomials of triply graded knot homology, but this agreement breaks down for the multi-component torus links and for other representations.)
Whether a Schur-to-Macdonald refinement extends to all of quantum Chern-Simons theory continues to be debated. A dogmatist about locality properties of quantum field theories will quickly answer that Chern-Simons TQFT cannot be deformed. A less dogmatic person will ask not ``whether'' Chern-Simons theory is deformable, but ``how much'' of it is deformable, perhaps in a not-fully-local way. For example, are the higher-genus MCG representations deformable?
This paper contributes significant evidence supporting the existence of refined higher-genus MCG representations. Specifically, the paper proposes refinements of the genus-2 representations. The refinements owe their existence to two curious facts about quantum groups and Chern-Simons TQFT. First, the quantum \(6j\)-symbols do not admit any reasonable refinements, but their squares do; these refined squared \(6j\)-symbols are the main algebraic contribution of this paper. Second, the quantum \(6j\)-symbol is the essential local information in Chern-Simons TQFT, but it never appears by itself in final MCG representations (at least in genus \(\leq 2\)): only its square does.
Combining these facts, the authors propose refined representations of genus-\(2\) mapping class groups, and compute the corresponding refined Jones and HOMFLYPT invariants of a number of genus-\(2\) knots. They do not prove that their proposals are truly topological invariants, but they do provide a number of explicit nontrivial calculations confirming topological invariance in many examples. The authors deal with a loss of synchronization for a system of two coupled Rössler oscillators. If \(u_1\) and \(u_2\) represent the states of the two interacting subsystems and \(u_1=u_2\) is a synchronization manifold, then from a mathematical point of view the problem is reduced to examining the transverse stability of solutions embedded in this manifold.
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\textit{Y. Mizusawa} [J. Théor. Nombres Bordx. 22, No. 1, 115--138 (2010; Zbl 1221.11215)] computed a pro-\(2\) presentation of the Galois group of the maximal unramified pro-\(2\)-extension of the cyclotomic \(\mathbb Z_2\)-extension of certain complex quadratic number fields. In this article it is shown that the same methods allow to discuss some maximal \textit{tamely ramified} pro-\(2\)-extensions. The author's main result is the following: let \(p \equiv \pm 3 \bmod 8\) and \(q \equiv -p \bmod 8\) be prime numbers and put \(S = \{q\}\). Let \(k = \mathbb Q(\sqrt{-p}\,)\), let \(k_\infty\) be the cyclotomic \(\mathbb Z_2\)-extension of \(k\), and \(L_S^\infty\) the maximal pro-\(2\)-extension of \(k_\infty\) unramified outside of \(S\). Then the Galois group \(G\) of \(L_S^\infty/k_\infty\) has rank \(2\), its abelianization is isomorphic to \(\mathbb Z_2 \oplus \mathbb Z/2\mathbb Z\) as a \(\mathbb Z_2\)-module, and \(G\) has the presentation \(G = \langle a, b| [a,b]a^2 \rangle\), where \([a,b] = a^{-1}b^{-1}ab\) is the commutator of \(a\) and \(b\). Let \(p\) be a prime number, \(K\) a number field, \(K_\infty/K\) the cyclotomic \({\mathbb Z}_p\)-extension of \(K\), and \(G(K_\infty)\) the Galois group of the maximal unramified pro-\(p\)-extension of \(K_\infty\). In this article, the case \(p = 2\) is studied. One of the main results is the following theorem: let \(K = \mathbb Q(\sqrt{-q_1q_2})\) be a complex quadratic number field, where \(q_1 \equiv 3 \bmod 8\) and \(q_2 \equiv 7 \bmod 16\) are prime numbers. Let \(K_\infty\) be the cyclotomic \(\mathbb Z_2\)-extension of \(K\), and let \(\Gamma = \text{Gal}(K_\infty/K)\) denote its Galois group. Then the Galois group \(G(K_\infty)\) of the maximal unramified pro-\(2\)-extension of \(K_\infty\) is the pro-\(2\) completion of the group with presentation
\[
\langle a, b, c: [a,b] = a^{-2}, [b,c] = a^2, [a,c] = 1 \rangle,
\]
with the action of the topological generator \(\gamma\) of \(\Gamma\) given by
\[
a^\gamma = a, \quad b^\gamma = bc, \quad c^\gamma = a^{C_1} b^{-C_0} c^{1-C_1},
\]
where the \(2\)-adic integers \(C_0\) and \(C_1\) are the coefficients of the Iwasawa polynomial \(P(T) = T^2 + C_1 T + C_0\).
In addition it is shown that for a complex quadratic number field \(K = \mathbb Q(\sqrt{-m}\,)\), the Galois group \(G(K_\infty)\) is a nonabelian metacyclic pro-\(2\) group if and only if either \(m = \ell\) for some prime \(\ell \equiv 9 \bmod 16\) with \(2^{(\ell-1)/4} \equiv -1 \bmod \ell\), or \(m = pq\) for prime numbers \(p \equiv q \equiv 5 \bmod 8\).
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\textit{Y. Mizusawa} [J. Théor. Nombres Bordx. 22, No. 1, 115--138 (2010; Zbl 1221.11215)] computed a pro-\(2\) presentation of the Galois group of the maximal unramified pro-\(2\)-extension of the cyclotomic \(\mathbb Z_2\)-extension of certain complex quadratic number fields. In this article it is shown that the same methods allow to discuss some maximal \textit{tamely ramified} pro-\(2\)-extensions. The author's main result is the following: let \(p \equiv \pm 3 \bmod 8\) and \(q \equiv -p \bmod 8\) be prime numbers and put \(S = \{q\}\). Let \(k = \mathbb Q(\sqrt{-p}\,)\), let \(k_\infty\) be the cyclotomic \(\mathbb Z_2\)-extension of \(k\), and \(L_S^\infty\) the maximal pro-\(2\)-extension of \(k_\infty\) unramified outside of \(S\). Then the Galois group \(G\) of \(L_S^\infty/k_\infty\) has rank \(2\), its abelianization is isomorphic to \(\mathbb Z_2 \oplus \mathbb Z/2\mathbb Z\) as a \(\mathbb Z_2\)-module, and \(G\) has the presentation \(G = \langle a, b| [a,b]a^2 \rangle\), where \([a,b] = a^{-1}b^{-1}ab\) is the commutator of \(a\) and \(b\). The geometry of conjugate connections is a natural generalization of the geometry of Levi-Civita connections from the theory of Riemannian manifolds. Conjugate connections arise in affine differential geometry and in the geometric theory of statistical inferences.
The generalized conjugate connection is introduced in Weyl geometry to characterize Weyl connections. The semi-conjugate connection arises naturally in affine hypersurface theory. In the present paper, the notion of dual semi-conjugate connection is introduced in order to study the relations between these connections.
Let \((M,g)\) be a semi-Riemannian manifold, \(\nabla \) an affine connection on \(M\) and \(\tau \) a 1-form. The dual semi-conjugate connection \(\tilde {\nabla }\) of \(\nabla \) with respect to \(g\) by \(\tau \) is defined by
\[
Xg(Y,Z)=g(\nabla _XY,Z)+g(Y,\tilde {\nabla}_XZ)-\tau (X)g(Y,Z)-\tau (Y)g(X,Z).
\]
Also, local triviality of generalized conjugate connections and equiaffine structures on a manifold are considered.
It is shown that, for a connected and simply connected \(n\)-dimensional semi-Riemannian manifold \((M,g)\), if \(\nabla \) is a torsion-free affine connection and \(\overline {\nabla }\) is the generalized conjugate connection of \(\nabla ,\) then any two of the following conditions imply the third one:
(1) \(\tau \) is an exact 1-form,
(2) \(\nabla \) is equiaffine,
(3) \(\overline {\nabla }\) is equiaffine.
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The book consists of two fairly independent parts, I. Bernhard Riemann and II. Topological themes in contemporary physics. The concepts of Riemann (manifolds, topology, analysis) and his view of a mathematical physics can be seen as a spiritual link. The Russian originals appeared in 1976 and 1979. Part II has been considerably expanded in this second English edition. (First edition of 1987; Zbl 0626.01033.)
Part I is an essay on Riemann's life and his work, as Riemann surfaces, complex analysis, and some of his achievements in physics (shock waves, liquid ellipsoids). There are new sections on recent results on the Riemann-Hilbert Problem, the Riemann-Roch Theorem, and on the Schottky Problem. These pave the way for reports on soliton particles in Part II.
Part II is a collection of survey articles (no proofs) on significant subjects of contemporary physical research, accessible for readers with some background in physics and mathematics (third university year). All of these subjects are more or less concerned with topology in a general sense, including here algebraic and differential geometry. There is a brief introduction to topology. Topics include: systems with spontaneous symmetry breaking; liquid crystals; gauge fields (here topology makes it possible to explain the complex structure of solutions of equations of gauge fields by means of fiber bundles); topological particles (instantons, Donaldson); soliton particles (the complete story); knots and links (including V. Jones' results and their physical consequences, and condensed matter theory); and an outlook on the Quantum Hall Effect, strings and membranes, and quasi-crystals.
Presentation in Part II is precise and clear. This is a well-readable text aimed at a wide audience. The same may be said about Part I, though the historian must mention at least the graver of the numerous blunders and inaccuracies in the biographical part which have survived in the new translation (by R. Cooke). None of the three topics prepared by Riemann for his Habilitationsvortrag was `associated with his investigations on electricity' (p. 28). Maßbestimmungen cannot be translated as `dimension of space' (p. 33). The function described on p. 26 is not the famous `Riemann function'. Fakultät cannot be translated as `university council' (passim). There are many historical inaccuracies on p. 43, etc.
Part I rests mainly on the biography by Dedekind (1876) and on Klein's lectures (publ. 1925). There are new facsimile reprints of two pages from Riemann's hand. This does not justify the lines which the publishers choose for the back cover: ``The author\dots takes into account his own research at the Riemann archives of Göttingen university\(\dots\)''. [Das russische Original (Znanie Moskva 1979) wurde nicht im Zbl MATH referiert.]
Das Buch bringt die Übersetzung zweier an sich unabhängiger Arbeiten des Verf. I ist eine Biographie Riemanns mit besonderer Hervorhebung seines geometrischen Zugangs zur Funktionentheorie, der Riemannschen n- dimensionalen Mannigfaltigkeiten und der Benutzung des Dirichletschen Prinzips bei partiellen Differentialgleichungen. Die damaligen historischen Voraussetzungen, die zunächst geringe Wirksamkeit und die späteren bedeutenden Folgen für die theoretische Physik werden gezeigt. II, topologische Themen der gleichzeitigen Physik zeigen, zunächst an einfachen Beispielen, was Topologie ist und wie sie in der theoretischen Physik genutzt werden kann, manchmal nur andeutend: beim quantisierten magnetischen Fluß durch mehrfach zusammenhängende supraleitende Gebiete, bei Phasenübergängen, bei verschiedenen Fällen der Symmetriebrechung, auch bei Solitonen, ausführlicher bei flüssigen Kristallen und bei den Eichtheorien der Wechselwirkungen zwischen Elementarteilchen. (Bei den Anmerkungen sind die Seiten 68 und 137 zu vertauschen.)
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The book consists of two fairly independent parts, I. Bernhard Riemann and II. Topological themes in contemporary physics. The concepts of Riemann (manifolds, topology, analysis) and his view of a mathematical physics can be seen as a spiritual link. The Russian originals appeared in 1976 and 1979. Part II has been considerably expanded in this second English edition. (First edition of 1987; Zbl 0626.01033.)
Part I is an essay on Riemann's life and his work, as Riemann surfaces, complex analysis, and some of his achievements in physics (shock waves, liquid ellipsoids). There are new sections on recent results on the Riemann-Hilbert Problem, the Riemann-Roch Theorem, and on the Schottky Problem. These pave the way for reports on soliton particles in Part II.
Part II is a collection of survey articles (no proofs) on significant subjects of contemporary physical research, accessible for readers with some background in physics and mathematics (third university year). All of these subjects are more or less concerned with topology in a general sense, including here algebraic and differential geometry. There is a brief introduction to topology. Topics include: systems with spontaneous symmetry breaking; liquid crystals; gauge fields (here topology makes it possible to explain the complex structure of solutions of equations of gauge fields by means of fiber bundles); topological particles (instantons, Donaldson); soliton particles (the complete story); knots and links (including V. Jones' results and their physical consequences, and condensed matter theory); and an outlook on the Quantum Hall Effect, strings and membranes, and quasi-crystals.
Presentation in Part II is precise and clear. This is a well-readable text aimed at a wide audience. The same may be said about Part I, though the historian must mention at least the graver of the numerous blunders and inaccuracies in the biographical part which have survived in the new translation (by R. Cooke). None of the three topics prepared by Riemann for his Habilitationsvortrag was `associated with his investigations on electricity' (p. 28). Maßbestimmungen cannot be translated as `dimension of space' (p. 33). The function described on p. 26 is not the famous `Riemann function'. Fakultät cannot be translated as `university council' (passim). There are many historical inaccuracies on p. 43, etc.
Part I rests mainly on the biography by Dedekind (1876) and on Klein's lectures (publ. 1925). There are new facsimile reprints of two pages from Riemann's hand. This does not justify the lines which the publishers choose for the back cover: ``The author\dots takes into account his own research at the Riemann archives of Göttingen university\(\dots\)''. No review copy delivered.
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A family of linear state equations \({\mathcal L}(h)\omega =q\) in a Banach space is considered. Where the linear, invertible operator \({\mathcal L}(h)\) depends on a (possibly functional) parameter h. A simple formula on sensitivity of \(\omega\) with respect to h is given. The appliation of this formula to design of mechanical structures is suggested and illustrated by some simple examples. [For the entire collection see Zbl 0539.00022.]
This paper deals with optimization of constrained functionals with distributed parameters. The functionals may be chosen for identifying parameters that optimize performance or minimize energy forms, or for computation of eigenvalues via the Rayleigh quotient. Constraints may represent design limitations, extremes of operating conditions, or state equations for dynamical systems. The intrinsic nature of iterative solution methods for functional minimization, the functional sensitivity analysis, and state function sensitivity analysis have been the subject of extensive research. Using simple examples from engineering, this paper points out some pitfalls for gradient-type computational methods particularly in connection with computing of unstable processes or eigenvalues via minimization of a constrained Rayleigh quotient. Auxiliary conditions involving energy levels of the system for constrained problems are suggested as indicators of existence of multiple gradient directions.
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A family of linear state equations \({\mathcal L}(h)\omega =q\) in a Banach space is considered. Where the linear, invertible operator \({\mathcal L}(h)\) depends on a (possibly functional) parameter h. A simple formula on sensitivity of \(\omega\) with respect to h is given. The appliation of this formula to design of mechanical structures is suggested and illustrated by some simple examples. Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of complex nondifferentiable fractional programming problems containing generalized convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing one parametric and two other parameter-free dual models with appropriate duality theorems.
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The author explores how much information about a compact Lie group action can be extracted from information about the restriction of the action to finite subgroups.
Theorem 1. Suppose that f: M'\(\to M\) is a G homotopy equivalence between two smooth simply connected G-manifolds, G a connected Lie group, which is H-homotopic to an H-diffeomorphism for all finite subgroups H of G, then if dim M-dim \(G\neq 3\), 4, f is G-homotopic to PL G-homeomorphism which is a G-diffeomorphism in the complement of an orbit. If dim M-dim G is even, then f is G-homotopic to a G-diffeomorphism.
Corollary. Two free smooth actions of \(G=S^ 1\) or SU(2) on homotopy spheres \(\Sigma^ n\), \(n\neq 5\), 7 respectively, which are conjugate for their finite subgroups are conjugate.
The results were announced in the author's earlier paper in Geometry and topology, Proc. Conf. Athens/Ga. 1985, Lect. Notes Pure Appl. Mat. 105, 319-323 (1987; Zbl 0629.57024), and the necessity of some of the assumptions is explained there as well. The proof of Theorem 1 follows from surgery theory and from a cohomological result: Theorem 2. Suppose G is a compact Lie group acting freely on a space X of finite type and h is a cohomology theory of finite type, then
\[
h*(X/G)\to \prod h*(X/H)
\]
is injective as H ranges over the finite subgroups of G. If h is ordinary cohomology, or the identity component of G is a torus, then one need not assume that the action is free. (One does need some technical hypotheses, easily holding for G-CW complexes.) If h is p-local then the subgroups can be chosen to be the p-groups. [For the entire collection see Zbl 0594.00013.]
The following theorems are announced. (1) Any two homotopy equivalent free absolute neighbourhood retract actions of a compact Lie group on a simply connected manifold conjugate for each finite subgroup can be connected by a one-parameter family of such actions. (2) Let H be a simply connected finite-dimensional homotopy type. If for each n there is an \(X_ n\in H\) having a free \({\mathbb{Z}}_ n\)-action, then there is an \(X\in H\) having a free \({\mathbb{Z}}_ n\)-action for all n. If the \(X_ n's\) are finite, X can be chosen finite. (3) Under the hypotheses of (2) there is an \(X\in H\) having a free \({\mathbb{Q}}/ {\mathbb{Z}}\)-action.
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The author explores how much information about a compact Lie group action can be extracted from information about the restriction of the action to finite subgroups.
Theorem 1. Suppose that f: M'\(\to M\) is a G homotopy equivalence between two smooth simply connected G-manifolds, G a connected Lie group, which is H-homotopic to an H-diffeomorphism for all finite subgroups H of G, then if dim M-dim \(G\neq 3\), 4, f is G-homotopic to PL G-homeomorphism which is a G-diffeomorphism in the complement of an orbit. If dim M-dim G is even, then f is G-homotopic to a G-diffeomorphism.
Corollary. Two free smooth actions of \(G=S^ 1\) or SU(2) on homotopy spheres \(\Sigma^ n\), \(n\neq 5\), 7 respectively, which are conjugate for their finite subgroups are conjugate.
The results were announced in the author's earlier paper in Geometry and topology, Proc. Conf. Athens/Ga. 1985, Lect. Notes Pure Appl. Mat. 105, 319-323 (1987; Zbl 0629.57024), and the necessity of some of the assumptions is explained there as well. The proof of Theorem 1 follows from surgery theory and from a cohomological result: Theorem 2. Suppose G is a compact Lie group acting freely on a space X of finite type and h is a cohomology theory of finite type, then
\[
h*(X/G)\to \prod h*(X/H)
\]
is injective as H ranges over the finite subgroups of G. If h is ordinary cohomology, or the identity component of G is a torus, then one need not assume that the action is free. (One does need some technical hypotheses, easily holding for G-CW complexes.) If h is p-local then the subgroups can be chosen to be the p-groups. Corrects an erreor in the authors' paper [ibid. 392, 49--62 (2018; Zbl 1390.81654)] in the derivation of the determination of
\(P_{EM}\) for the system ``point-like charged particle in an external EM field'' as a function of the scalar \(\varphi\) and vector \(A\) potentials of the external EM field.
The reported corrections do not influence the conclusions of [loc. cit.] and open a way to some new insights in the physical meaning of quantum phases.
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In the paper under review the following theorem is proved.
Let \(\Omega_1\), \(\Omega_2\) be irreducible bounded symmetric domains of rank at least 2 and let \(F:\Omega_1\longrightarrow\Omega_2\) be a holomorphic map. If \(F\) maps the minimal disks of \(\Omega_1\) properly into the rank-1 characteristic symmetric subspaces of \(\Omega_2\), then \(F\) is a totally geodesic isometric embedding (up to a normalization constant) with respect to the Bergman metrics.
By proving the above theorem one obtains a much simpler proof of the result proved by \textit{I. H. Tsai} in [J. Differ. Geom. 37, No. 1, 123--160 (1993; Zbl 0799.32027)]. The rigidity problem for holomorphic maps between bounded symmetric domains is considered.
The main result is as follows: Let \((\Omega_ 1, g_ 1)\) and \((\Omega_ 2, g_ 2)\) be bounded symmetric domains. Suppose that \(\Omega_ 1\) is irreducible and \(\text{rank} (\Omega_ 1) \geq \text{rank} (\Omega_ 2) \geq 2\). Then any proper holomorphic map \(f:\Omega_ 1 \to \Omega_ 2\) is necessarily a totally geodesic isometric embedding (up to normalizing constants).
This theorem resolves a conjecture of Mok.
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In the paper under review the following theorem is proved.
Let \(\Omega_1\), \(\Omega_2\) be irreducible bounded symmetric domains of rank at least 2 and let \(F:\Omega_1\longrightarrow\Omega_2\) be a holomorphic map. If \(F\) maps the minimal disks of \(\Omega_1\) properly into the rank-1 characteristic symmetric subspaces of \(\Omega_2\), then \(F\) is a totally geodesic isometric embedding (up to a normalization constant) with respect to the Bergman metrics.
By proving the above theorem one obtains a much simpler proof of the result proved by \textit{I. H. Tsai} in [J. Differ. Geom. 37, No. 1, 123--160 (1993; Zbl 0799.32027)]. The new model of turbulence with small addition of polymers, leading to a system of four differential equations, is applied to the case of plane channel flows. These equations are derived for the kinetic energy of turbulence, the rate of dissipation and the Reynolds stresses (two equations). The numerical calculations revealed many important effects, for example, an enlargement of the viscous sublayer thickness, a reduction of the drag coefficient depending on concentration, an increase of the turbulence energy in the transition zone, etc. Some results are discussed in a graphical form.
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The authors consider the equation
\[
aB_m(x)=bB_n(y)+C(y),
\]
where \(C(y)\in\mathbb{Q}[y]\) and \(m\geq n>\text{deg}\;C+2\). Applying the method of \textit{Y. F. Bilu} and \textit{R. F. Tichy} [Acta Arith. 95, No.~3, 261--288 (2000; Zbl 0958.11049)] they show that this equation has only finitely many rational solutions \((x,y)\) with bounded denominators except when \(m=n,a=\pm b\) and \(C(y)\equiv 0\). Let \(K\) be a number field and \(S\) be a finite set of places of \(K\) containing all archimedean places. Denote by \(O_S\) the ring of \(S\)-integers of \(K\). Let \(F(x,y)\in K[x,y]\). We say that the equation \(F(x,y)= 0\) has infinitely many solutions with a bounded \(O_S\)-denominator if there exists a non-zero \(\Delta\in O_S\) such that \(F(x,y)= 0\) has infinitely many solutions \((x,y)\in K^2\) with \(\Delta x,\Delta y\in O_S\).
In this paper the case of the equation \(f(x)= g(y)\), where \(f(x), g(x)\in K[x,y] \setminus K\), is studied. The authors determine an explicit set \(E\) of \(K[x] \times K[x]\) and prove that the equation \(f(x)= g(y)\) has infinitely many solutions with a bounded \(O_S\)-denominator if and only if \(f= \varphi\circ f_1\circ \lambda\) and \(g= \varphi\circ g_1\circ \mu\), where \(\lambda(x), \mu(x)\in K[x]\) are linear polynomials, \(\varphi(x)\in K[x]\), \((f_1(x), g_1(x))\in E\) and the equation \(f_1(x)= g_1(y)\) has infinitely many solutions with a bounded \(O_S\)-denominator.
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The authors consider the equation
\[
aB_m(x)=bB_n(y)+C(y),
\]
where \(C(y)\in\mathbb{Q}[y]\) and \(m\geq n>\text{deg}\;C+2\). Applying the method of \textit{Y. F. Bilu} and \textit{R. F. Tichy} [Acta Arith. 95, No.~3, 261--288 (2000; Zbl 0958.11049)] they show that this equation has only finitely many rational solutions \((x,y)\) with bounded denominators except when \(m=n,a=\pm b\) and \(C(y)\equiv 0\). In their book ''An introduction to bilinear time series models.'' (1978; Zbl 0379.62074), p. 43, \textit{C. W. J. Granger} and \textit{A. P. Andersen}, dismiss the use of third order moments for identifying models on the grounds that for some bilinear models they will all be zero and hence are of no use in discriminating between true white noise and some bilinear models.
However, in this paper it is shown that some of the third order moments do not vanish for some superdiagonal and diagonal bilinear models and the pattern of non zero moments can be used to discriminate between true white noise and these bilinear models and also between different bilinear models. Simulation experiments are used to study the applicability of theoretical results.
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The \(n\)-dimensional Hankel transform of distributions considered earlier by the authors in [Integral Transforms Spec.\ Funct.\ 18, No.\,12, 897--911 (2007; Zbl 1184.46040)] for Hankel functions of order \(\mu\in\mathbb{R}^n\), \(\mu_i\geq-\tfrac 12\), as kernel is extended to all \(\mu\in\mathbb{R}^n\) without any restriction. The methods are well known. The authors study the \(n\)-dimensional Hankel transform defined by
\[
h_\mu \phi (y)=\int_{\mathbb{R}_+^n}\phi (x_1,\dots,x_n)\prod _{i=1}^n\sqrt{x_iy_i}J_{\mu _i}(x_iy_i)\,dx_1\dots dx_n, \quad y=(y_1,\dots,y_n)\in \mathbb{R}_+^n,
\]
where \(\mu =(\mu _1,\dots,\mu _n)\), with \(\mu _i\geq -\frac{1}{2}, i=1,\dots,n\), and \(J_\nu\) represents the Bessel function of the first kind and order \(\nu\).
They deal with certain \(n\)-dimensional spaces, \(H_\mu\) and its dual \(H_\mu '\), which generalize the ones studied by \textit{A.\,H.\thinspace Zemanian} [SIAM J.~Appl.\ Math.\ 14, 561--576 (1966; Zbl 0154.13803)] in the one-dimensional case. One of the main results establishes that \(h_\mu\), and also the ge\-ne\-ra\-li\-zed Hankel transform defined on \(H_\mu '\) by transposition, are automorphisms on \(H_\mu\) and on \(H_\mu '\), respectively.
On the other hand, the authors introduce an \(n\)-dimensional generalization of the Bessel operator, \(S_\mu\), and establish useful properties which allow them to find solutions to some partial differential equations involving the Bessel operator.
Finally, they investigate the part of the Bessel operator in \(L^2(\mathbb{R}_+^n)\), \(S_{\mu ,2}\), and apply the results to study the fractional complex powers and to obtain a representation of \((-S_{\mu ,2})^\alpha \), \(\alpha \in \mathbb{C}\), in terms of the Hankel transformation.
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The \(n\)-dimensional Hankel transform of distributions considered earlier by the authors in [Integral Transforms Spec.\ Funct.\ 18, No.\,12, 897--911 (2007; Zbl 1184.46040)] for Hankel functions of order \(\mu\in\mathbb{R}^n\), \(\mu_i\geq-\tfrac 12\), as kernel is extended to all \(\mu\in\mathbb{R}^n\) without any restriction. The methods are well known. The order of simultaneous approximation and Voronovskaja kind results with quantitative estimates for the complex genuine Durrmeyer polynomials attached to analytic functions on compact disks are obtained. In this way, we show the overconvergence phenomenon for the genuine Durrmeyer polynomials, namely the extension of the approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.
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Let \(m_j,k_j>0\) for all \(j\in\mathbb{N}\). Consider a self-adjoint extension \(J\) of the minimal symmetric operator \(J_0\) naturally associated in \(l^2(\mathbb{N})\) with the Jacobi (tri-diagonal) matrix
\[
J_0=\left(\begin{matrix} q_1 & b_1 & 0 & \dots\\ b_1 & q_2 & b_2 & \dots\\ 0 & b_2 & q_3 & \dots\\ \dots & \dots & \dots & \dots \end{matrix} \right),\quad q_j=-\frac{k_j+k_{j+1}}{m_j},\;b_j=\frac{k_{j+1}}{\sqrt{m_jm_{j+1}}}.
\]
Note that the corresponding difference equation allows a useful mechanical interpretation. Namely, it describes the motion of a string (sometimes called the Krein-Stieltjes string) with the mass distribution \(M(x)=\sum_{x_j<x}m_j\), where \(x_0:=0\) and \(x_j:=x_{j-1}+k_j\), \(j\in\mathbb{N}\).
Fix \(\theta>0\) and \(h\in\mathbb{R}\) and consider the following operator \(\tilde{J}\), which is a rank 2 perturbation of \(J\):
\[
\tilde{J}=J+(q_1(\theta^2-1)+\theta^2h)(e_1,\cdot)e_1+b_1(\theta-1)\big((e_1,\cdot)e_2+(e_2,\cdot)e_1\big).
\]
Under the assumption that the spectrum of \(J\) (and hence \(\tilde{J}\)) is discrete, the authors investigate the problem of reconstruction of the operator \(J\) from the spectra of \(J\) and \(\tilde{J}\). Also, they present necessary and sufficient conditions for two sequences to be the spectra of \(J\) and \(\tilde{J}\).
For Part I, see [\textit{R. del Rio} and \textit{M. Kudryavtsev}, Inverse Probl. 28, No.~5, Article ID 055007 (2012; Zbl 1259.47040), \url{arXiv:1106.1691}]. For Part II, see [\textit{R. del Rio}, \textit{M. Kudryavtsev} and \textit{L. O. Silva}, ``Inverse problems for Jacobi operators. II: Mass perturbations of semi-infinite mass-spring systems'', \url{arXiv:1106.4598}]. The paper deals with the problem of the reconstruction of a finite Jacobi matrix by its spectrum, and the spectrum of its 3-rank perturbation of a special type. The question is generated by the problem of finding characteristics of a system of point masses and connective springs by its frequencies, and the frequencies of a perturbed system with an interior mass changed and a new spring attached to this mass. The authors find a constructive algorithm for the recovery of the matrix by two spectra and an additional parameter. They give necessary and sufficient conditions on the spectral data and solve completely the corresponding inverse problem.
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Let \(m_j,k_j>0\) for all \(j\in\mathbb{N}\). Consider a self-adjoint extension \(J\) of the minimal symmetric operator \(J_0\) naturally associated in \(l^2(\mathbb{N})\) with the Jacobi (tri-diagonal) matrix
\[
J_0=\left(\begin{matrix} q_1 & b_1 & 0 & \dots\\ b_1 & q_2 & b_2 & \dots\\ 0 & b_2 & q_3 & \dots\\ \dots & \dots & \dots & \dots \end{matrix} \right),\quad q_j=-\frac{k_j+k_{j+1}}{m_j},\;b_j=\frac{k_{j+1}}{\sqrt{m_jm_{j+1}}}.
\]
Note that the corresponding difference equation allows a useful mechanical interpretation. Namely, it describes the motion of a string (sometimes called the Krein-Stieltjes string) with the mass distribution \(M(x)=\sum_{x_j<x}m_j\), where \(x_0:=0\) and \(x_j:=x_{j-1}+k_j\), \(j\in\mathbb{N}\).
Fix \(\theta>0\) and \(h\in\mathbb{R}\) and consider the following operator \(\tilde{J}\), which is a rank 2 perturbation of \(J\):
\[
\tilde{J}=J+(q_1(\theta^2-1)+\theta^2h)(e_1,\cdot)e_1+b_1(\theta-1)\big((e_1,\cdot)e_2+(e_2,\cdot)e_1\big).
\]
Under the assumption that the spectrum of \(J\) (and hence \(\tilde{J}\)) is discrete, the authors investigate the problem of reconstruction of the operator \(J\) from the spectra of \(J\) and \(\tilde{J}\). Also, they present necessary and sufficient conditions for two sequences to be the spectra of \(J\) and \(\tilde{J}\).
For Part I, see [\textit{R. del Rio} and \textit{M. Kudryavtsev}, Inverse Probl. 28, No.~5, Article ID 055007 (2012; Zbl 1259.47040), \url{arXiv:1106.1691}]. For Part II, see [\textit{R. del Rio}, \textit{M. Kudryavtsev} and \textit{L. O. Silva}, ``Inverse problems for Jacobi operators. II: Mass perturbations of semi-infinite mass-spring systems'', \url{arXiv:1106.4598}]. Let \(G\) be a simple undirected graph. Then the signless Laplacian matrix of \(G\) is defined as \(D_G + A_G\) in which \(D_G\) and \(A_G\) denote the degree matrix and the adjacency matrix of \(G\), respectively. The graph \(G\) is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as \(G\) is isomorphic to \(G\). We show that \(G\sqcup rK_2\) is determined by its signless Laplacian spectra under certain conditions, where \(r\) and \(K_2\) denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.
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Let \(A=\bigoplus_{n\geq 0}A_n\) be a tower of algebras as defined by \textit{N. Bergeron} and \textit{H. Li} [J. Algebra 321, No. 8, 2068-2084 (2009; Zbl 1185.16008)], where they show that for such a tower the Grothendieck groups of categories of finitely generated projective left \(A\)-modules, respectively of all left \(A\)-modules, form a pair of graded dual Hopf algebras. In the paper under review it is proved that in the above framework one has \(\dim(A_n)=\dim(A_1)^n\cdot n!\) for any \(n\). A quantum version of this result, involving \(q\)-twisted Hopf algebras, is also proved. A tower of algebras is a graded algebra \(A=\bigoplus_nA_n\) over the field of complex numbers, such that for each \(n\) the \(n\)-component \(A_n\) is a finite-dimensional algebra with unit \(1_n\). Moreover \(A_0=\mathbb{C}\), the external product \(A_m\otimes A_n\to A_{n+m}\) is an injective homomorphism of algebras satisfying two more technical conditions.
The main example is the graded algebra \(\bigoplus_n\mathbb{C}[\mathfrak S_n]\), where \(\mathfrak S_n\) is the symmetric group and the product \(\mathfrak S_m\otimes\mathfrak S_n\to\mathfrak S_{n+m}\) is concatenation. It is proved that induction and restriction yields a graded Hopf algebra structure on the Grothendieck groups \(\bigoplus_nG_0(A_n)\) and \(\bigoplus_nK_0(A_n)\).
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Let \(A=\bigoplus_{n\geq 0}A_n\) be a tower of algebras as defined by \textit{N. Bergeron} and \textit{H. Li} [J. Algebra 321, No. 8, 2068-2084 (2009; Zbl 1185.16008)], where they show that for such a tower the Grothendieck groups of categories of finitely generated projective left \(A\)-modules, respectively of all left \(A\)-modules, form a pair of graded dual Hopf algebras. In the paper under review it is proved that in the above framework one has \(\dim(A_n)=\dim(A_1)^n\cdot n!\) for any \(n\). A quantum version of this result, involving \(q\)-twisted Hopf algebras, is also proved. In ibid. 1982, No.4, 33-36 (1982; Zbl 0508.54023) the author described m- proximities for a \(T_ 1\) space X with a given map \(f: X\to Y\) and their induced bicompactifications \(\delta_ fX\) and \(f_{\delta}\). Here he uses m-proximities to define a covering type of dimension for X and f and proves them equal, respectively to the usual covering dimension of the remainder \(N=\delta_ fX\setminus X\) and of \(f_{\delta}| N\). This extends to mappings some results of \textit{Yu. M. Smirnov} [Mat. Sb., Nov. Ser. 69(111), 141-160 (1966; Zbl 0161.424)].
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This paper studies the catenary degree of a finitely generated commutative cancellative monoid \(S\). This invariant has received much attention in the recent literature of nonunique factorizations. The catenary degree of an element \(n\) is a nonnegative integer that gives a measure of the distance between any two factorizations of \(n\). The catenary degree of a monoid \(S\) is the supremum of the catenary degrees of the elements of \(S\). The goal of this paper is to understand more than just this supremum. The authors study the set \(C(S)\) of catenary degree of elements of \(S\).
A finitely generated monoid \(S\) has a special set of elements called Betti elements, and it is known that the maximum element of \(C(S)\) is equal to the maximum catenary degree achieved by a Betti element of \(S\) [\textit{S. T. Chapman} et al., Manuscr. Math. 120, No. 3, 253--264 (2006; Zbl 1117.20045)]. A main result of this paper is to show that the minimum nonzero value of \(C(S)\) occurs as the catenary degree of a Betti element of \(S\). The authors give an example of a monoid \(S\) and an element \(x\) of \(C(S)\) such that no Betti element has catenary degree equal to \(x\). The authors also give several nice examples of computations of \(C(S)\) for families of numerical monoids. They show that \(|C(S)|\) can be arbitrarily large for numerical monoids generated by three elements. Let \(S\) be a commutative cancellative atomic monoid. The catenary degree and the tame degree of \(S\) are (somewhat technical) combinatorial invariants of \(S\) which describe the behavior of chains of factorizations in \(S\) (see for example, the book Non-unique factorizations. Algebraic, combinatorial and analytic theory by \textit{A. Geroldinger} and \textit{F. Halter-Koch} [Pure Appl. Math. 278. Boca Raton, FL: Chapman \& Hall/CRC (2006; Zbl 1113.11002)]).
In this paper, the authors give methods to compute both invariants when \(S\) is finitely generated. These methods are based on the computation of a minimal presentation of \(S\), and there are algorithms to compute this. Several examples are given to illustrate the theory.
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This paper studies the catenary degree of a finitely generated commutative cancellative monoid \(S\). This invariant has received much attention in the recent literature of nonunique factorizations. The catenary degree of an element \(n\) is a nonnegative integer that gives a measure of the distance between any two factorizations of \(n\). The catenary degree of a monoid \(S\) is the supremum of the catenary degrees of the elements of \(S\). The goal of this paper is to understand more than just this supremum. The authors study the set \(C(S)\) of catenary degree of elements of \(S\).
A finitely generated monoid \(S\) has a special set of elements called Betti elements, and it is known that the maximum element of \(C(S)\) is equal to the maximum catenary degree achieved by a Betti element of \(S\) [\textit{S. T. Chapman} et al., Manuscr. Math. 120, No. 3, 253--264 (2006; Zbl 1117.20045)]. A main result of this paper is to show that the minimum nonzero value of \(C(S)\) occurs as the catenary degree of a Betti element of \(S\). The authors give an example of a monoid \(S\) and an element \(x\) of \(C(S)\) such that no Betti element has catenary degree equal to \(x\). The authors also give several nice examples of computations of \(C(S)\) for families of numerical monoids. They show that \(|C(S)|\) can be arbitrarily large for numerical monoids generated by three elements. A compressed full-text self-index for a text \(T\) is a data structure requiring reduced space and able to search for patterns \(P\) in \(T\). It can also reproduce any substring of \(T\), thus actually replacing \(T\). Despite the recent explosion of interest on compressed indexes, there has not been much progress on functionalities beyond the basic exact search. In this paper we focus on indexed approximate string matching (ASM), which is of great interest, say, in bioinformatics. We study ASM algorithms for Lempel-Ziv compressed indexes and for compressed suffix trees/arrays. Most compressed self-indexes belong to one of these classes. We start by adapting the classical method of partitioning into exact search to self-indexes, and optimize it over a representative of either class of self-index. Then, we show that a Lempel-Ziv index can be seen as an extension of the classical \(q\)-samples index. We give new insights on this type of index, which can be of independent interest, and then apply them to a Lempel-Ziv index. Finally, we improve hierarchical verification, a successful technique for sequential searching, so as to extend the matches of pattern pieces to the left or right. Most compressed suffix trees/arrays support the required bidirectionality, thus enabling the implementation of the improved technique. In turn, the improved verification largely reduces the accesses to the text, which are expensive in self-indexes. We show experimentally that our algorithms are competitive and provide useful space-time tradeoffs compared to classical indexes.
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Let \({\mathfrak g}\) be a finite dimensional Lie algebra over a field \(\Phi\). If the characteristic of \(\Phi\) is zero a theorem of Tits which goes back to Lie asserts that each hyperplane subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) contains a codimension two ideal \({\mathfrak c}\) such that \({\mathfrak g}/{\mathfrak c}\) is either \(\Phi\), \(\text{sl}(2,\Phi)\) or the two dimensional non- abelian algebra \({\mathfrak s}_ 2\). A computational approach to this fact was given by \textit{K. H. Hofmann} in [Geom. Dedicata 36, 207-224 (1990; Zbl 0718.17006)]. There, it was left open if the intersection \(\Delta_{aff}({\mathfrak g})\) of all the \({\mathfrak c}\)'s for which \({\mathfrak g}/{\mathfrak c}\cong{\mathfrak s}_ 2\) is a characteristic ideal. In the paper under review an elementary proof for this fact is given. On the way the author presents a new proof of Lie's theorem and an alternative construction of \(\Delta_{aff}({\mathfrak g})\). Moreover he gives an example for the fact that \(\Delta_{aff}({\mathfrak g})\) is not a characteristic ideal if \(\Phi\) has finite characteristic. Eincodimensionale Unteralgebren einer reellen Lieschen Algebra \({\mathfrak g}\) sind seit längerem im Prinzip bekannt: Man hat die Quotienten von \({\mathfrak g}\) zu bestimmen, welche eindimensional, zweidimensional nicht- abelsch oder zu \({\mathfrak sl}_ 2\) isomorph sind. Die gesuchten Algebren ergeben sich dann durch Zurückziehen der entsprechenden (wohlbekannten) Unteralgebren in den Quotienten. Neuere Untersuchungen in der Theorie der Lieschen Halbgruppen/Halbalgebren erfordern eine genaue Beschreibung sämtlicher eincodimensionaler Unteralgebren in einer gegebenen Lieschen Algebra. Diese Aufgabe wird in dem vorliegenden Aufsatz vollständig behandelt. Dabei wird ein neues ``Radikal'', der Durchschnitt \(\Delta\) (\({\mathfrak g})\) aller eincodimensionalen Unteralgebren eingeführt und systematisch studiert. \(\Delta\) (\({\mathfrak g})\) ist ein charakteristisches Ideal in \({\mathfrak g}\). Der Artikel endet mit einer Klassifikation der sogenannten Durchschnittshalbalgebren.
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Let \({\mathfrak g}\) be a finite dimensional Lie algebra over a field \(\Phi\). If the characteristic of \(\Phi\) is zero a theorem of Tits which goes back to Lie asserts that each hyperplane subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) contains a codimension two ideal \({\mathfrak c}\) such that \({\mathfrak g}/{\mathfrak c}\) is either \(\Phi\), \(\text{sl}(2,\Phi)\) or the two dimensional non- abelian algebra \({\mathfrak s}_ 2\). A computational approach to this fact was given by \textit{K. H. Hofmann} in [Geom. Dedicata 36, 207-224 (1990; Zbl 0718.17006)]. There, it was left open if the intersection \(\Delta_{aff}({\mathfrak g})\) of all the \({\mathfrak c}\)'s for which \({\mathfrak g}/{\mathfrak c}\cong{\mathfrak s}_ 2\) is a characteristic ideal. In the paper under review an elementary proof for this fact is given. On the way the author presents a new proof of Lie's theorem and an alternative construction of \(\Delta_{aff}({\mathfrak g})\). Moreover he gives an example for the fact that \(\Delta_{aff}({\mathfrak g})\) is not a characteristic ideal if \(\Phi\) has finite characteristic. The authors are motivated by the mathematical foundations of quantum computing and they define (i) an expansion of \(\sqrt{'}\) quasi-MV algebras by lattice operations, and (ii) Gödel-like implications. These notions are used in a new class of algebras, called \textit{Gödel quantum computational algebras}. The authors show that every such algebra arises as a pair algebra out of a Heyting-Wajsberg algebra. The variety of such algebras is an arithmetical variety as well as a discriminator variety.
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The integral form of the classical Schur algebra has a \(\mathbb{Z}\)-basis consisting of codeterminants, as defined by \textit{J. A. Green} [J. Pure Appl. Algebra 88, No. 1-3, 89-106 (1993; Zbl 0794.20053)]. The author investigates the quantized version of this fact. He considers the quantized Schur algebras and defines quantized codeterminants. He shows that an analogous result also holds in this case and provides a straightening formula which allows one to express any codeterminant of dominant shape as a sum of standard codeterminants. The author studies the structure of the Schur algebra \(S_ R(n,r)\) over a commutative ring \(R\). A basis of codeterminants for this algebra is given which is analogous to the one constructed by \textit{G.-C. Rota} [Théorie combinatoire des invariants classiques, Sér. Math. Pures Appl. 1\(\setminus\)S-01 (IRMA, Strasbourg, 1976/1977)] for the \(R\)-module \(A_ R(n,r)\) of all homogeneous polynomials of degree \(r\) in \(n^ 2\) indeterminates over \(R\). Towards the end of the paper (\S8) a connection is made with the Weyl and Schur modules for the general linear groups.
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The integral form of the classical Schur algebra has a \(\mathbb{Z}\)-basis consisting of codeterminants, as defined by \textit{J. A. Green} [J. Pure Appl. Algebra 88, No. 1-3, 89-106 (1993; Zbl 0794.20053)]. The author investigates the quantized version of this fact. He considers the quantized Schur algebras and defines quantized codeterminants. He shows that an analogous result also holds in this case and provides a straightening formula which allows one to express any codeterminant of dominant shape as a sum of standard codeterminants. We determine the small Bjorken \(x\) asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton in the double-logarithmic approximation (DLA), which resums powers of \(${\alpha}$\)\_{}\{\(s\)\} ln2(1\textit{/x}) with \(${\alpha}$\)\_{}\{\(s\)\} the strong coupling constant. Starting with the operator definitions for the quark and gluon OAM, we simplify them at small \(x\), relating them, respectively, to the polarized dipole amplitudes for the quark and gluon helicities defined in our earlier works. Using the small-\(x\) evolution equations derived for these polarized dipole amplitudes earlier we arrive at the following small-\(x\) asymptotics of the quark and gluon OAM distributions in the large-\(N\)\_{}\{\(c\)\} limit: \(\( {L}_{q+\overline{q}}\left(x,{Q}^2\right)=-\Delta \Sigma \left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{\frac{4}{\sqrt{3}}\kern0.5em \sqrt{\frac{\alpha_s\kern0.5em {N}_c}{2\pi }}}, \)\)\(\( {L}_G\left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{\frac{13}{4\sqrt{3}}\kern0.5em \sqrt{\frac{\alpha_s\kern0.5em {N}_c}{2\pi }}}. \)\)
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Most results on topological groups cannot be literally extended to topological hypergroups due to the lack of an algebraic operation on the hypergroup. To extend such results alternative methods are generally used. In this paper the authors define means of partial derivatives of a composition of polynomials generating the polynomial hypergroup and an analytic function.
In a previous paper [\textit{Ż.~Fechner} et al., Result. Math. 76, No.~4, Paper No.~171, 16~p. (2021; Zbl 07380206)]
the authors proved the characterization of generalized
moment functions on a commutative group as the product of an exponential and a
composition of multivariate Bell polynomials and a sequence additive functions.
In essence the authors generalize moment functions of higher order on commutative groups to include polynomial hypergroups. An example of an explicit formula for moment generating functions of rank at most two on the Chebyshev hypergroup is provided. In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.
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Most results on topological groups cannot be literally extended to topological hypergroups due to the lack of an algebraic operation on the hypergroup. To extend such results alternative methods are generally used. In this paper the authors define means of partial derivatives of a composition of polynomials generating the polynomial hypergroup and an analytic function.
In a previous paper [\textit{Ż.~Fechner} et al., Result. Math. 76, No.~4, Paper No.~171, 16~p. (2021; Zbl 07380206)]
the authors proved the characterization of generalized
moment functions on a commutative group as the product of an exponential and a
composition of multivariate Bell polynomials and a sequence additive functions.
In essence the authors generalize moment functions of higher order on commutative groups to include polynomial hypergroups. An example of an explicit formula for moment generating functions of rank at most two on the Chebyshev hypergroup is provided. Elementary school students learn two types of division scenarios: partitive and quotitive. Previous researchers have assumed that the partitive scenario is easier because it reflects the everyday notion of sharing, whereas the quotitive scenario, which represents grouping, is more difficult and is understood gradually in the course of mathematics learning. However, this assumption has not been adequately investigated in empirical studies. The present study examines the assumption in a cross-sectional study. Participants were 336 elementary school students (98 in grade 3, 82 in grade 4, 88 in grade 5, and 68 in grade 6) and 70 university students who performed two tasks. In the preference task, they generated a division scenario of any type consistent with a given numerical equation. In the problem-posing task, they generated a division scenario consistent with both a numerical equation and a picture representing a partitive or quotitive scenario. On the preference task, students at all grade levels preferred the partitive to the quotitive scenario, and this preference increased with students' grade level. On the problem-posing task, younger students (grades 3, 4, and 5) had equivalent success in the partitive and quotitive scenarios, but older students (grade 6 and university) found the partitive scenario to be easier than the quotitive. Implications for mathematics education are discussed.
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The authors extend results of their previous paper [Adv. Differ. Equ. 11, No. 1, 91--119 (2006; Zbl 1132.35427)]. Precisely, define
\[
G(x,u,\nabla u, D^2u)=F(x,\nabla u, D^2u)+b(x)\cdot \nabla u| \nabla u| ^\alpha+c(x)| u| ^\alpha u,
\]
where \(F\) is continuous on \(\Omega\times \mathbb{R}^N\times S\), \(S\) being the space of \(N\times N\) symmetric matrices. Moreover, \(F\) satisfies, for all real number \(t\) and all non negative \(\mu\)
\[
F(x,t p,\mu X)=| t| ^\alpha\mu F(x,p,X),
\]
and for all \(p\in \mathbb{R}^N\setminus \{0\}\), \(M\in S\), \(N\in S\) with \(N\geq 0\), \(\alpha>-1\), \(0<a\leq A\)
\[
a| p| ^\alpha \mathrm{tr}(N)\leq F(x,p,M+N)-F(x,p,M)\leq A| p| ^\alpha \mathrm{tr}(N).
\]
Moreover, suitable conditions about the uniform continuity of \(F(x,p,X)\) are assumed. The functions \(b:\Omega\to \mathbb{R}^N\) and \(c:\Omega\to \mathbb{R}\), are supposed to be continuous and bounded. Additional conditions are assumed to prove specific theorems.
The main object of the work is the following definition of eigenvalue
\[
\overline\lambda=\sup\{\lambda\in \mathbb{R}:\exists \phi>0 \text{ in } \Omega,\;G(x,\phi,\nabla \phi,D^2\phi)+\lambda \phi^{1+\alpha}\leq 0\},
\]
where the inequality is understood in the viscosity sense. The following results are proved.
(i) If \(f\leq 0\) is bounded and continuous in \(\Omega\), and if \(\lambda<\overline\lambda\) then there exists a nonnegative solution of the problem
\[
G(x,u,\nabla u, D^2u)+\lambda u^{1+\alpha}=f \text{ in }\;\Omega,\;u=0 \text{ on } \;\partial\Omega.
\]
(ii) There exists \(\phi>0\) in \(\Omega\) such that \(\phi\) is a viscosity solution of
\[
G(x,\phi,\nabla \phi, D^2\phi)+\lambda \phi^{1+\alpha}=f \text{ in }\;\Omega,\;\phi=0\;\text{ on }\;\partial\Omega.
\]
(iii) \(\phi\) is \(\gamma\)-Hölder continuous for all \(\gamma\in (0,1)\) and locally Lipschitz continuous.
To get the previous results, the authors prove a comparison principle and some boundary estimates which may have a own interest. Also, they prove regularity results which give the required relative compactness used to prove the main existence theorem. The authors give a definition of the first eigenvalue for a class of fully nonlinear elliptic operators which are nonvariational but homogeneous. The main idea is to exploit the property that an elliptic operator satisfies a maximum principle if the involved parameter is less than the first eigenvalue. In the linear case it is well known that for the operator \(Mu:= -\Delta u+\lambda u\) the maximum principle holds if \(\lambda<\lambda_0\), where \(\lambda_0\) is the first eigenvalue of \(-\Delta u=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\). Thus, \(\lambda_0\) is the supremum of all \(\lambda\in{\mathbb R}\) such that the maximum principle holds. The authors extend this view to a wide class of operators of the form \(F(\nabla u,D^2 u)\). Let \(\lambda_0\) be defined through the supremum of all \(\lambda\in{\mathbb R}\) such that there exists \(\Phi>0\) in \(\Omega\) such that \(F(\nabla\Phi,D^2\Phi)+\lambda\Phi^ {\alpha+1}\leq 0\) in the viscosity sense, where \(\alpha>-1\). The authors show under additional assumptions that this \(\lambda_0\) is the first eigenvalue of \(-F\) in \(\Omega\) in the sense that \(F(\nabla u,D^2u)+\lambda|u|^\alpha u\) satisfies the maximum principle in \(\Omega\) if \(\lambda<\lambda_0\).
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The authors extend results of their previous paper [Adv. Differ. Equ. 11, No. 1, 91--119 (2006; Zbl 1132.35427)]. Precisely, define
\[
G(x,u,\nabla u, D^2u)=F(x,\nabla u, D^2u)+b(x)\cdot \nabla u| \nabla u| ^\alpha+c(x)| u| ^\alpha u,
\]
where \(F\) is continuous on \(\Omega\times \mathbb{R}^N\times S\), \(S\) being the space of \(N\times N\) symmetric matrices. Moreover, \(F\) satisfies, for all real number \(t\) and all non negative \(\mu\)
\[
F(x,t p,\mu X)=| t| ^\alpha\mu F(x,p,X),
\]
and for all \(p\in \mathbb{R}^N\setminus \{0\}\), \(M\in S\), \(N\in S\) with \(N\geq 0\), \(\alpha>-1\), \(0<a\leq A\)
\[
a| p| ^\alpha \mathrm{tr}(N)\leq F(x,p,M+N)-F(x,p,M)\leq A| p| ^\alpha \mathrm{tr}(N).
\]
Moreover, suitable conditions about the uniform continuity of \(F(x,p,X)\) are assumed. The functions \(b:\Omega\to \mathbb{R}^N\) and \(c:\Omega\to \mathbb{R}\), are supposed to be continuous and bounded. Additional conditions are assumed to prove specific theorems.
The main object of the work is the following definition of eigenvalue
\[
\overline\lambda=\sup\{\lambda\in \mathbb{R}:\exists \phi>0 \text{ in } \Omega,\;G(x,\phi,\nabla \phi,D^2\phi)+\lambda \phi^{1+\alpha}\leq 0\},
\]
where the inequality is understood in the viscosity sense. The following results are proved.
(i) If \(f\leq 0\) is bounded and continuous in \(\Omega\), and if \(\lambda<\overline\lambda\) then there exists a nonnegative solution of the problem
\[
G(x,u,\nabla u, D^2u)+\lambda u^{1+\alpha}=f \text{ in }\;\Omega,\;u=0 \text{ on } \;\partial\Omega.
\]
(ii) There exists \(\phi>0\) in \(\Omega\) such that \(\phi\) is a viscosity solution of
\[
G(x,\phi,\nabla \phi, D^2\phi)+\lambda \phi^{1+\alpha}=f \text{ in }\;\Omega,\;\phi=0\;\text{ on }\;\partial\Omega.
\]
(iii) \(\phi\) is \(\gamma\)-Hölder continuous for all \(\gamma\in (0,1)\) and locally Lipschitz continuous.
To get the previous results, the authors prove a comparison principle and some boundary estimates which may have a own interest. Also, they prove regularity results which give the required relative compactness used to prove the main existence theorem. We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone.
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Nonuniform multiwavelets associated with a nonuniform multiresolution analysis in \(L^2({\mathbb R})\) were introduced by \textit{J.-P. Gabardo} and \textit{M. Z. Nashed} [J. Funct. Anal. 158, No. 1, 209--241 (1998; Zbl 0910.42018)]. Here translated functions \(f(\cdot - \lambda)\) with \(\lambda \in \{0,r/N\} + 2 {\mathbb Z}\) are used, where \(N\geq 1\) is an integer and \(r\) is an odd integer with \(1 \leq r \leq 2N-1\). In the case \(N = r =1\), one obtains the classic theory of orthonormal wavelets.
In this paper, the authors characterize nonuniform multiwavelet sets and nonuniform multiscaling sets associated with nonuniform multiresolution analysis of finite multiplicity. The authors consider a generalization of the notion of multiresolution analysis, based on the theory of spectral pairs: If \(A\) is a measurable subset of \({\mathbb R}^n\) with measure \(| A | > 0\) and \(\Lambda \subset {\mathbb R}^n\) is discrete, \((A, \Lambda)\) forms a spectral pair if the collection \(\{| A |^{-1/2} e^{2 \pi i \xi \cdot \lambda} \chi_{A}(\xi), \lambda \in \Lambda\}\) is a complete orthonormal system for \(L^2_{A}\). In contrast to the standard setting, the associated subspace \(V_0\) of \(L^2({\mathbb R})\) has, as an orthonormal basis, a collection of translates of the scaling function of the form \(\{\phi(x - \lambda); \lambda \in \Lambda\}\), where \(\Lambda = \{0, r/N\} + 2 {\mathbb Z}\), \(N \geq 1\) is an integer, and \(r\) is an odd integer with \(1 \leq r \leq 2N-1\) such that \(r\) and \(N\) are relatively prime and \({\mathbb Z}\) is the set of all integers. Furthermore, the corresponding dilation factor is \(2N\), the case where \(N=1\) corresponds to the usual definition of a multiresolution analysis with dilation factor \(2\). A necessary and sufficient condition for the existence of associated wavelets, which is always satisfied when \(N=1\) or \(2\), is obtained and is shown to always hold if the Fourier transform of \(\phi\) is a constant multiple of the characteristic function of a set.
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Nonuniform multiwavelets associated with a nonuniform multiresolution analysis in \(L^2({\mathbb R})\) were introduced by \textit{J.-P. Gabardo} and \textit{M. Z. Nashed} [J. Funct. Anal. 158, No. 1, 209--241 (1998; Zbl 0910.42018)]. Here translated functions \(f(\cdot - \lambda)\) with \(\lambda \in \{0,r/N\} + 2 {\mathbb Z}\) are used, where \(N\geq 1\) is an integer and \(r\) is an odd integer with \(1 \leq r \leq 2N-1\). In the case \(N = r =1\), one obtains the classic theory of orthonormal wavelets.
In this paper, the authors characterize nonuniform multiwavelet sets and nonuniform multiscaling sets associated with nonuniform multiresolution analysis of finite multiplicity. We present results concerning the learning of Monotone DNF (MDNF) from Incomplete Membership Queries and Equivalence Queries. Our main result is a new algorithm that allows efficient learning of MDNF using Equivalence Queries and Incomplete Membership Queries with probability of \(p = 1-1/\text{poly}(n,t)\) of failing. Our algorithm is expected to make
\[
O\left(\left(\frac{tn}{1-p}\right)^2\right)
\]
queries, when learning a MDNF formula with \(t\) terms over \(n\) variables. Note that this is polynomial for any failure probability \(p=1-1/\text{poly}(n,t)\). The algorithm's running time is also polynomial in \(t\), \(n\), and \(1/(1-p)\). In a sense this is the best possible, as learning with \(p =1-1/\omega(\text{poly}(n,t))\) would imply learning MDNF, and thus also DNF, from equivalence queries alone.
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Steinhaus three gap theorem states that the gaps in the fractional parts of \(\alpha,2\alpha,\dots,N\alpha\) take at most three distinct values. The Littlewood conjecture states that for every \(\alpha_1,\alpha_2\in\mathbb R\), \(\liminf_{n\to\infty}n||n\alpha_1||||n\alpha_2||=0\), where \(||x||\) denotes the distance to the nearest integer. In this paper, the authors demonstrate the close connection betwen multi-dimensional Steinhaus problem and the multi-dimensional version of the Littlewood conjecture.
They prove: (Theorem 3.) For \(\alpha\in\mathbb R^d\), let \(\mathcal D\subset \mathbb R^d\) and let \(G(\alpha,\mathcal D)\) be number of distinct gaps between the elements of \(\{m\alpha\bmod 1\mid m\in \mathbb Z^d\cap\mathcal D\}\). Consider dilation \(\mathcal D_T=\{xT\mid x\in\mathcal D\}\), where \(T=\text{diag}(T_1,\dots,T_d)\) is a diagonal matrix with, \(\mathcal D\subset\mathbb R^d\) is bounded convex and contains the cube \([0,\epsilon)^d\) for some \(\epsilon >0\). If \(\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d\) is such that \(\sup_{T_1,\dots,T_d}G(\alpha,\mathcal D_T)=\infty\), then \(\liminf_{n\to\infty}n||n\alpha_1||\dots||n\alpha_d||=0\).
For \(q\in\mathcal D\), the first return time to \(\mathcal D\) is given by \(\min\{n\in\mathbb N\mid q+n\alpha\in\mathcal D+\mathbb Z^d\}\) and \(L(\alpha,\mathcal D)\) is the number of distinct values. Whether the number \(L(\alpha,\mathcal D)\) remains finite is Slater problem. In the higher dimensional Slater problems the authors proved: (Theorem 7.) Let \(d\ge2\) and \(\mathcal D\subset\mathbb R^d\) be bounded and convex with non-empty interior. If \(\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d\) such that \(\sup_{T_1,\ldots,T_d\ge1}L(\alpha,\mathcal D_{T^{-1}})=\infty\), then \(\liminf_{n\to\infty}n||n\alpha_1||\dots||n\alpha_d||=0\).
The authors define the function \(F(M,t)=\min\{y>0\mid (x,y)\in\mathbb Z^{d+1}M, x+t\in\mathcal D\}\), where \(G=\mathrm{SL}(d+1,\mathbb R)\) and \(\Gamma=\mathrm{SL}(d+1,\mathbb Z)\) and \(M\in G\) and this used to derive some theorems from dynamical properties of the diagonal action on the space of lattices. The authors used 30 items in References among other \textit{N. Chevallier} [Discrete Math. 223, No. 1--3, 355--362 (2000; Zbl 1034.11018)]. Let \(\theta\) be in \(\mathbb{T}^1=\mathbb{R}/ \mathbb{Z}\), the points \(0\), \(\theta\), \(2\theta,\dots, n\theta\) divide \(\mathbb{T}^1\) into \(n+1\) intervals having at most three distinct lengths. This property is known as the three distance theorem conjectured by Steinhaus. A first generalisation was conjectured by Graham, first proved by \textit{F. R. K. Chung} and \textit{R. L. Graham} [Ars Combin. 1, 57--76 (1976; Zbl 0352.10006)] and three years later by \textit{F. M. Liang} [Discrete Math. 28, 325--326 (1979; Zbl 0427.05014)]. \textit{J. F. Geelen} and \textit{R. J. Simpson} [Australas. J. Comb. 8, 169--197 (1993; Zbl 0804.11020)] proved a two-dimensional version of the three distance theorem. They also conjectured the following \(d\)-dimensional generalisation. Let \(\theta_1,\dots, \theta_d\) be in \(\mathbb{T}^1\) and \(n_1,\dots, n_d\) be positive integers. Then the points \(k_1\theta_1+ \cdots+k_d \theta_d\), \(k_1=0,\dots ,n_1-1,\dots, k_d=0,\dots, n_d-1\), divide \(\mathbb{T}^1\) into intervals having at most \(\prod_{i=1}^{d-1} n_i+ C_d\) distincts lengths where \(C_d\) depends only on \(d\).
Here we prove a result which is close to this conjecture.
Theorem 1. Suppose \(d\geq 2\). Let \(\theta_1, \dots,\theta_d\) be in \(\mathbb{T}^1\) and \(n_1,\dots,n_d\) be positive integers. Then the points \(k_1\theta_1+ \cdots+ k_d\theta_d\), \(k_1=0, \dots,n_1-1, \dots,k_d=0, \dots,n_d-1\), divide \(\mathbb{T}^1\) into intervals having at most \(\prod^{d-1}_{i=1} n_i+3\prod^{d-2}_{i=1} n_i+1\) distinct lengths. If \(d=2\) our result is not as good as Geelen and Simpson's; the upper bound is \(n_1+4\) instead of \(n_1+3\).
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Steinhaus three gap theorem states that the gaps in the fractional parts of \(\alpha,2\alpha,\dots,N\alpha\) take at most three distinct values. The Littlewood conjecture states that for every \(\alpha_1,\alpha_2\in\mathbb R\), \(\liminf_{n\to\infty}n||n\alpha_1||||n\alpha_2||=0\), where \(||x||\) denotes the distance to the nearest integer. In this paper, the authors demonstrate the close connection betwen multi-dimensional Steinhaus problem and the multi-dimensional version of the Littlewood conjecture.
They prove: (Theorem 3.) For \(\alpha\in\mathbb R^d\), let \(\mathcal D\subset \mathbb R^d\) and let \(G(\alpha,\mathcal D)\) be number of distinct gaps between the elements of \(\{m\alpha\bmod 1\mid m\in \mathbb Z^d\cap\mathcal D\}\). Consider dilation \(\mathcal D_T=\{xT\mid x\in\mathcal D\}\), where \(T=\text{diag}(T_1,\dots,T_d)\) is a diagonal matrix with, \(\mathcal D\subset\mathbb R^d\) is bounded convex and contains the cube \([0,\epsilon)^d\) for some \(\epsilon >0\). If \(\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d\) is such that \(\sup_{T_1,\dots,T_d}G(\alpha,\mathcal D_T)=\infty\), then \(\liminf_{n\to\infty}n||n\alpha_1||\dots||n\alpha_d||=0\).
For \(q\in\mathcal D\), the first return time to \(\mathcal D\) is given by \(\min\{n\in\mathbb N\mid q+n\alpha\in\mathcal D+\mathbb Z^d\}\) and \(L(\alpha,\mathcal D)\) is the number of distinct values. Whether the number \(L(\alpha,\mathcal D)\) remains finite is Slater problem. In the higher dimensional Slater problems the authors proved: (Theorem 7.) Let \(d\ge2\) and \(\mathcal D\subset\mathbb R^d\) be bounded and convex with non-empty interior. If \(\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d\) such that \(\sup_{T_1,\ldots,T_d\ge1}L(\alpha,\mathcal D_{T^{-1}})=\infty\), then \(\liminf_{n\to\infty}n||n\alpha_1||\dots||n\alpha_d||=0\).
The authors define the function \(F(M,t)=\min\{y>0\mid (x,y)\in\mathbb Z^{d+1}M, x+t\in\mathcal D\}\), where \(G=\mathrm{SL}(d+1,\mathbb R)\) and \(\Gamma=\mathrm{SL}(d+1,\mathbb Z)\) and \(M\in G\) and this used to derive some theorems from dynamical properties of the diagonal action on the space of lattices. The authors used 30 items in References among other \textit{N. Chevallier} [Discrete Math. 223, No. 1--3, 355--362 (2000; Zbl 1034.11018)]. In this paper, optimal algorithms are proposed for hybrid flow shop schedule of identical jobs on two machines (named as \({M_1}\) and \({M_2}\)) with learning effect. In our problem, each job has two tasks, named as task \(A\) and task \(B\) respectively, and two optional processing modes. The task \(B\) can start to be processed only if after task \(A\) has been finished. The first processing mode, named as mode 1, is to assign both task \(A\) and \(B\) to machine \({M_2}\). The second processing mode, named mode 2, is to assign task \(A\) and \(B\) to machine \({M_1}\) and \({M_2}\), respectively. It is assumed that each machine has learning effect when processing the job, in other words, the actual processing time of the job is related to the processing position of the job. Our objective is to minimize the makespan. Optimal algorithms are given for two systems with no buffer and infinite buffer respectively.
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The goals of this paper include the generalization of Maddox' results in [Math. Slovaca 42, 211--214 (1992; Zbl 0746.40003)]. The authors accomplish this by presenting the notions of \(\mathcal I\)-convergence field and bounded \(\mathcal I\)-convergence field of an infinite matrix \(A\) denoted \(C_{A}^{\mathcal I}\) and \(M_{A}^{\mathcal I}\) respectively. By restricting their study to diagonal matrices they were able to present a series of necessary and/or sufficient conditions on the entries of a matrix for the solidity of \(C_{A}^{\mathcal I}\) and \(M_{A}^{\mathcal I}\). We investigate Köthe-Toeplitz solidity of the summability field of an infinite matrix of operators in Banach spaces.
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The goals of this paper include the generalization of Maddox' results in [Math. Slovaca 42, 211--214 (1992; Zbl 0746.40003)]. The authors accomplish this by presenting the notions of \(\mathcal I\)-convergence field and bounded \(\mathcal I\)-convergence field of an infinite matrix \(A\) denoted \(C_{A}^{\mathcal I}\) and \(M_{A}^{\mathcal I}\) respectively. By restricting their study to diagonal matrices they were able to present a series of necessary and/or sufficient conditions on the entries of a matrix for the solidity of \(C_{A}^{\mathcal I}\) and \(M_{A}^{\mathcal I}\). We present an alternative cosmology based on conformal gravity, as originally introduced by H. Weyl and recently revisited by P. Mannheim and D. Kazanas. Unlike past similar attempts our approach is a purely kinematical application of the conformal symmetry to the Universe, through a critical reanalysis of fundamental astrophysical observations, such as the cosmological redshift and others. As a result of this novel approach we obtain a closed-form expression for the cosmic scale factor \(R(t)\) and a revised interpretation of the space-time coordinates usually employed in cosmology. New fundamental cosmological parameters are introduced and evaluated. This emerging new cosmology does not seem to possess any of the controversial features of the current standard model, such as the presence of dark matter, dark energy or of a cosmological constant, the existence of the horizon problem or of an inflationary phase. Comparing our results with current conformal cosmologies in the literature, we note that our kinematic cosmology is equivalent to conformal gravity with a cosmological constant at late (or early) cosmological times. The cosmic scale factor and the evolution of the Universe are described in terms of several dimensionless quantitites, among which a new cosmological variable \(\delta \) emerges as a natural cosmic time. The mathematical connections between all these quantities are described in details and a relationship is established with the original kinematic cosmology by L. Infeld and A. Schild. The mathematical foundations of our kinematical conformal cosmology will need to be checked against current astrophysical experimental data, before this new model can become a viable alternative to the standard theory.
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