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\textit{J. Rhodes} and \textit{B. Tilson} [J. Pure Appl. Algebra 2, 13-71 (1972; Zbl 0257.20059)] introduced the notion of Type II elements in a finite monoid, showing that in the regular case these elements form the smallest submonoid that admits ``conjugation''. In doing so, they confirmed for finite regular monoids what has become known as the Type II conjecture. The present paper advances the general study of the Type II conjecture by showing that it can be reduced to the case of block groups, monoids in which every element has at most one inverse. Using this reduction procedure, the authors are able to prove certain special cases of the Type II conjecture, and to simplify existing proofs of some special cases.
\{Reviewer's remark: the type II conjecture is now a theorem, having been verified in the general case by \textit{C. J. Ash} [Semigroup Theory, Proc. Monash Univ. Conf. Semigroup Theory in Honour of G. B. Preston, eds. T. E. Hall, J. C. Meakin and P. R. Jones (World Scientific, Singapore) (1991)].\} If \(S\) and \(T\) are semigroups then \(S| T\) means that \(S\) is a homomorphic image of a subsemigroup of \(T\). A semigroup is called combinatorial if it contains trivial subgroups only. \((X,T)\) is a semigroup \(T\) of transformations of a set \(X\). All semigroups considered are finite. For every semigroup \(S\) there exists a sequence of transformation semigroups such that \(S| (X_n,T_n)\wr \cdots \wr(X_1,T_1)\) where \(\wr\) denotes the wreath product of transformation semigroups. Factors in the above wreath product are alternately transformation groups and combinatorial semigroups. The minimal possible number of groups in the above wreath product is called the group complexity of \(S\) and is denoted by \(\#_C(S)\). The authors define two distinctly different types of semigroups: if \(T\) is a subsemigroup of type II of a semigroup \(S\) of type I, then \(\#_C(T)<\#_C(S)\). Clearly, this device helps in evaluating the group complexity of semigroups. The semigroups of both types are studied in the paper (which, in a sense, may be considered as a continuation to the previous paper by the authors [J. Pure Appl. Algebra 1, 79--95 (1971; Zbl 0259.20051)]. There are given new lower bounds of group complexity; it is shown that there exist semigroups of arbitrary complexity whose idempotents generate a combinatorial semigroup.
| 1 |
\textit{J. Rhodes} and \textit{B. Tilson} [J. Pure Appl. Algebra 2, 13-71 (1972; Zbl 0257.20059)] introduced the notion of Type II elements in a finite monoid, showing that in the regular case these elements form the smallest submonoid that admits ``conjugation''. In doing so, they confirmed for finite regular monoids what has become known as the Type II conjecture. The present paper advances the general study of the Type II conjecture by showing that it can be reduced to the case of block groups, monoids in which every element has at most one inverse. Using this reduction procedure, the authors are able to prove certain special cases of the Type II conjecture, and to simplify existing proofs of some special cases.
\{Reviewer's remark: the type II conjecture is now a theorem, having been verified in the general case by \textit{C. J. Ash} [Semigroup Theory, Proc. Monash Univ. Conf. Semigroup Theory in Honour of G. B. Preston, eds. T. E. Hall, J. C. Meakin and P. R. Jones (World Scientific, Singapore) (1991)].\} Seit 1981 wird der Adam-Ries-Wettbewerb (kurz ARW) in Annaberg-Buchholz, Land Sachsen, für Schüler der Klassenstufe 5 durchgeführt. Die Ziele und die Organisationsstrukturen haben sich in den vergangenen 17 Jahren (nicht zuletzt auch durch die gesellschaftlichen Veränderungen) geändert, das Grundanliegen ist geblieben: Entwicklung von Freude bei der Beschäftigung mit Mathematik und Förderung mathematischer Begabungen, wie in diesem Beitrag diskutiert wird. Es werden ausgewählte Klausuraufgaben mit Musterlösungen vorgestellt.
| 0 |
Let \(\mathbb H\) be the upper half-plane, let \(\Gamma\) be a Fuchsian subgroup of the first kind of \(\text{PSL}_2({\mathbb R})\), and let \(M=\Gamma\setminus {\mathbb H}\). Assume the compactification of \(M\) has genus \(g_M>0.\) Let \(\{f_1,\ldots,f_{g_M}\}\) be an orthonormal basis of cusp forms of weight \(2\) with respect to \(\Gamma\), normalized to have \(L^2\)-norm equal to 1. Assume \(M\) is a finite degree cover of the hyperbolic Riemann surface \(M_0=\Gamma_0\setminus{\mathbb H}\). The main result is
\[
\sup_{z\in{\mathbb H}}\sum_{j=1}^{g_M}\text{Im}(z)^2| f_j(z)| ^2=O_{M_0}(1).
\]
Corollary: Let \(f_H\) be a new form for \(\Gamma_0(N)\), which is Hecke-normalized, i.e. the first Fourier coefficient in its \(q\)-expansion is 1. For any \(\varepsilon>0\) it holds
\[
\sup_{z\in{\mathbb H}}| \text{Im}(z)f_H(z)| =O_{\varepsilon} (N^{\frac{1}{2}+\varepsilon}).
\]
This corollary reproves theorem A from \textit{A. Abbes} and \textit{E. Ullmo} [Duke Math. J. 80, No. 2, 295--307 (1995; Zbl 0895.14007)]. The authors speculate that the optimal bound is of the form
\[
\sup_{z\in{\mathbb H}}| \text{Im}(z)f_H(z)| =O_{\varepsilon}(N^{\varepsilon}).
\]
Let \(N\) be a square-free positive integer. This paper proves an estimate comparing the Arakelov metric for the smooth projective curve \(X_0(N)\) with the Poincaré metric \((dx\wedge dy)/y^2\) on its universal covering space, the upper half plane \(\mathcal H\). More specifically, for \(N\) as above let \(g\) be the genus of \(X_0(N)\), let \(f_1,\dots,f_g\) be an orthonormal basis for \(\mathcal S(2,\Gamma_0(N))\) under the Petersson inner product, and let \(F_N=\frac{y^2}g\sum_{i=1}^g| f_i| ^2\). This is the ratio between the metrics of Arakelov and Poincaré. The paper shows that for all \(\varepsilon>0\) there exists \(A_\varepsilon>0\) such that for all \(N\) as above and all \(z\in\mathcal H\), the inequality \(F_N(z)\leq A_\varepsilon N^{1+\varepsilon}\) holds.
An intermediate result in the above proof is the following. For all \(\varepsilon>0\) there is a constant \(C_\varepsilon>0\) such that for all \(N\) as above, all newforms \(f\) for \(\Gamma_0(N)\) (required to be normalized eigenforms for the Hecke operators), and all \(z\in\mathcal H\), the inequality \(| yf(z)| \leq C_\varepsilon N^{1/2+\varepsilon}\) holds.
Finally, as a corollary of the main theorem, the following two assertions are equivalent:
(i) there is an upper bound, polynomial in the conductor, for the degree of a strong modular parametrization of a semi-stable elliptic curve \(E\) over \(\mathbb{Q}\) (strong means that the induced map \(H_1(X_0(N),\mathbb{Z})\to H_1(E,\mathbb{Z})\) is surjective); and
(ii) the modular height of such an elliptic curve satisfies an upper bound that is linear in \(\log N\).
| 1 |
Let \(\mathbb H\) be the upper half-plane, let \(\Gamma\) be a Fuchsian subgroup of the first kind of \(\text{PSL}_2({\mathbb R})\), and let \(M=\Gamma\setminus {\mathbb H}\). Assume the compactification of \(M\) has genus \(g_M>0.\) Let \(\{f_1,\ldots,f_{g_M}\}\) be an orthonormal basis of cusp forms of weight \(2\) with respect to \(\Gamma\), normalized to have \(L^2\)-norm equal to 1. Assume \(M\) is a finite degree cover of the hyperbolic Riemann surface \(M_0=\Gamma_0\setminus{\mathbb H}\). The main result is
\[
\sup_{z\in{\mathbb H}}\sum_{j=1}^{g_M}\text{Im}(z)^2| f_j(z)| ^2=O_{M_0}(1).
\]
Corollary: Let \(f_H\) be a new form for \(\Gamma_0(N)\), which is Hecke-normalized, i.e. the first Fourier coefficient in its \(q\)-expansion is 1. For any \(\varepsilon>0\) it holds
\[
\sup_{z\in{\mathbb H}}| \text{Im}(z)f_H(z)| =O_{\varepsilon} (N^{\frac{1}{2}+\varepsilon}).
\]
This corollary reproves theorem A from \textit{A. Abbes} and \textit{E. Ullmo} [Duke Math. J. 80, No. 2, 295--307 (1995; Zbl 0895.14007)]. The authors speculate that the optimal bound is of the form
\[
\sup_{z\in{\mathbb H}}| \text{Im}(z)f_H(z)| =O_{\varepsilon}(N^{\varepsilon}).
\]
An image registration algorithm is developed to estimate dense motion vectors between two images using the coarse-to-fine wavelet-based motion model. This motion model is described by a linear combination of hierarchical basis functions proposed by Cai and Wang (1996). The coarser-scale basis function has larger support while the finer-scale basis function has smaller support. With these variable supports in full resolution, the basis functions serve as large-to-small windows so that the global and local information can be incorporated concurrently for image matching, especially for recovering motion vectors containing large displacements. To evaluate the accuracy of the wavelet-based method, two sets of test images were experimented using both the wavelet-based method and a leading pyramid spline-based method by Szeliski et al. (1996). One set of test images, taken from Barron et al. (1994), contains small displacements. The other set exhibits low texture or spatial aliasing after image blurring and contains large displacements. The experimental results showed that our wavelet-based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation.
| 0 |
In the introduction, the authors define expander sequences of manifolds as follows.
``Let \(M_n^k\) be a sequence of closed \(k\)-dimensional Riemannian manifolds all of which have volume \(1\). We say that it is an expander sequence (or an expander manifold) if for any \(n\) any smooth hypersurface \(S_n^{k-1}\) separating \(M_n^k\) in two open sets both of which have volume at least \(1/n\), has \((k-1)\)-volume at least equal to \(n\).''
They also define the isoperimetric profile function of a manifold as follows.
``If \((M^n,\,g)\) is a Riemannian manifold the isoperimetric profile function of \(M^n\) is a function \(I_M : (0,\,\operatorname{vol}(M))\rightarrow\mathbb{R}^+\) defined by:
\[
I_M(t)=\inf_\Omega \{\operatorname{vol}_{n-1}(\partial \Omega):\Omega\subset M^n,\,\operatorname{vol}_n(\Omega)=t\}
\]
where \(\Omega\) ranges over all open sets of \(M^n\) with smooth boundary. We note that we don't require \(\Omega\) to be connected.''
It turns out that \(I_M(t)\) is an upper semi-continuous but not necessarily continuous function.
In Section 2, the authors use the concept of expander manifolds to show that there exists a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile. This answers a question of \textit{S. Nardulli} and \textit{P. Pansu} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18, No. 2, 537--549 (2018; Zbl 1396.53058)].
In Section 3, the authors show that, for every \(\epsilon,\,M>0\) there is a Riemannian sphere \(S\) of volume \(1\), such that every (not necessarily connected) surface separating \(S\) in two regions of volume greater than \(\epsilon\) has area greater than \(M\), again with the help of expander manifolds.
The article ends with a discussion and open questions (Section 4). Let \((M,g)\) be a Riemannian manifold. Fix \(0<\nu<\text{Vol}(M,g)\). Let \(\Omega\subset M\) be a compact subdomain of \(M\). Let \(I(\nu)=\inf_{\Omega\subset M:\text{Vol}(\Omega)=\nu}\text{vol}(d\Omega)\). This is the \textit{isoperimetric profile}. \textit{S. Gallot} [in: On the geometry of differentiable manifolds. Workshop, Rome, June 23-27, 1986. Paris: Société Mathématique de France. (1988; Zbl 0674.53001)] showed that the isoperimetric profile \(I\) is continuous on \((0,\text{Vol}(M,g))\) if \(M\) is compact. The first author and \textit{F. G. Russo} [``On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume'', Preprint, \url{arXiv:1502.05903}] extended this to the finite volume setting. Similarly, the isoperimetric profile is continuous if \((M,g)\) is homogeneous, \((M,g)\) is an unbounded convex Euclidean domain, and in many other settings. In this paper the authors establish the following result:
Theorem. There exists a complete connected non-compact 3-dimensional Riemannian manifold such that the isoperimetric profile is discontinuous.
Section 1 contains an introduction to the problem. Section 2 treats isoperimetry in nilmanifolds. The desired result is then obtained in Section 3 by piecing together compact nilmanifolds.
| 1 |
In the introduction, the authors define expander sequences of manifolds as follows.
``Let \(M_n^k\) be a sequence of closed \(k\)-dimensional Riemannian manifolds all of which have volume \(1\). We say that it is an expander sequence (or an expander manifold) if for any \(n\) any smooth hypersurface \(S_n^{k-1}\) separating \(M_n^k\) in two open sets both of which have volume at least \(1/n\), has \((k-1)\)-volume at least equal to \(n\).''
They also define the isoperimetric profile function of a manifold as follows.
``If \((M^n,\,g)\) is a Riemannian manifold the isoperimetric profile function of \(M^n\) is a function \(I_M : (0,\,\operatorname{vol}(M))\rightarrow\mathbb{R}^+\) defined by:
\[
I_M(t)=\inf_\Omega \{\operatorname{vol}_{n-1}(\partial \Omega):\Omega\subset M^n,\,\operatorname{vol}_n(\Omega)=t\}
\]
where \(\Omega\) ranges over all open sets of \(M^n\) with smooth boundary. We note that we don't require \(\Omega\) to be connected.''
It turns out that \(I_M(t)\) is an upper semi-continuous but not necessarily continuous function.
In Section 2, the authors use the concept of expander manifolds to show that there exists a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile. This answers a question of \textit{S. Nardulli} and \textit{P. Pansu} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18, No. 2, 537--549 (2018; Zbl 1396.53058)].
In Section 3, the authors show that, for every \(\epsilon,\,M>0\) there is a Riemannian sphere \(S\) of volume \(1\), such that every (not necessarily connected) surface separating \(S\) in two regions of volume greater than \(\epsilon\) has area greater than \(M\), again with the help of expander manifolds.
The article ends with a discussion and open questions (Section 4). We present two-stage stochastic mixed 0-1 optimization models to hedge against uncertainty in production planning of typical small-scale Brazilian furniture plants under stochastic demands and setup times. The proposed models consider cutting and drilling operations as the most limiting production activities, and synchronize them to avoid intermediate work-in-process. To design solutions less sensitive to changes in scenarios, we propose four models that perceive the risk reductions over the scenarios differently. The first model is based on the minimax regret criteria and optimizes a worst-case scenario perspective without needing the probability of the scenarios. The second formulation uses the conditional value-at-risk as the risk measure to avoid solutions influenced by a bad scenario with a low probability. The third strategy is a mean-risk model based on the upper partial mean that aggregates a risk term in the objective function. The last approach is a restricted recourse approach, in which the risk preferences are directly considered in the constraints. Numerical results indicate that it is possible to achieve significant risk reductions using the risk-averse strategies, without overly sacrificing average costs.
| 0 |
Der Erstautor hat bisher in einer Reihe von Arbeiten gezeigt, wie man explizite Polynome über Körpern positiver Charakteristik mit vorgegebener ''geometrischer'' Galoisgruppe konstruiert.
In der vorliegenden umfangreichen Untersuchung wird dies für symplektische Gruppen durch\-geführt. Dabei werden insbesondere geometrische Charakterisierungen dieser Gruppen durch Cameron und Kantor verwendet bzw. vereinfacht. Let \(b\) be a nondegenerate symplectic form on a vector space \(V\) over a finite field. It is well-known that every intermediate group between \(\text{Sp}(V,b)\) and \(\text{}\Gamma\)Sp
| 1 |
Der Erstautor hat bisher in einer Reihe von Arbeiten gezeigt, wie man explizite Polynome über Körpern positiver Charakteristik mit vorgegebener ''geometrischer'' Galoisgruppe konstruiert.
In der vorliegenden umfangreichen Untersuchung wird dies für symplektische Gruppen durch\-geführt. Dabei werden insbesondere geometrische Charakterisierungen dieser Gruppen durch Cameron und Kantor verwendet bzw. vereinfacht. ``The one-dimensional motion of any number \({\mathcal N}\) of particles in the field of many independent waves (with strong spatial correlation) is formulated as a second-order system of stochastic differential equations, driven by two Wiener processes. In the limit of vanishing particle mass \({\mathfrak m}\to 0\), or, equivalently, of large noise intensity, we show that the momenta of all \({\mathcal N}\) particles converge weakly to \({\mathcal N}\) independent Brownian motions, and this convergence holds even if the noise is periodic. This justifies the usual application of the diffusion equation to a family of particles in a unique stochastic force field. The proof rests on the ergodic properties of the relative velocity of two particles in the scaling limit.''
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This article is devoted to multi-sided surface patches, particularly to toric surface patches, which constitute a generalization of the classical rational Bézier surface, and were introduced in [\textit{R. Krasauskas}, Adv. Comput. Math. 17, No. 1--2, 89--113 (2002; Zbl 0997.65027)]. Toric surface patches are defined over arbitrary polygonal domains which are the convex hull of a set of integer lattice points. Thus, they enjoy a nature-endowed advantage in geometric modeling of polygonal regions.
In this article, the authors present a subdivision algorithm for toric surface patches. Since rational Bézier surface patches are compatible with computer-aided design (CAD) systems, after subdivision, a computational domain parameterized by toric surface patches can be compatible with CAD systems. The latter is convenient to perform analysis and processing, and so it yields a new path to the refinement of toric surface patches in isogeometric analysis (IGA). This refinement procedure follows two steps: first, subdivide the given toric surface patch (which is the parameterization of a planar multi-sided computational domain) into several rational tensor product Bézier surface patches. Then the classical non-uniform rational B-spline (NURBS) knot insertion algorithm is used on each rational Bézier surface patch.
The main contributions of the paper are summarized in the following:
\begin{itemize}
\item[--] An improved subdivision algorithm of toric surface patches which is convenient for application in geometric modeling is proposed.
\item[--] The improved subdivision algorithm is introduced into the partition of the planar multi-sided computational domain in IGA.
\item[--] A new refinement scheme of IGA with planar multi-sided toric surface patches is proposed by combining the new computational domain partition method and the classical NURBS knot insertion algorithm.
\end{itemize} A rational surface patch generalizing the classical Bézier surface is defined. It is associated with a convex polygon with vertices at integer coordinates. The main properties of toric patches: multisided forms, singular corner points, a wider variety of implicit degrees. A method is described for subdivision of a toric patch into smaller tensor product patches. A definition of real projective varieties of arbitrary dimension via global coordinates is also introduced. Moreover, the concept of Bézier polytope is presented which develops a multidimensional variant of the theory.
| 1 |
This article is devoted to multi-sided surface patches, particularly to toric surface patches, which constitute a generalization of the classical rational Bézier surface, and were introduced in [\textit{R. Krasauskas}, Adv. Comput. Math. 17, No. 1--2, 89--113 (2002; Zbl 0997.65027)]. Toric surface patches are defined over arbitrary polygonal domains which are the convex hull of a set of integer lattice points. Thus, they enjoy a nature-endowed advantage in geometric modeling of polygonal regions.
In this article, the authors present a subdivision algorithm for toric surface patches. Since rational Bézier surface patches are compatible with computer-aided design (CAD) systems, after subdivision, a computational domain parameterized by toric surface patches can be compatible with CAD systems. The latter is convenient to perform analysis and processing, and so it yields a new path to the refinement of toric surface patches in isogeometric analysis (IGA). This refinement procedure follows two steps: first, subdivide the given toric surface patch (which is the parameterization of a planar multi-sided computational domain) into several rational tensor product Bézier surface patches. Then the classical non-uniform rational B-spline (NURBS) knot insertion algorithm is used on each rational Bézier surface patch.
The main contributions of the paper are summarized in the following:
\begin{itemize}
\item[--] An improved subdivision algorithm of toric surface patches which is convenient for application in geometric modeling is proposed.
\item[--] The improved subdivision algorithm is introduced into the partition of the planar multi-sided computational domain in IGA.
\item[--] A new refinement scheme of IGA with planar multi-sided toric surface patches is proposed by combining the new computational domain partition method and the classical NURBS knot insertion algorithm.
\end{itemize} The purpose of this paper is to study pair of left centralizers in prime rings with involution satisfying certain identities.
| 0 |
Vinogradov started the study of short exponential sums over primes of the form
\[
\sum\limits_{x-y<p\leq x} e\left(\alpha p^k\right),
\]
which are closely related to weighted exponential sums of the form
\[
\sum\limits_{x-y<n\leq x} \Lambda(n)\cdot e\left(\alpha n^k\right),
\]
where \(\Lambda(n)\) is the van Mangoldt function. The authors look at short cubic exponential sums of this form \((k=3)\), for which there is a long record of research (see [\textit{J. Liu} and \textit{T. Zhan}, Monatsh. Math. 127, No. 1, 27--41 (1999; Zbl 0922.11070)], for example). They obtain new minor arcs estimates for the case when
\[
\alpha=\frac{a}{q}+\frac{\theta}{q^2}, \quad |\theta|\leq 1, \quad (a,q)=1, \quad y\geq x^{4/5}\log(xq)^{8B+151},
\]
\[
\log(xq)^{32(B+20)}<q\leq y^5x^{-2}\log(xq)^{-32(B+20)},
\]
where \(B\) is an absolute constant. The key innovation is a non-trivial estimate for bilinear exponential sums of the form
\[
\sum\limits_{M<m\leq 2M} a_m \sum\limits_{\overset{U<n\leq 2N}{ x-y<mn\leq x}} b_n e\left(\alpha(mn)^k\right)
\]
coming from Vinogradov's method for the cases when the parameter \(N\) is large or of similar order as \(M\). Let \(\Lambda(n)\) and \(\mu(n)\) be the von Mangoldt and Möbius functions, respectively. A non-trivial estimate for \(\sum_{n\leq x}\Lambda(n)e^{2\pi in\alpha}\), where \(\alpha\) is real, was first obtained by I. M. Vinogradov in his famous solution to the ternary Goldbach problem. Recently much effort has been devoted to such sums over the interval \(x\leq n\leq x+y\), with \(y\) being small relative to \(x\), thereby giving solutions to the ternary Goldbach problem with almost equal primes.
The authors now establish estimates for the sum
\[
S=\sum_{x<n\leq x+y}\Lambda(n)e^{2\pi in^2\alpha},
\]
valid for \(x^\theta\leq y\leq x\) with \(\theta>{11\over 16}\). Because of the nonlinear nature of \(n^2\), the sum \(S\) has a more complicated dominating term than that for the linear case when \(\alpha\) lies in the major arcs of the circle method. It is too complicated to state here these estimates, which depend on diophantine properties associated with \(\alpha\), but they can be applied to show that, for any fixed \(A>0\),
\[
\sum_{x<n\leq x+y}\mu(n)e^{2\pi in^2\alpha}\ll y(\log x)^{-A},
\]
uniformly in \(\alpha\) and in the same range of \(y\). The short interval can be reduced to \(\theta>{2\over 3}\) under the Generalised Riemann Hypothesis.
The main problem in the estimation of \(S\) lies in the case of the minor arcs, which have to be further subdivided into two sets in order to apply a combination of elementary and analytic methods, and the argument involves the use of Weyl's method and Vaughan's identity.
| 1 |
Vinogradov started the study of short exponential sums over primes of the form
\[
\sum\limits_{x-y<p\leq x} e\left(\alpha p^k\right),
\]
which are closely related to weighted exponential sums of the form
\[
\sum\limits_{x-y<n\leq x} \Lambda(n)\cdot e\left(\alpha n^k\right),
\]
where \(\Lambda(n)\) is the van Mangoldt function. The authors look at short cubic exponential sums of this form \((k=3)\), for which there is a long record of research (see [\textit{J. Liu} and \textit{T. Zhan}, Monatsh. Math. 127, No. 1, 27--41 (1999; Zbl 0922.11070)], for example). They obtain new minor arcs estimates for the case when
\[
\alpha=\frac{a}{q}+\frac{\theta}{q^2}, \quad |\theta|\leq 1, \quad (a,q)=1, \quad y\geq x^{4/5}\log(xq)^{8B+151},
\]
\[
\log(xq)^{32(B+20)}<q\leq y^5x^{-2}\log(xq)^{-32(B+20)},
\]
where \(B\) is an absolute constant. The key innovation is a non-trivial estimate for bilinear exponential sums of the form
\[
\sum\limits_{M<m\leq 2M} a_m \sum\limits_{\overset{U<n\leq 2N}{ x-y<mn\leq x}} b_n e\left(\alpha(mn)^k\right)
\]
coming from Vinogradov's method for the cases when the parameter \(N\) is large or of similar order as \(M\). Motivated by the construction of holographic duals to five-dimensional superconformal quantum field theories, we obtain global solutions to Type IIB supergravity invariant under the superalgebra \(F(4)\) on a space-time of the form \(\mathrm{AdS}_{6} \times S^2 \) warped over a two-dimensional Riemann surface \(\Sigma\). In our earlier work, Part I [the authors with \textit{A. Karch}, ``Warped \(\mathrm{AdS}_{6} \times S^2 \) in type IIB supergravity. I: Local solutions'', ibid. 2016, No. 8, Paper No. 046, 52 p. (2016; \url{doi:10.1007/JHEP08(2016)046})], the general local solutions were expressed in terms of two locally holomorphic functions \(\mathcal{A}_{\pm}\) on \(\Sigma\) and global solutions were sketched when \(\Sigma\) is a disk. In the present paper, the physical regularity conditions on the supergravity fields required for global solutions are implemented on \(\mathcal{A}_{\pm}\) for arbitrary \(\Sigma\). Global solutions exist only when \(\Sigma\) has a non-empty boundary \(\partial \Sigma\). The differentials \(\partial \mathcal{A}_{\pm}\) are allowed to have poles only on \(\partial \Sigma\) and each pole corresponds to a semi-infinite \((p, q)\) five-brane. The construction for the disk is carried out in detail and the conditions for the existence of global solutions are articulated for surfaces with more than one boundary and higher genus.
| 0 |
Let \(X=G/K\) be a Riemannian symmetric space of noncompact type. The notion of the complex crown \(\Xi\) of \(X\) was introduced by Akhiezer and Gindikin in 1990. Then it was studied in works of Gindikin, Bernstein, Reznikov, Krötz, Stanton. This notion turns out to be an important tool for using complex methods in harmonic analysis on \(G/K\). For example, spherical functions \(\varphi_\lambda\) on \(G/K\) can be holomorphically extended to the crown \(\Xi\), and the crown is maximal with respect to this property. The definition of the crown is the following. Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be the canonical decomposition of the Lie algebra \({\mathfrak g}\) of \(G\) (we can consider \({\mathfrak g}\) simple), \({\mathfrak k}= \operatorname{Lie} K\), and \({\mathfrak a}\) a maximal Abelian subspace in \({\mathfrak p}\). Then the crown is a complex submanifold \(\Xi=G\exp (i\pi\Omega/2).x_0\) of \(X_{\mathbb C}\), where \(x_0=\{K\}\) and \(\Omega\) the set of \(Y\in{\mathfrak a}\) such that \(|\alpha(Y)|<1\) for all restricted roots \(\alpha\). Of special interest is the so-called distinguished boundary \(\partial_d\Xi\) of the crown (introduced in a paper of \textit{S. G. Gindikin} and \textit{B. Krötz} [Trans. Am. Math. Soc. 354, 3299--3327 (2002; Zbl 0999.22019)]. For the definition, one has to replace \(\Omega\) in the definition of \(\Xi\) by \(\partial_e\Omega\), the set of extremal points of the (compact) polyhedron \(\overline \Omega\).
In the paper under review a number of interesting and useful results concerning the crown is given. In particular, a unipotent model of the crown is presented: \(\Xi=G\exp (i\Lambda).x_0\), where \(\Lambda\) is some subset in the complexification \(N_{\mathbb C}\) of the unipotent subgroup \(N\) of \(G\) (from the Iwasawa decomposition \(G=NAK\)). The determination of the precise shape of \(\Lambda\) is a difficult problem. In the paper under review, a solution of this problem is given for rank one spaces.
The case of the Lobachevsky plane (\(G=\text{SL}(2,\mathbb R)\)) is a crucial example, which is considered very thoroughly.
The space \(X=G/K\) can be realized as a \(G\)-orbit of a (normalized) vector \(v_K\) fixed under \(K\) in a spherical representation \((\pi,\mathcal H)\) of \(G\). This orbit can holomorphically be extended to the image of the crown under the map \(z\mapsto \pi(z)v_K\), \(z\in\Xi\).
The main problem, to which the paper is devoted, is the study of the behaviour of the norm \(\|\pi(z)v_K\|\), when \(z\in\Xi\) goes to a point of the distinguished boundary \(\partial_d\Xi\) of the crown; namely, let \(\eta \in\Omega_e\), then \(z=t_\varepsilon=\exp(i(1-\varepsilon)\pi/2)\eta.x_0\) goes to \(t=\exp(i\pi/2)\eta.x_0\) when \(\varepsilon\to 0\). The problem can be translated into the growth behaviour of spherical functions \(\varphi_\lambda\) on \(\Xi\). For that, lower and upper bounds are found. The determination of the lower bound is reduced to estimates from below of the values \(\varphi_\lambda(t_\varepsilon^2)\) by means of an integral representation. The determination of the upper bound is based on the analysis of the radial system of eigenfunction equations for \(\varphi_\lambda\) with respect to the commutative algebra of \(G_{\mathbb C}\)-invariant differential operators on \(\Xi\).
As an application of the estimates, it is proved that Maass cusp forms have exponential decay. Let \(X= G/K\) be a non-compact Riemannian symmetric space contained in its complexification \(X_{\mathbb{C}}= G_{\mathbb{C}}/K_{\mathbb{C}}\). If \(G= NAK\) is an Iwasawa decomposition and \({\mathfrak a}\) the Lie algebra of \(A\), then one defines \(\Omega\subseteq{\mathfrak a}\) to be the convex domain on which all restricted roots take values in \([-{\pi\over 2},{\pi\over 2}]\). Then \(\Xi:= G\exp(i\Omega) K_{\mathbb{C}}/K_{\mathbb{C}}\subseteq X_{\mathbb{C}}\) is called the complex crown of \(X\). The complex crown has been studied intensely by a number of authors in the last year because of its non-trivial but natural appearance in complex geometry and harmonic analysis.
In the paper under review the authors study \(G\)-orbits in the boundary of \(\Xi\). More precisely, they only consider the orbits through the exponential image of extremal points in \(i\Omega\). They show that the union of those orbits has properties reminiscent of a Shilov boundary and thus call it the distinguished boundary \(\partial_d\Xi\) of \(\Xi\). Further, they show that for simply connected \(G_{\mathbb{C}}\) the distinguished boundary is the disjoint union of homogeneous spaces which are symmetric if and only if they carry an invariant order structure, i.e. are non-compactly causal symmetric spaces. Moreover, they show that this is also equivalent to the total reality of the homogeneous space as a submanifold of \(X_{\mathbb{C}}\).
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Let \(X=G/K\) be a Riemannian symmetric space of noncompact type. The notion of the complex crown \(\Xi\) of \(X\) was introduced by Akhiezer and Gindikin in 1990. Then it was studied in works of Gindikin, Bernstein, Reznikov, Krötz, Stanton. This notion turns out to be an important tool for using complex methods in harmonic analysis on \(G/K\). For example, spherical functions \(\varphi_\lambda\) on \(G/K\) can be holomorphically extended to the crown \(\Xi\), and the crown is maximal with respect to this property. The definition of the crown is the following. Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be the canonical decomposition of the Lie algebra \({\mathfrak g}\) of \(G\) (we can consider \({\mathfrak g}\) simple), \({\mathfrak k}= \operatorname{Lie} K\), and \({\mathfrak a}\) a maximal Abelian subspace in \({\mathfrak p}\). Then the crown is a complex submanifold \(\Xi=G\exp (i\pi\Omega/2).x_0\) of \(X_{\mathbb C}\), where \(x_0=\{K\}\) and \(\Omega\) the set of \(Y\in{\mathfrak a}\) such that \(|\alpha(Y)|<1\) for all restricted roots \(\alpha\). Of special interest is the so-called distinguished boundary \(\partial_d\Xi\) of the crown (introduced in a paper of \textit{S. G. Gindikin} and \textit{B. Krötz} [Trans. Am. Math. Soc. 354, 3299--3327 (2002; Zbl 0999.22019)]. For the definition, one has to replace \(\Omega\) in the definition of \(\Xi\) by \(\partial_e\Omega\), the set of extremal points of the (compact) polyhedron \(\overline \Omega\).
In the paper under review a number of interesting and useful results concerning the crown is given. In particular, a unipotent model of the crown is presented: \(\Xi=G\exp (i\Lambda).x_0\), where \(\Lambda\) is some subset in the complexification \(N_{\mathbb C}\) of the unipotent subgroup \(N\) of \(G\) (from the Iwasawa decomposition \(G=NAK\)). The determination of the precise shape of \(\Lambda\) is a difficult problem. In the paper under review, a solution of this problem is given for rank one spaces.
The case of the Lobachevsky plane (\(G=\text{SL}(2,\mathbb R)\)) is a crucial example, which is considered very thoroughly.
The space \(X=G/K\) can be realized as a \(G\)-orbit of a (normalized) vector \(v_K\) fixed under \(K\) in a spherical representation \((\pi,\mathcal H)\) of \(G\). This orbit can holomorphically be extended to the image of the crown under the map \(z\mapsto \pi(z)v_K\), \(z\in\Xi\).
The main problem, to which the paper is devoted, is the study of the behaviour of the norm \(\|\pi(z)v_K\|\), when \(z\in\Xi\) goes to a point of the distinguished boundary \(\partial_d\Xi\) of the crown; namely, let \(\eta \in\Omega_e\), then \(z=t_\varepsilon=\exp(i(1-\varepsilon)\pi/2)\eta.x_0\) goes to \(t=\exp(i\pi/2)\eta.x_0\) when \(\varepsilon\to 0\). The problem can be translated into the growth behaviour of spherical functions \(\varphi_\lambda\) on \(\Xi\). For that, lower and upper bounds are found. The determination of the lower bound is reduced to estimates from below of the values \(\varphi_\lambda(t_\varepsilon^2)\) by means of an integral representation. The determination of the upper bound is based on the analysis of the radial system of eigenfunction equations for \(\varphi_\lambda\) with respect to the commutative algebra of \(G_{\mathbb C}\)-invariant differential operators on \(\Xi\).
As an application of the estimates, it is proved that Maass cusp forms have exponential decay. The paper considers a procedure to discretize the governing differential-algebraic equations. This enables the use of a new, energy-consistent time-stepping procedure which is applicable to general nonholonomic systems. The procedure is applicable to both rigid and flexible systems.
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The authors proved sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields. Namely, let \(\gamma\not=1-N/2\) be a real number and let \(u\in C^{\infty}_{c}(\mathbb{R}^{N})^{N}\) be a curl-free vector field. Then, there are optimal constants \(H_{N,\gamma}\) and \(R_{N,\gamma}\) (given explicitly) satisfying:
\begin{itemize}
\item [\(i)\)] If \(u(0)=0\) and \(\gamma<1-N/2\), then
\[
H_{N,\gamma}\int_{\mathbb{R}^{N}}\frac{|u|^2}{|x|^2}|x|^{2\gamma}dx\le \int_{\mathbb{R}^{N}}|\nabla u|^{2}|x|^{2\gamma}dx.
\]
\item [\(ii)\)] If \(\int_{\mathbb{R}^N}|x|^{2\gamma-4}|u|^{2}dx<\infty\), then
\[
R_{N,\gamma}\int_{\mathbb{R}^{N}}\frac{|u|^2}{|x|^4}|x|^{2\gamma}dx\le \int_{\mathbb{R}^{N}}|\Delta u|^{2}|x|^{2\gamma}dx.
\]
\end{itemize}
These results complement the former work by [\textit{O. Costin} and \textit{V. Maz'ya}, Calc. Var. Partial Differ. Equ. 32, No. 4, 523--532 (2008; Zbl 1147.35122)] on the sharp Hardy-Leray inequality for axisymmetric divergence-free vector fields. We show that the sharp constant in the classical \(n\)-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for \(n = 2\) without the axisymmetry assumption.
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The authors proved sharp Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields. Namely, let \(\gamma\not=1-N/2\) be a real number and let \(u\in C^{\infty}_{c}(\mathbb{R}^{N})^{N}\) be a curl-free vector field. Then, there are optimal constants \(H_{N,\gamma}\) and \(R_{N,\gamma}\) (given explicitly) satisfying:
\begin{itemize}
\item [\(i)\)] If \(u(0)=0\) and \(\gamma<1-N/2\), then
\[
H_{N,\gamma}\int_{\mathbb{R}^{N}}\frac{|u|^2}{|x|^2}|x|^{2\gamma}dx\le \int_{\mathbb{R}^{N}}|\nabla u|^{2}|x|^{2\gamma}dx.
\]
\item [\(ii)\)] If \(\int_{\mathbb{R}^N}|x|^{2\gamma-4}|u|^{2}dx<\infty\), then
\[
R_{N,\gamma}\int_{\mathbb{R}^{N}}\frac{|u|^2}{|x|^4}|x|^{2\gamma}dx\le \int_{\mathbb{R}^{N}}|\Delta u|^{2}|x|^{2\gamma}dx.
\]
\end{itemize}
These results complement the former work by [\textit{O. Costin} and \textit{V. Maz'ya}, Calc. Var. Partial Differ. Equ. 32, No. 4, 523--532 (2008; Zbl 1147.35122)] on the sharp Hardy-Leray inequality for axisymmetric divergence-free vector fields. This paper solves the problem of optimal dynamic investment, consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz law and investment opportunities are constant. Healthcare slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Healthcare spending steadily increases with age, both in absolute terms and relative to total spending. The optimal stochastic control problem reduces to a nonlinear ordinary differential equation with a unique solution, which has an explicit expression in the old-age limit. The main results are obtained through a novel version of Perron's method.
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Employing umbral calculus [see \textit{S. Roman}, The umbral calculus, Academic Press, Boston (1984; Zbl 0536.33001)] generalizations of Bernstein polynomials and Bézier curves are introduced. The basic idea is to introduce a finite sequence of parameters in the definition of Bernstein polynomials which allows a great deal of freedom to the design of the corresponding generalized Bézier curves. Such curves continue to preserve some basic properties of the usual Bézier curves. Several illustrations are also given. A Sheffer sequence is a set of polynomials \(\{s_n(x)\}\) with the generating function
\[
\sum^{\infty}_{n=0}s_n(x)z^n/n!=A(z)\exp(xB(z)), \tag{1}
\]
where \(A(0)\neq 0\), \(B(0)=0\), \(B'(0)\neq 0\), and \(A(z)\) and \(B(z)\) are analytic in a neighborhood of the origin, or more generally have formal power series expansions about zero. Most of this book is a study of Sheffer sequences via an operational method known as umbral calculus. Following Rota, the author treats the umbral calculus via linear functionals and operators. If \(f(t)\) has the formal power series \(\sum a_n t^n/n!\) then \(\langle f(t)| x_n\rangle=a_n\). I would rather have seen the author use \(D\) as the variable since that is the usual notation for a derivative, but with a bit of effort one can get used to thinking of \(t\) as \(d/dx\). A linear operator associated with the formal power series of \(f\) is defined by \(f(t)x^n=\sum \binom nk a_k x^{n-k}\). In this setting Sheffer sequences can be defined as follows. Assume \(a_0=0\), \(a_1\neq 0\) and \(g(0)\neq 0\). Then there is a unique sequence of polynomials \(s_n(x)\) satisfying \(\langle g(t)[f(t)]^k| s_n(x)\rangle =n!\delta_{n,k}\), \(n,k\geq 0\). These polynomials are the Sheffer sequence for the pair \((g(t),f(t))\). The generating function (1) can be written as
\[
[g(z)]^{-1}e^{xz}=\sum^{\infty}_{n=0}s_n(x)[f(z)]^ n/n!\, .
\]
Many expansions and other facts are derived for Sheffer sequences, including special cases such as Appell sequences (those when \(f(t)=t)\). Among the examples given to illustrate the general theory are \(x^n\), \(x(x-1)\cdots(x-n+1)\), the polynomials discovered by Euler that are called Abel polynomials, those found by Lambert and called Gould polynomials, Hermite polynomials, Laguerre polynomials, Meixner polynomials, Charlier polynomials, Bessel polynomials, Bell polynomials, Bernoulli polynomials, Euler polynomials, and many miscellaneous polynomials mentioned earlier by Boas and Buck and in the Bateman project.
The last two chapters are an introduction to special topics including the Lagrange inversion formula, expansions of one Sheffer sequence in terms of another if they are sufficiently similar, a proof of Meixner's theorem giving all sets of orthogonal polynomials which are Sheffer sequences, and a brief introduction to other umbral calculi, including a very brief introduction to \(q\)-umbral calculus. The most satisfying \(q\)-umbral calculus has recently been discovered by \textit{A. M. Garsia} and \textit{J. Remmel} [Houston J. Math. 12, 503--523 (1986; Zbl 0616.05006)]. Here is a test question for those who study \(q\)-umbral methods. The right \(q\)-extension of (1) is probably
\[
\sum s_n(x)t^n=A(t)\prod^{\infty}_{0}[1-xB(tq^n)]
\]
with \(A(t)\) and \(B(t)\) as given above. It is likely that all orthogonal polynomials in this set were found by \textit{W. A. Al-Salam} and \textit{T. S. Chihara} [SIAM J. Math. Anal. 7, 16--28 (1976; Zbl 0323.33007)]. An adequate \(q\)-umbral calculus should be able to show this.
There are some misprints. A few noted are: \(\binom yk\) should be \(y^k/k!\) twice on page 55, a minus sign is missing in the first formula on page 61, \(e\nu^{t^ 2/2}\) should be \(e^{\nu t^2/2}\) on page 158. The infinite \(q\)-binomial theorem on page 179 is stated incorrectly twice. In the standard notation the only error is \(z\) instead of \(y\) on the left hand side. In the alternative notation used in the middle formula of this page, \(z\) should be \(x\) on the right hand side and the author forgot to replace \(t\) by \(t/(1-q)\) in the series, as he said to in the previous line. Also on page 180, the formula for \(H_n(x;y)\) in the middle of the page is at most a \(q\)-analogue of \((x+y)^n\), not the \(q\)-analog. A better \(q\)- analog was given on the previous page.
A couple of statements that can be misinterpreted were given. On page 78 it is said that \(y_n(x)\) and their derivatives form an orthogonal sequence of polynomials. What is true is that \(\{y_n(x)\}\) are orthogonal with respect to a signed measure. That is also true for \(\{y^{(k)}_{n+k}(x)\}\) for each fixed \(k\), and even more generally for \(\{{}_2F_1(-n,n+a;-;-x/2)\}.\) On page 159 the author says ``From these examples the reader should have no difficulty solving other recurrence relations (whose solutions are not known in advance).'' A novice might conclude from this that one can exactly solve most or all linear recurrence relations. That is far from the case, although there are quite a few interesting ones that are not given in the standard handbooks.
Much of the notation annoyed me. The use of \(t\) instead of \(D\) was a minor instance. Much more annoying was the use of the standard letters to denote standard functions with a different normalization. Thus \(L_n^{\alpha}(x)\) does not denote the usual Laguerre polynomials but \(n!\) times the usual ones. \(H_n(x)\) does not denote the Hermite polynomials as given by Szegő and Erdélyi in the two standard references for the classical orthogonal polynomials, but polynomials orthogonal with respect to \(\exp(-x^2/2).\) I know why the author changed these functions, but do not understand why he did not change the notation as well. \(\ell_n^{\alpha}(x)\) would have been better than \(L_n^{\alpha}(x)\). Hypergeometric functions occur in many of the examples, but the standard notation was not used.
The worst case of confusion occurs in the use of \((x)_ n\) to denote \(x(x-1)\cdots(x-n+1),\) as many combinatorialists do, rather than \(x(x+1)\cdots(x+n-1),\) as almost everyone else does. The standard \(q\)-notation \((a;q)_ n=(1-a)(1-aq)\cdots(1-aq^{n-1})\) is introduced and used in the statement of the \(q\)-binomial theorem (due to Rothe, not Heine), but the notation that was used in most of the \(q\)-formulas, \([x]_{y,n}\) for \(x^n(y/x;q)_n\) is an abomination and should not be complied by others.
The bibliography is probably relatively good when dealing with umbral calculus, but is lacking in references to other treatments of operational calculus and any of the deeper formulas for special functions related to additional formulas. Two examples are \textit{I. I. Hirschman} and \textit{D. V. Widder}'s very important and unjustly neglected book ``The convolution transform.'' Princeton: Princeton University Press (1955; Zbl 0065.09301) for some operational results and \textit{Ch. F. Dunkl}'s work on Krawtchouk polynomials [Indiana Univ. Math. J. 25, 335--358 (1976; Zbl 0326.33008)]. Without some comments on other work a novice could get a wrong idea about how much can be done with the present methods. They work fine for a limited class of problems, and for some they seem to be the best method. But there are many very important sets of polynomials which can not be treated yet, and most of the more complicated formulas for some Sheffer polynomials do not seem to be amenable to umbral methods.
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Employing umbral calculus [see \textit{S. Roman}, The umbral calculus, Academic Press, Boston (1984; Zbl 0536.33001)] generalizations of Bernstein polynomials and Bézier curves are introduced. The basic idea is to introduce a finite sequence of parameters in the definition of Bernstein polynomials which allows a great deal of freedom to the design of the corresponding generalized Bézier curves. Such curves continue to preserve some basic properties of the usual Bézier curves. Several illustrations are also given. We compare the semantical and syntactical definitions of extensions for open default theories. We prove that, over monadic languages, these definitions are equivalent and do not depend on the cardinality of the underlying infinite world. We also show that, under the domain closure assumption, one-free-variable open default theories are decidable.
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Let \(f\in\mathbb{Q}[X]\) and let us consider the Diophantine equation
\[
z^2 = f(x)^2 \pm f (y)^2.
\]
The authors are interested in the existence of infinitely many rational solutions \((x, y, z)\) of the equation above. A similar problem was studied in \textit{M. Ulas} [Colloq. Math. 107, No. 1, 1--6 (2007; Zbl 1153.11016)]. Ulas considered the Diophantine equation
\[
f(x)f(y)=f(z)^2,
\]
where \(f\in\mathbb{Q}[X]\) is a polynomial function of \(\deg f \leq 3\). Ulas proved that if \(f\) is a quadratic function, then the Diophantine equation \(f(x)f(y) = f(z)^2\) has infinitely many nontrivial solutions in \(\mathbb{Q}(t)\).
In this paper, Ulas and Togbé show that if \(\deg f = 2\) and there exists a rational number \(t\) such that on the quartic curve \(V^2 = f(U)^2 + f(t)^2\) there are infinitely many rational points, then the set of rational parametric solutions of the equation \(z^2 = f(x)^2 + f(y)^2\) is non-empty. Without any assumptions they show that the surface related to the Diophantine equation \(z^2 = f(x)^2 - f (y)^2\) is unirational over the field \(\mathbb{Q}\) in this case. If \(\deg f = 3\) and \(f\) has the form \(f (x) = x(x^2 + ax + b)\) with \(a \neq 0\) then both of the equations \(z^2 = f(x)^2 \pm f (y)^2\) have infinitely many rational parametric solutions. A similar result is proved for the equation \(z^2 = f (x)^2 - f (y)^2\) with \(f (X) = X^3 + aX^2 + b\) and \(a \neq 0\). Let \(f\in{\mathbb Q}[X]\) and \(\deg f\leq 3\). We prove that if \(\deg f=2\), then the Diophantine equation \(f(x)f(y)=f(z)^2\) has infinitely many nontrivial solutions in \({\mathbb Q}(t)\). In the case when \(\deg f=3\) and \(f(X)=X(X^2+aX+b)\) we show that for all but finitely many \(a,b\in{\mathbb Z}\) satisfying \(ab\neq 0\) and additionally, if \(p| a\), then \(p^2 \nmid b\), the equation \(f(x)f(y)=f(z)^2\) has infinitely many nontrivial solutions in rationals.
| 1 |
Let \(f\in\mathbb{Q}[X]\) and let us consider the Diophantine equation
\[
z^2 = f(x)^2 \pm f (y)^2.
\]
The authors are interested in the existence of infinitely many rational solutions \((x, y, z)\) of the equation above. A similar problem was studied in \textit{M. Ulas} [Colloq. Math. 107, No. 1, 1--6 (2007; Zbl 1153.11016)]. Ulas considered the Diophantine equation
\[
f(x)f(y)=f(z)^2,
\]
where \(f\in\mathbb{Q}[X]\) is a polynomial function of \(\deg f \leq 3\). Ulas proved that if \(f\) is a quadratic function, then the Diophantine equation \(f(x)f(y) = f(z)^2\) has infinitely many nontrivial solutions in \(\mathbb{Q}(t)\).
In this paper, Ulas and Togbé show that if \(\deg f = 2\) and there exists a rational number \(t\) such that on the quartic curve \(V^2 = f(U)^2 + f(t)^2\) there are infinitely many rational points, then the set of rational parametric solutions of the equation \(z^2 = f(x)^2 + f(y)^2\) is non-empty. Without any assumptions they show that the surface related to the Diophantine equation \(z^2 = f(x)^2 - f (y)^2\) is unirational over the field \(\mathbb{Q}\) in this case. If \(\deg f = 3\) and \(f\) has the form \(f (x) = x(x^2 + ax + b)\) with \(a \neq 0\) then both of the equations \(z^2 = f(x)^2 \pm f (y)^2\) have infinitely many rational parametric solutions. A similar result is proved for the equation \(z^2 = f (x)^2 - f (y)^2\) with \(f (X) = X^3 + aX^2 + b\) and \(a \neq 0\). Nonlinear physics continues to be an area of dynamic modern research, with applications to physics, engineering, chemistry, mathematics, computer science, biology, medicine and economics. In this second edition extensive use is made of the computer algebra system, Maple V. No prior knowledge of Maple or of programming is assumed. The authors have provided 74 Maple files on a CD-ROM, all classroom tested, as well as 60 annotated Maple worksheets. These files and worksheets may be used to both solve and explore the text's 400 problems. The book includes 30 experimental activities which are intended to deepen and broaden the reader's understanding of the nonlinear physics. These activities are correlated with Part I, the theoretical framework of the text.
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In recent years, the connection between coding theory and algebraic geometry has generated considerable interest, ever since it was shown that this allows the explicit construction of a family of good codes (i.e., codes exceeding the Gilbert-Varshomov bound). This goes back to ideas of Goppa: accessible introductions for the nonspecialist are given by \textit{J. H. van Lint} and \textit{G. v. d. Geer}: ``Introduction to coding theory and algebraic geometry'' Basel (1988; Zbl 0639.00048)) and by \textit{J. H. van Lint}: ``Algebraic geometric codes'', In: Coding theory and design theory, Part I (ed. by D. Ray-Chaudhuri), IMA Vol. 20, 137-162 (1990), Springer. The present author considers the problem of using this approach to construct good (formally) self-dual codes. His main result is as follows: For every prime power q, there exists a family of formally self-dual Goppa-codes for which the ratio d(C)/n(C) is bounded away from 0. If q is even, the codes are in fact self-dual. The two articles of this volume will be reviewed individually: Jacobus H. van Lint, Introduction to coding theory and algebraic geometry. I: Coding theory (9--33) [Zbl 0648.94011], Gerard van der Geer, Introduction to coding theory and algebraic geometry. II: Algebraic geometry (35--81) [Zbl 0648.94012].
| 1 |
In recent years, the connection between coding theory and algebraic geometry has generated considerable interest, ever since it was shown that this allows the explicit construction of a family of good codes (i.e., codes exceeding the Gilbert-Varshomov bound). This goes back to ideas of Goppa: accessible introductions for the nonspecialist are given by \textit{J. H. van Lint} and \textit{G. v. d. Geer}: ``Introduction to coding theory and algebraic geometry'' Basel (1988; Zbl 0639.00048)) and by \textit{J. H. van Lint}: ``Algebraic geometric codes'', In: Coding theory and design theory, Part I (ed. by D. Ray-Chaudhuri), IMA Vol. 20, 137-162 (1990), Springer. The present author considers the problem of using this approach to construct good (formally) self-dual codes. His main result is as follows: For every prime power q, there exists a family of formally self-dual Goppa-codes for which the ratio d(C)/n(C) is bounded away from 0. If q is even, the codes are in fact self-dual. We describe the fundamental constructions and properties of determinantal probability measures and point processes, giving streamlined proofs. We illustrate these with some important examples. We pose several general questions and conjectures.
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Let \(\mathrm{SF}(\omega)\) denote the group of all finitary permutations of \(\omega\). The research of this paper is motivated by a result of \textit{P. Matet} [J. Symb. Log. 51, 12--21 (1986; Zbl 0588.04011)], which is associated with a lattice consisting of certain subgroups of \(\mathrm{SF}(\omega)\), modulo an ideal consisting of certain finite subgroups.
This paper considers the lattice \((\mathrm{LF},\mathrm{IF})\) (under almost containedness and almost orthogonality), where \(\mathrm{LF}\) is the lattice of all subgroups of \(\mathrm{SF}(\omega)\) and \(\mathrm{IF}\) is the ideal of all finite subgroups.
The author compares (in \S 2.1) this lattice with the classical analogous \(({\mathcal P}(\omega), \mathrm{Fin})\), computes the six van Douwen cardinals for this lattice (see Theorem 2.6), and further proves (in \S3) two variants of Matet's theorem for this lattice, one under MA and the other under CH. The purpose of this paper is to study the properties of those filters that can be associated with the dual Galvin-Prikry theorem of \textit{T. J. Carlson} and \textit{S. G. Simpson} [Adv. Math. 53, 265-290 (1984; Zbl 0564.05005)]. Given \(X,Y\in (\omega)^{\omega}\) (the collection of all infinite partitions of \(\omega)\), write \(X\leq Y\) if X is coarser than Y, and denote by \(X\cap Y\) the finest partition Z that is coarser than both X and Y. Identify each partition of \(\omega\) with the associated equivalence relation, and put a topology on \((\omega)^{\omega}\) by restricting the product topology on \(2^{\omega \times \omega}\), 2 itself considered discrete. It is shown that assuming the continuum hypothesis, there exists an \(F\subset (\omega)^{\omega}\) such that: i) if \(X\in F\) and \(X\leq Y\), then \(Y\in F\); ii) if X,Y\(\in F\), then \(X\cap Y\in F\); and iii) for every Borel subset A of \((\omega)^{\omega}\), there is an \(X\in F\) such that the set \(\{Y\in (\omega)^{\omega}:\) \(Y\leq X\}\) is either included in A or else disjoint from A.
| 1 |
Let \(\mathrm{SF}(\omega)\) denote the group of all finitary permutations of \(\omega\). The research of this paper is motivated by a result of \textit{P. Matet} [J. Symb. Log. 51, 12--21 (1986; Zbl 0588.04011)], which is associated with a lattice consisting of certain subgroups of \(\mathrm{SF}(\omega)\), modulo an ideal consisting of certain finite subgroups.
This paper considers the lattice \((\mathrm{LF},\mathrm{IF})\) (under almost containedness and almost orthogonality), where \(\mathrm{LF}\) is the lattice of all subgroups of \(\mathrm{SF}(\omega)\) and \(\mathrm{IF}\) is the ideal of all finite subgroups.
The author compares (in \S 2.1) this lattice with the classical analogous \(({\mathcal P}(\omega), \mathrm{Fin})\), computes the six van Douwen cardinals for this lattice (see Theorem 2.6), and further proves (in \S3) two variants of Matet's theorem for this lattice, one under MA and the other under CH. Shot peening is a mechanical surface treatment with the purpose to modify the surface state of a material in order to improve fatigue strength of a component subjected to cyclic loading. The material state after a shot peening treatment is governed by various shot peening parameters. In order to comprehend the complex interaction between process parameters and material state time and money consuming experiments are usually accomplished. Promising alternatives are numerical simulation methods such as FEM combined with similarity mechanics having the possibility to predict the shot peening results for arbitrary combinations of shot peening parameters. In this work the residual stress development during shot peening for various shot peening parameters was simulated successfully with a FEM model. Additionally the method of similarity mechanics was used in combination with the FEM simulation results in order to predict residual stress states for a wide parameter field with almost no additional computing effort.
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The paper deals with small perturbations of the controlled system \(M\ddot q + L\dot q+Pq = N_1w + N_2u,\;z = Q_1q + R_1u,\;y = Q_2q + R_2w\), where \( q\in X\), \( X\) is the \(n\)-dimensional configuration space, \( z\), \(y\), \(w\) and \(u\) are the output, measurement, perturbation and control vectors. Presented are asymptotic decoupling conditions for perturbations of this system [see \textit{S. Weiland} and \textit{J. C. Willems}, IEEE Trans. Autom. Control 34, 277-286 (1989; Zbl 0674.93014)]. We solve, in terms of the geometric concepts of linear system theorem, the almost disturbance decoupling problem with internal stability. The solution gives necessary and sufficient conditions for the existence of a dynamic output feedback controller such that in the closed-loop system the disturbances are quenched, say in the \(H_{\infty}\)-sense, up to any degree of accuracy while maintaining a stable system matrix.
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The paper deals with small perturbations of the controlled system \(M\ddot q + L\dot q+Pq = N_1w + N_2u,\;z = Q_1q + R_1u,\;y = Q_2q + R_2w\), where \( q\in X\), \( X\) is the \(n\)-dimensional configuration space, \( z\), \(y\), \(w\) and \(u\) are the output, measurement, perturbation and control vectors. Presented are asymptotic decoupling conditions for perturbations of this system [see \textit{S. Weiland} and \textit{J. C. Willems}, IEEE Trans. Autom. Control 34, 277-286 (1989; Zbl 0674.93014)]. The problem of two-dimensional bubble rising at a constant velocity in an inclined slit is considered in two different cases: a) the bubble extends downwards without limits as in the case of an infinite jet limited on the right by a wall of the slit and on the left by a free surface, and b) the jet emerges from a nozzle and falls down along an inclined wall. The inviscid fluid is subject to the gravitational force and to superficial forces on the free boundary; in previous studies these forces were neglected. As the fluid motion is two-dimensional, the authors use the complex variable technique and find exact solutions which can be applied to different cases and are discussed in the final part of the paper. The authors show that under the action of superficial forces the free surface behaves completely differently from the case in which the superficial forces are neglected.
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In the paper under review, the authors consider the odd unitary groups \(U(V,q)\) of odd quadratic spaces \((V,q)\) over an associative unital ring \(R\). These groups, introduced by \textit{V. A. Petrov} [J. Math. Sci., New York 130, No. 3, 4752--4766 (2005; Zbl 1144.20316)], provide a uniform generalization of the classical groups over \(R\). In particluar, the authors study those subgroups of \(U(V,q)\) that are normalized by the \textit{odd elementary subgroup} \(\mathrm{EU}(V,q)\) (which is generated by certain transvections) and contain an element that is non-central upon restriction to the even part of \(V\). They prove that, under the so-called stable rank condition, any such subgroup \(H\) contains an elementary transvection and satisfies the sandwich property \(E\leq H\leq C\), where \(E\) and \(C\) are explicitly constructed. Specifically, \(E\) (\(C\)) is the relative elementary subgroup (the odd general unitary congruence subgroup) of level \(J\), the level being an explicitly constructed odd form ideal. We introduce a new type of classical-like groups, the so-called odd unitary groups. This notion covers the cases of Bak's quadratic groups and Hermitian groups. The normality of the elementary subgroup and the surjective stability of the \(K_1\)-functor are proved in this context.
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In the paper under review, the authors consider the odd unitary groups \(U(V,q)\) of odd quadratic spaces \((V,q)\) over an associative unital ring \(R\). These groups, introduced by \textit{V. A. Petrov} [J. Math. Sci., New York 130, No. 3, 4752--4766 (2005; Zbl 1144.20316)], provide a uniform generalization of the classical groups over \(R\). In particluar, the authors study those subgroups of \(U(V,q)\) that are normalized by the \textit{odd elementary subgroup} \(\mathrm{EU}(V,q)\) (which is generated by certain transvections) and contain an element that is non-central upon restriction to the even part of \(V\). They prove that, under the so-called stable rank condition, any such subgroup \(H\) contains an elementary transvection and satisfies the sandwich property \(E\leq H\leq C\), where \(E\) and \(C\) are explicitly constructed. Specifically, \(E\) (\(C\)) is the relative elementary subgroup (the odd general unitary congruence subgroup) of level \(J\), the level being an explicitly constructed odd form ideal. At one level of abstraction neural tissue can be regarded as a medium for turning local synaptic activity into output signals that propagate over large distances via axons to generate further synaptic activity that can cause reverberant activity in networks that possess a mixture of excitatory and inhibitory connections. This output is often taken to be a firing rate, and the mathematical form for the evolution equation of activity depends upon a spatial convolution of this rate with a fixed anatomical connectivity pattern. Such formulations often neglect the metabolic processes that would ultimately limit synaptic activity. Here we reinstate such a process, in the spirit of an original prescription by \textit{H. R. Wilson} and \textit{J. D. Cowan} [``Excitatory and inhibitory interactions in localized populations of model neurons'', Biophys. J. 12, No. 1, 1--24 (1972; \url{doi:10.1016/S0006-3495(72)86068-5})], using a term that multiplies the usual spatial convolution with a moving time average of local activity over some \textit{refractory} time-scale. This modulation can substantially affect network behaviour, and in particular give rise to periodic travelling waves in a purely excitatory network (with exponentially decaying anatomical connectivity), which in the absence of refractoriness would only support travelling fronts. We construct these solutions numerically as stationary periodic solutions in a co-moving frame (of both an equivalent delay differential model as well as the original delay integro-differential model). Continuation methods are used to obtain the dispersion curve for periodic travelling waves (speed as a function of period), and found to be reminiscent of those for spatially extended models of excitable tissue. A kinematic analysis (based on the dispersion curve) predicts the onset of wave instabilities, which are confirmed numerically.
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Semilinear, infinite-dimensional, second-order integrodifferential control systems, with distributed and point delays in the state variables are considered. Using the Schaefer fixed-point theorem, sufficient conditions for exact controllability in a given finite time interval are formulated and proved. In the proofs the theory of semigroups of linear operators is used. Several remarks and comments on the relationships between the results in the paper and those existing in the literature are presented. The Schaefer fixed-point theorem and the exact controllability definition are also recalled for the completeness of the considerations. Moreover, it should be pointed out that similar exact controllability problems for second-order infinite-dimensional control systems have been recently investigated in the paper [\textit{J. Y. Park} and \textit{H. K. Han}, Bull. Korean Math. Soc. 34, No. 3, 411-419 (1997; Zbl 0889.93008)]. The paper studies the exact controllability of a semilinear control system of second order in a Banach space \(X\). The system is described by
\[
x''(t)= Ax(t)+ f\bigl(t,x(t)\bigr) +Bu(t), \quad x(0)= x_0, \quad x'(0)= y_0.
\]
Here, \(A\) denotes the infinitesimal generator of a strongly continuous cosine family \(C(t) (\in{\mathcal L} (X))\), \(t\in \mathbb{R}^1\) and the associated uniformly bounded and compact sine family is \(S(t)= \int^t_0 C(s)ds\); \(f\) is a uniformly bounded nonlinear function; \(u\) is a control belonging to \(U\); and \(B\) is a bounded linear operator. One assumes the exact controllability of the linear equation in time \(T\), that is, the mapping \(W\) from \(U\) to \(X\) defined by
\[
Wu= \int^T_0 S(T-s) Bu(s)ds
\]
admits the bounded inverse \(W^{-1}\). Under these assumptions, it is shown that the above semilinear system is exactly controllable in time \(T\). A nonlinear mapping \(G\) is introduced via \(W^{-1}\) in a bounded set of continuous functions with values in \(X\). The result is proven by showing that \(G\) has a fixed point (the Schauder fixed point theorem).
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Semilinear, infinite-dimensional, second-order integrodifferential control systems, with distributed and point delays in the state variables are considered. Using the Schaefer fixed-point theorem, sufficient conditions for exact controllability in a given finite time interval are formulated and proved. In the proofs the theory of semigroups of linear operators is used. Several remarks and comments on the relationships between the results in the paper and those existing in the literature are presented. The Schaefer fixed-point theorem and the exact controllability definition are also recalled for the completeness of the considerations. Moreover, it should be pointed out that similar exact controllability problems for second-order infinite-dimensional control systems have been recently investigated in the paper [\textit{J. Y. Park} and \textit{H. K. Han}, Bull. Korean Math. Soc. 34, No. 3, 411-419 (1997; Zbl 0889.93008)]. The statement `no non-Abelian simple group can be obtained from a non-simple one by adding one generator and one relator' first is equivalent to the Kervaire-Laudenbach conjecture, and second, becomes true under the additional assumption that the initial non-simple group is either finite or torsion-free.
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Let \(\rho \equiv 3(\text{mod }4)\) and let \(\mathcal S\) be an extension of the symplectic group \(\text{Sp}(2n,\rho)\) by the diagonal automorphism of order 2. \textit{R. Gow} [J. Algebra 122, No. 2, 510-519 (1989; Zbl 0679.20039)] considered integer lattices in the space of the irreducible representations of \(\mathcal S\) over \(\mathbb{Q}\) of degree \((\rho^ n - (-1)^ n)/2\). The author calculates the full automorphism groups \(\mathcal G\) of these lattices. In particular, for \(n>4\) \(\mathcal S\) is shown to have index 3 in \(G\). The irreducible components of the Weyl representation of the group \(G=Sp(2n,q)\) where \(q=p^ d\), \(p>2\) is a prime are considered. The Weyl representation W of G is a complex representation of degree \(q^ n\) obtained from the action of G on an extraspecial group of order \(pq^{2n}\). See, for example, the articles of \textit{I. M. Isaacs} [Am. J. Math. 95, 594-635 (1973; Zbl 0277.20008)], \textit{G. M. Seitz} [J. Lond. Math. Soc., II. Ser. 10, 115-120 (1975; Zbl 0333.20039)] and \textit{H. N. Ward} [J. Algebra 20, 182-195 (1972; Zbl 0239.20013)]. W is the sum of two irreducible components of degrees \((q^ n-1)/2\) and \((q^ n+1)/2\), one of them is faithful and the other is not. Define the characters of the faithful and unfaithful components by \(\psi_ 1\) and \(\psi_ 2\), respectively. Let \(q=p\equiv 3 (mod 4)\), \(H\supset G\) be the group of skew symplectic transformations, Q be the rational field. Then \(| H:G| =2\), it is proved that the induced character \(\psi^ H_ 1\) is a character of an absolutely irreducible Q-representation \(\theta\) of H. Let M be the QH-module corresponding to \(\theta\). The main result of the paper (Theorem 3.2) shows that for even n there exists an H-invariant integral lattice in M that supports an even symmetric positive definite unimodular H-invariant form.
The author points out that the argument in Theorem 3.2 is derived from \textit{J. G. Thompson}'s work [J. Algebra 38, 523-524 (1976; Zbl 0344.20001)]. It is crucial that \(\psi_ 1\) defines an irreducible Brauer character modulo any prime r. This was shown by \textit{Seitz} (cited above) for \(r\neq p\) and by \textit{A. E. Zalesskij} and the reviewer [Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk 1987, No.6, 9-15 (1987)] for \(r=p.\)
The paper under review also gives some information of independent interest on Schur indices of irreducible characters of Sp(2n,q). Theorem 1.4. Let \(q\equiv 3 (mod 4)\), \(\lambda\in Irr G\) be a not real-valued character, and let \(\lambda\) (1) be relatively prime to p. Then \(\lambda\) has Schur index 1 over Q. Lemma 1.6. Let \(q\equiv 3 (mod 4)\). Then \(Q(\psi_ i)=Q(\sqrt{-p})\), \(i=1,2\), and the \(\psi_ i's\) both have Schur index 1 over Q.
In [J. Algebra 96, 249-274 (1985; Zbl 0576.20026)] the author has shown that if \(q\equiv 1 (mod 4)\), then any \(\lambda\in Irr G\) is real-valued and if \(\lambda\) is faithful, it has Schur index 2 over Q. The author emphasizes that an analogue of theorem 3.2 for \(p\equiv 1 (mod 4)\) and arbitrary n cannot be proved by his methods because in this case \(\psi_ 1\) has Schur index 2 over Q. For \(G=Sp(2,p)\), \(p\equiv -1 (mod 4)\) Theorem 4.1 gives such an analogue.
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Let \(\rho \equiv 3(\text{mod }4)\) and let \(\mathcal S\) be an extension of the symplectic group \(\text{Sp}(2n,\rho)\) by the diagonal automorphism of order 2. \textit{R. Gow} [J. Algebra 122, No. 2, 510-519 (1989; Zbl 0679.20039)] considered integer lattices in the space of the irreducible representations of \(\mathcal S\) over \(\mathbb{Q}\) of degree \((\rho^ n - (-1)^ n)/2\). The author calculates the full automorphism groups \(\mathcal G\) of these lattices. In particular, for \(n>4\) \(\mathcal S\) is shown to have index 3 in \(G\). It has been known for more than five years that the author was preparing a second edition of his well-known text, it entirely fulfilles our expectations; it is a very useful text for introductory courses in probability and statistics. It is written in a very precise style and very agreeable to read. This first volume comprising chapters 1 to 8 is devoted to a detailed acquaintance with probability models which have to be used as the basis for the statistical analysis of data. It is similar in content and organization to the 1979 edition. Some sections have been rewritten and expanded - for example, the discussions of independent random variables and conditional probability. Many new exercises have been added.
Seven main headings are needed to classify the contents of volume 1: a) combinatorial analysis and probability of events that may be determined by the number of points which the events contain; b) probability models on non equi-probable outcomes; c) discrete random variables, distributions; d) moments; e) continuous random variables, distributions; f) limit theorems; g) transformations of random variables.
Students who read usual texts on probability theory might come away with the impression that probability theory is a special branch of measure theory. The main topics of the book focus on the applications of probability theory. Students may well benefit from this approach feeling the physical roots of the probability.
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The authors prove a corona theorem for infinitely many functions with norm control for the space \(\mathbb C+BH^\infty\), where \(B\) is a Blaschke product. This generalizes the corresponding result for \(n\)-tuples in [\textit{R. Mortini, A. Sasane} and \textit{B. D. Wick} [Houston J. Math. 36, No. 1, 289--302 (2010; Zbl 1218.46030)]. Developing a variant of the usual Koszul complex, they give a new representation of the set of all solutions to \(F(z)U(z)^T=1\), where \(F\) and \(U\) are infinite vectors in \(H^\infty_{\ell^2}(\mathbb D)\). In this paper, the corona theorem for the algebra \(H_B^\infty:={\mathbb C}+BH^\infty\) is proved in \(4\) different ways: (a) using a classical Carleson Corona Theorem; (b) using a subalgebra \(A({\mathbb D})_B\) of \(H_B^\infty\) consisting of all elements having a continuous extension to the closed unit disk \(\overline{\mathbb D}\); (c) using estimates for the solution to the Bezout equation; (d) using the Bass stable rank. In the last section, another result is included as a conjecture, which has already been proved by [\textit{J. S. Ryle} and \textit{T. T. Trent}, ``A corona theorem for certain subalgebras of \(H^\infty(D)\)'', Houston J. Math. (to appear)].
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The authors prove a corona theorem for infinitely many functions with norm control for the space \(\mathbb C+BH^\infty\), where \(B\) is a Blaschke product. This generalizes the corresponding result for \(n\)-tuples in [\textit{R. Mortini, A. Sasane} and \textit{B. D. Wick} [Houston J. Math. 36, No. 1, 289--302 (2010; Zbl 1218.46030)]. Developing a variant of the usual Koszul complex, they give a new representation of the set of all solutions to \(F(z)U(z)^T=1\), where \(F\) and \(U\) are infinite vectors in \(H^\infty_{\ell^2}(\mathbb D)\). This paper gives a method of quantifying small visual differences between 3D mesh models with conforming topology, based on the theory of strain fields. Our experiments show that our difference estimates are well correlated with human perception of differences. This work has applications in the evaluation of 3D mesh watermarking, 3D mesh compression reconstruction, and 3D mesh filtering.
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A ring extension \(R \subseteq S\) is said to be maximal non-Manis if \(R\) is not a Manis subring of \(S\) and each proper \(S\)-overring of \(R\) is a Manis subring of \(S\). We study properties of maximal non-Manis extensions. We show that if \(R\subseteq S\) is maximal non-Manis extension and \(R\) is integrally closed in \(S\), then \(R\subseteq S\) is a Prüfer extension. We investigate conditions under which the extension \(R[X]\subseteq S[X]\) (respectively \(R(X) \subseteq S(X))\) is maximal non-Manis. These notes give a careful study of relative Prüfer rings and Manis valuations with an eye towards applications to real and \(p\)-adic algebraic geometry. The main topic is Prüfer ring extensions, where an extension \(A\subset R\) of commutative rings is called Prüfer if \(A\) is \(R\)-Prüfer in the sense of M. Griffin that \((A_{[P]}, P_{[P]})\) is a Manis pair in \(R\) for every maximal ideal \(P\) of \(A\).
This volume has three chapters. The first chapter develops the basic properties of Manis valuations and Prüfer extensions; the second chapter studies Prüfer extensions from a ideal-theoretic rather than a valuation point of view; and the final chapter studies several special types of Manis valuations. As indicated by the title, a second volume is also planned.
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A ring extension \(R \subseteq S\) is said to be maximal non-Manis if \(R\) is not a Manis subring of \(S\) and each proper \(S\)-overring of \(R\) is a Manis subring of \(S\). We study properties of maximal non-Manis extensions. We show that if \(R\subseteq S\) is maximal non-Manis extension and \(R\) is integrally closed in \(S\), then \(R\subseteq S\) is a Prüfer extension. We investigate conditions under which the extension \(R[X]\subseteq S[X]\) (respectively \(R(X) \subseteq S(X))\) is maximal non-Manis. New results are obtained concerning the analysis of the storage allocation algorithm which permits one to maintain stacks inside a shared (continuous) memory area of fixed size \(m\) and of the banker's algorithm (a deadlock avoidance policy). The formulation of these problems is in terms of random walks inside polygonal domains in a \(k\)-dimensional lattice space with several reflecting barriers and one absorbing barrier. Several open problems related to hitting place and hitting time are solved with tools such as diffusion, random walk techniques and Poisson clumping heuristic.
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For dynamical systems of the form \(y'= f(y)\), \(y_ 0= y\) \((t=0)\) with \(y=(y_ 1,\dots,y_ m)^ T\), \(f= (f_ 1,\dots,f_ m)^ T\) the authors construct multistage higher order explicit symplectic difference schemes by using self-adjoint schemes and a composing method studied by the authors earlier [Computing 47, No. 3/4, 309-321 (1992; Zbl 0751.65046)]. They apply those self-adjoint schemes to get high order accuracy in the time direction for wave and heat equation. The authors compare numerical results obtained for several concrete problems with those by Crank-Nicolson scheme. The authors discuss systems of ordinary differential equations (1) \(y'=f(y)\), \(f: \mathbb{R}^ n\to\mathbb{R}^ n\), \(y=y(t)\), \(t\) is the independent variable. A one-step compatible difference scheme approximating (1) can be formally written as (2) \(y_{n+1}=s(\tau)y_ n\) where \(\tau\) is the stepsize and \(s(\tau)\) is called the integrator. Now the authors introduce the concept of adjoint methods by defining: an integrator \(s^*(\tau)\) is called the adjoint integrator of \(s(\tau)\), if (3) \(s^*(-\tau)s(\tau)=I\) and \(s(\tau)s^*(-\tau)=I\). Furthermore: \(s(\tau)\) is called self-adjoint, if \(s^*(\tau)=s(\tau)\) i.e. \(s(- \tau)s(\tau)=I\).
The authors show some properties of this concept, e.g. (i) there is a self-adjoint scheme of even order corresponding to every method, (ii) every self-adjoint integrator has an even order of accuracy, (iii) using self adjoint schemes with lower order, one can construct higher order schemes by ``composing'' and this constructing process can be continued to get arbitrary even order schemes.
The authors show the self-adjointness of some schemes of Runge-Kutta form and the way one can decide this looking at the Butcher-tableau of those forms.
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For dynamical systems of the form \(y'= f(y)\), \(y_ 0= y\) \((t=0)\) with \(y=(y_ 1,\dots,y_ m)^ T\), \(f= (f_ 1,\dots,f_ m)^ T\) the authors construct multistage higher order explicit symplectic difference schemes by using self-adjoint schemes and a composing method studied by the authors earlier [Computing 47, No. 3/4, 309-321 (1992; Zbl 0751.65046)]. They apply those self-adjoint schemes to get high order accuracy in the time direction for wave and heat equation. The authors compare numerical results obtained for several concrete problems with those by Crank-Nicolson scheme. There is a class of the geometrical problems which deals with features of plane figures (triangles or tetragons) built on the sides of another triangle (tetragon). Here a vector transformation method is applied to solve above-mentioned problems. Vector turn operation is considered with respect to linearity, additivity, differentiability and integrability.
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Vector coherent state (VCS) theory uses a \({\mathbb{Z}}\)-gradation of a Lie (super) algebra \({\mathfrak g}=\sum_{i\in {\mathbb{Z}}}{\mathfrak g}_ i\) in order to study irreducible representations of \({\mathfrak g}\). The VCS representation \(\Gamma\) (X) of the Lie (super) algebra elements X corresponds to a differential operator realization; using this and induction of irreducible representations of \({\mathfrak g}_ 0\) to \({\mathfrak g}\), irreducible representations of \({\mathfrak g}\) can be analyzed. In part I [J. Phys. A, Math. Gen. 23, 5383-5409 (1990; Zbl 0715.17003)], this technique was applied to study the (infinite-dimensional) positive discrete series irreducible representations of the noncompact orthosymplectic Lie superalgebra \({\mathfrak osp}(P/2N,{\mathbb{R}})\) in a basis \({\mathfrak osp}(P/2N,{\mathbb{R}})\supset {\mathfrak so}(P)\oplus {\mathfrak sp}(2N,{\mathbb{R}})\supset {\mathfrak so}(P)\oplus {\mathfrak su}(N)\). In the present paper this technique is illustrated in more detail for the Lie superalgebras \({\mathfrak osp}(1/2N,{\mathbb{R}})\), \({\mathfrak osp}(2/2,{\mathbb{R}})\), \({\mathfrak osp}(3/2,{\mathbb{R}})\), \({\mathfrak osp}(4/2,{\mathbb{R}})\) and \({\mathfrak osp}(2/4,{\mathbb{R}})\), as well as for certain representations of \({\mathfrak osp}(2/2N,{\mathbb{R}})\). Vector Coherent State (VCS) theory uses a \({\mathbb{Z}}\)-gradation of a Lie (super) algebra \({\mathfrak g}=\sum_{i\in {\mathbb{Z}}}{\mathfrak g}_ i\) in order to study irreducible representations of \({\mathfrak g}\). The VCS representation \(\Gamma\) (X) of the Lie (super)algebra elements X corresponds to a differential operator realisation; using this and induction of irreducible representations of \({\mathfrak g}_ 0\) to \({\mathfrak g}\), irreducible representations of \({\mathfrak g}\) can be analysed.
In this paper, this technique is applied to study the (infinite- dimensional) positive discrete series irreducible representations of the noncompact orthosymplectic Lie superalgebra osp(P/2N,\({\mathbb{R}})\) in a basis osp(P/2N,\({\mathbb{R}})\supset so(P)\otimes sp(2N,{\mathbb{R}})\supset so(P)\oplus su(N)\). The conditions for star representations are investigated, and the branching rule according to this chain of subalgebras is described. The VCS theory is shown to provide an easy way of calculating matrix elements for the generators of \({\mathfrak g}\). Finally, the case \(P=2\) is analysed in more detail.
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Vector coherent state (VCS) theory uses a \({\mathbb{Z}}\)-gradation of a Lie (super) algebra \({\mathfrak g}=\sum_{i\in {\mathbb{Z}}}{\mathfrak g}_ i\) in order to study irreducible representations of \({\mathfrak g}\). The VCS representation \(\Gamma\) (X) of the Lie (super) algebra elements X corresponds to a differential operator realization; using this and induction of irreducible representations of \({\mathfrak g}_ 0\) to \({\mathfrak g}\), irreducible representations of \({\mathfrak g}\) can be analyzed. In part I [J. Phys. A, Math. Gen. 23, 5383-5409 (1990; Zbl 0715.17003)], this technique was applied to study the (infinite-dimensional) positive discrete series irreducible representations of the noncompact orthosymplectic Lie superalgebra \({\mathfrak osp}(P/2N,{\mathbb{R}})\) in a basis \({\mathfrak osp}(P/2N,{\mathbb{R}})\supset {\mathfrak so}(P)\oplus {\mathfrak sp}(2N,{\mathbb{R}})\supset {\mathfrak so}(P)\oplus {\mathfrak su}(N)\). In the present paper this technique is illustrated in more detail for the Lie superalgebras \({\mathfrak osp}(1/2N,{\mathbb{R}})\), \({\mathfrak osp}(2/2,{\mathbb{R}})\), \({\mathfrak osp}(3/2,{\mathbb{R}})\), \({\mathfrak osp}(4/2,{\mathbb{R}})\) and \({\mathfrak osp}(2/4,{\mathbb{R}})\), as well as for certain representations of \({\mathfrak osp}(2/2N,{\mathbb{R}})\). In this paper, we propose LZ\_MSA, a novel method for progressive multiple sequence alignment based on Lempel-Ziv. The vector space is constructed by 10 types of copy modes. Under this approach, sequence alignment is converted into vector alignment and the guide tree can be dynamically amended. Finally we use five subsets in the standard dataset of BAliBASE to validate the proposed algorithm. Compared to ClusatalW, MAFFT, LZ\_MSA reduces the alignment time without sacrificing accuracy.
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Following a method developed by \textit{M. Haase} [J. Aust. Math. Soc. 84, No. 1, 73--83 (2008; Zbl 1149.47031)], the authors establish sufficient conditions to guarantee the validity of the complex inversion formula in UMD Banach spaces for the wide class of \((a, k)\)- regularized families. Some preliminaries on vector-valued Fourier transform are given. The validity of the complex inversion formula for \((a, k)\)-regularized families, under certain conditions on the scalar kernels \(a(t)\) and \(k(t)\) is proved. The result of Haase for resolvent families follows as a special case for \(k(t) = 1\). In the special case \(a(t) = 1\), they provide a wide class of kernels \(k(t)\) such that the complex inversion formula holds. In this interesting paper, the author uses (elementary) Fourier analysis to give a simplified proof of the integral form of the complex inversion formula for \(C_0\)-semigroups of linear operators on a Banach space \(X\). If, moreover, \(X\) is an UMD (i.e., unconditional martingale differences) space (in particular, a Hilbert space or an \(L^p\)-space), he also proves the strong convergence of the complex inversion formula. The results apply, more generally, to solution families for scalar type Volterra equations.
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Following a method developed by \textit{M. Haase} [J. Aust. Math. Soc. 84, No. 1, 73--83 (2008; Zbl 1149.47031)], the authors establish sufficient conditions to guarantee the validity of the complex inversion formula in UMD Banach spaces for the wide class of \((a, k)\)- regularized families. Some preliminaries on vector-valued Fourier transform are given. The validity of the complex inversion formula for \((a, k)\)-regularized families, under certain conditions on the scalar kernels \(a(t)\) and \(k(t)\) is proved. The result of Haase for resolvent families follows as a special case for \(k(t) = 1\). In the special case \(a(t) = 1\), they provide a wide class of kernels \(k(t)\) such that the complex inversion formula holds. The authors use Lévy random fields to model the term structure of forward default intensity. This allows them to describe the contagion risk. They consider the pricing of the defaultable bonds. The main result is to prove the pricing kernels as the unique solution of a parabolic integro-differential equation by constructing a suitable contracting operator and then considering the limit case for an unbounded terminal condition. The impact of contagious jump risks on the defaultable bond price is demonstrated by numerical examples.
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Let \(H\) be a separable Hilbert space; \(T,T',A\in L(H)\) are contractions such that \(AT=T'A\). The commutant lifting theorem of Sz.-Nagy and Foias states that there exists a contractive lifting \(B\) of \(A\) which intertwines the isometric dilations of \(T\) and \(T'\). Such a \(B\) is called an intertwining lifting of the triple \(\{T, T', A\}\): the collection of all intertwining liftings can be parametrized by a certain type of sequences of contractions, called ``choice sequences''. If the choice sequence is identically null, the corresponding lifting is called central, or maximum entropy, and has been the object of extensive study.
The present paper deals with an opposite case called minimum entropy by \textit{C. Foias, A. Frazho} and \textit{A. Tannenbaum} [Linear Algebra Appl. 137/138, 213-238 (1990; Zbl 0709.47001)] corresponding to sequences with one isometry at the beginning. The main theorem shows that if \(T\in C_{\cdot 0}\) and \(I-A^*A\) has closed range, then any minimum entropy lifting is isometric. The result generalizes [loc. cit.], where it was proved under the supplementary assumption that \(I-TT^*\) has finite rank. Let T and \(T'\) be contractions on Hilbert spaces H and \(H'\), respectively and let U and \(U'\) be the minimal unitary dilations of T and \(T'\) to K and \(K'\), respectively. Recall that \(K=(...\oplus D_{T^*}\oplus D_{T^*})\oplus H\oplus (D_ T\oplus D_ T\oplus...)\) and similarly for \(K'\). Set
\[
T_ 1=\left[ \begin{matrix} T\\ (I-T^*T)^{1/2}\end{matrix} \begin{matrix} 0\\ 0\end{matrix} \right],\quad H_ 1=H\oplus D_ T\text{ and } K_+=H\oplus (D_ T\oplus D_ T\oplus...)
\]
and similarly for \(T'_ 1\), \(H'_ 1\) and \(K'_+\). Denote by P and \(P'\) the orthogonal projections from \(H_ 1\) and \(H'_ 1\) onto H and \(H'\), respectively. Let A be a contraction that satisfies \(AT=T'A\) and let \(A_ 1\) be a contraction that satisfies \(A_ 1T_ 1=T'_ 1A_ 1\) and \(P'A_ 1=AP\). Any contractive intertwining dilation \(B_+: K_+\to K'_+\) of \(A_ 1\) has a contractive extension B: \(K\to K'\) satisfying \(BU-U'B\). The main result of this paper is to prove that, under some further assumptions, B is a coisometry or an isometry. The uniqueness of the extension B is also considered. An application of this result to the classical Carathéodory extension problem is given.
| 1 |
Let \(H\) be a separable Hilbert space; \(T,T',A\in L(H)\) are contractions such that \(AT=T'A\). The commutant lifting theorem of Sz.-Nagy and Foias states that there exists a contractive lifting \(B\) of \(A\) which intertwines the isometric dilations of \(T\) and \(T'\). Such a \(B\) is called an intertwining lifting of the triple \(\{T, T', A\}\): the collection of all intertwining liftings can be parametrized by a certain type of sequences of contractions, called ``choice sequences''. If the choice sequence is identically null, the corresponding lifting is called central, or maximum entropy, and has been the object of extensive study.
The present paper deals with an opposite case called minimum entropy by \textit{C. Foias, A. Frazho} and \textit{A. Tannenbaum} [Linear Algebra Appl. 137/138, 213-238 (1990; Zbl 0709.47001)] corresponding to sequences with one isometry at the beginning. The main theorem shows that if \(T\in C_{\cdot 0}\) and \(I-A^*A\) has closed range, then any minimum entropy lifting is isometric. The result generalizes [loc. cit.], where it was proved under the supplementary assumption that \(I-TT^*\) has finite rank. The authors present a new method for approximating the eignevalues and the subregions which support such localized functions. The present method is based on the recent theoretical results of the localized landscape and effective potentials. This is a deterministic approach and enables to evaluate the locations and shapes of the approximate eigenvalues, their support or the density of states at the costs of solving numerically one elliptic equation.
Extensive numerical experiments confirm the robustness and efficiency of the proposed method. A variety of piecewise constant potentials with values sampled from correlated and uncorrelated random distributions has been tested. Consequently, the authors demonstrated numerically that the effective potential, defined as a reciprocal of the localization landscape function, accurately captures an important portion of information on the localization properties of a random potential.
| 0 |
The authors adapt the classical notion of Lusternik-Schnirelmann-category to (sectioned) spaces over a space \(B\). Given maps \(p_ i: X_ i\to B\), \(i=1,2\), a fibrewise map \(f: X_ 1\to X_ 2\) is fibrewise constant if there exists a section \(s: B\to X_ 2\) such that \(f=sp_ 1\). A fibrewise homotopy into a fibrewise constant map is called a nullhomotopy. Call a subspace \(U\subset X\) fibrewise categorical if \(U\to X\) is fibrewise nullhomotopic, and define \(\hbox{cat}_ B(X)\) as the minimal number of fibrewise categorical open sets needed to cover \(X\). (The corresponding notion is defined in the case of sectioned spaces). Then the classical results are generalized to this setting, some examples are discussed. It would be interesting to know how this theory fits into the general framework of LS-category in model categories studied by \textit{J.-P. Doeraene} [J. Pure Appl. Algebra 84, No. 3, 215--261 (1993; Zbl 0777.55007)]. The notion of category was introduced by \textit{L. Lusternik} and \textit{I. Schnirelmann} [Méthodes topologiques dans les problèmes variationnels (1934; Zbl 0011.02803)] to study the stationary points of functions on a manifold. For a path-connected, normal and categorically well based space, \(X\), the LS-category can be defined by two equivalent ways:
(1) as the least integer, \(n\), such that there exists a (homotopy) factorization of the diagonal, \(\Delta: X\to X^{n+1}\), through the fat wedge \(T^ n(X)\) of \(X\), [\textit{G. W. Whitehead}, Centre Belge Rech. math., Colloque de Topologie algébrique, Louvain les 11, 12 et 13 juin 1956, 89-95 (1957; Zbl 0079.391)],
(2) or as the least integer, \(n\), such that Ganea's fibration, \(G_ n X\to X\), admits a section up to homotopy [\textit{T. Ganea}, Illinois J. Math. 11, 417-427 (1967; Zbl 0149.407)].
Fat wedge and Ganea's fibrations can be realized in Quillen's closed model category but, in general, the previous definitions give two distinct integers as that can be noted for the dual notion of cocategory [Ganea, loc. cit.]. In this paper the author introduced the notion of \(J\)-category, as a closed model category enriched with a cube axiom (which is not autodual!). The author proves that the Whitehead and Ganea definitions coincide in a \(J\)-category, \(C\), and calls \(C\)-cat this common notion. He studies the basic properties of \(C\)-cat, proves an intrinsic version of the mapping theorem of \textit{Y. Félix} and \textit{S. Halperin} [Trans. Am. Math. Soc. 273, 1-37 (1982; Zbl 0508.55004)] and gives examples of \(J\)-categories. Finally, if \(F: C\to D\) is an ``appropriate'' functor between \(J\)-categories, the author shows that \(C\)-\(\text{cat}(X)\), \(X\in C\), can be approximated by \(D\)- \(\text{cat}(F(X))\).
| 1 |
The authors adapt the classical notion of Lusternik-Schnirelmann-category to (sectioned) spaces over a space \(B\). Given maps \(p_ i: X_ i\to B\), \(i=1,2\), a fibrewise map \(f: X_ 1\to X_ 2\) is fibrewise constant if there exists a section \(s: B\to X_ 2\) such that \(f=sp_ 1\). A fibrewise homotopy into a fibrewise constant map is called a nullhomotopy. Call a subspace \(U\subset X\) fibrewise categorical if \(U\to X\) is fibrewise nullhomotopic, and define \(\hbox{cat}_ B(X)\) as the minimal number of fibrewise categorical open sets needed to cover \(X\). (The corresponding notion is defined in the case of sectioned spaces). Then the classical results are generalized to this setting, some examples are discussed. It would be interesting to know how this theory fits into the general framework of LS-category in model categories studied by \textit{J.-P. Doeraene} [J. Pure Appl. Algebra 84, No. 3, 215--261 (1993; Zbl 0777.55007)]. In an attempt to better understand structural benefits and generalization power of deep neural networks, we first present a novel graph theoretical formulation of neural network models, including fully connected, residual network (ResNet) and densely connected networks (DenseNet). Second, we extend the error analysis of the population risk for a two-layer network [\textit{E. Weinan} et al., Commun. Math. Sci. 17, No. 5, 1407--1425 (2019; Zbl 1427.68277)] and ResNet [\textit{W. E} et al., Commun. Math. Sci. 18, No. 6, 1755--1774 (2020; Zbl 1467.62158)] to DenseNet, and show further that for neural networks satisfying certain mild conditions, similar estimates can be obtained. These estimates are a priori in nature since they depend solely on the information prior to the training process, in particular, the bounds for the estimation errors do not suffer from the curse of dimensionality.
| 0 |
Let \({\mathcal {B(H)}}\) be the algebra of all bounded linear operators on a complex Hilbert space \({\mathcal H}\). Given a \(C^*\)-subalgebra \({\mathcal A} \subset {\mathcal {B(H)}}\), denote by \({\mathcal A}^{pr}\) the set of all projections in \({\mathcal A}\). For a unital \(C^*\)-algebra \({\mathcal A}\), let
\[
{\mathcal L}_1 = {\mathcal A}^{pr}, \dots, {\mathcal L}_{n+1} = \{ap+bp^{\perp}: a, b \in {\mathcal L}_n,\;p \in {\mathcal A}^{pr} \}, \dots\,.
\]
It is clear that \({\mathcal L}_1 \subset {\mathcal L}_2 \subset\dots\,\). Set \({\mathcal L} = \bigcup_{n=1}^{\infty} {\mathcal L}_n\). In [\textit{A.\,M.\thinspace Bikchentaev} and \textit{A.\,N.\thinspace Sherstnev}, Math.\ Notes 76, No.\,4, 578--581 (2004; Zbl 1080.46037)], it was proved that if a von Neumann algebra \({\mathcal A}\) doesn't have a direct summand of finite type I, then \({\mathcal L} ={\mathcal A}\).
The main result of the present paper is the following Theorem. If \({\mathcal A} = {\mathcal {B(H)}}\), \(\dim({\mathcal H}) < \infty\), then \({\mathcal A} = {\mathcal L}\). In particular, one obtains the following Corollary. If \({\mathcal A}\) is a von~Neumann algebra of finite type~I without Abelian direct summands, then \({\mathcal A} ={\mathcal L}\). A unital \(C^*\)-algebra \(\mathcal{A}\) is said to have the unitary factorization property (written as \(\mathcal A \in(\text{UF})\)) if each unitary \(u\) is a finite product of symmetries (self-adjoint unitaries). For example, each unital purely infinite simple \(C^*\)-algebra \(A\) belongs to \((\text{UF})\). A von Neumann algebra \(\mathcal{M}\) belongs to \((\text{UF})\) if and only if \(\mathcal{M}\) has no direct summands of finite type \(\text{I}\).
For a unital \(C^*\)-algebra \(\mathcal{A}\), the authors define the following increasing sequence
\[
{\mathcal L}_1={\mathcal A}^{\text{pr}},\;\dots, {\mathcal L}_{n+1} =\{ap+bp^\bot:a,b \in {\mathcal L}_n, p \in {\mathcal A}^{\text{pr}}\},\;\dots,
\]
where \({\mathcal A}^{\text{pr}}\) is the set of all projections from \(\mathcal A\), and put \({\mathcal L} = \bigcup_{n=1}^\infty {\mathcal L}_n\).
The main result is the following Theorem: If \(\mathcal A\in(\text{UF})\), then \(\mathcal A = \mathcal L\).
As a corollary, they obtain that any operator from the \(C^*\)-algebra \(\mathcal A \in (\text{UF})\) can be represented as a finite sum of finite products of its projections.
| 1 |
Let \({\mathcal {B(H)}}\) be the algebra of all bounded linear operators on a complex Hilbert space \({\mathcal H}\). Given a \(C^*\)-subalgebra \({\mathcal A} \subset {\mathcal {B(H)}}\), denote by \({\mathcal A}^{pr}\) the set of all projections in \({\mathcal A}\). For a unital \(C^*\)-algebra \({\mathcal A}\), let
\[
{\mathcal L}_1 = {\mathcal A}^{pr}, \dots, {\mathcal L}_{n+1} = \{ap+bp^{\perp}: a, b \in {\mathcal L}_n,\;p \in {\mathcal A}^{pr} \}, \dots\,.
\]
It is clear that \({\mathcal L}_1 \subset {\mathcal L}_2 \subset\dots\,\). Set \({\mathcal L} = \bigcup_{n=1}^{\infty} {\mathcal L}_n\). In [\textit{A.\,M.\thinspace Bikchentaev} and \textit{A.\,N.\thinspace Sherstnev}, Math.\ Notes 76, No.\,4, 578--581 (2004; Zbl 1080.46037)], it was proved that if a von Neumann algebra \({\mathcal A}\) doesn't have a direct summand of finite type I, then \({\mathcal L} ={\mathcal A}\).
The main result of the present paper is the following Theorem. If \({\mathcal A} = {\mathcal {B(H)}}\), \(\dim({\mathcal H}) < \infty\), then \({\mathcal A} = {\mathcal L}\). In particular, one obtains the following Corollary. If \({\mathcal A}\) is a von~Neumann algebra of finite type~I without Abelian direct summands, then \({\mathcal A} ={\mathcal L}\). The SOSEMANUK stream cipher is one of the finalists of the eSTREAM project. In this paper, we improve the linear cryptanalysis of SOSEMANUK presented in Asiacrypt 2008. We apply the generalized linear masking technique to SOSEMANUK and derive many linear approximations holding with the correlations of up to \(2^{ - 25.5}\). We show that the data complexity of the linear attack on SOSEMANUK can be reduced by a factor of \(2^{10}\) if multiple linear approximations are used. Since SOSEMANUK claims 128-bit security, our attack would not be a real threat on the security of SOSEMANUK.
| 0 |
If M is a subspace of a Banach space X, let D denote the set of points in X which admit nearest points in M and, if \(x\in D\), let \(P_ M(x)\) be the nonempty set of nearest points. M has property * in X if for all x in D and m in M \(\| x-m\| =d(m,P_ M(x))+d(x,M)\); (d for distance). M has the 1\(\tfrac12\) ball property if whenever \(r_ 1,r_ 2\geq 0\), \(x\in X\), \(m\in M\), \(\| x-m\| <r_ 1+r_ 2\) and \(M\cap B(x,r_ 2)\neq \emptyset\) then, also, \(M\cap B(x,r_ 2)\cap B(m,r_ 1)\neq \emptyset\). The author answers a question of \textit{G. Godini} [Lect. Notes Math. 991, 44-54 (1983; Zbl 0535.46007)] by exhibiting a Banach space X and a subspace M such that \(D\neq M\), while M has property * but does not have the 1\(\tfrac12\) ball property. [For the entire collection see Zbl 0504.00015.]
Author's abstract: For a linear subspace G of the normed linear space E and \(x\in E\), let \(P_ G(x)\) be the set of all best approximations of x out of G. Observing that for each x,\(y\in E\) we always have \(dist(y,P_ G(x))\geq \| x-y\| -dist(x,G),\) we study the subspaces G with the property - which we call property (*) - that this inequality is an equality for each \(x\in E\) with \(P_ G(x)\neq \emptyset\) and each \(g\in G\). This property generalizes the notion of semi L-summand studied by \textit{A. Lima} [Trans. Am. Math. Soc. 227, 1-62 (1977; Zbl 0347.46017)]. For a subspace G with property (*), the one-sided Gateaux differential of the norm at \(x\in E\) with \(0\in P_ G(x)\), in the direction \(g\in G\) equals the distance of -g to the cone spanned by \(P_ G(x)\). Using this result, we obtain a characterization of those \(x\in E\) with \(0\in P_ G(x)\) in order that the cone spanned by \(P_ G(x)\) to be norm-dense in G. When G is proximinal, property (*) is equivalent with 1\(frac{1}{2}\)- ball property studied by \textit{D. Yost} [Bull. Aust. Math. Soc. 20, 297- 312 (1979; Zbl 0407.41015)]. We give geometrical characterizations of the subspaces with property (*), as well as with 1\(frac{1}{2}\)-ball property.
| 1 |
If M is a subspace of a Banach space X, let D denote the set of points in X which admit nearest points in M and, if \(x\in D\), let \(P_ M(x)\) be the nonempty set of nearest points. M has property * in X if for all x in D and m in M \(\| x-m\| =d(m,P_ M(x))+d(x,M)\); (d for distance). M has the 1\(\tfrac12\) ball property if whenever \(r_ 1,r_ 2\geq 0\), \(x\in X\), \(m\in M\), \(\| x-m\| <r_ 1+r_ 2\) and \(M\cap B(x,r_ 2)\neq \emptyset\) then, also, \(M\cap B(x,r_ 2)\cap B(m,r_ 1)\neq \emptyset\). The author answers a question of \textit{G. Godini} [Lect. Notes Math. 991, 44-54 (1983; Zbl 0535.46007)] by exhibiting a Banach space X and a subspace M such that \(D\neq M\), while M has property * but does not have the 1\(\tfrac12\) ball property. Given topological spaces \(X\), \(Y\), a fundamental problem of algebraic topology is understanding the structure of all continuous maps \(X \to Y\). We consider a computational version, where \(X,Y\) are given as finite simplicial complexes, and the goal is to compute \([X,Y]\), that is, all homotopy classes of such maps. We solve this problem in the stable range, where for some \(d \geq 2\), we have dim \(X \leq 2d-2\) and \(Y\) is \((d-1)\)-connected; in particular, \(Y\) can be the \(d\)-dimensional sphere \(S^{d}\). The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, \([X,Y]\) is known to be uncomputable for general \(X,Y\), since for \(X=S^{1}\) it includes a well known undecidable problem: testing triviality of the fundamental group of \(Y\).
In follow-up papers, the algorithm is shown to run in polynomial time for \(d\) fixed, and extended to other problems, such as the extension problem, where we are given a subspace \(A \subset X\) and a map \(A \to Y\) and ask whether it extends to a map \(X \to Y\), or computing the \(\mathbb{Z}_{2}\)-index -- everything in the stable range. Outside the stable range, the extension problem is undecidable.
| 0 |
Various models have been proposed for \((\infty,1)\)-categories, namely, \textit{simplicial categories}, \textit{quasi-categories}, \textit{relative categories}, \textit{Segal categories}, \textit{complete Segal spaces} and \(1\)\textit{-complicial sets}. In parallel with the development of models of \((\infty,1)\)-categories and the construction of comparisons between them, AndréJoyal pioneered and Jacob Lurie as well as many others expanded a wildly successful project to extend basic category theory from ordinary \(1\)-categories to \((\infty ,1)\)-categories modeled as quasi-categories in such a way that the new quasi-categorical notions restrict along the standard embedding
\[
\mathcal{C}at\rightarrow\mathcal{QC}at
\]
to the classical \(1\)-categorical concepts. For practical, aesthetic and moral reasons, the ultimate goal of practitioners is to work model independently in the sense that theorems with any of the models of \((\infty,1)\)-categories would apply to them at all, with the technical details inherent to any particular model never entering the discussion.
This book develops the theory of \(\infty\)-categories from first principles in a model independent fashion using a common axiomatic framework that is satisfied by a variety of models. In contrast with prior \textit{analytic} treatment of the theory of \(\infty\)-categories, where the central categorical notions are defined in reference to the coordinates of a particular model, this book adopts a \textit{synthetic} approach proceeding from definitions that can be interpreted simultaneously in many models to which the proofs in the book then apply. While synthetic, this book is not schematic or hand-wavy with the details of how to make things fully precise left to the experts and turtles all the way down, but rather establishes theorems beginning with a short list of clearly enumerated axioms so that the conclusions are valid in any model of \(\infty\)-categories hewing to these axioms.
A less rigorous model-independent presentation of \(\infty\)-category theory might confront a problem of infinite regress, since infinite-dimensional categories are themselves the objects of an ambient infinite-dimensional category so that in developing the theory of the former one is tempted to use the theory of the latter. This book avoids the problem by using a very concise model for the ambient \((\infty,2)\)-category of \(\infty\)-categories arising frequently in practice and is designed to facilitate relatively simple proofs. While the theory of \((\infty,2)\)-categories remains in its infancy, the authors are content to cut the Gordian knot in this way.
A synopsis of the book goes as follows.
Part I Basic \(\infty\)-Category Theory
This part defines and develops the notions of equivalence and adjunction between \(\infty\)-categories, limits and colimits in \(\infty\)-categories, and cartesian and cocartesian fibrations and their discrete variants, for which a version of the Yoneda lemma is established. The majority of these results are developed from the comfort of the homotopy \(\infty\)-category.
Chapter 1. \(\infty\)-Cosmos and Their Homotopy \(2\)-Categories
This chapter introduces a framework to develop the formal category theory of \(\infty\)-categories, which goes by the name of an \(\infty\)\textit{-cosmos}.
Chapter 2. Adjunctions, Limits, and Colimits I
This chapter uses \(2\)-categorical techniques to define \textit{adjunction} between \(\infty\)-categories and \textit{limits} and \textit{colimits} of diagrams valued in an \(\infty\)-category, establishing that these notions interact in the expected way.
Chapter 3. Comma \(\infty\)-Categories
This chapter makes use of the axiomatized limits in an \(\infty\)-cosmos to exhibit a general construction that specializes to define both this \(\infty\)-category of cones as well as hom-spces, permitting of representing a functor between \(\infty\)-categories as an \(\infty\)-category, in dual ``left'' or ``right'' fashions, so that an adjunction consists of a pair of functors \(f:B\rightarrow A\)\ and \(u:A\rightarrow B\)\ so that the left representation of \(f\)\ is equivalent to the right representation of \(u\)\ over \(A\times B\) (Proposition 4.1.1).
Chapter 4. Adjunctions, Limits, and Colimits II
This chapter uses the comma \(\infty\)-categories of the previous chapter as a vehicle to give precise expressions to these universal properties, establishing that these new characterizations are equivalent to the previous definitions. The main results in this chapter are mere special cases of the general theorems characterizing representable comma \(\infty\)-categories.
Chapter 5. Fibrations and Yoneda's Lemma
This chapter aims to describe an \(\infty\)-categorical encoding of the contravariant functor represented by an element \(b:1\rightarrow B\)\ of an \(\infty\)-category \(B\)\ informally defined to send an element \(x\)\ of \(B\)\ to the mapping space \(\mathrm{Hom}_{B}(x,b)\)\ (Definition 3.4.9).
An Interlude on \(\infty\)-Cosmology
In this interlude, the authors digress into abstract \(\infty\)-cosmology to give a more careful account of the full class of limit construction present in any \(\infty\)-cosmos. This analysis is used to develop further examples of \(\infty\)-cosmoi, whose objects are pointed \(\infty\)-categories, or stable \(\infty\)-categories, or cartesian or cocartesian fibrations in a given \(\infty\)-cosmos.
Chapter 6. Exotic \(\infty\)-Cosmoi
Part II The Calculus of Modules
This part develops the calculus of modules between \(\infty\)-categories, which is applied to define and study pointwise Kan extensions.
Chapter 7. Two-Sided Fibrations and Modules
This chapter considers the codomain domain projection functors as a span
\[
A\overset{p_{1}}{\leftarrow}A^{2}\overset{p_{0}}{\rightarrow}A
\]
defining a \textit{two-sided fibration} from \(A\)\ to \(A\)\ and a \textit{discrete two-sided fibration} or a \textit{module} from \(A\)\ to \(A\).
Chapter 8. The Calculus of Modules
This chapter aims to establish Theorem 8.2.6 claiming that \(\infty\)-categories, functors, modules and module maps in any \(\infty\)-cosmos define a \textit{virtual equipment} in the sense of [\textit{G. S. H. Cruttwell} and \textit{M. A. Shulman}, Theory Appl. Categ. 24, 580--655 (2010; Zbl 1220.18003)].
Chapter 9. Formal \(\infty\)-Category Theory in a Virtual Equipment
This chapter deploys the calculus of the previous chapter to further develop the formal category theory of \(\infty\)-categories, specifically by defining and developing pointwise right and left Kan extensions, which are notably missing in Part I.
Part III Model Independence
This part establishes that the category theory of \(\infty\)-categories is also model independent in the precise sense that all categorical notions are preserved, reflected, and created by any change-of-model functor defining what is called a \textit{cosmological biequivalence}.
Chapter 10. Change-of-Model Functors
This chapter studies a certain class of cosmological functors between \(\infty\)-cosmoi that do not merely preserve \(\infty\)-categorical structure but also reflect and create it. These functors are referred to as \textit{cosmological biequivalences}, since the \(2\)-functors they induce between homotopy \(2\)-categories are equivalences.
Chapter 11. Model Independence
This chapter aims to establish that the study of \(\infty\)-categories is invariant under change of model, claiming a stronger result that statements about \(\infty\)-categories proven by analytic techniques indigenous to a single \(\infty\)-cosmos are independent from models, providing that the statements are expressible in a suitable equivalence invariant language.
Chapter 12. Applications of Model Independence
This chapter establishes some special properties of a certain class of \(\infty\)-cosmoi, called \(\infty\)-cosmoi of \((\infty,1)\)-categories, which mean \(\infty\)-cosmoi that are biequivalent to the \(\infty\)-cosmos of quasi-categories. The chapter aims also to illustrate how the model independence theorem can be used, by combining analytic and synthetic techniques, to establish results concerning any family of biequivalent \(\infty\)-cosmoi.
Appendix of Abstract Nonsense
Here the authors review all of the material one needs on enriched category theory, \(2\)-category theory and abstract homotopy theory in Appendices A, B and C, respectively.
Appendix A Basic Concepts of Enriched Category Theory
Appendix B An Introduction to \(2\)-Category Theory
Appendix C Abstract Homotopy Theory
Appendix of Concrete Constructions
Appendix D The Combinatorics of (Marked) Simplicial Sets
This appendix explores the combinatorics of simplicial sets, proving results stated in Chapters 1 and 4.
Appendix E \(\infty\)-Cosmoi Found in Nature
This appendix establishes concrete examples of \(\infty\)-cosmoi found in nature.
Appendix F The Analytic Theory of Quasi-Categories
This appendix aims to establish that the synthetic theory of quasi-categories is compatible with the analytic theory pioneered by André Joyal, Jacob Lurie, and many other. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the ``lax algebras'' or ``Kleisli monoids'' relative to a ``monad'' on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.
| 1 |
Various models have been proposed for \((\infty,1)\)-categories, namely, \textit{simplicial categories}, \textit{quasi-categories}, \textit{relative categories}, \textit{Segal categories}, \textit{complete Segal spaces} and \(1\)\textit{-complicial sets}. In parallel with the development of models of \((\infty,1)\)-categories and the construction of comparisons between them, AndréJoyal pioneered and Jacob Lurie as well as many others expanded a wildly successful project to extend basic category theory from ordinary \(1\)-categories to \((\infty ,1)\)-categories modeled as quasi-categories in such a way that the new quasi-categorical notions restrict along the standard embedding
\[
\mathcal{C}at\rightarrow\mathcal{QC}at
\]
to the classical \(1\)-categorical concepts. For practical, aesthetic and moral reasons, the ultimate goal of practitioners is to work model independently in the sense that theorems with any of the models of \((\infty,1)\)-categories would apply to them at all, with the technical details inherent to any particular model never entering the discussion.
This book develops the theory of \(\infty\)-categories from first principles in a model independent fashion using a common axiomatic framework that is satisfied by a variety of models. In contrast with prior \textit{analytic} treatment of the theory of \(\infty\)-categories, where the central categorical notions are defined in reference to the coordinates of a particular model, this book adopts a \textit{synthetic} approach proceeding from definitions that can be interpreted simultaneously in many models to which the proofs in the book then apply. While synthetic, this book is not schematic or hand-wavy with the details of how to make things fully precise left to the experts and turtles all the way down, but rather establishes theorems beginning with a short list of clearly enumerated axioms so that the conclusions are valid in any model of \(\infty\)-categories hewing to these axioms.
A less rigorous model-independent presentation of \(\infty\)-category theory might confront a problem of infinite regress, since infinite-dimensional categories are themselves the objects of an ambient infinite-dimensional category so that in developing the theory of the former one is tempted to use the theory of the latter. This book avoids the problem by using a very concise model for the ambient \((\infty,2)\)-category of \(\infty\)-categories arising frequently in practice and is designed to facilitate relatively simple proofs. While the theory of \((\infty,2)\)-categories remains in its infancy, the authors are content to cut the Gordian knot in this way.
A synopsis of the book goes as follows.
Part I Basic \(\infty\)-Category Theory
This part defines and develops the notions of equivalence and adjunction between \(\infty\)-categories, limits and colimits in \(\infty\)-categories, and cartesian and cocartesian fibrations and their discrete variants, for which a version of the Yoneda lemma is established. The majority of these results are developed from the comfort of the homotopy \(\infty\)-category.
Chapter 1. \(\infty\)-Cosmos and Their Homotopy \(2\)-Categories
This chapter introduces a framework to develop the formal category theory of \(\infty\)-categories, which goes by the name of an \(\infty\)\textit{-cosmos}.
Chapter 2. Adjunctions, Limits, and Colimits I
This chapter uses \(2\)-categorical techniques to define \textit{adjunction} between \(\infty\)-categories and \textit{limits} and \textit{colimits} of diagrams valued in an \(\infty\)-category, establishing that these notions interact in the expected way.
Chapter 3. Comma \(\infty\)-Categories
This chapter makes use of the axiomatized limits in an \(\infty\)-cosmos to exhibit a general construction that specializes to define both this \(\infty\)-category of cones as well as hom-spces, permitting of representing a functor between \(\infty\)-categories as an \(\infty\)-category, in dual ``left'' or ``right'' fashions, so that an adjunction consists of a pair of functors \(f:B\rightarrow A\)\ and \(u:A\rightarrow B\)\ so that the left representation of \(f\)\ is equivalent to the right representation of \(u\)\ over \(A\times B\) (Proposition 4.1.1).
Chapter 4. Adjunctions, Limits, and Colimits II
This chapter uses the comma \(\infty\)-categories of the previous chapter as a vehicle to give precise expressions to these universal properties, establishing that these new characterizations are equivalent to the previous definitions. The main results in this chapter are mere special cases of the general theorems characterizing representable comma \(\infty\)-categories.
Chapter 5. Fibrations and Yoneda's Lemma
This chapter aims to describe an \(\infty\)-categorical encoding of the contravariant functor represented by an element \(b:1\rightarrow B\)\ of an \(\infty\)-category \(B\)\ informally defined to send an element \(x\)\ of \(B\)\ to the mapping space \(\mathrm{Hom}_{B}(x,b)\)\ (Definition 3.4.9).
An Interlude on \(\infty\)-Cosmology
In this interlude, the authors digress into abstract \(\infty\)-cosmology to give a more careful account of the full class of limit construction present in any \(\infty\)-cosmos. This analysis is used to develop further examples of \(\infty\)-cosmoi, whose objects are pointed \(\infty\)-categories, or stable \(\infty\)-categories, or cartesian or cocartesian fibrations in a given \(\infty\)-cosmos.
Chapter 6. Exotic \(\infty\)-Cosmoi
Part II The Calculus of Modules
This part develops the calculus of modules between \(\infty\)-categories, which is applied to define and study pointwise Kan extensions.
Chapter 7. Two-Sided Fibrations and Modules
This chapter considers the codomain domain projection functors as a span
\[
A\overset{p_{1}}{\leftarrow}A^{2}\overset{p_{0}}{\rightarrow}A
\]
defining a \textit{two-sided fibration} from \(A\)\ to \(A\)\ and a \textit{discrete two-sided fibration} or a \textit{module} from \(A\)\ to \(A\).
Chapter 8. The Calculus of Modules
This chapter aims to establish Theorem 8.2.6 claiming that \(\infty\)-categories, functors, modules and module maps in any \(\infty\)-cosmos define a \textit{virtual equipment} in the sense of [\textit{G. S. H. Cruttwell} and \textit{M. A. Shulman}, Theory Appl. Categ. 24, 580--655 (2010; Zbl 1220.18003)].
Chapter 9. Formal \(\infty\)-Category Theory in a Virtual Equipment
This chapter deploys the calculus of the previous chapter to further develop the formal category theory of \(\infty\)-categories, specifically by defining and developing pointwise right and left Kan extensions, which are notably missing in Part I.
Part III Model Independence
This part establishes that the category theory of \(\infty\)-categories is also model independent in the precise sense that all categorical notions are preserved, reflected, and created by any change-of-model functor defining what is called a \textit{cosmological biequivalence}.
Chapter 10. Change-of-Model Functors
This chapter studies a certain class of cosmological functors between \(\infty\)-cosmoi that do not merely preserve \(\infty\)-categorical structure but also reflect and create it. These functors are referred to as \textit{cosmological biequivalences}, since the \(2\)-functors they induce between homotopy \(2\)-categories are equivalences.
Chapter 11. Model Independence
This chapter aims to establish that the study of \(\infty\)-categories is invariant under change of model, claiming a stronger result that statements about \(\infty\)-categories proven by analytic techniques indigenous to a single \(\infty\)-cosmos are independent from models, providing that the statements are expressible in a suitable equivalence invariant language.
Chapter 12. Applications of Model Independence
This chapter establishes some special properties of a certain class of \(\infty\)-cosmoi, called \(\infty\)-cosmoi of \((\infty,1)\)-categories, which mean \(\infty\)-cosmoi that are biequivalent to the \(\infty\)-cosmos of quasi-categories. The chapter aims also to illustrate how the model independence theorem can be used, by combining analytic and synthetic techniques, to establish results concerning any family of biequivalent \(\infty\)-cosmoi.
Appendix of Abstract Nonsense
Here the authors review all of the material one needs on enriched category theory, \(2\)-category theory and abstract homotopy theory in Appendices A, B and C, respectively.
Appendix A Basic Concepts of Enriched Category Theory
Appendix B An Introduction to \(2\)-Category Theory
Appendix C Abstract Homotopy Theory
Appendix of Concrete Constructions
Appendix D The Combinatorics of (Marked) Simplicial Sets
This appendix explores the combinatorics of simplicial sets, proving results stated in Chapters 1 and 4.
Appendix E \(\infty\)-Cosmoi Found in Nature
This appendix establishes concrete examples of \(\infty\)-cosmoi found in nature.
Appendix F The Analytic Theory of Quasi-Categories
This appendix aims to establish that the synthetic theory of quasi-categories is compatible with the analytic theory pioneered by André Joyal, Jacob Lurie, and many other. We present some lower bounds for regular solutions of Schrödinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, we prove that this mass can be observed if one looks at the solution and its gradient in space-time regions outside of that ball.
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A Chebyshev system \(F=\{f_ i\}^ n_ 1\subseteq C[a,b]\) is called extendable if there exist an interval [c,d] containing strictly [a,b] and the extensions \(\phi_ i\in C[c,d]\) of the functions \(f_ i\), \(i=1,...,n\), such that \(\{\phi_ i\}^ n_ 1\) is a Chebyshev system. The aim of this paper is to give sufficient conditions for the nonextendability of a Chebyshev system. We give here a sample: If the span of F contains m linearly independent elements \(P_ 1,...,P_ m\), m an integer, \(2\leq m\leq n-1\), such that every linear combination \(P(x)=\sum^{m}_{i=1}\lambda_ iP_ i(x),\) with \(\lambda_ 1\neq 0\), has at least \(n-m+1\) distinct zeros in [a,b], then F is nonextendable. The obtained results are applied to five concrete Chebyshev systems, two of which are new and the other three are known. The results of this paper were announced in Dokl. Akad. Nauk SSSR 276, 277-281 (1984; Zbl 0592.41034). Let \(C[a,b]\) be the space of real continuous functions defined on [a,b] and \(\{f_ i\}^ n_ 1\) be a Chebyshev system on \([a,b](T^ n[a,b]\)- system). A \(T^ n[a,b]\)-system \(\{f_ i\}^ n_ 1\) is called extendable outside \([a,b]\) if there is an interval \([c,d]\supset [a,b],[c,d]\neq [a,b]\) and \(T^ n[c,d]\) system \(\phi_ i(x)\) such that \(\phi_ i(x)=f_ i(x)\), \(x\in [a,b]\), \(i=1,...,h\). It is known that for \(h=1,2\) every \(T^ n[a,b]\) system is an extendable one and for every \(n\geq 3\) there are examples of nonextendable Chebyshev systems of n functions; the first example has been constructed for \(h=3\) by \textit{V. I. Volkov} [Kalinin. Gos. Ped. Inst. Uchen. Zap. 26, 41-48 (1958)].
In this paper sufficient conditions for the nonextendability of the \(T^ n[a,b]\) system for \(h\geq 3\) and criteria for the existence in a \(T^ n[a,b]\) system of polynomials with preassigned zeros are given. Sufficient conditions for a \(T^ n[a,b]\) system (for odd \(h\geq 3)\) to have no polynomials with arbitrary distributed zeros are also given. From the last result nonextendability of such a system also follows.
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A Chebyshev system \(F=\{f_ i\}^ n_ 1\subseteq C[a,b]\) is called extendable if there exist an interval [c,d] containing strictly [a,b] and the extensions \(\phi_ i\in C[c,d]\) of the functions \(f_ i\), \(i=1,...,n\), such that \(\{\phi_ i\}^ n_ 1\) is a Chebyshev system. The aim of this paper is to give sufficient conditions for the nonextendability of a Chebyshev system. We give here a sample: If the span of F contains m linearly independent elements \(P_ 1,...,P_ m\), m an integer, \(2\leq m\leq n-1\), such that every linear combination \(P(x)=\sum^{m}_{i=1}\lambda_ iP_ i(x),\) with \(\lambda_ 1\neq 0\), has at least \(n-m+1\) distinct zeros in [a,b], then F is nonextendable. The obtained results are applied to five concrete Chebyshev systems, two of which are new and the other three are known. The results of this paper were announced in Dokl. Akad. Nauk SSSR 276, 277-281 (1984; Zbl 0592.41034). This paper builds bridges between the two main families of modal logics of belief change, both based on plausibility pre-orders: dynamic doxastic logics computing stepwise updates, and temporal doxastic logics describing global system evolutions. Following earlier results linking dynamic-epistemic and epistemic-temporal logics, we prove representation theorems showing under which conditions a doxastic temporal model can be represented as the stepwise evolution of a doxastic model under successive `priority updates'. This allows for merges, where, in particular, the notion of a `temporal protocol' defining a global information process (for instance of communication or learning) can be introduced into the more local dynamic perspective.
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This impressive monograph is devoted to two interesting methods for the analysis of multi-dimensional queueing systems. The main ideas of both methods have been described in the introductory chapter.
In Chapters 2-4, the author extends the compensation approach, as developed by \textit{I. J. B. F. Adan}, \textit{J. Wessels} and \textit{W. H. M. Zijm} [Adv. Appl. Probab. 25, No. 4, 783-817 (1993; Zbl 0798.60081)] for the class of two-dimensional, homogeneous, nearest, nearest-neighbouring random walks to the corresponding class of multi-dimensional random walks. An additional property, called the projection property, is introduced to avoid complex notations and to simplify the analysis. It is shown that the equilibrium distribution of this random walk can be expressed as an alternating sum of infinitely many, pure product form distributions. Chapters 2 and 3 respectively deal with two and three dimensions.
In Chapter 4, an extensive analysis of the structure of the solution is presented. This leads to the development of efficient numerical procedures for the computation of the equilibrium distribution and related quantities.
Chapters 5-7 are devoted to the precedence relation method which leads to simpler and intuitively clearer proofs than other existing methods. This method is developed in Chapter 5 and in the Chapters 6 and 7, it is applied to the Symmetric Shortest Queue System (SSQS) and SSQS with a Job-Dependent Parallelism. The author derives flexible bound models for both of these systems. Some interesting extensions and suggestions for future work are discussed in Chapter 8. An up-to-date comprehensive bibliography is given in the end.
The precedence relation can be applied to any Markov Process, but it depends on the structure of the state space and the transition probabilities of a particular model.
This is a high quality contribution to the scope, theory and methodology of multi-dimensional queueing systems in terms of its lucid presentation, insights and directions. Researchers in the fields of manufacturing, communication, computing and service industries will find this outstanding monograph very useful. Some queueing problems (symmetric shortest queue one, for instance) may be treated as random walks on multidimensional grids. The main purpose of the paper is to explore the basic conditions for two-dimensional random walks to have stationary distribution as infinite series of some product form. It is considered continuous time Markov process on the pairs of nonnegative integers, and it is assumed that transitions can only take place to adjacent points and rates for transitions do not depend on the starting points (both in inner area and on boundaries). It is assumed that stationary probabilities exist and satisfy usual balance linear equations.
The essence of compensation approach is to choose the product form coefficients in such way that the ``initial'' solution for inner area turns to be one for boundaries also. This compensation approach generates an infinite, in general, sequence, so-called product form compensation terms to remove the errors of ``initial'' solution on horizontal and vertical boundaries. The result compensation is expressed as infinite sum of compensation terms (here the linear structure of the equations is used). The recursion relations for compensation terms are found and the problem of absolutely convergence of compensation series is solved (in the terms of solutions of equation for inner area). The convergence criterion is obtained as the restriction on the set of ``initial'' solutions (feasible ones) and the mean boundary drifts. Eventually there are obtained the conditions which guarantee the product form solution for probabilities having sufficiently large sum of (both) coordinates. It is mentioned that compensation method is a numerically oriented one and leads to efficient numerical procedures.
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This impressive monograph is devoted to two interesting methods for the analysis of multi-dimensional queueing systems. The main ideas of both methods have been described in the introductory chapter.
In Chapters 2-4, the author extends the compensation approach, as developed by \textit{I. J. B. F. Adan}, \textit{J. Wessels} and \textit{W. H. M. Zijm} [Adv. Appl. Probab. 25, No. 4, 783-817 (1993; Zbl 0798.60081)] for the class of two-dimensional, homogeneous, nearest, nearest-neighbouring random walks to the corresponding class of multi-dimensional random walks. An additional property, called the projection property, is introduced to avoid complex notations and to simplify the analysis. It is shown that the equilibrium distribution of this random walk can be expressed as an alternating sum of infinitely many, pure product form distributions. Chapters 2 and 3 respectively deal with two and three dimensions.
In Chapter 4, an extensive analysis of the structure of the solution is presented. This leads to the development of efficient numerical procedures for the computation of the equilibrium distribution and related quantities.
Chapters 5-7 are devoted to the precedence relation method which leads to simpler and intuitively clearer proofs than other existing methods. This method is developed in Chapter 5 and in the Chapters 6 and 7, it is applied to the Symmetric Shortest Queue System (SSQS) and SSQS with a Job-Dependent Parallelism. The author derives flexible bound models for both of these systems. Some interesting extensions and suggestions for future work are discussed in Chapter 8. An up-to-date comprehensive bibliography is given in the end.
The precedence relation can be applied to any Markov Process, but it depends on the structure of the state space and the transition probabilities of a particular model.
This is a high quality contribution to the scope, theory and methodology of multi-dimensional queueing systems in terms of its lucid presentation, insights and directions. Researchers in the fields of manufacturing, communication, computing and service industries will find this outstanding monograph very useful. The author studies the approximation of the nonlinear Dirichlet problem \(\Delta u+\lambda e^{\lambda u}=0\) in D (\(\lambda\in {\mathbb{R}})\), \(u=0\) on \(\partial D\), by means of the perturbation and conformal mapping method. Here D is nearly circular domain, including starlike domains. First order approximations and an application are also appended.
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Let \(F\) be a field of characteristic \(2\). Let \(\mathrm{WF}\) denote the Witt ring of symmetric bilinear forms over \(F\), and \(\mathrm{W}_q\mathrm{F}\) the \(\mathrm{WF}\)-module of nonsingular quadratic forms over \(F\). A bilinear \(n\)-fold Pfister form \(\langle\!\langle a_1,\dots ,a_n\rangle\!\rangle\) is a tensor product of \(n\) binary symmetric bilinear forms \(\langle 1,a_i\rangle\), and a quadratic \(n\)-fold Pfister form \(\langle\!\langle a_1,\dots ,a_n;b]]\) is the product of such a bilinear Pfister form by the binary nonsingular quadratic form \(x^2+xy+by^2\). Let \(I^nF\) be the ideal in \(\mathrm{WF}\) generated by \(n\)-fold bilinear Pfister forms, and let \(I^n\mathrm{W}_q\mathrm{F}\) denote the \(\mathrm{WF}\)-submodule of \(\mathrm{W}_q\mathrm{F}\) generated by quadratic \(n\)-fold Pfister forms. One defines \(\overline{I}^nF=I^nF/I^{n+1}F\) and \(\overline{I}^n\mathrm{W}_q\mathrm{F}=I^nF\cdot \mathrm{W}_q\mathrm{F}/I^{n+1}F\cdot \mathrm{W}_q\mathrm{F}\). Furthermore, let \(\Omega^n_F\) be the \(F\)-vector space of absolute Kähler \(n\)-differentials. Then there is a well-defined homomorphism
\[
\wp: \Omega^n_F\mapsto \Omega^n_F/d\Omega^{n-1}_F\,\,\text{ with}\,\, \wp (a\frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n})= \overline{(a^2-a) \frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n}},
\]
whose kernel, resp. cokernel is denoted by \(\nu_F(n)\), resp. \(H^{n+1}_2(F)\). A famous result by Kato states that there are isomorphisms \(\overline{I}^nF\cong \nu_F(n)\) mapping \(\overline{\langle\!\langle a_1,\ldots ,a_n\rangle\!\rangle}\) to \(\frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n}\), and \(H^{n+1}_2(F)\cong\overline{I}^nW_q(F)\) mapping \(\overline{b\frac{da_1}{a_1}\wedge\ldots\wedge\frac{da_n}{a_n}}\) to \(\overline{\langle\!\langle a_1,\dots ,a_n;b]]}\). The graded Witt groups of \(F\) are then defined to be \(\mathrm{GWF}= (\overline{I}^0F, \overline{I}^1F, \overline{I}^2F,\ldots)\) and \(\mathrm{GW}_q\mathrm{F}=(\overline{I}^0\mathrm{W}_q\mathrm{F}, \overline{I}^1\mathrm{W}_q\mathrm{F}, \overline{I}^2\mathrm{W}_q\mathrm{F},\ldots)\).
The main purpose of the present paper is the determination of the kernel of \(\mathrm{GW}_q\mathrm{F}\to \mathrm{GW}_q\mathrm{E}\) for a biquadratic separable extension \(E/F\) given by \(E=F(\beta_1, \beta_2)\) where \(\beta_i\) is a root of \(X^2+X+b_i\) for some \(b_i\in F^*\). It is shown that there is an exact sequence \(\mathrm{GWF}\oplus \mathrm{GWF}\to \mathrm{GW}_q\mathrm{F}\to \mathrm{GW}_q\mathrm{E}\) where the first map is given by \((\eta_1,\eta_2)\mapsto \eta_1\otimes [1,b_1]+\eta_2\otimes [1,b_2]\). Actually, it is shown that there is an exact sequence \(\nu_F(n)\oplus\nu_F(n)\to H_2^{n+1}(F)\to H_2^{n+1}(E)\). The result on the graded Witt groups then follows from Kato's isomorphisms. A crucial ingredient in the proof is the study of \textit{O. Izhboldin}'s groups \(Q^n(F,m)\), see [Algebraic \(K\)-theory, Pap. Semin., Leningrad/USSR, Adv. Sov. Math. 4, 129--144 (1991; Zbl 0746.19002)]. These groups are defined as certain quotients of \(W_m(F)\otimes F^{*\otimes n}\) where \(W_m(F)\) denotes the Witt vectors over \(F\) of length \(m\). It is sketched how to prove the above exactness in the case \(n=1\) solely within the theory of differential forms, and it is also indicated why this approach doesn't work anymore for arbitrary \(n\) and why then one has to use the groups \(Q^n(F,m)\), \(m=1,2\). Applications to algebras of index and exponent \(4\) are also given. [For the entire collection see Zbl 0723.00020.]
The author shows that for an arbitrary field \(F\) of characteristic \(p\) the Milnor \(K\)-groups \(K^ M_ n(F)\) have no \(p\)-torsion, a result due to Suslin in the special case \(n=2\) [cf. \textit{A. A. Suslin}, K-Theory 1, 5-29 (1987; Zbl 0635.12015)]. As a consequence the \(K_ n\)-analogue of Hilbert's Theorem 90 is proved for cyclic extensions of degree \(p\).
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Let \(F\) be a field of characteristic \(2\). Let \(\mathrm{WF}\) denote the Witt ring of symmetric bilinear forms over \(F\), and \(\mathrm{W}_q\mathrm{F}\) the \(\mathrm{WF}\)-module of nonsingular quadratic forms over \(F\). A bilinear \(n\)-fold Pfister form \(\langle\!\langle a_1,\dots ,a_n\rangle\!\rangle\) is a tensor product of \(n\) binary symmetric bilinear forms \(\langle 1,a_i\rangle\), and a quadratic \(n\)-fold Pfister form \(\langle\!\langle a_1,\dots ,a_n;b]]\) is the product of such a bilinear Pfister form by the binary nonsingular quadratic form \(x^2+xy+by^2\). Let \(I^nF\) be the ideal in \(\mathrm{WF}\) generated by \(n\)-fold bilinear Pfister forms, and let \(I^n\mathrm{W}_q\mathrm{F}\) denote the \(\mathrm{WF}\)-submodule of \(\mathrm{W}_q\mathrm{F}\) generated by quadratic \(n\)-fold Pfister forms. One defines \(\overline{I}^nF=I^nF/I^{n+1}F\) and \(\overline{I}^n\mathrm{W}_q\mathrm{F}=I^nF\cdot \mathrm{W}_q\mathrm{F}/I^{n+1}F\cdot \mathrm{W}_q\mathrm{F}\). Furthermore, let \(\Omega^n_F\) be the \(F\)-vector space of absolute Kähler \(n\)-differentials. Then there is a well-defined homomorphism
\[
\wp: \Omega^n_F\mapsto \Omega^n_F/d\Omega^{n-1}_F\,\,\text{ with}\,\, \wp (a\frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n})= \overline{(a^2-a) \frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n}},
\]
whose kernel, resp. cokernel is denoted by \(\nu_F(n)\), resp. \(H^{n+1}_2(F)\). A famous result by Kato states that there are isomorphisms \(\overline{I}^nF\cong \nu_F(n)\) mapping \(\overline{\langle\!\langle a_1,\ldots ,a_n\rangle\!\rangle}\) to \(\frac{dx_1}{x_1}\wedge\ldots\wedge\frac{dx_n}{x_n}\), and \(H^{n+1}_2(F)\cong\overline{I}^nW_q(F)\) mapping \(\overline{b\frac{da_1}{a_1}\wedge\ldots\wedge\frac{da_n}{a_n}}\) to \(\overline{\langle\!\langle a_1,\dots ,a_n;b]]}\). The graded Witt groups of \(F\) are then defined to be \(\mathrm{GWF}= (\overline{I}^0F, \overline{I}^1F, \overline{I}^2F,\ldots)\) and \(\mathrm{GW}_q\mathrm{F}=(\overline{I}^0\mathrm{W}_q\mathrm{F}, \overline{I}^1\mathrm{W}_q\mathrm{F}, \overline{I}^2\mathrm{W}_q\mathrm{F},\ldots)\).
The main purpose of the present paper is the determination of the kernel of \(\mathrm{GW}_q\mathrm{F}\to \mathrm{GW}_q\mathrm{E}\) for a biquadratic separable extension \(E/F\) given by \(E=F(\beta_1, \beta_2)\) where \(\beta_i\) is a root of \(X^2+X+b_i\) for some \(b_i\in F^*\). It is shown that there is an exact sequence \(\mathrm{GWF}\oplus \mathrm{GWF}\to \mathrm{GW}_q\mathrm{F}\to \mathrm{GW}_q\mathrm{E}\) where the first map is given by \((\eta_1,\eta_2)\mapsto \eta_1\otimes [1,b_1]+\eta_2\otimes [1,b_2]\). Actually, it is shown that there is an exact sequence \(\nu_F(n)\oplus\nu_F(n)\to H_2^{n+1}(F)\to H_2^{n+1}(E)\). The result on the graded Witt groups then follows from Kato's isomorphisms. A crucial ingredient in the proof is the study of \textit{O. Izhboldin}'s groups \(Q^n(F,m)\), see [Algebraic \(K\)-theory, Pap. Semin., Leningrad/USSR, Adv. Sov. Math. 4, 129--144 (1991; Zbl 0746.19002)]. These groups are defined as certain quotients of \(W_m(F)\otimes F^{*\otimes n}\) where \(W_m(F)\) denotes the Witt vectors over \(F\) of length \(m\). It is sketched how to prove the above exactness in the case \(n=1\) solely within the theory of differential forms, and it is also indicated why this approach doesn't work anymore for arbitrary \(n\) and why then one has to use the groups \(Q^n(F,m)\), \(m=1,2\). Applications to algebras of index and exponent \(4\) are also given. In this paper, the applicability of the entropy method for the trend towards equilibrium for reaction-diffusion systems arising from first order chemical reaction networks is studied. In particular, we present a suitable entropy structure for weakly reversible reaction networks without detail balance condition.
We show by deriving an entropy-entropy dissipation estimate that for any weakly reversible network each solution trajectory converges exponentially fast to the unique positive equilibrium with computable rates. This convergence is shown to be true even in cases when the diffusion coefficients of all but one species are zero.
For non-weakly reversible networks consisting of source, transmission and target components, it is shown that species belonging to a source or transmission component decay to zero exponentially fast while species belonging to a target component converge to the corresponding positive equilibria, which are determined by the dynamics of the target component and the mass injected from other components. The results of this work, in some sense, complete the picture of trend to equilibrium for first order chemical reaction networks.
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The author introduces and thoroughly studies some functionals defined on groups which resemble the notions of an inner product or a norm (especially the norm generated by an inner product). They are called a semi-inner product and a (para)(pre)seminorm, respectively. One should be aware that the names \textit{semi-inner product} and \textit{seminorm} are used here in a different sense than elsewhere in the literature but consistently with author's previous papers. Having defined the above mentioned functionals and structures, the author considers some functional equations and establishes their connections with the Cauchy functional equation.
In particular the result of \textit{G. Maksa} and \textit{P. Volkmann} [Publ. Math. Debrecen 56, 197--200 (2000; Zbl 0991.39015)] is generalized to the following one.
Theorem. Let \(f\colon X\to Y\) be a mapping between two groups \(X,Y\) (not necessarily commutative) and let \(q\colon Y\to [0,\infty)\) be a paraprenorm on \(Y\). If \(f\) satisfies the functional inequality
\[
q(f(x)+f(y))\leq q(f(x+y)),\quad x,y\in X,
\]
then \(f\) has to be additive.
The introduced concepts and results are broadly compared with similar issues appearing in the literature. The list of references is impressive. The authors prove in a very elegant way that, for a function \(f\) mapping from a group \(G\) into an inner product space, the inequality \(\|f(xy)\|\geq\|f(x)+f(y)\|\;(x,y\in G)\) implies \(f(xy)=f(x)+f(y)\;(x,y\in G)\).
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The author introduces and thoroughly studies some functionals defined on groups which resemble the notions of an inner product or a norm (especially the norm generated by an inner product). They are called a semi-inner product and a (para)(pre)seminorm, respectively. One should be aware that the names \textit{semi-inner product} and \textit{seminorm} are used here in a different sense than elsewhere in the literature but consistently with author's previous papers. Having defined the above mentioned functionals and structures, the author considers some functional equations and establishes their connections with the Cauchy functional equation.
In particular the result of \textit{G. Maksa} and \textit{P. Volkmann} [Publ. Math. Debrecen 56, 197--200 (2000; Zbl 0991.39015)] is generalized to the following one.
Theorem. Let \(f\colon X\to Y\) be a mapping between two groups \(X,Y\) (not necessarily commutative) and let \(q\colon Y\to [0,\infty)\) be a paraprenorm on \(Y\). If \(f\) satisfies the functional inequality
\[
q(f(x)+f(y))\leq q(f(x+y)),\quad x,y\in X,
\]
then \(f\) has to be additive.
The introduced concepts and results are broadly compared with similar issues appearing in the literature. The list of references is impressive. In this paper, fuzzy h-bi-ideals, fuzzy h-quasi-ideals and fuzzy h-interior ideals of a \(\Gamma\)-hemiring are studied and some related properties are investigated. The notions of h-intra-hemiregularity and h-quasi-hemiregularity of a \(\Gamma\)-hemiring are introduced and studied along with h-hemiregularity, and their characterization in terms of fuzzy h-ideals is also obtained. The concept of fuzzy h-duo \(\Gamma\)-hemiring is introduced and some characterizations of such hemirings are obtained.
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Bei \(\Lambda:=(\lambda_n)\in\mathbb{Z}^{\mathbb{N}_0}\) mit \(0=:\lambda_0<\lambda_1<\dots\) besagt der Müntzsche Approximationssatz, daß\ \(\text{span}\{z^\lambda \mid \lambda\in\Lambda\}\) genau dann dicht in \(C[0,1]\) ist, wenn \(\sum_{n=1}^\infty \lambda_n^{-1}\) divergiert. Dieser Satz erfährt hier eine Verallgemeinerung, zu deren Formulierung folgendes vorausgeschickt sei. Bei \(\ell,p\in \mathbb{Z}\) mit \(1\leq\ell\leq p\) bezeichne \(H(\ell,p)\) die Menge aller Teilmengen \(\{m_0,\dots,m_{\ell-1}\}\) von \(P:=\{0,1,\dots,p-1\}\) derart, daß jede \(\ell\)-reihige quadratische Untermatrix der Matrix \((\omega^{jm_{k-1}})_{(j,k)\in P\times\{1,\dots,\ell\}}\) regulär ist, wobei \(\omega:=e^{2\pi i/p}\) gesetzt ist. Verff. beweisen nun für \(\Lambda, \ell,p,H(\ell,p)\) wie oben: Bei \(a_0,\dots,a_{\ell-1}\in\mathbb{R}_+\) und \(\{m_0,\dots,m_{\ell-1}\} \in H(\ell,p)\) bezeichne \(\Gamma:=\Gamma(\underline{m},\underline{a}):= \bigcap_{j=0}^{\ell-1} \{r\omega^{m_j}\mid 0\leq r\leq a_j\}\); dann gilt: \(\text{span}\{z^\lambda \mid \lambda\in\Lambda\}\) ist genau dann dicht in \(C[\Gamma]\), wenn die Doppelsumme \(\sum_{q\in Q}\sum_{\lambda\in\Lambda\setminus\{0\}, \lambda\equiv q \,(\text{mod} p)} \lambda^{-1}\) für jede aus \(p-\ell+1\) Elementen von \(P\) bestehende Teilmenge \(Q\) divergiert. Durch Kombination dieses Ergebnisses mit einer Methode von \textit{A. Pinkus} [Surveys Approx. Theory 1, 1--45 (2005; Zbl 1067.41003)] wird die Dichte (in \(C[\Lambda]\)) des von einer gewissen Familie ganzer Funktionen aufgespannten Vektorraums charakterisiert. Approximation theory is concerned with the ability to approximate functions by simpler and more easy calculated functions. It is important that the set of functions from which one plans to approximate is dense in the set of continuous functions. In this interesting work, the author surveys some of the main density results and density methods. Starting with the Weierstrass approximation theorems, he discusses numerous generalizations (as Hahn-Banach theorem, Stone-Weierstrass theorem, Bohman-Korovkin theorem, Müntz theorem, Mergelyan theorem). Many historical hints and nice proofs of main results are presented. The author mainly considers univariate functions. Finally, some multivariate density results are given.
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Bei \(\Lambda:=(\lambda_n)\in\mathbb{Z}^{\mathbb{N}_0}\) mit \(0=:\lambda_0<\lambda_1<\dots\) besagt der Müntzsche Approximationssatz, daß\ \(\text{span}\{z^\lambda \mid \lambda\in\Lambda\}\) genau dann dicht in \(C[0,1]\) ist, wenn \(\sum_{n=1}^\infty \lambda_n^{-1}\) divergiert. Dieser Satz erfährt hier eine Verallgemeinerung, zu deren Formulierung folgendes vorausgeschickt sei. Bei \(\ell,p\in \mathbb{Z}\) mit \(1\leq\ell\leq p\) bezeichne \(H(\ell,p)\) die Menge aller Teilmengen \(\{m_0,\dots,m_{\ell-1}\}\) von \(P:=\{0,1,\dots,p-1\}\) derart, daß jede \(\ell\)-reihige quadratische Untermatrix der Matrix \((\omega^{jm_{k-1}})_{(j,k)\in P\times\{1,\dots,\ell\}}\) regulär ist, wobei \(\omega:=e^{2\pi i/p}\) gesetzt ist. Verff. beweisen nun für \(\Lambda, \ell,p,H(\ell,p)\) wie oben: Bei \(a_0,\dots,a_{\ell-1}\in\mathbb{R}_+\) und \(\{m_0,\dots,m_{\ell-1}\} \in H(\ell,p)\) bezeichne \(\Gamma:=\Gamma(\underline{m},\underline{a}):= \bigcap_{j=0}^{\ell-1} \{r\omega^{m_j}\mid 0\leq r\leq a_j\}\); dann gilt: \(\text{span}\{z^\lambda \mid \lambda\in\Lambda\}\) ist genau dann dicht in \(C[\Gamma]\), wenn die Doppelsumme \(\sum_{q\in Q}\sum_{\lambda\in\Lambda\setminus\{0\}, \lambda\equiv q \,(\text{mod} p)} \lambda^{-1}\) für jede aus \(p-\ell+1\) Elementen von \(P\) bestehende Teilmenge \(Q\) divergiert. Durch Kombination dieses Ergebnisses mit einer Methode von \textit{A. Pinkus} [Surveys Approx. Theory 1, 1--45 (2005; Zbl 1067.41003)] wird die Dichte (in \(C[\Lambda]\)) des von einer gewissen Familie ganzer Funktionen aufgespannten Vektorraums charakterisiert. We show that a free graded commutative Banach algebra over a (purely odd) Banach space \(E\) is a Banach-Grassmann algebra in the sense of Jadczyk and Pilch if and only if \(E\) is infinite-dimensional. Thus, a large amount of new examples of separable Banach-Grassmann algebras arise in addition to the only one example previously known due to A. Rogers.
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A topological space \(X\) is said to be \textit{cellular-Lindelöf}\, if for every family \(\mathcal{U}\) of pairwise disjoint non-empty open sets of \(X\) there is a Lindelöf subspace \(L \subseteq X\) such that \(U \cap L \neq \emptyset\) for every \(U \in \mathcal{U}\). This class of topological spaces was recently introduced by \textit{A. Bella} and \textit{S. Spadaro} [Monatsh. Math. 186, No. 2, 345--353 (2018; Zbl 1398.54008)] and generalizes both Lindelöf spaces and c.c.c. spaces.
If \(A\) is a subset of \(X\) and \(\mathcal{U}\) is a family of subsets of \(X\), the \textit{star} of \(A\) with respect to \(\mathcal{U}\) is given by \(\mathrm{St}(A,\mathcal{U}) := \bigcup\{U \in \mathcal{U}: U \cap A \neq \emptyset\}\). We put \(\mathrm{St}^0(A,\mathcal{U}) = A\) and, for every natural number \(n\), \(\mathrm{St}^{n+1}(A,\mathcal{U}) = \mathrm{St}(\mathrm{St}^{n}(A,\mathcal{U}),\mathcal{U})\). For simplicity, we write \(\mathrm{St}^n(x,\mathcal{U})\) instead of \(\mathrm{St}^n(\{x\},\mathcal{U})\).
A topological space \(X\) is said to have a \textit{rank 2-diagonal}\, if there is a sequence \(\{\mathcal{U}_n: n \in \omega\}\) of open covers of \(X\) such that for each \(x \in X\) one has \(\{x\} = \bigcap\{\mathrm{St}^2(x,\mathcal{U}_n): n \in \omega\}\).
A topological space \(X\) is said to have a \textit{\(G_\delta\)-diagonal} if the diagonal \(\Delta_X = \{(x,x):x \in X\}\) is a \(G_\delta\) subset of the topological product \(X \times X\), and it is said to have a \textit{regular \(G_\delta\)-diagonal} if there is a countable family \(\{U_n: n \in \omega\}\) of open neighbourhoods of the diagonal \(\Delta_X\) in the square \(X \times X\) such that \(\Delta_X = \bigcap\{\overline{U_n}: n \in \omega\}\).
If \(X\) is a Hausdorff space, the \textit{Hausdorff pseudocharacter} of \(X\), denoted \(H\psi(X)\), is the smallest infinite cardinal \(\kappa\) such that for each \(x \in X\) there is a collection \(\{V(\alpha,x):\alpha < \kappa\}\) of open neighbourhoods of \(X\) such that if \(x \neq y\) then there exist \(\alpha, \beta < \kappa\) such that \(V(\alpha,x) \cap V(\beta,y) = \emptyset\).
In the paper under review, the authors first discuss some basic properties of cellular-Lindelöf spaces such as the behaviour with respect to products and subspaces. The authors also prove that a normal, cellular-Lindelöf quasitopological group with countable Hausdorff pseudocharacter has cardinality at most \(\mathfrak{c}\). Finally, they introduce and investigate the class of cellular-compact (cellular \(\sigma\)-compact) spaces and prove that every cellular \(\sigma\)-compact Hausdorff space having either a rank 2-diagonal or a regular \(G_\delta\)-diagonal has cardinality at most \(\mathfrak{c}\). Some new questions are also posed. For instance: is it true that every first-countable cellular-compact (or, cellular-\(\sigma\)-compact) Hausdorff space has cardinality at most \(\mathfrak{c}\)? A topological space \(X\) is said to be \textit{discretely Lindelöf}, if the closure of every discrete subset is Lindelöf, and it is said to be \textit{almost discretely Lindelöf} if for every discrete subset \(D \subseteq X\) there is a Lindelöf subspace \(L\) of \(X\) such that \(D \subseteq L\), that is, almost discrete Lindelöfness means that every discrete subset can be covered by a Lindelöf subspace.
It was shown by \textit{I. Juhász} et al. [Stud. Sci. Math. Hung. 54, No. 4, 523--535 (2017; Zbl 1413.54082)] that every almost discretely Lindelöf first-countable space has cardinality at most \(2^{\mathfrak c}\). It was also asked, in the paper we have just referred to, whether every almost discretely Lindelöf first-countable \textit{Hausdorff} space has its cardinality bounded by the continuum. In the paper under review, the authors prove that this is the case under \(2^{< \mathfrak c} = \mathfrak c\), which is a set theoretical hypothesis which holds, for instance, under Martin's Axiom. They also provide a \(\mathbf{ZFC}\) proof of the analogous result for \textit{Urysohn spaces} (recall that a topological space is said to be a Urysohn space if any two distinct points are separated by disjoint closed neighbourhoods). It is worthwhile remarking that this \(\mathbf{ZFC}\) result on Urysohn spaces strengthens a recently published result of \textit{I. Juhász} et al. [Topology Appl. 241, 145--149 (2018; Zbl 1396.54006)], where it has been shown that every almost discretely Lindelöf first countable \textit{regular} space has cardinality bounded by the continuum. The paper finishes by exploring further generalizations and related results.
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A topological space \(X\) is said to be \textit{cellular-Lindelöf}\, if for every family \(\mathcal{U}\) of pairwise disjoint non-empty open sets of \(X\) there is a Lindelöf subspace \(L \subseteq X\) such that \(U \cap L \neq \emptyset\) for every \(U \in \mathcal{U}\). This class of topological spaces was recently introduced by \textit{A. Bella} and \textit{S. Spadaro} [Monatsh. Math. 186, No. 2, 345--353 (2018; Zbl 1398.54008)] and generalizes both Lindelöf spaces and c.c.c. spaces.
If \(A\) is a subset of \(X\) and \(\mathcal{U}\) is a family of subsets of \(X\), the \textit{star} of \(A\) with respect to \(\mathcal{U}\) is given by \(\mathrm{St}(A,\mathcal{U}) := \bigcup\{U \in \mathcal{U}: U \cap A \neq \emptyset\}\). We put \(\mathrm{St}^0(A,\mathcal{U}) = A\) and, for every natural number \(n\), \(\mathrm{St}^{n+1}(A,\mathcal{U}) = \mathrm{St}(\mathrm{St}^{n}(A,\mathcal{U}),\mathcal{U})\). For simplicity, we write \(\mathrm{St}^n(x,\mathcal{U})\) instead of \(\mathrm{St}^n(\{x\},\mathcal{U})\).
A topological space \(X\) is said to have a \textit{rank 2-diagonal}\, if there is a sequence \(\{\mathcal{U}_n: n \in \omega\}\) of open covers of \(X\) such that for each \(x \in X\) one has \(\{x\} = \bigcap\{\mathrm{St}^2(x,\mathcal{U}_n): n \in \omega\}\).
A topological space \(X\) is said to have a \textit{\(G_\delta\)-diagonal} if the diagonal \(\Delta_X = \{(x,x):x \in X\}\) is a \(G_\delta\) subset of the topological product \(X \times X\), and it is said to have a \textit{regular \(G_\delta\)-diagonal} if there is a countable family \(\{U_n: n \in \omega\}\) of open neighbourhoods of the diagonal \(\Delta_X\) in the square \(X \times X\) such that \(\Delta_X = \bigcap\{\overline{U_n}: n \in \omega\}\).
If \(X\) is a Hausdorff space, the \textit{Hausdorff pseudocharacter} of \(X\), denoted \(H\psi(X)\), is the smallest infinite cardinal \(\kappa\) such that for each \(x \in X\) there is a collection \(\{V(\alpha,x):\alpha < \kappa\}\) of open neighbourhoods of \(X\) such that if \(x \neq y\) then there exist \(\alpha, \beta < \kappa\) such that \(V(\alpha,x) \cap V(\beta,y) = \emptyset\).
In the paper under review, the authors first discuss some basic properties of cellular-Lindelöf spaces such as the behaviour with respect to products and subspaces. The authors also prove that a normal, cellular-Lindelöf quasitopological group with countable Hausdorff pseudocharacter has cardinality at most \(\mathfrak{c}\). Finally, they introduce and investigate the class of cellular-compact (cellular \(\sigma\)-compact) spaces and prove that every cellular \(\sigma\)-compact Hausdorff space having either a rank 2-diagonal or a regular \(G_\delta\)-diagonal has cardinality at most \(\mathfrak{c}\). Some new questions are also posed. For instance: is it true that every first-countable cellular-compact (or, cellular-\(\sigma\)-compact) Hausdorff space has cardinality at most \(\mathfrak{c}\)? After a brief review of the use of latent variables to accommodate the correlation among multiple outcomes of mixed types, through theoretical and numerical calculations, the consequences of such a construction are quantified. The effects of including latent variables on marginal inference in these models are contrasted with the situation for jointly normal outcomes. A simulation study illustrates the efficiency and reduction in bias gains possible in using joint models, and analysis of an example from the field of osteoarthritis illustrates the potential practical differences.
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For \(n,q \in \mathbb{N}\), \(q \geq 2\) let \(v_ q(n)\) be the sum of digits of \(n\) in its \(q\)-ary representation. For \(r \in \mathbb{N}\) let \(M_{r,q} (N): = \sum_{n<N} v^ r_ q(n)\) be the \(r\)-th moment of the sum of digit function. The Trollope-Delange formula states that
\[
M_{1,2} (N) = {1 \over 2} N \cdot \log_ 2N + N\cdot \delta_ 1 (\log_ 2N)
\]
with a continuous, periodic, nowhere differentiable function \(S_ 1\). In a recent paper \textit{R. E. Kennedy} and \textit{C. Cooper} [Fibonacci Q. 31, 341-345 (1993; Zbl 0790.11006)] dealt with \(M_{2,10}(N)\).
In the present paper Delange's approach is used to compute \(M_{3,2}(N)\), the approach to \(M_{r,2} (N)\) for \(r \geq 3\) is sketched, and finally the Mellin transform is used to establish exact and asymptotic formulae in various cases of digital sums. Further the mean and the Fourier coefficients of the periodic functions involved are computed. Denote by \(s(n)\) the digital sum of the positive integer \(n\) and let \(\log x\) be the base 10 logarithm of \(x\). In an earlier paper the authors showed for positive integers \(k\) the asymptotic formula
\[
{1\over x} \sum_{n\leq x} s(n)^ k= \left({9\over 2}\right)^ k \log^ k x+ O(\log^{k-1/3} x)
\]
and they conjectured the better error term \(O(\log^{k-1} x)\). In the paper under review the authors prove this result in the special case that \(x\) is a power of ten.
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For \(n,q \in \mathbb{N}\), \(q \geq 2\) let \(v_ q(n)\) be the sum of digits of \(n\) in its \(q\)-ary representation. For \(r \in \mathbb{N}\) let \(M_{r,q} (N): = \sum_{n<N} v^ r_ q(n)\) be the \(r\)-th moment of the sum of digit function. The Trollope-Delange formula states that
\[
M_{1,2} (N) = {1 \over 2} N \cdot \log_ 2N + N\cdot \delta_ 1 (\log_ 2N)
\]
with a continuous, periodic, nowhere differentiable function \(S_ 1\). In a recent paper \textit{R. E. Kennedy} and \textit{C. Cooper} [Fibonacci Q. 31, 341-345 (1993; Zbl 0790.11006)] dealt with \(M_{2,10}(N)\).
In the present paper Delange's approach is used to compute \(M_{3,2}(N)\), the approach to \(M_{r,2} (N)\) for \(r \geq 3\) is sketched, and finally the Mellin transform is used to establish exact and asymptotic formulae in various cases of digital sums. Further the mean and the Fourier coefficients of the periodic functions involved are computed. [For the entire collection see Zbl 0699.00041.]
We study the filtering problem of a homogeneous Markov chain in a singular case, according to the time scale of the observed process.
Under proper assumptions, we obtain a new asymptotic expansion of the unnormalized conditional distribution of the Zakai equation, by introducing time scale and boundary layer terms. The terms of this expansion are calculated easier by decentralization and aggregation [see the author, ``Méthodes de perturbations singulières en filtrage non linéaire'', Thesis, Univ. Provence (1986)].
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The author shows how to construct a very well-behaved \(p\)-adic cohomology theory, called continuous cohomology by deriving the left exact functor
\[
\{\text{inverse system }(F_n)\text{ of étale sheaves on }X\} \to \text{ abelian groups } (F_n) \mapsto \lim_{\overset \leftarrow n}H_0(X,F_n).
\]
The construction, when applied to locally constant sheaves, \(F_n\), gives the continuous étale cohomology theory of \textit{W. G. Dwyer} and \textit{E. M. Friedlander} [Trans. Am. Math. Soc. 292, 247--280 (1985; Zbl 0581.14012)]. However, the author's construction applies to arbitrary sheaves while enjoying all the desirable properties of a cohomology theory (e.g. Hochschild-Serre spectral sequences, Chern classes, a Milnor \(\lim_{\leftarrow}^ 1\) sequence to relate it to \(\ell\)-adic cohomology). All in all, continuous cohomology looks to be one way around a number of technical difficulties in \(\ell\)-adic cohomology. The authors develop étale \(K\)-theory for a noetherian \({\mathbb Z}[1/\ell]\)- algebra \(A\) and smooth schemes over \(A\). This extends the work of \textit{E. M. Friedlander} [Invent. Math. 60, 105--134 (1980; Zbl 0519.14010) and Ann. Sci. Éc. Norm. Supér. (4) 15, 231--256 (1982; Zbl 0537.14011)]. The importance of étale \(K\)-theory is that it provides a computable target for algebraic \(K\)-theory. In fact there is a natural map
\[
\phi: K_ i(A;{\mathbb Z}/\ell^{\nu})\to K_ i^{et}(A;{\mathbb Z}/\ell^{\nu})
\]
which is expected (the Lichtenbaum-Quillen conjecture) to be an isomorphism for ``nice'' \(A\) when \(i\) is large. In fact \(\phi\) is onto in these circumstances [the authors, the reviewer and \textit{R. W. Thomason}, Invent. Math. 66, 481--491 (1982; Zbl 0501.14013)].
The main theorem in this subject is that \(\phi\) made ``Bolt periodic'' is an isomorphism [\textit{R. W. Thomason}, Ann. Sci. Éc. Norm. Supér. (4) 18, 437--552 (1985; Zbl 0596.14012)]. - The authors' main application of étale \(K\)-theory is to show that if \(A\) is the ring of \(S\)-integers in a number field then \(\phi\) is surjective if \(i\geq 1.\)
During the gestation period of this paper other authors -- for example, J. F. Jardine, A. A. Suslin, R. W. Thomason -- have increased our knowledge of algebraic \(K\)-theory and our understanding of étale \(K\)-theory. Nonetheless, although thereby partially superannuated, it is important from a historical point of view that this paper has finally appeared.
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The author shows how to construct a very well-behaved \(p\)-adic cohomology theory, called continuous cohomology by deriving the left exact functor
\[
\{\text{inverse system }(F_n)\text{ of étale sheaves on }X\} \to \text{ abelian groups } (F_n) \mapsto \lim_{\overset \leftarrow n}H_0(X,F_n).
\]
The construction, when applied to locally constant sheaves, \(F_n\), gives the continuous étale cohomology theory of \textit{W. G. Dwyer} and \textit{E. M. Friedlander} [Trans. Am. Math. Soc. 292, 247--280 (1985; Zbl 0581.14012)]. However, the author's construction applies to arbitrary sheaves while enjoying all the desirable properties of a cohomology theory (e.g. Hochschild-Serre spectral sequences, Chern classes, a Milnor \(\lim_{\leftarrow}^ 1\) sequence to relate it to \(\ell\)-adic cohomology). All in all, continuous cohomology looks to be one way around a number of technical difficulties in \(\ell\)-adic cohomology. Auszug aus einer Abhandlung, die in Lond. Phil. Trans. 1891 (siehe JFM 23.0982.01) erschienen ist.
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A family \({\mathcal F}\) of linear operators on a vector space \(V\) over an algebraically closed field is said to be irreducible if no non-trivial subspace of \(V\) is invariant under every member of \({\mathcal F}\). The author of this notice propose for discussions and studies several open problems concerning the irreducibility property for families of linear operators with additional algebraic structure (such as a semigroup, a group, a Lie or Jordan algebra, etc.) or additional properties (such as permutability, submultiplicativity or sublinearity of spectra, etc.). One of interesting fields of possible discussions is related to various analogues of the classical Burnside theorem which asserts that the only irreducible algebra of operators on a finite-dimensional \(V\) is the full algebra \({\mathcal L}(V)\) of all operators on \(V\). For example, the paper [\textit{L. Grunenfelder, M. Omladič} and \textit{H. Radjavi}, Pac. J. Math. 161, 335-346 (1993; Zbl 0811.46052)] considers the Jordan-algebra analogue of this theorem and gives a satisfactory solution. The Lie algebra analogue is still open. If \({\mathcal A}\) is an (associative) algebra of linear operators on a vector space, it is well known that 2-transitivity for \({\mathcal A}\) implies density and, in certain situations, transitivity guarantees 2- transitivity. In this paper we consider analogs of these results for Jordan algebras of linear operators with the standard Jordan product.
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A family \({\mathcal F}\) of linear operators on a vector space \(V\) over an algebraically closed field is said to be irreducible if no non-trivial subspace of \(V\) is invariant under every member of \({\mathcal F}\). The author of this notice propose for discussions and studies several open problems concerning the irreducibility property for families of linear operators with additional algebraic structure (such as a semigroup, a group, a Lie or Jordan algebra, etc.) or additional properties (such as permutability, submultiplicativity or sublinearity of spectra, etc.). One of interesting fields of possible discussions is related to various analogues of the classical Burnside theorem which asserts that the only irreducible algebra of operators on a finite-dimensional \(V\) is the full algebra \({\mathcal L}(V)\) of all operators on \(V\). For example, the paper [\textit{L. Grunenfelder, M. Omladič} and \textit{H. Radjavi}, Pac. J. Math. 161, 335-346 (1993; Zbl 0811.46052)] considers the Jordan-algebra analogue of this theorem and gives a satisfactory solution. The Lie algebra analogue is still open. A category of combinatorial objects, called combinatorial cell complexes, is defined and discussed, and a functor \(T\) is considered from this category to the category of topological spaces with cell structure, the image of which being closely related to the category of CW-complexes. This formalism was developed to study finite group actions on topological spaces.
A combinatorial cell complex is based on a poset \(X\). The functor \(T\) assigns to each \(X\) its geometrical realization \(T(X)\). It is shown that \(T\) defines an equivalence of categories between the category of combinatorial cell complexes the cell boundaries of which are spheres. A certain subcategory of CW-complexes are called normal CW-complexes.
Especially one considers the subcategory of the restricted combinatorial cell complexes. The corresponding restricted CW-complexes include regular CW-complexes but also many other classical examples like the torus, the Klein bottle, and the Poincaré dodecahedron, which are discussed as illustrations.
A simplicial complex \(K(X)\) is associated to each restricted complex \(X\), which gives a canonical triangulation of \(T(X)\). Then cellular homology is defined combinatorially. It is shown that if \(X\) is restricted and the boundary of each cell is homologically spherical, then the homology of \(T(X)\) is the cellular homology of \(X\).
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The main aim of the paper is to prove new affine isoperimetric inequalities for the so-called \(L_p\)-affine surface area \(as_p(K)\) of a convex body \(K\subset{\mathbb R}^n\), for all \(p\in[-\infty,1)\). The \(L_p\)-affine surface area was introduced by \textit{E. Lutwak} in [Adv. Math. 118, No.~2, 244--294 (1996; Zbl 0853.52005)], where he established \(L_p\)-affine isoperimetric inequalities for \(p>1\).
In this paper the authors start by giving new geometric interpretations of the \(L_p\)-affine surface areas, obtaining as a consequence an interesting duality formula relating the \(L_p\)-affine surface area of a convex body \(K\) with the one of its polar body \(K^{\circ}\): \(as_p(K)=as_{n^2/p}(K^{\circ})\). Then they prove several affine isoperimetric inequalities: for a convex body \(K\) with centroid at the origin, if \(p\geq 0\) then
\[
\frac{as_p(K)}{as_p(B^n_2)}\leq\left(\frac{| K| }{| B^n_2| }\right)^{(n-p)/(n+p)},
\]
and the opposite inequality when \(-n<p<0\); here \(| K| \) denotes the volume of \(K\) and \(B^n_2\) the Euclidean unit ball. In both inequalities the equality holds if and only if \(K\) is an ellipsoid. If \(K\) is in addition sufficiently smooth and \(p<-n\), then the lower bound for the quotient of the \(L_p\)-affine surface areas is scaled by a certain constant.
Finally, nice relations among the \(L_p\)-affine surface areas of \(K\) and its dual are obtained: \(as_p(K)\,as_p(K^{\circ})\leq n^2| K| | K^{\circ}| \) when \(p\geq 0\), and the opposite inequality if \(-n<p<0\). The aim of the article is to extend the known inequalities involving (extended) affine and geominimal surface areas to the so-called \(p\)-affine and \(p\)-geominimal surface areas for \(p >1\). All of these latter notions refer to the Firey linear combination \(\lambda \cdot K +_p \mu \cdot L\), given by
\[
h(\lambda \cdot K +_p \mu \cdot L,\cdot) := (\lambda h(K,\cdot)^p + \mu h(L,\cdot)^p)^{1\over p}
\]
\((\lambda \geq 0\), \(\mu \geq 0\), not both zero), where \(h(K,u)\) resp. \(h(L,u)\) \((u \in S^{n-1})\) is the support function of a convex body \(K\) resp. \(L\) in euclidean \(n\)-space containing the origin \(o\) in its interior. From this a \(p\)-mixed volume \(V_p (K,L)\) may be derived by
\[
{n\over p} V_p(K,L) := \lim_{\varepsilon \to 0} {V(K+_p \varepsilon \cdot L) - V(K) \over \varepsilon} = {1\over p} \int_{S^{n-1}} h(L,u)^p dS_p (K,u)
\]
\((S_p(K,\cdot) =\) suitable Borel measure on \(S^{n-1})\) and moreover
\(V_p(K,L^*) := {1\over n} \int_{S^{n-1}} \rho_L (u)^{-p} dS_p (K,u)\) for a star body \(L\) about \(o\) with the radial function \(\rho_L(u)\).
Now in continuation of part I of the author's paper [part I: Mixed volumes and the Minkowski problem, J. Differ. Geom. 38, No. 1, 131-150 (1993; Zbl 0788.52007)] a \(p\)-affine surface area \(\Omega_p(K)\) is defined by \(\Omega_p(K)^{n+p \over n} = \inf_L nV_p (K,L^*) \cdot (nV(L))^{p \over n}\), in a certain sense generalizing a \(p\)-geominimal surface area \(G_p(K)\) with \(G_p(K) = \inf_{L'} nV_p (K,L') \cdot (\omega_n^{-1} V(L'{}^*))^{p\over n}\) where the infimum is only taken for the convex star bodies \(L'\) with polar bodies \(L'{}^*\) \((\omega_n = \) volume of the unit \(n\)-ball). As one of the big varieties of results the following \(p\)-extension of the affine isoperimetric inequality may be stated: \(\Omega_p(K)^{n+p} \leq n^{n+p} \omega^{2p}_n V(K)^{n-p}\) with equality if and only if \(K\) is an ellipsoid. This extension represents an improvement of the classical affine isoperimetric inequality because the \(p\)-th root of the \(p\)-affine isoperimetric ratio is shown to be monotone nondecreasing in \(p\). A similar result is proved for the \(p\)-geominimal surface area. As in the classical case there exists an inequality between \(p\)-affine and \(p\)-geominimal surface area.
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The main aim of the paper is to prove new affine isoperimetric inequalities for the so-called \(L_p\)-affine surface area \(as_p(K)\) of a convex body \(K\subset{\mathbb R}^n\), for all \(p\in[-\infty,1)\). The \(L_p\)-affine surface area was introduced by \textit{E. Lutwak} in [Adv. Math. 118, No.~2, 244--294 (1996; Zbl 0853.52005)], where he established \(L_p\)-affine isoperimetric inequalities for \(p>1\).
In this paper the authors start by giving new geometric interpretations of the \(L_p\)-affine surface areas, obtaining as a consequence an interesting duality formula relating the \(L_p\)-affine surface area of a convex body \(K\) with the one of its polar body \(K^{\circ}\): \(as_p(K)=as_{n^2/p}(K^{\circ})\). Then they prove several affine isoperimetric inequalities: for a convex body \(K\) with centroid at the origin, if \(p\geq 0\) then
\[
\frac{as_p(K)}{as_p(B^n_2)}\leq\left(\frac{| K| }{| B^n_2| }\right)^{(n-p)/(n+p)},
\]
and the opposite inequality when \(-n<p<0\); here \(| K| \) denotes the volume of \(K\) and \(B^n_2\) the Euclidean unit ball. In both inequalities the equality holds if and only if \(K\) is an ellipsoid. If \(K\) is in addition sufficiently smooth and \(p<-n\), then the lower bound for the quotient of the \(L_p\)-affine surface areas is scaled by a certain constant.
Finally, nice relations among the \(L_p\)-affine surface areas of \(K\) and its dual are obtained: \(as_p(K)\,as_p(K^{\circ})\leq n^2| K| | K^{\circ}| \) when \(p\geq 0\), and the opposite inequality if \(-n<p<0\). We consider a linear model of rotating Timoshenko beam, which is clamped at one end to a disk, the other end being free. The motion of the beam is controlled by angular acceleration of the disk. We study the minimization problem of mean square deviation of the Timoshenko beam from a given position. For the minimization problem of the first mode, we prove that optimal control is the chattering control, i.e., it has an infinite number of switches in a finite time interval. We then construct a suboptimal control with a finite number of switches.
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[For part I see ibid. 281, 507-527 (1984; see the preceding review).]
The simple 2q-knots, \(q\geq 4\), are classified in terms of modules and pairings which depend only on the knot, and not on the Seifert surface which is used to define them. This result includes as special cases the classification theorems of S. Kojima for the odd \({\mathbb{Z}}\)-torsion case and of the reviewer for the \({\mathbb{Z}}\)-torsion-free case. The author begins by classifying the r-simple n-knots, \(3r\geq n+1\geq 6\), in terms of the stable homotopy theory associated with the set of Seifert surfaces of a knot. He then applies this to classify the simple 2q-knots, \(q\geq 4\), in terms of modules and pairings derived from a (q-1)-connected Seifert surface, modulo an equivalence relation generated by ambient surgery on the Seifert surface.
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[For part I see ibid. 281, 507-527 (1984; see the preceding review).]
The simple 2q-knots, \(q\geq 4\), are classified in terms of modules and pairings which depend only on the knot, and not on the Seifert surface which is used to define them. This result includes as special cases the classification theorems of S. Kojima for the odd \({\mathbb{Z}}\)-torsion case and of the reviewer for the \({\mathbb{Z}}\)-torsion-free case. We construct an optimal quadrature formula in the sense of Sard in the Hilbert space \(K_2(P_3)\). Using Sobolev's method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space \(L_2^{(3)}(0,1)\). The obtained optimal quadrature formula is exact for the trigonometric functions \(\sin x\), \(\cos x\) and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.
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Consider a stationary Poisson process \(\Xi\) in \(\mathbb R^d\), \(d\geq 2\), of unit intensity. It forms a set of centers claiming a territory of volume \(\alpha \leq 1\) according to some rule which is called allocation. Any point \(x\in\mathbb R^d\) may be allocated to some center \(\xi\in\Xi\), stay unclaimed or lie on the border between several influence regions of points of \(\Xi\). For allocation rules, a notion of stability is introduced as in [\textit{C. Hoffman, A. E. Holroyd} and \textit{Y. Peres}, Ann. Probab. 34, No.~4, 1241--1272 (2006; Zbl 1111.60008)].
The model percolates if the total territory claimed by \(\Xi\) has an unbounded connected subset. Let \(\alpha_p(d)\) be the critical volume \(\alpha\) of allocation regions at which a switch from no percolation to percolation occurs. The following results are proved in the paper: for any \(d\geq 2\),
1) \(\alpha_p(d)>0\) (no percolation for small \(\alpha\)),
2) \(\limsup_{d\to\infty}\alpha_p(d) 2^d\leq 1\),
3) unclaimed sites of \(\mathbb R^d\) percolate for \(\alpha\) small enough. Let \(\Xi\) be a discrete set in the \(d\)-dimensional space whose elements are called centers. In the paper, ``fair'' allocations of the underlying space to the points of \(\Xi\) are studied in which the regions allocated to different centers have equal volumes \(\alpha\in [0,\infty]\). Assume that sites and centers both prefer to be allocated as close as possible. An allocation is called unstable if some site and center both prefer each other over their current allocation. Otherwise, it is called stable. For stable allocations, an existence result is proved for any discrete set of centers. The stable allocation is unique if the configuration \(\Xi\) is recurrent in some sense. If \(\Xi\) is a translation-invariant point process with finite intensity \(\lambda\), then there exists the unique stable allocation a.s. Furthermore, phase transition problems are studied depending on the value of \(\lambda \alpha\). Namely, the question whether all sites are claimed or all centers are sated is investigated in detail.
It is shown that the region allocated to each center is a union of finitely many bounded connected components. If \(\Xi\) is a homogeneous Poisson point process and \(\lambda=\alpha=1\) (critical case), then every site is a.s. ``coveted'' by infinitely many centers and vice versa, every center is ``desired'' a.s. by an infinite volume of sites. In this case, power law lower bounds on the allocation distance of a typical site are given.
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Consider a stationary Poisson process \(\Xi\) in \(\mathbb R^d\), \(d\geq 2\), of unit intensity. It forms a set of centers claiming a territory of volume \(\alpha \leq 1\) according to some rule which is called allocation. Any point \(x\in\mathbb R^d\) may be allocated to some center \(\xi\in\Xi\), stay unclaimed or lie on the border between several influence regions of points of \(\Xi\). For allocation rules, a notion of stability is introduced as in [\textit{C. Hoffman, A. E. Holroyd} and \textit{Y. Peres}, Ann. Probab. 34, No.~4, 1241--1272 (2006; Zbl 1111.60008)].
The model percolates if the total territory claimed by \(\Xi\) has an unbounded connected subset. Let \(\alpha_p(d)\) be the critical volume \(\alpha\) of allocation regions at which a switch from no percolation to percolation occurs. The following results are proved in the paper: for any \(d\geq 2\),
1) \(\alpha_p(d)>0\) (no percolation for small \(\alpha\)),
2) \(\limsup_{d\to\infty}\alpha_p(d) 2^d\leq 1\),
3) unclaimed sites of \(\mathbb R^d\) percolate for \(\alpha\) small enough. This book provides a broad overview of the topic bioinformatics with focus on data, information and knowledge. From data acquisition and storage to visualization, ranging through privacy, regulatory and other practical and theoretical topics, the author touches several fundamental aspects of the innovative interface between medical and technology domains that is biomedical informatics. Each chapter starts by providing a useful inventory of definitions and commonly used acronyms for each topic and throughout the text, the reader finds several real-world examples, methodologies and ideas that complement the technical and theoretical background. This new edition includes new sections at the end of each chapter, called ``future outlook and research avenues,'' providing pointers to future challenges. At the beginning of each chapter a new section called ``key problems'', has been added, where the author discusses possible traps and unsolvable or major problems.
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Generalizing a construction of \textit{J. H. Conway}, \textit{P. B. Kleidman} and \textit{R. A. Wilson} [ibid. 26, No. 2, 157-170 (1988; Zbl 0643.51015)] the author obtains new families of ovoids in \(O^ +_ 8(p)\) for \(p\) prime using the \(E_ 8\) root lattice. For distinct primes \(r\) and \(p\), the construction gives ovoids \(O_{r,p}(x)\) which are those of the paper cited above when \(r\in\{2,3\}\). Certain (not necessarily full) automorphism groups in \(PGO_ 8^ +(p)\) are given for families with \(r \leq 7\), and some isomorphisms between members of different families are obtained. An ovoid in the 8-dimensional orthogonal geometry \(O\) \(+_ 8(q)\) (q a prime-power) may be thought of as a set of q \(3+1\) points (isotropic 1- spaces) no two of which are orthogonal. The authors construct four infinite families of ovoids in \(O\) \(+_ 8(p)\), and determine their automorphism groups. They show that four previously known ovoids, including two found recently, are members of the families.
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Generalizing a construction of \textit{J. H. Conway}, \textit{P. B. Kleidman} and \textit{R. A. Wilson} [ibid. 26, No. 2, 157-170 (1988; Zbl 0643.51015)] the author obtains new families of ovoids in \(O^ +_ 8(p)\) for \(p\) prime using the \(E_ 8\) root lattice. For distinct primes \(r\) and \(p\), the construction gives ovoids \(O_{r,p}(x)\) which are those of the paper cited above when \(r\in\{2,3\}\). Certain (not necessarily full) automorphism groups in \(PGO_ 8^ +(p)\) are given for families with \(r \leq 7\), and some isomorphisms between members of different families are obtained. A single server retrial queue with Bernoulli feedback and FIFO discipline is studied, where the server is subjected to starting failure. The paper presents some necessary and sufficient conditions for the stability of the system. Some performance measures, such as mean system size, server utilization, mean orbit size (between trials, the blocked customer joins a pool of unsatisfied customers called orbit), probability that the server is under repair, probability that the orbit is empty, are obtained. The paper is accomplished by some numerical examples.
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A few years ago H. Kawanoue announced a program (the \textit{Idealistic Filtration Program}) to resolve singularities of algebraic varieties which might work even if the base field is of positive characteristic. It is expressed in terms of \textit{idealistic filtrations}. If \(U\) is an open set of an affine variety \(W\), with coordinate ring \(R\), an idealistic filtration on \(W\) (or on \(R\)) is an indexed family of ideals \(I_a\) of \(R\), where \(a\) is a non-negative real number, such that \(I_a \subset I_b\) if \(a \geq b\), \(I_a I_b \subset I_{a+b}\) and \(I_0=R\). One defines the support, or singular locus, of an idealistic filtration. Then the goal is to give an algorithm allowing us to resolve the filtration, i.e., to find blowing-ups with smooth centres such that, taking suitable transforms, eventually the support becomes empty. The centres will be given as the set of points where certain \textit{algorithmic resolution functions}, with values in a suitable totally ordered set, reach the maximum value. From this, a similar algorithm to resolve singularities of varieties will follow.
To construct such resolutions functions, given an idealistic filtration \(\mathcal I\) on the variety \(W\) a \textit{strand of invariants} is attached to each point \(P\) of \(W\). Each step of the strand involves a triplet \(\sigma\), \(\tilde {\mu}\) and \(s\) of invariants. In the definition of \(\sigma\) and \(\tilde \mu\) a fundamental role is played by \(L(\mathcal I)\), a certain graded sub-algebra of \(Gr_{M}(A)\), the graded ring of \((A,M)\) (the local ring of \(W\) at a closed point \(P\)). Of particular importance is the notion of \textit{leading generator system}. These are elements in suitable ideals \(I_j\) of \(\mathcal I\) whose initial forms in the graded ring \(Gr_{M}(A)\) are generators of \(L(\mathcal I)\), satisfying certain ``minimality'' conditions. This notion seems essential to avoid the explicit use of hypersurfaces of maximal contact, a usual tool in desingularization techniques which is not always available in positive characteristic. Leading generator systems exist when the filtration \(\mathcal I\) is \(D\)-saturated, that is closed under the application of differential operators.
The resolution functions thus obtained have the fundamental property necessary in problems of this type. Namely, their values strictly decrease after performing a blowing-up with the appropriate algorithmic center. But still, when the characteristic is positive, there is no warranty that the process terminates.
Kawanoue, working in cooperation with K. Matsuki, plans to present the details of the program in a series of four papers. In the first one, \textit{H. Kawanoue} [Publ. Res. Inst. Math. Sci. 43, 819--909 (2007; Zbl 1170.14012)], discussed the main lines of the program, in particular what was mentioned so far. In the second part of the series (the article under review) the authors carefully study properties of the invariants \(\sigma\) and \(\tilde \mu\) (at ``time zero'', i.e., at the beginning of the process). For instance, they check the upper-semicontinuity of the invariants \(\sigma\) and \(\tilde \mu\). They also discuss certain power series expansions, but nor necessarily in terms of a regular system of parameters, but rather relative to a leading generating system; something that will be important later. Many of these results had been announced (without proof) in Part I.
In an Appendix, they include an important improvement of a result in Part I. Over there, they prove what they call \textit{the new non-singularity principle}. It says (among other things) that given an idealistic filtration \(\mathcal I\) which is both \(D\)-saturated and \(R\)-saturated (i.e., closed under the operations of differentiation and extraction of roots) and satisfies, for some closed point, \(\tilde \mu (z) = \infty\), then on a neighbourhood of \(z\) the support of \(\mathcal I\) is non-singular. From this result it follows that the centres involved in the application of the algorithm are indeed regular, which is a non-trivial result. In this Appendix a similar result is proved, but avoiding the hypothesis of \(R\)-saturation. This improvement could be an important step toward proving that their algorithm (or a slight variation thereof) will terminate after a finite number of steps, thus concluding a proof of resolution in positive characteristic. Indeed, the assumption on \(R\)-saturation seems to be an obstacle to prove the termination of the algorithm.
A subsection (numbered 0.3) resumes the current status of the Idealistic Filtration Program. This is the first part of a series of papers devoted to the resolution of singularities (and containing joint work with K. Matsuki, as mentioned in 0.6). The goal is presenting a program towards the construction of a resolution algorithm which works for an algebraic variety over a perfect field in positive characteristic. The author's intention is however, to develop a program working in full generality, i.e. including characteristic 0 as well.
The intended parts of the complete work are:
I. Foundation; the language of the idealistic filtration
II. Basic invariants associated to the idealistic filtration and their properties
III. Transformations and modifications of the idealistic filtration
IV. Algorithm in the framework of the idealistic filtration
Part I (under review) establishes the notion and fundamental properties of the \textit{idealistic filtration}, which is considered the main language of this program.
Chapter 0 starts with a crash course on the existing algorithms in characteristic 0 and introduces the author's program as a ``new approach to overcome the main source of troubles in the language of the \textit{idealistic filtration}, which is a refined extension of such classical notions as the idealistic exponent by Hironaka, the presentation by Bierstone-Milman, the basic object by Villamayor, and the marked ideal by Wlodarczyk.''
Section 0.2.3 introduces the idealistic filtration and mentions some of its distinguished features as there are: leading generator systems as substitutes for hypersurfaces of maximal contact, construction of the strand of invariants through enlargements of an idealistic filtration, saturation and a ``new nonsingularity principle''.
In 0.3 (``Algorithm constructed according to the program'') the author refers to the forthcomimg part IV of the paper as far as termination of the algorithm (in case of positive characteristic) is concerned; he mentions that this question is not yet settled.
In 0.5 a brief account of (mainly references to) the history of the problem is given, as well as hints to recent announcements and approaches.
The remaining about 70 pages of the paper (Part I) contain essentially the ``local'' ingredients of the program. Below follows (a part of) the author's outline, taken from 0.8:
``In Chapter 1, we recall some basic facts on the differential operators, especially those in positive characteristic. Both in the description of the preliminaries and in Chapter 1, our purpose is not exhaustively cover all the material, bu only to minimally summarize what is needed to present our program and to fix our notation.''
``Chapter 2 is devoted to establishing the notion of an idealistic filtration and its fundamental properties. The most important ingredient of Chapter 2 is the analysis of the \(\mathcal D\)-saturation and the \(\mathcal R\)-saturation and that of their interaction. In our algorithm, given an idealistic filtration, we always look for its bi-saturation, called the \(\mathcal B\)-saturation, which is both \(\mathcal D\)-saturated and \(\mathcal R\)-saturated and which is minimal among such containing the original idealistic filtration. The existence of the \(\mathcal B\)-saturation is theoretically clear. However, we do not know a priori whether we can reach the \(\mathcal B\)-saturation by a repetition of \(\mathcal D\)-saturations and \(\mathcal R\)-saturations starting from the given idealistic filtration, even after infinitely many times. The main result here is that the \(\mathcal B\)-saturation is actually realized if we take the \(\mathcal D\)-saturation and then the \(\mathcal R\)-saturation of the given one, each just once in this order. In our algorithm, we do not deal with an arbitrary idealistic filtration, but only with those which are generated by finitely many elements with rational levels. We say they are of r.f.g. type (short for `rationally and finitely generated'). It is then a natural and crucial question if the propery of being of r.f.g. type is stable under \(\mathcal D\)-saturation and \(\mathcal R\)-saturation.''
``In Chapter 3, through the analysis of the leading terms of an idealistic filtration (which is \(\mathcal D\)-saturated), we define the notion of a leading generator system, which \dots plays the role of a collective substitute for the notion of a hypersurface of maximal contact.
Chapter 4 is the culmination of part I, establishing the new nonsingularity principle of the center for an idealistic filtration which is \(\mathcal B\)-saturated. Its proof is given via three somewhat technical but important lemmas, which we will use again later in the series of papers.''
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A few years ago H. Kawanoue announced a program (the \textit{Idealistic Filtration Program}) to resolve singularities of algebraic varieties which might work even if the base field is of positive characteristic. It is expressed in terms of \textit{idealistic filtrations}. If \(U\) is an open set of an affine variety \(W\), with coordinate ring \(R\), an idealistic filtration on \(W\) (or on \(R\)) is an indexed family of ideals \(I_a\) of \(R\), where \(a\) is a non-negative real number, such that \(I_a \subset I_b\) if \(a \geq b\), \(I_a I_b \subset I_{a+b}\) and \(I_0=R\). One defines the support, or singular locus, of an idealistic filtration. Then the goal is to give an algorithm allowing us to resolve the filtration, i.e., to find blowing-ups with smooth centres such that, taking suitable transforms, eventually the support becomes empty. The centres will be given as the set of points where certain \textit{algorithmic resolution functions}, with values in a suitable totally ordered set, reach the maximum value. From this, a similar algorithm to resolve singularities of varieties will follow.
To construct such resolutions functions, given an idealistic filtration \(\mathcal I\) on the variety \(W\) a \textit{strand of invariants} is attached to each point \(P\) of \(W\). Each step of the strand involves a triplet \(\sigma\), \(\tilde {\mu}\) and \(s\) of invariants. In the definition of \(\sigma\) and \(\tilde \mu\) a fundamental role is played by \(L(\mathcal I)\), a certain graded sub-algebra of \(Gr_{M}(A)\), the graded ring of \((A,M)\) (the local ring of \(W\) at a closed point \(P\)). Of particular importance is the notion of \textit{leading generator system}. These are elements in suitable ideals \(I_j\) of \(\mathcal I\) whose initial forms in the graded ring \(Gr_{M}(A)\) are generators of \(L(\mathcal I)\), satisfying certain ``minimality'' conditions. This notion seems essential to avoid the explicit use of hypersurfaces of maximal contact, a usual tool in desingularization techniques which is not always available in positive characteristic. Leading generator systems exist when the filtration \(\mathcal I\) is \(D\)-saturated, that is closed under the application of differential operators.
The resolution functions thus obtained have the fundamental property necessary in problems of this type. Namely, their values strictly decrease after performing a blowing-up with the appropriate algorithmic center. But still, when the characteristic is positive, there is no warranty that the process terminates.
Kawanoue, working in cooperation with K. Matsuki, plans to present the details of the program in a series of four papers. In the first one, \textit{H. Kawanoue} [Publ. Res. Inst. Math. Sci. 43, 819--909 (2007; Zbl 1170.14012)], discussed the main lines of the program, in particular what was mentioned so far. In the second part of the series (the article under review) the authors carefully study properties of the invariants \(\sigma\) and \(\tilde \mu\) (at ``time zero'', i.e., at the beginning of the process). For instance, they check the upper-semicontinuity of the invariants \(\sigma\) and \(\tilde \mu\). They also discuss certain power series expansions, but nor necessarily in terms of a regular system of parameters, but rather relative to a leading generating system; something that will be important later. Many of these results had been announced (without proof) in Part I.
In an Appendix, they include an important improvement of a result in Part I. Over there, they prove what they call \textit{the new non-singularity principle}. It says (among other things) that given an idealistic filtration \(\mathcal I\) which is both \(D\)-saturated and \(R\)-saturated (i.e., closed under the operations of differentiation and extraction of roots) and satisfies, for some closed point, \(\tilde \mu (z) = \infty\), then on a neighbourhood of \(z\) the support of \(\mathcal I\) is non-singular. From this result it follows that the centres involved in the application of the algorithm are indeed regular, which is a non-trivial result. In this Appendix a similar result is proved, but avoiding the hypothesis of \(R\)-saturation. This improvement could be an important step toward proving that their algorithm (or a slight variation thereof) will terminate after a finite number of steps, thus concluding a proof of resolution in positive characteristic. Indeed, the assumption on \(R\)-saturation seems to be an obstacle to prove the termination of the algorithm.
A subsection (numbered 0.3) resumes the current status of the Idealistic Filtration Program. For each cone in \(\mathbb R^{n+1}\), there exists a complete affine hypersphere of hyperbolic type asymptotic to it. On the other hand, complete affine hyperspheres of elliptic or parabolic types are necessarily hyperquadrics. In this paper, the author defines the concept of affine hemispheres, which are (incomplete) affine hyperspheres \(M\) of elliptic type, locally strongly convex, whose boundary is contained in a hyperplane \(L\) and encloses a convex region \(K\subset L\), called the anchor of \(M\), such that \(K\cup M\) is the boundary of the convex region \(\tilde K\subset\mathbb R^{n+1}\).
By using techniques of convex geometry, the author proves that given a compact convex set \(K\) contained in a hyperplane \(L\), there exists an affine hemisphere \(M\) with anchor \(K\), unique up to an affine transformation. Moreover, the centre of \(M\) is the Santaló point of \(K\). This result shows that, although complete affine spheres of elliptic type are rare, there are plenty of affine hemispheres of elliptic type.
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L'auteur étudie le choc entre deux systèmes variables tels que: deux boules de glace en fusion, des fusées du modèle Ward I et II à propellant solide ou liquide; de tels systèmes sont appelés systèmes variables du type \({\mathcal F}(t)\). Cette recherche montre que le choc entre tels systèmes présente des aspects beaucoup plus variés que celui de deux solides rigides constants; aspects qui peuvent être exploités dans le domaine des servo-mécanismes par example. L'auteur rappelle tout d'abord la théorie des chocs et percussions sur les systèmes variables de particules, en particulier les théorèmes d'Euler établis dans [\textit{E. Ligan}, Afr. Mat., Sér. III 9, 47-66 (1998; Zbl 0927.70016)]. Dans l'application des théorèmes d'Euler l'auteur utilise l'écriture sous forme matricielle des équations de mouvement. The paper extends the general theory of collisions and percussions from non-variable systems (constant systems) to the more general variable systems of particles, that is, systems where there is a transfer of particles between the initial system and exterior media. The author establishes Euler's laws for collisions and percussions for this kind of systems of particles.
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L'auteur étudie le choc entre deux systèmes variables tels que: deux boules de glace en fusion, des fusées du modèle Ward I et II à propellant solide ou liquide; de tels systèmes sont appelés systèmes variables du type \({\mathcal F}(t)\). Cette recherche montre que le choc entre tels systèmes présente des aspects beaucoup plus variés que celui de deux solides rigides constants; aspects qui peuvent être exploités dans le domaine des servo-mécanismes par example. L'auteur rappelle tout d'abord la théorie des chocs et percussions sur les systèmes variables de particules, en particulier les théorèmes d'Euler établis dans [\textit{E. Ligan}, Afr. Mat., Sér. III 9, 47-66 (1998; Zbl 0927.70016)]. Dans l'application des théorèmes d'Euler l'auteur utilise l'écriture sous forme matricielle des équations de mouvement. In the present study the authors use positivity properties to prove a general comparison principle for equations of the form
\[
-\Delta_p u+ V|u|^{p-2} u= 0\quad\text{in }\Omega,
\]
where \(p\in (1,\infty)\), \(\Delta_p(u):= \nabla\cdot(|\nabla u|^{p-2}\nabla u)\) and \(V\in L^\infty_{\text{loc}}(\Omega;\mathbb{R})\).
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Topological field theories can increase their structure, intrinsically, when we add topological or geometric structure to the bordisms. In dimension 2, we have the \(r\)-spin structures, \(0<r<\infty\), where the bordisms are equipped with a reduction of structure in the \(r\)-fold cover \(\mathrm{Spin}_r(n)\rightarrow \mathrm{SO}(n)\). These are called 2-dimensional \(r\)-spin topological field theories. There are two points of view: a combinatorial model which takes a polygonal cell decomposition of an oriented surface, called PLCW by \textit{A. Kirillov jun.} [Algebr. Geom. Topol. 12, No. 1, 95--108 (2012; Zbl 1283.57026)], equipped with a marking associated to the edges providing the corresponding \(r\)-spin structure; the second combinatorial model considers a surface \(S\) with a finite set of punctures \(M\) and an embedded graph \(\Gamma\) in \(S\setminus M\), with \(|\Gamma|\simeq S\setminus M\), together with torsor trivialization for vertices and edges. The present paper establishes the connection between the two combinatorial models for \(r\)-spin surfaces. This connection is used to proceed to classify open-closed \(r\)-spin topological field theories. The main result of the paper is an equivalence between the symmetric monoidal categories of open-closed \(r\)-spin topological field theories and knowledgeable Frobenius algebras. We introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be obtained from each other by a sequence of certain ``elementary'' moves. This definition is motivated by the needs of Topological Quantum Field Theory, especially extended theories as defined by Lurie.
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Topological field theories can increase their structure, intrinsically, when we add topological or geometric structure to the bordisms. In dimension 2, we have the \(r\)-spin structures, \(0<r<\infty\), where the bordisms are equipped with a reduction of structure in the \(r\)-fold cover \(\mathrm{Spin}_r(n)\rightarrow \mathrm{SO}(n)\). These are called 2-dimensional \(r\)-spin topological field theories. There are two points of view: a combinatorial model which takes a polygonal cell decomposition of an oriented surface, called PLCW by \textit{A. Kirillov jun.} [Algebr. Geom. Topol. 12, No. 1, 95--108 (2012; Zbl 1283.57026)], equipped with a marking associated to the edges providing the corresponding \(r\)-spin structure; the second combinatorial model considers a surface \(S\) with a finite set of punctures \(M\) and an embedded graph \(\Gamma\) in \(S\setminus M\), with \(|\Gamma|\simeq S\setminus M\), together with torsor trivialization for vertices and edges. The present paper establishes the connection between the two combinatorial models for \(r\)-spin surfaces. This connection is used to proceed to classify open-closed \(r\)-spin topological field theories. The main result of the paper is an equivalence between the symmetric monoidal categories of open-closed \(r\)-spin topological field theories and knowledgeable Frobenius algebras. The paper presents an algebraic analysis for the correctness of prefix-based adders. In contrast to using higher-order functions and rewriting systems previously, we harness first-order recursive equations for correctness proof. A new carry operator is defined in terms of a semi-group with the set of binary bits. Both sequential and parallel addition algorithms are formalized and analyzed. The formal analysis on some special prefix adder circuits demonstrates the effectiveness of our algebraic approach. This study lays an underpinning for further understanding on computer arithmetic systems.
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The results of this interesting and well-written paper are concentrated around the following question: Under which conditions is an ordered field viewed as a topological field with respect to the interval topology metrizable?
For an ordered field \(K\) the following conditions are equivalent: (i) \(K\) is metrizable; (ii) \(K\) contains an infinite convergent sequence; (iii) \(K\) is a \(k\)-space (in particular, first countable space); (iv) \(K\) contains an infinite compact subspace; \((v)\) \(K\) contains a non-discrete countable subspace.
Furthermore, the following hold: (1) \(K\) is metrizable if and only if it contains a countable subset having an accumulation point. (2) \(K\) is Archimedean if and only if it contains a subset \(Q\) having an accumulation point in \(K\). (3) \(K\) is Dedekind-complete if and only if every bounded infinite subset of \(Q\) has an accumulation point in \(K\).
Related results can be found in [\textit{D. B. Shakhmatov}, Trans. Mosc. Math. Soc. 1988, 251--261 (1988); translation from Tr. Mosk. Mat. O.-va 50, 249--259 (1987; Zbl 0671.54019)]. In this very interesting paper the author deals with cardinal functions on topological fields and their subspaces. He proves the results which were announced in his note [Dokl. Akad. Nauk SSSR 271, 1332-1336 (1983; Zbl 0543.54005)]. In particular, he shows Theorem 5. A pseudocomact subspace of a topological field is metrizable. He also gives a survey of (new) open problems related to topics considered in his papers.
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The results of this interesting and well-written paper are concentrated around the following question: Under which conditions is an ordered field viewed as a topological field with respect to the interval topology metrizable?
For an ordered field \(K\) the following conditions are equivalent: (i) \(K\) is metrizable; (ii) \(K\) contains an infinite convergent sequence; (iii) \(K\) is a \(k\)-space (in particular, first countable space); (iv) \(K\) contains an infinite compact subspace; \((v)\) \(K\) contains a non-discrete countable subspace.
Furthermore, the following hold: (1) \(K\) is metrizable if and only if it contains a countable subset having an accumulation point. (2) \(K\) is Archimedean if and only if it contains a subset \(Q\) having an accumulation point in \(K\). (3) \(K\) is Dedekind-complete if and only if every bounded infinite subset of \(Q\) has an accumulation point in \(K\).
Related results can be found in [\textit{D. B. Shakhmatov}, Trans. Mosc. Math. Soc. 1988, 251--261 (1988); translation from Tr. Mosk. Mat. O.-va 50, 249--259 (1987; Zbl 0671.54019)]. We present a proof of the exponential convergence to equilibrium of single-spin-flip stochastic dynamics for the two-dimensional Ising ferromagnet in the low-temperature case with not too small external magnetic field \(h\) uniformly in the volume and in the boundary conditions.
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For a polynomial \(P\) of degree \(n\) and an \(m\)-tuple \(\Lambda = (\lambda_1, \dots, \lambda_m)\) of distinct complex numbers, the dope matrix of \(P\) with respect to \(\Lambda\) is \(D_P(\Lambda) = (\delta_{ij})_{i \in [1, m], j \in [0, n]}\), where \(\delta_{ij} = 1\) if \(P^{(j)}(\lambda_i) = 0\), and \(\delta_{ij} = 0\) otherwise. Our first result is a combinatorial characterization of the 2-row dope matrices (for all pairs \(\Lambda)\); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of \(m \times(n + 1)\) dope matrices, and we show that the number of \(m \times(n + 1)\) dope matrices for a fixed \(m\)-tuple \(\Lambda\) is maximized when \(\Lambda\) is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems. An \(m \times(n + 1)\) multiplicity matrix is a matrix \(M = \begin{pmatrix} \mu_{i , j} \end{pmatrix}\) with rows enumerated by \(i \in \{1, 2, \ldots, m \}\) and columns enumerated by \(j \in \{0, 1, \ldots, n \}\) whose coordinates are nonnegative integers satisfying the following two properties: (1) If \(\mu_{i , j} \geq 1\), then \(j \leq n - 1\) and \(\mu_{i , j + 1} = \mu_{i , j} - 1\), and (2) \(\text{colsum}_j(M) = \sum_{i = 1}^m \mu_{i , j} \leq n - j\) for all \(j\).
Let \(K\) be a field of characteristic 0 and let \(f(x)\) be a polynomial of degree \(n\) with coefficients in \(K\). Let \(f^{( j )}(x)\) be the \(j\)th derivative of \(f(x)\). Let \({\Lambda} = (\lambda_1, \ldots, \lambda_m)\) be a sequence of distinct elements of \(K\). For \(i \in \{1, 2, \ldots, m\}\) and \(j \in \{1, 2, \ldots, n\}\), let \(\mu_{i, j}\) be the multiplicity of \(\lambda_i\) as a zero of the polynomial \(f^{(j)}(x)\). The \(m \times(n + 1)\) matrix \(M_f({\Lambda}) = \begin{pmatrix} \mu_{i, j} \end{pmatrix}\) is called the \textit{multiplicity matrix of the polynomial}\(f(x)\)\textit{with respect to \(\Lambda \)}. Conditions for a multiplicity matrix to be the multiplicity matrix of a polynomial are established, and examples are constructed of multiplicity matrices that are not multiplicity matrices of polynomials. An open problem is to classify the multiplicity matrices that are multiplicity matrices of polynomials in \(K [x]\) and to construct multiplicity matrices that are not multiplicity matrices of polynomials.
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For a polynomial \(P\) of degree \(n\) and an \(m\)-tuple \(\Lambda = (\lambda_1, \dots, \lambda_m)\) of distinct complex numbers, the dope matrix of \(P\) with respect to \(\Lambda\) is \(D_P(\Lambda) = (\delta_{ij})_{i \in [1, m], j \in [0, n]}\), where \(\delta_{ij} = 1\) if \(P^{(j)}(\lambda_i) = 0\), and \(\delta_{ij} = 0\) otherwise. Our first result is a combinatorial characterization of the 2-row dope matrices (for all pairs \(\Lambda)\); using this characterization, we solve the associated enumeration problem. We also give upper bounds on the number of \(m \times(n + 1)\) dope matrices, and we show that the number of \(m \times(n + 1)\) dope matrices for a fixed \(m\)-tuple \(\Lambda\) is maximized when \(\Lambda\) is generic. Finally, we resolve an ``extension'' problem of Nathanson and present several open problems. H. A. Schwarz proved stability of a minimal surface in 3-dimensional Euclidean
space \(E^3\), when this minimal surface could be included in a regular family of
minimal surfaces [Collected mathematical works. Berlin: Springer (1890; JFM 22.0031.04)]. It follows from this theorem that every compact domain on a minimal surface \(z = z(x^1, x^2)\) is stable. Notice, that the question of minimal surface stability was considered in several articles as well as the existence and applications of stable minimal surfaces.
Here we give the generalizations of this theorem for the cases of minimal hypersurfaces in a Riemannian space and for 2-dimensional surfaces in 4-dimensional Riemannian space.
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Let \((a_{jk})_{1\leq j,k\leq m}\) be a Hermitian matrix whose entries are functions on \([0,b)\) with \(a_{mm}=1\), and consider the ordinary differential operator \({\mathcal P}[u]=\sum_{j,k=0}^mD^ja_{jk}D^ku\) on \([0,b)\), with \(D=i(d/dx)\), \(0<b\leq\infty\). Consider a selfadjoint operator \(T\) (on a space of vector functions) naturally defined by this differential operator with suitable boundary conditions and suitable local integrability assumptions on the function matrix \((a_{jk})_{1\leq j,k\leq m}\).
The main result here is that the operator \(T\) is uniquely determined by its spectral measure up to unitary equivalence by a unitary multiplication operator. The method of proof relies on an appropriate version of the Paley-Wiener theorem, which is also obtained in the present paper. This method is similar to the one used by \textit{C.~Bennewitz} [Y. Karpeshina (ed.) et al., Advances in differential equations and mathematical physics. Proceedings of the 9th UAB international conference, University of Alabama, Birmingham, AL, USA, March 26--30, 2002. Providence, RI.: American Mathematical Society (AMS). Contemp. Math. 327, 21--31 (2003; Zbl 1070.34019)] in the case of second-order ordinary differential operators. Consider the boundary value problem
\[
\begin{cases} -(pu')' +q u =\lambda u & \text{on } [0, b), \\ u(0) \cos \alpha +p u'(0) \sin\alpha =0, \end{cases} \tag{1}
\]
where \(1/p\) ands \(q\) are real-valued, locally integrable functions on \([0, b)\), and \(\alpha\in [0, \pi]\) is fixed; and denote by \(T\) one of the selfadjoint realizations of (1). The main result of the paper is a generalization of the Paley-Wiener theorem to the case of generalized Fourier transform associated to \(T\).
One of the main ingredients is the asymptotic formulas for the Weyl solution and Titchmarsh-Weyl \(m\)-function associated to \(T\), obtained by the author in his paper [Proc. Lond. Math. Soc., III. Ser. 59, No. 2, 294--338 (1989; Zbl 0681.34023)]. Indeed, the relation between the support of a function and the behaviour of its generalized Fourier transform is formulated in terms related to these asymptotics. As an application, the author obtains a uniqueness (modulo unitary Liouville transforms) result for the inverse spectral problem associated to (1). Some other generalizations are discussed briefly.
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Let \((a_{jk})_{1\leq j,k\leq m}\) be a Hermitian matrix whose entries are functions on \([0,b)\) with \(a_{mm}=1\), and consider the ordinary differential operator \({\mathcal P}[u]=\sum_{j,k=0}^mD^ja_{jk}D^ku\) on \([0,b)\), with \(D=i(d/dx)\), \(0<b\leq\infty\). Consider a selfadjoint operator \(T\) (on a space of vector functions) naturally defined by this differential operator with suitable boundary conditions and suitable local integrability assumptions on the function matrix \((a_{jk})_{1\leq j,k\leq m}\).
The main result here is that the operator \(T\) is uniquely determined by its spectral measure up to unitary equivalence by a unitary multiplication operator. The method of proof relies on an appropriate version of the Paley-Wiener theorem, which is also obtained in the present paper. This method is similar to the one used by \textit{C.~Bennewitz} [Y. Karpeshina (ed.) et al., Advances in differential equations and mathematical physics. Proceedings of the 9th UAB international conference, University of Alabama, Birmingham, AL, USA, March 26--30, 2002. Providence, RI.: American Mathematical Society (AMS). Contemp. Math. 327, 21--31 (2003; Zbl 1070.34019)] in the case of second-order ordinary differential operators. The secondary instability was investigated in high-speed boundary layers over flat plates by numerical methodology. The numerical simulation suggests that the main streaky structures in 2D hypersonic boundary transition process are resulted from the secondary instability theory caused by the primary 2D Mack mode. The secondary instability analysis found that a new family called fundamental family of solutions was found which is the least stable secondary instability when the amplitude of the primary mode instability reaches a threshold value. It is helpful for us to understand the fundamental breakdown in hypersonic boundary layers.
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For a given ring \(R\), neither the injective hull \(E(R_R)\) of \(R_R\) nor the maximal right ring of quotients \(Q(R)\) of \(R\) provides us with an overlying structure which allows for an effective transfer of some types of information between \(R\) and itself. But, there exists clear disparity between \(R\) and \(Q(R)\). It was this kind of disparity that motivated us to consider rings from a ``distinguished'' class that are intermediate between the base ring \(R\) and \(Q(R)\) or \(E(R_R)\).
Such an intermediate ring \(T\) from a distinguished class possesses the (desirable) properties that identify the class, and since it is ``intermediate'' it is generally closer to \(R\) than either \(Q(R)\) or \(E(R_R)\). Hence there is some hope that the desirable properties of the specific class and the closeness of \(T\) to \(R\) will enable a significant transfer of information. Usually this information transfer can be enhanced by: (1) choosing a distinguished class that generalizes some property (or properties) or is related to the class of right self-injective rings; (2) finding (if it exists) a ``minimal'' element (right ring hull) from the distinguished class; or (3) finding (if they exist) elements of the distinguished class that are ``minimally'' generated by \(R\) and some subset of \(E(R_R)\) (pseudo right ring hull).
In the paper under review, the authors use the general approach and the theory of ring hulls introduced earlier [\textit{G. F. Birkenmeier, J. K. Park} and \textit{S. T. Rizvi}, J. Algebra 304, No. 2, 633-665 (2006; Zbl 1161.16002)] to focus on ring hulls belonging to the class of quasi-Baer rings or the class of right FI-extending rings and investigate \(R\mathbf B(Q (R))\).
Let \(R\) be an associative ring with unit. The authors investigate and determine ``minimal'' right essential overrings belonging to certain classes which are generated by \(R\) and subsets of the central idempotents of \(Q(R)\), where \(Q(R)\) is the maximal right ring of quotients of \(R\). It is shown the existence of and characterization of a quasi-Baer hull and a right FI-extending hull for every semiprime ring.
Their results include: (i) \(R\mathbf B(Q(R))\) (i.e., the subring of \(Q(R)\) generated by \(\{re\mid r\in R\) and \(e\in\mathbf B(Q(R))\}\), where \(\mathbf B(Q(R))\) is the set of all central idempotents of \(Q(R)\)) is the smallest quasi-Baer and the smallest right FI-extending right ring of quotients of a semiprime ring \(R\) with unity. In this case, various overrings of \(R\mathbf B(Q(R))\), including all right essential overrings of \(R\) containing \(\mathbf B(Q(R))\), are also quasi-Baer and right FI-extending; (ii) lying over, going up, and incomparability of prime ideals, various regularity conditions, and classical Krull dimension transfer between \(R\) and \(R\mathbf B(Q(R))\); and (iii) the existence of a boundedly centrally closed hull for every \(C^*\)-algebra and a complete characterization for an intermediate \(C^*\)-algebra between a \(C^*\)-algebra \(A\) and its local multiplier \(C^*\)-algebra \(M_{\text{loc}}(A)\) to be boundedly centrally closed. In this very lengthy and quite interesting paper, the authors consider the following two problems.
Problem I: For a given ring \(R\) and a class \(\mathbf k\) of rings: (i) to find conditions to ensure the existence of right rings of quotients and that of right essential overrings of \(R\) which are in some sense ``minimal'' with respect to belonging to the class \(\mathbf k\). (ii) Characterizing right rings of quotients and right essential overrings of \(R\) which lie in the class \(\mathbf k\), possibly using ``minimal'' ones obtained in part (i).
Problem II: Given a ring \(S\) and a class \(\mathbf k\), to find rings \(T\in\mathbf k\) such that \(S=Q(T)\) (the maximal right ring of quotients of \(T\)).
In Section 2 the authors develop theory to tackle problems I and II mentioned above. Their main theorem in this section is the following. Let \(R\) be a ring such that its maximal right ring of quotients \(Q(R)\) is Abelian. If further \(R\) is a right Ore ring such that \(r_R(x)=0\) implies \(l_R(x)=0\) for each \(x\in R\) and \(Z(R_R)\) has finite right uniform dimension, then \(Q(R)\) is right extending if and only if \(\widehat Q_{Con}(R)\) exists and is equal to \(H_1\oplus H_2\) for some right continuous strongly regular ring \(H_1\) and a direct sum \(H_2\) of right continuous local rings.
They use the theory developed in Section 2, answer Problems I and II stated above when the class \(\mathbf k\) is the class of Baer rings, (FI) extending rings or the class of right (semi-)hereditary rings and the ring \(R\) (in Problem I) or the ring \(T\) (in Problem II) is a subring of a \(2\times 2\) matrix ring. In this direction, they characterize a right extending ring whose maximal ring of quotients is \(M_2(D)\) for a division ring \(D\). They provide answer to Problem I for the case when \(\mathbf k\) is the class of all right extending rings and \(R=T_2(W)\) by characterizing the right extending right rings of quotients which are intermediate between \(T_2(W)\) and \(M_2(W)\) where \(W\) is a local right finitely \(\Sigma\)-extending ring. They conclude the paper with some open problems.
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For a given ring \(R\), neither the injective hull \(E(R_R)\) of \(R_R\) nor the maximal right ring of quotients \(Q(R)\) of \(R\) provides us with an overlying structure which allows for an effective transfer of some types of information between \(R\) and itself. But, there exists clear disparity between \(R\) and \(Q(R)\). It was this kind of disparity that motivated us to consider rings from a ``distinguished'' class that are intermediate between the base ring \(R\) and \(Q(R)\) or \(E(R_R)\).
Such an intermediate ring \(T\) from a distinguished class possesses the (desirable) properties that identify the class, and since it is ``intermediate'' it is generally closer to \(R\) than either \(Q(R)\) or \(E(R_R)\). Hence there is some hope that the desirable properties of the specific class and the closeness of \(T\) to \(R\) will enable a significant transfer of information. Usually this information transfer can be enhanced by: (1) choosing a distinguished class that generalizes some property (or properties) or is related to the class of right self-injective rings; (2) finding (if it exists) a ``minimal'' element (right ring hull) from the distinguished class; or (3) finding (if they exist) elements of the distinguished class that are ``minimally'' generated by \(R\) and some subset of \(E(R_R)\) (pseudo right ring hull).
In the paper under review, the authors use the general approach and the theory of ring hulls introduced earlier [\textit{G. F. Birkenmeier, J. K. Park} and \textit{S. T. Rizvi}, J. Algebra 304, No. 2, 633-665 (2006; Zbl 1161.16002)] to focus on ring hulls belonging to the class of quasi-Baer rings or the class of right FI-extending rings and investigate \(R\mathbf B(Q (R))\).
Let \(R\) be an associative ring with unit. The authors investigate and determine ``minimal'' right essential overrings belonging to certain classes which are generated by \(R\) and subsets of the central idempotents of \(Q(R)\), where \(Q(R)\) is the maximal right ring of quotients of \(R\). It is shown the existence of and characterization of a quasi-Baer hull and a right FI-extending hull for every semiprime ring.
Their results include: (i) \(R\mathbf B(Q(R))\) (i.e., the subring of \(Q(R)\) generated by \(\{re\mid r\in R\) and \(e\in\mathbf B(Q(R))\}\), where \(\mathbf B(Q(R))\) is the set of all central idempotents of \(Q(R)\)) is the smallest quasi-Baer and the smallest right FI-extending right ring of quotients of a semiprime ring \(R\) with unity. In this case, various overrings of \(R\mathbf B(Q(R))\), including all right essential overrings of \(R\) containing \(\mathbf B(Q(R))\), are also quasi-Baer and right FI-extending; (ii) lying over, going up, and incomparability of prime ideals, various regularity conditions, and classical Krull dimension transfer between \(R\) and \(R\mathbf B(Q(R))\); and (iii) the existence of a boundedly centrally closed hull for every \(C^*\)-algebra and a complete characterization for an intermediate \(C^*\)-algebra between a \(C^*\)-algebra \(A\) and its local multiplier \(C^*\)-algebra \(M_{\text{loc}}(A)\) to be boundedly centrally closed. \(L^2\)-methods were introduced into topology by Atiyah in the 1970's, from an analytic perspective (elliptic operators on infinite covering spaces of Riemannian manifolds). He defined \(L^2\)-Betti numbers by means of the von Neumann dimension associated to the von Neumann algebra \({\mathcal N}(\pi)\) of the covering group \(\pi\). Subsequently, two further types of \(L^2\)-invariants were introduced: the Novikov-Shubin invariants, which measure the asymptotic behaviour of the spectrum of the Laplace operator, and the \(L^2\)-torsion, which is an analogue of the Reidemeister torsion. Although these invariants were all originally defined in terms of elliptic operators on the \(L^2\)-completions of spaces of differential forms on the covering space, the analysis has been stripped away and the method given a more algebraic appearance, notably by Lück and Farber.
The von Neumann algebra \({\mathcal N}(\pi)\) is the algebra of bounded (left) \(\mathbb{C}[\pi]\)-linear operators on \(\ell^2(\pi)\), the Hilbert space completion of \(\mathbb{C}[\pi]\). The von Neumann trace gives rise to a dimension function on finitely generated projective \({\mathcal N}(\pi)\)-modules, with values in \([0,\infty)\), and which extends to a \([0,\infty]\)-valued function on arbitrary \({\mathcal N}(\pi)\)-modules. The \(L^2\)-cohomology modules of a cell-complex \(X\) with finite skeleta and \(\pi_1(X)=\pi\) are determined by the complex of cellular cocycles with square summable values on the universal covering of \(X\), and the \(L^2\)-Betti numbers of \(X\) are the dimensions of these modules. They are in general non-negative real numbers, rather than integers, but their alternating sum is the Euler characteristic \(\chi(X)\), if \(X\) is finitely dominated, and they satisfy Poincaré duality, if \(X\) is a Poincaré duality complex. In addition, they are multiplicative in finite covers, and in degrees 0 and 1 depend only on the fundamental group. In Farber's version of the theory the Novikov-Shubin invariants are invariants of the ``torsion'' of the \(L^2\)-cohomology modules. (The \(L^2\)-Betti numbers are invariants of the reduced \(L^2\)-cohomology, obtained by factoring out the torsion part). The \(L^2\)-torsion is defined when all the \(L^2\)-Betti numbers are 0. (We shall not attempt detailed definitions here).
The above-mentioned properties of the \(L^2\)-Betti numbers all hold for formal reasons, and the proofs are straightforward. The hard work lies in determing the nature of these invariants, in particular whether they are rational, integral or 0. The Atiyah conjecture is that the \(L^2\)-Betti numbers of a finite complex are rational, and are integral if the fundamental group is torsion-free. This is known to imply that the complex group rings of torsion-free groups have no zero-divisors. It is similarly conjectured that the Novikov-Shubin invariants of a finite complex are rational numbers (when finite). The known properties of Riemannian manifolds of constant negative curvature suggest the Singer conjecture, that the \(L^2\)-Betti numbers of an aspherical closed manifold should be nonzero only in the middle dimension. In particular, if the dimension is odd the \(L^2\)-Betti numbers should all be 0. There is then a related conjecture for the sign of the \(L^2\)-torsion of such a manifold. Vanishing criteria for some or all of the \(L^2\)-Betti numbers have surprisingly strong consequences, particularly in combinatorial group theory and low-dimensional topology.
The present book is the first substantial monograph on this topic. There are 17 numbered chapters: \S 0. Introduction; \S 1. \(L^2\)-Betti numbers; \S 2. Novikov-Shubin invariants; \S 3. \(L^2\)-torsion; \S 4. \(L^2\)-Invariants of 3-manifolds; \S 5. \(L^2\)-Invariants of symmetric spaces; \S 6. \(L^2\)-Invariants for general spaces with group action; \S 7. Application to groups; \S 8. The algebra of affiliated operators; \S 9. Middle \(K\)- and \(L\)-theory of von Neumann algebras; \S 10. The Atiyah conjecture; \S 11. The Singer conjecture; \S 12. The zero-in-the-spectrum conjecture; \S 13. The approximation and determinant conjectures; \S 14. \(L^2\)-Invariants and the simplicial volume; \S 15. Survey on other topics related to \(L^2\)-invariants; and \S 16. Solutions to the exercises; and a list of over 500 references.
This is an impressive account of much of what is presently known about these invariants, focused on the major applications and a number of key conjectures, with strong implications for other areas of mathematics. It combines features of a text and a reference work; to a considerable degree the chapters can be read independently, and there are numerous nontrivial exercises, with nearly 50 pages of detailed hints at the end. As the author states, it would require another book to treat in detail the topics outlined in \S 15. In particular, the applications to the classical knot concordance group (due to Cochran, Orr and Teichner) are probably still evolving too rapidly to be incorporated in a reference work. However, this reviewer would have liked to have seen a more substantial discussion of Farber's approach, here confined to one paragraph in \S 6.8. (It should be noted also that operator algebra methods have had significant impact on other areas of manifold topology and \(K\)-theory, notably in connection with the Baum-Connes conjecture and the Novikov conjecture, but these applications involve \(C^*\)-algebras rather than von Neumann algebras).
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The Torelli group \(\mathcal{T}_g^b\) of a surface of genus \(g\) with \(b\) boundary components is the subgroup of the mapping class group \(\mathrm{Mod}_g^b\) acting trivially on the homology of the surface, or the kernel of the canonical map from \(\mathrm{Mod}_g^b\) to the symplectic group \(\mathrm{Sp}_{2g}(\mathbb{Z})\). It is known that the Torelli group \(\mathcal{T}_g^b\) is finitely generated for \(g \ge 3\) but not for \(g = 2\); it is not known whether \(\mathcal{T}_g^b\) is finitely presentable for \(g \ge 3\). The second homology \(H_2(G)\) of a finitely presentable group \(G\) is a finitely generated abelian group. In the present paper, the authors prove that
the second homology \(H_2(\mathcal{T}_g^b)\) of the Torelli group is finitely generated as a \(\mathbb{Z}(\mathrm{Sp}_{2g} (\mathbb{Z}))\)-module, for \(g \ge 3\) and \(b = 0\) and \(b = 1\).
For the proof, the authors define the notion of a \(\Gamma\)-equivariant presentation of a group \(G\) with an action of a group \(\Gamma\). They prove that, under some technical conditions, for a group \(G\) with a finite \(\Gamma\)-equivariant presentation, the second homology \(H_2(G)\) is a finitely generated \(\mathbb{Z}(\Gamma)\)-module. Since \(\mathcal{T}_g^b\) has such a finite \(\mathrm{Mod}_g^b\)-equivariant presentation for \(g \ge 3\) and \(b = 0\) and 1, this implies the result stated above (which is the case \(k = 2\) of a conjecture of Church and Farb that \(H_k(\mathcal{T}_g^b)\) is a finitely generated \(\mathbb{Z}(\mathrm{Sp}_{2g} (\mathbb{Z}))\)-module for large \(k\)). An analogous result for the Torelli subgroup of the automorphism group \(\mathrm{Aut}(F_n)\) of a free group \(F_n\) is proved in a paper by \textit{M. Day} and \textit{A. Putman} [Geom. Topol. 21, No. 5, 2851--2896 (2017; Zbl 1429.20035)]. Let \(\mathrm{IA}_n\) be the Torelli subgroup of \(\Aut(F_n)\). We give an explicit finite set of generators for \(H_2(\mathrm{IA}_n)\) as a \(\mathrm{GL}_n(\mathbb Z)\)-module. Corollaries include a version of surjective representation stability for \(H_2(\mathrm{IA}_n)\), the vanishing of the \(\mathrm{GL}_n(\mathbb Z)\)-coinvariants of \(H_2(\mathrm{IA}_n)\), and the vanishing of the second rational homology group of the level \(\ell\) congruence subgroup of \(\Aut(F_n)\). Our generating set is derived from a new group presentation for \(\mathrm{IA}_n\) which is infinite but which has a simple recursive form.
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The Torelli group \(\mathcal{T}_g^b\) of a surface of genus \(g\) with \(b\) boundary components is the subgroup of the mapping class group \(\mathrm{Mod}_g^b\) acting trivially on the homology of the surface, or the kernel of the canonical map from \(\mathrm{Mod}_g^b\) to the symplectic group \(\mathrm{Sp}_{2g}(\mathbb{Z})\). It is known that the Torelli group \(\mathcal{T}_g^b\) is finitely generated for \(g \ge 3\) but not for \(g = 2\); it is not known whether \(\mathcal{T}_g^b\) is finitely presentable for \(g \ge 3\). The second homology \(H_2(G)\) of a finitely presentable group \(G\) is a finitely generated abelian group. In the present paper, the authors prove that
the second homology \(H_2(\mathcal{T}_g^b)\) of the Torelli group is finitely generated as a \(\mathbb{Z}(\mathrm{Sp}_{2g} (\mathbb{Z}))\)-module, for \(g \ge 3\) and \(b = 0\) and \(b = 1\).
For the proof, the authors define the notion of a \(\Gamma\)-equivariant presentation of a group \(G\) with an action of a group \(\Gamma\). They prove that, under some technical conditions, for a group \(G\) with a finite \(\Gamma\)-equivariant presentation, the second homology \(H_2(G)\) is a finitely generated \(\mathbb{Z}(\Gamma)\)-module. Since \(\mathcal{T}_g^b\) has such a finite \(\mathrm{Mod}_g^b\)-equivariant presentation for \(g \ge 3\) and \(b = 0\) and 1, this implies the result stated above (which is the case \(k = 2\) of a conjecture of Church and Farb that \(H_k(\mathcal{T}_g^b)\) is a finitely generated \(\mathbb{Z}(\mathrm{Sp}_{2g} (\mathbb{Z}))\)-module for large \(k\)). An analogous result for the Torelli subgroup of the automorphism group \(\mathrm{Aut}(F_n)\) of a free group \(F_n\) is proved in a paper by \textit{M. Day} and \textit{A. Putman} [Geom. Topol. 21, No. 5, 2851--2896 (2017; Zbl 1429.20035)]. Es handelt sich um Differenzengleichungen der Form
\[
\varPhi(y(x + m), y(x+ m - 1),\dots, y (x), x) = 0,
\]
wo \(\varPhi\) ein Polynom seiner \(m + 2\) Argumente ist. Verf. denkt sich die Werte
\[
y (a), y(a + 1),\dots, y(a + m - 1)
\]
gegeben und untersucht im ersten Abschnitt das Verhalten von \(y (a +n)\) für \(n\to\infty\). Seine diesbezüglichen Resultate sind aber alle als Spezialfälle in älteren Sätzen anderer Autoren enthalten und werden auch mit wesentlicher Benutzung solcher Sätze her\-geleitet. In den folgenden Abschnitten wird nach Lösungen der Gleichung
\[
\varPhi(S_{n+m}, S_{n+m-1},\dots, S_n,n) = 0\quad (n = 1, 2, 3,\dots)
\]
gefragt, für welche der \({\lim\limits_{n\to\infty}} S_n =\alpha\) existiert. Wenn man durch die höchste vorkom\-mende Potenz von \(n\) dividiert, kann man (im allgemeinen) sofort eine algebraische Gleichung angeben, der die Zahl \(\alpha\) genügt. Wenn \(\varPhi\) insbesondere rationale Koeffi\-zienten hat, was im folgenden durchweg vorausgesetzt wird, ist also \(\alpha\) eine algebraische Zahl. Es wird dann gezeigt, daß man für jede algebraische Zahl \(\alpha\) Gleichungen \(\varPhi= 0\) angeben kann, bei welchen (neben anderen auch) Lösungen vorhanden sind mit \({\lim\limits_{n\to\infty}} S_n =\alpha\). Sodann wird die Güte der Approximation \(S_n\to\alpha\) untersucht. Dabe werden Sätze von folgender Art bewiesen: Wenn eine natürliche Zahl \(k\) und eine alge\-braische Zahl \(\alpha\) gegeben sind, so gibt es Gleichungen \(\varPhi= 0\), welche eine rationale Zahlenfolge \(S_n\) definieren, die die Zahl \(\alpha\) mit der Konvergenzordnung \(k\) approximiert. Konvergenzordnung \(k\) bedeutet dabei, daß die Größenordnung des Fehlers \(|S_{n+1}-\alpha|\) die \(k\)-te Potenz von \(|S_n-\alpha|\) ist. Von weiteren Sätzen aus diesem Ideenkreis sei noch folgender als Beispiel angeführt: Es gibt Gleichungen \(\varPhi= 0\) von höherer als erster Ordnung, die rationale \(S_n\) definieren, die gegen eine algebraische Zahl \(\alpha\) mit höherer Konvergenzordnung konvergieren als die \(S_n\) einer jeden Gleichung erster Ordnung und gleichen Grades.
| 0 |
In this paper under review, the authors study the following non-linear differential equation
\[
f^{n}+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z},
\]
where \(n\geq 2\) is an integer, \(p_{1},\) \(p_{2}\) and \(\alpha _{1},\) \(\alpha _{2}\) are non-zero constants and \(P_{d}\left( z,f\right) \) is a differential polynomial in \(f\) of degree \(d\). The authors find the forms of meromorphic solutions with few poles of the above equation when \(d=n-1\) under some restrictions on \(\alpha _{1},\) \(\alpha _{2},\) The Theorems 1.1 and 1.2 obtained extend the result established by \textit{P. Li} [J. Math. Anal. Appl. 375, No. 1, 310--319 (2011; Zbl 1206.30046)] provided \(\alpha _{1}\neq \) \(\alpha _{2}\) and \(d\leq n-2\). Some examples are given to illustrate the results. We analyze the transcendental entire solutions of nonlinear differential equations of the type
\[
f^n(z)+P(f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}
\]
in the complex plane, where \(p_{1}, p_{2}\) and \(\alpha _{1}, \alpha _{2}\) are nonzero constants, and \(P(f)\) denotes a differential polynomial in \(f\), of degree at most \(n - 1\) and with small functions of \(f\) as coefficients.
| 1 |
In this paper under review, the authors study the following non-linear differential equation
\[
f^{n}+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z},
\]
where \(n\geq 2\) is an integer, \(p_{1},\) \(p_{2}\) and \(\alpha _{1},\) \(\alpha _{2}\) are non-zero constants and \(P_{d}\left( z,f\right) \) is a differential polynomial in \(f\) of degree \(d\). The authors find the forms of meromorphic solutions with few poles of the above equation when \(d=n-1\) under some restrictions on \(\alpha _{1},\) \(\alpha _{2},\) The Theorems 1.1 and 1.2 obtained extend the result established by \textit{P. Li} [J. Math. Anal. Appl. 375, No. 1, 310--319 (2011; Zbl 1206.30046)] provided \(\alpha _{1}\neq \) \(\alpha _{2}\) and \(d\leq n-2\). Some examples are given to illustrate the results. Clinical decision science is concerned with rational clinical decisions. All branches of medical research contribute here, but controlled clinical trials of the pragmatic variety carry a particular responsibility. Usually, however, they are not conducted and reported so that they can be used directly as input to a decision analysis. We suggest that the forces of the two methodologies should be united, and point out some areas where this ``marriage'' will have a non-trivial impact: choice of end points, style of outcome recording, adaptive designs, and style of result presentation. Special attention is given to the decision-analytic setting of research priorities, the role of utility calculus in quantifying the ethical dilemmas that surround clinical trials, and the use of patient attitude towards outcomes of treatment as a covariate in its own right.
| 0 |
A partially ordered (p.o.) semigroup \((S,\cdot,\leq)\) is called pseudo-commutative if for any \(x,y\in S\) there exists \(n\geq 1\) such that \((xy)^n\leq yx^n\). If \(n\geq 2\) for all \(x,y\in S\) then these p.o. semigroups form a proper subclass of the class of all weakly commutative p.o. semigroups (which are defined by the property that for all \(x,y\in S\) there exists \(n\geq 1\) such that \((xy)^n\leq yax\) for some \(a\in S\)). A subsemigroup \(T\) of a p.o. semigroup \((S,\cdot,\leq)\) is called Archimedean if for all \(a,b\in T\) there exists some \(k\geq 1\) such that \(a^k\leq xby\) for some \(x,y\in S\). The main theorem of the paper shows that every pseudo-commutative p.o. semigroup is a semilattice of Archimedean p.o. subsemigroups and that such a decomposition is not necessarily unique. This extends results by \textit{N. Kehayopulu} [Math. Jap. 36, No. 3, 427-432 (1991; Zbl 0728.06011)] on weakly commutative p.o. semigroups with greatest element \(e\) satisfying \(a=e\) for all \(x,y\in S\) in the above definition. Furthermore, some relations with other concepts of commutativity in p.o. semigroups are investigated and several examples are provided. The weakly commutative poe-semigroups S are semilattices of archimedean semigroups, i.e. they are decomposable into archimedean subsemigroups of S [the author, \textit{P. Kiriakuli, S. Hanumantha Rao} and \textit{P. Lakshmi}, Semigroup Forum 41, No.3, 373-376 (1990; Zbl 0708.06011)]. Here we show that the same semigroups are not semilattices of archimedean poe- semigroups, in general, which means that there is no, in general, decomposition of S into archimedean po-subsemigroups \(S_{\alpha}\), \(\alpha\in Y\), of S each of which has a greatest element, say \(``e_{\alpha}''\).
| 1 |
A partially ordered (p.o.) semigroup \((S,\cdot,\leq)\) is called pseudo-commutative if for any \(x,y\in S\) there exists \(n\geq 1\) such that \((xy)^n\leq yx^n\). If \(n\geq 2\) for all \(x,y\in S\) then these p.o. semigroups form a proper subclass of the class of all weakly commutative p.o. semigroups (which are defined by the property that for all \(x,y\in S\) there exists \(n\geq 1\) such that \((xy)^n\leq yax\) for some \(a\in S\)). A subsemigroup \(T\) of a p.o. semigroup \((S,\cdot,\leq)\) is called Archimedean if for all \(a,b\in T\) there exists some \(k\geq 1\) such that \(a^k\leq xby\) for some \(x,y\in S\). The main theorem of the paper shows that every pseudo-commutative p.o. semigroup is a semilattice of Archimedean p.o. subsemigroups and that such a decomposition is not necessarily unique. This extends results by \textit{N. Kehayopulu} [Math. Jap. 36, No. 3, 427-432 (1991; Zbl 0728.06011)] on weakly commutative p.o. semigroups with greatest element \(e\) satisfying \(a=e\) for all \(x,y\in S\) in the above definition. Furthermore, some relations with other concepts of commutativity in p.o. semigroups are investigated and several examples are provided. Annotated bibliography.
| 0 |
An instance of list coloring consists of a graph \(G\) and a list \(L(v)\) of colors for each vertex \(v\) of \(G\). The list-chromatic number \(\chi^l (G)\) of \(G\) is the minimum integer \(k\) such that for every assignment of a list \(L(v)\) of size at least \(k\) to every vertex \(v\) of \(G\) there exists an acceptable coloring (without a monochromatic edge).
\textit{K. Ohba} [J. Graph Theory 40, 130-135 (2002; Zbl 1004.05030)] conjectured that if the graph \(G\) has at most \(2\chi(G) +1\) vertices then the list-chromatic number and chromatic number of \(G\) are equal. The authors prove in several steps that this conjecture is asymptotically correct for graphs \(G\) with less then \(2\chi(G)\) vertices. More precisely, using the probabilistic method they prove that for any \(0< \varepsilon <1\) there exists an \(n_0\) such that the list-chromatic number of \(G\) equals its chromatic number, if \(n_0\leq |V(G)|\leq (2-\varepsilon)\chi (G)\) holds. By the choice number \(\text{ch}(G)\) of a graph \(G\) we mean the smallest integer \(k\) such that \(G\) is \(k\)-choosable. A graph \(G\) is chromatic-choosable, if \(\text{ch}(G)\) is equal to the chromatic number \(\chi(G)\) of \(G\). It is shown, that if the chromatic number of a graph \(G\) is close enough to the order, then \(G\) is chromatic-choosable. The author proposes the following conjecture. If the order of a graph \(G\) is at most \(2\chi(G)+1\), then \(\chi(G)= \text{ch}(G)\).
| 1 |
An instance of list coloring consists of a graph \(G\) and a list \(L(v)\) of colors for each vertex \(v\) of \(G\). The list-chromatic number \(\chi^l (G)\) of \(G\) is the minimum integer \(k\) such that for every assignment of a list \(L(v)\) of size at least \(k\) to every vertex \(v\) of \(G\) there exists an acceptable coloring (without a monochromatic edge).
\textit{K. Ohba} [J. Graph Theory 40, 130-135 (2002; Zbl 1004.05030)] conjectured that if the graph \(G\) has at most \(2\chi(G) +1\) vertices then the list-chromatic number and chromatic number of \(G\) are equal. The authors prove in several steps that this conjecture is asymptotically correct for graphs \(G\) with less then \(2\chi(G)\) vertices. More precisely, using the probabilistic method they prove that for any \(0< \varepsilon <1\) there exists an \(n_0\) such that the list-chromatic number of \(G\) equals its chromatic number, if \(n_0\leq |V(G)|\leq (2-\varepsilon)\chi (G)\) holds. However, because of the limited size of the training image, only a very limited amount of multiple-point statistics can be actually inferred from the training image. Therefore, in practice, only a very few conditioning data close to the node to be simulated are used whereas farther away data carrying important large-scale information are generally ignored. Such approximation leads to inaccurate facies probability estimates, which may create ''anomalies'', for example channel disconnections, in the simulated realizations. In this paper, a method is proposed to use more data for conditioning, especially data located farther away from the node to be simulated. A measure of consistency between simulated realizations and training image is then defined, based on the number of times each simulated value, although initially identified as a conditioning datum to simulate a nearby node, had to be ignored eventually to be able to infer from the training image the conditional probability distribution at that node. Re-simulating the most inconsistent node values according to that measure enables improvement in the reproduction of training patterns without any significant increase of computation time. As an application, that post-processing process is used to remove channel disconnections from a fluvial reservoir simulated model.
| 0 |
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