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The authors introduce two types of multidimensional integral means of arbitrary real order with power weights, and prove corresponding mixed-means inequalities. The idea of introducing mixed-means with applications to derive Hardy or Levin-Cochran-Lee type inequalities has been used for the one-dimensoinal case (integrals, finite and infinite series) in a paper by the first two authors [Math. Inequal. Appl. 1, No. 4, 491-506 (1998; Zbl 0921.26015)]. Now the case of \(n\)-dimensional balls is considered, with best possible constants. The results are too complicated to be explicitely stated here. For \(a<b\) and \(f:[a,b]\to\mathbb{R}\), a non-negative, integrable function, and any \(\alpha>0\), let \[ \begin{aligned} M^{[r]}(f;a,b,\alpha) &=\left[{\alpha \over (b-a)^\alpha}\int^b_a (x-a)^{\alpha-1} f^r(x)dx \right]^{1/r}\\ \text{for \(r\neq 0;\)} &= \exp \left({\alpha \over(b-a)^\alpha} \int^b_a (x-a)^{\alpha-1} \ln f(x)dx \right), \end{aligned} \] for \(r=0\). The authors prove, e.g., the mixed-mean inequality \[ M^{[ r]}(M^{[s]}(f;a,x,\beta); a,b,\alpha)\geq M^{[s]}(M^{[r]} (f;a,x, \alpha); a,b, \beta) \] for any \(r<s\) \((r,s\neq 0)\) and \(\alpha,\beta>0\). Similar results are obtained for \(r=0\), too. These inequalities are then applied to deduce some classical results (or even improvement), as the Hardy and Carleman, or Levin-Cochran-Lee type theorems. The proofs are based mainly on Minkowski's integral inequality, properties of means, as well as certain basic theorems of real analysis.

The authors introduce two types of multidimensional integral means of arbitrary real order with power weights, and prove corresponding mixed-means inequalities. The idea of introducing mixed-means with applications to derive Hardy or Levin-Cochran-Lee type inequalities has been used for the one-dimensoinal case (integrals, finite and infinite series) in a paper by the first two authors [Math. Inequal. Appl. 1, No. 4, 491-506 (1998; Zbl 0921.26015)]. Now the case of \(n\)-dimensional balls is considered, with best possible constants. The results are too complicated to be explicitely stated here. New linear matrix inequalities-based conditions for robust stability and \(H_2\) performance analysis of linear parameter-varying uncertain systems under polytopic structure are presented in this paper. A new representation for the parameter's rate of variation is given. Besides, parameter-dependent Lyapunov functions of particular forms by considering suitable redundant system description are investigated. The proposed approach decreases the conservatism generated by the previous ones. A numerical example illustrates the efficiency of the method.

The classical Cauchy problem for the homogeneous wave equation with time dependent propagation speed \((\partial_t^2-a(t)^2\Delta)u=0\), \(a(t)>0\), is considered. The energy of the solution is expressed as the sum of elastic energy and kinetic energy. The aim of the paper is to prove an estimate of the elastic energy, similar to the one shown in an earlier paper [J. Hyperbolic Differ. Equ. 13, No. 2, 417--439 (2016; Zbl 1398.35135)], but without the assumption of monotonicity of \(a(t)\). We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is \[ u_{tt}-a(t)^2 \Delta u+b(t)u_t=0, \;u(0,x)=u_0(x), \;u_t(0,x)=u_1(x). \] We assume \(a \in (\mathbb R^+).\) Then we propose a classification of dissipation terms in non-effective and effective. In each case we derive estimates for kinetic and elastic type energies by developing a suitable WKB analysis. Moreover, we show optimality of results by the aid of scale-invariant models. Finally, we explain by an example that in some estimates a loss of regularity appears.

The classical Cauchy problem for the homogeneous wave equation with time dependent propagation speed \((\partial_t^2-a(t)^2\Delta)u=0\), \(a(t)>0\), is considered. The energy of the solution is expressed as the sum of elastic energy and kinetic energy. The aim of the paper is to prove an estimate of the elastic energy, similar to the one shown in an earlier paper [J. Hyperbolic Differ. Equ. 13, No. 2, 417--439 (2016; Zbl 1398.35135)], but without the assumption of monotonicity of \(a(t)\). A general finite-element model is proposed to deal with dynamic thermoelastic problems especially with longer transient period. The method consists of formulating and solving the problem in the Laplace transform domain by the finite element method (FEM) and then numerically inverting the transformed solution to obtain the time-domain response. Therefore, the transient solutions at any time could be evaluated directly. A number of examples are presented which demonstrate the accuracy, efficiency, and versatility of the proposed method, and show the effects of relaxation times, inertia, and thermoelastic coupling terms.

The author considers the convexity of the time maps of positive solutions of some nonlinear, two point boundary value problems \(u''+f(u)=0\), \(-L<x <L\), \(u(-L)=0=u(L)\), where i. \(2L>0\), the interval length, is a real bifurcation parameter; ii. \(f \in C^ 1[0,1]\) and satisfies \(f(x)>0\) in \((0,1)\), \(f(1)=0\). The work is a continuation of the author's paper with \textit{N. D. Kazarinoff} [J. Aust. Math. Soc., Ser. A 52, No. 3, 343-355 (1992; Zbl 0770.58027)]. This paper is a continuation of a previous paper by the same authors [ibid., 334-342 (1992; see the preceding paper)] where the time map for the two-point Dirichlet boundary value problem \(u''+f(u)=0\), \(u(- L)=u(L)=0\) is considered. Here, the time map is analyzed for the case when, in the phase plane, there are three rest points along the positive \(u\)-axis, a center between two saddle points. Again the main result concerns the number of critical points of the time map and may be stated as follows: If the \(u\)-coordinates of the rest points are \(0 < u_ 0 < u_ 1 < u_ 2\) with \(f''\) changing sign from positive to negative at \(u_ 1\) while \(\int^{u_ 1}_{u_ 0}f(s)ds > 0\) and \(\int^ a_{u_ 0}f(s)ds = 0\) for some \(a \in (u_ 1,u_ 2)\) such that \(2\int^ a_ 0 f(s)ds-af(a) > 0\), then the time map has exactly one critical point, a minimum. It follows that the boundary value problem has at most two positive solutions.

The author considers the convexity of the time maps of positive solutions of some nonlinear, two point boundary value problems \(u''+f(u)=0\), \(-L<x <L\), \(u(-L)=0=u(L)\), where i. \(2L>0\), the interval length, is a real bifurcation parameter; ii. \(f \in C^ 1[0,1]\) and satisfies \(f(x)>0\) in \((0,1)\), \(f(1)=0\). The work is a continuation of the author's paper with \textit{N. D. Kazarinoff} [J. Aust. Math. Soc., Ser. A 52, No. 3, 343-355 (1992; Zbl 0770.58027)]. No review copy delivered.

An effect algebra is a partial algebra with a commutative partial operation \(+\) of addition, which was introduced by \textit{D. J. Foulis} and \textit{M. K. Bennett} [Found. Phys. 24, No. 10, 1331--1352 (1994; Zbl 1213.06004)] and can model quantum-mechanical measurements. The main subject of the paper under review is some important substructures of Archimedean atomic lattice effect algebras like blocks, the set of sharp elements, center, etc. The main results are: (i) For every sharply dominating Archimedean atomic lattice effect algebra, every atomic block is gain sharply dominating, and every of its elements can be decomposed, i.e., it has the so-called basic decomposition property (BDP). (ii) The state smearing theorem for compactly generated Archimedean atomic lattice effect algebras is proved. The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

An effect algebra is a partial algebra with a commutative partial operation \(+\) of addition, which was introduced by \textit{D. J. Foulis} and \textit{M. K. Bennett} [Found. Phys. 24, No. 10, 1331--1352 (1994; Zbl 1213.06004)] and can model quantum-mechanical measurements. The main subject of the paper under review is some important substructures of Archimedean atomic lattice effect algebras like blocks, the set of sharp elements, center, etc. The main results are: (i) For every sharply dominating Archimedean atomic lattice effect algebra, every atomic block is gain sharply dominating, and every of its elements can be decomposed, i.e., it has the so-called basic decomposition property (BDP). (ii) The state smearing theorem for compactly generated Archimedean atomic lattice effect algebras is proved. Der Energieverlust eines sehr schnellen Elektrons (Höhenstrahlteilchens) beim Durchgang durch Materie beruht einerseits auf der Erzeugung von Lichtquanten, anderseits auf dem Zusammenstoß mit anderen Elektronen. Beide Arten von Energieverlusten sind von verschiedenen Autoren gesondert berechnet worden. Mit Benutzung dieser früheren Ergebnisse stellt Verf. die Funktionalgleichung der Wahrscheinlichkeitsfunktion für den gesamten Energieverlust infolge beider Ursachen zugleich auf und wertet diese nach analytischen Methoden aus.

The theory of finite fields finds its origins in the work of several eminent mathematicians of the 17th and 18th centuries with the general theory credited to Gauss and Galois. The emergence of discrete mathematics as an important mathematical discipline, both in theory and practice, makes this volume, the first devoted entirely to finite fields, an indispensable reference. Before embarking on a chapter by chapter outline, some important aspects and features of the book are noted. To quote from the editorial policy of the series: Books in the Encyclopedia of Mathematics and Its Applications will cover their subject comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes will be encyclopedic references or manageable guides to major subjects. It is remarkable how well the volume meets this statement. Comprehensive in content, the volume nonetheless depends on only a first level course in linear algebra with the occasional need for some abstract algebra and analysis. Throughout the book, every opportunity has been taken to provide simple and elegant proofs to deeper and complex treatments of more general results found elsewhere. Numerous and well chosen examples are worked out in detail and interesting exercises, designed to either illustrate or extend material covered, are given. The extensive historical notes at the end of each chapter make both fascinating reading and an important contribution to placing material in context. The 160 page bibliography is an invaluable resource, providing paths to sources which might have otherwise gone unreferenced. A few brief comments on the contents of each chapter are given. Chapter 1 on algebraic foundations summarizes the relevant properties of groups, rings, fields and field extensions that are required. The second chapter considers the structure of finite fields, including characterization of finite fields, roots of irreducible polynomials and the elementary properties of traces, norms and bases. Roots of unity are treated from the point of view of general field theory. Different ways of representing elements in the finite field are given and two proofs of Wederburn's theorem shown. Chapters 3 and 4 give a comprehensive look at properties and constructions of irreducible polynomials, as well as factoring algorithms for polynomials. Criteria for the irreducibility of binomials and trinomials are given, with a section on the properties of linearized polynomials. Chapter 5 considers exponential sums, including Gauss, Jacobi and Kloosterman sums, giving elementary proofs of many deep results, restricted to polynomials. Equations over finite fields are treated in chapter 6, using the estimates for character sums developed in the previous chapter. Several questions on permutation polynomials are explored in Chapter 7, both univariate and multivariate. Chapter 8 on linear recurring sequences, perhaps the most comprehensive treatment of the subject available, includes a treatment of the Berlekamp-Massey algorithm as well as distribution properties of sequences. The applications of finite fields to be found in Chapter 9 are limited to brief treatments of linear and cyclic codes over finite fields, affine and projective planes, certain questions of combinatorics and linear modular systems. The final chapter gives several tables of irreducible polynomials and field representations. This volume is an indispensable tool for the researcher in finite fields and their applications. It is a beautifully written and presented book, painstakingly compiled and thoroughly researched. As far as could be determined, it is a direct reprinting of the 1983 volume printed by Addison-Wesley (see the review in Zbl 0554.12010) which has been unavailable for several years. Its absence has been an impediment to the further development of the area, now corrected with this most welcome reprinting. The origins of the theory of finite fields and its connections with number theory reach back into the 17th and 18th century. In recent decades this subject has been rapidly growing in importance because of its diverse applications in such areas as coding theory, combinatorics, and the mathematical study of switching circuits. This book presents both the classical and the application-oriented aspect of the theory of finite fields. Chapters 1 and 2 cover the background and the general structure theory of finite fields. Chapters 3 and 4 are devoted to polynomials and factorization algorithms. Chapters 5, 6, and 7 deal with exponential sums, equations, and permutation polynomials. Chapters 8 and 9 are devoted to linear recurring sequence and other applications. The notes at the end of each chapter are excellent historical surveys. The bibliography is very comprehensive and up to date. This volume can be strongly recommended as the handbook of finite fields. In order to make the book useful as a text for advanced level courses the authors have inserted worked-out examples at appropriate points. These and the exercises at the end of each chapter are well-chosen.

The theory of finite fields finds its origins in the work of several eminent mathematicians of the 17th and 18th centuries with the general theory credited to Gauss and Galois. The emergence of discrete mathematics as an important mathematical discipline, both in theory and practice, makes this volume, the first devoted entirely to finite fields, an indispensable reference. Before embarking on a chapter by chapter outline, some important aspects and features of the book are noted. To quote from the editorial policy of the series: Books in the Encyclopedia of Mathematics and Its Applications will cover their subject comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes will be encyclopedic references or manageable guides to major subjects. It is remarkable how well the volume meets this statement. Comprehensive in content, the volume nonetheless depends on only a first level course in linear algebra with the occasional need for some abstract algebra and analysis. Throughout the book, every opportunity has been taken to provide simple and elegant proofs to deeper and complex treatments of more general results found elsewhere. Numerous and well chosen examples are worked out in detail and interesting exercises, designed to either illustrate or extend material covered, are given. The extensive historical notes at the end of each chapter make both fascinating reading and an important contribution to placing material in context. The 160 page bibliography is an invaluable resource, providing paths to sources which might have otherwise gone unreferenced. A few brief comments on the contents of each chapter are given. Chapter 1 on algebraic foundations summarizes the relevant properties of groups, rings, fields and field extensions that are required. The second chapter considers the structure of finite fields, including characterization of finite fields, roots of irreducible polynomials and the elementary properties of traces, norms and bases. Roots of unity are treated from the point of view of general field theory. Different ways of representing elements in the finite field are given and two proofs of Wederburn's theorem shown. Chapters 3 and 4 give a comprehensive look at properties and constructions of irreducible polynomials, as well as factoring algorithms for polynomials. Criteria for the irreducibility of binomials and trinomials are given, with a section on the properties of linearized polynomials. Chapter 5 considers exponential sums, including Gauss, Jacobi and Kloosterman sums, giving elementary proofs of many deep results, restricted to polynomials. Equations over finite fields are treated in chapter 6, using the estimates for character sums developed in the previous chapter. Several questions on permutation polynomials are explored in Chapter 7, both univariate and multivariate. Chapter 8 on linear recurring sequences, perhaps the most comprehensive treatment of the subject available, includes a treatment of the Berlekamp-Massey algorithm as well as distribution properties of sequences. The applications of finite fields to be found in Chapter 9 are limited to brief treatments of linear and cyclic codes over finite fields, affine and projective planes, certain questions of combinatorics and linear modular systems. The final chapter gives several tables of irreducible polynomials and field representations. This volume is an indispensable tool for the researcher in finite fields and their applications. It is a beautifully written and presented book, painstakingly compiled and thoroughly researched. As far as could be determined, it is a direct reprinting of the 1983 volume printed by Addison-Wesley (see the review in Zbl 0554.12010) which has been unavailable for several years. Its absence has been an impediment to the further development of the area, now corrected with this most welcome reprinting. Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of \(\omega_2 \cap \operatorname{cof}(\omega_1)\) in the approachability ideal \(I[\omega_2]\). In this paper we give a new proof of Mitchell's theorem, deriving it from an abstract framework of side condition methods.

For each positive integer \(n\), function \(f\), and point \(x\), the 1998 conjecture by Ginchev, Guerragio, and Rocca states that the existence of the \(n\) th Peano derivative \(f_{(n)}(x)\) is equivalent to the existence of all \(n(n+1)/2\) generalized Riemann derivatives, \[ D_{k,-j}f(x)=\lim\limits_{h\rightarrow 0}\frac{1}{h^n}\sum\limits_{i=0}^k(-1)^i\binom{k}{i}f(x+(k-i-j)h), \] for \(j,k\) with \(0\leq j<k\leq n\). A version of it for \(n\geq 2\) replaces all \(-j\) with \(j\) and eliminates all \(j=k-1\). Both the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on highly non-trivial combinatorial algorithms. This article provides a second, inductive, algebraic proof to each of these theorems, based on a reduction to (Laurent) polynomials. ``For a given real function of a real variable \(f\) the concept of Peano derivative of \(n\)th-order is reminded and some modification of the \(n\)-th-order Riemann derivative is introduced, the latter we call for brevity Riemann derivative. The conjecture is formulated that Peano derivative of \(n\)th-order exists if and only if all Riemann derivatives of order less or equal to \(n\) exist and then the \(n\)-th-order Peano and Riemann derivative coincide. This conjecture is reduced to a hypothesis referred as \(\mathbb{H}_n\). The verification of \(\mathbb{H}_n\) for \(n= 2,3,4\) is done''.

For each positive integer \(n\), function \(f\), and point \(x\), the 1998 conjecture by Ginchev, Guerragio, and Rocca states that the existence of the \(n\) th Peano derivative \(f_{(n)}(x)\) is equivalent to the existence of all \(n(n+1)/2\) generalized Riemann derivatives, \[ D_{k,-j}f(x)=\lim\limits_{h\rightarrow 0}\frac{1}{h^n}\sum\limits_{i=0}^k(-1)^i\binom{k}{i}f(x+(k-i-j)h), \] for \(j,k\) with \(0\leq j<k\leq n\). A version of it for \(n\geq 2\) replaces all \(-j\) with \(j\) and eliminates all \(j=k-1\). Both the GGR conjecture and its version were recently proved by the authors using non-inductive proofs based on highly non-trivial combinatorial algorithms. This article provides a second, inductive, algebraic proof to each of these theorems, based on a reduction to (Laurent) polynomials. In this paper, we present a method for factor analysis of discrete data. This is accomplished by fitting a dependent Poisson model with a factor structure. To be able to analyze ordinal data, we also consider a truncated Poisson distribution. We try to find the model with the lowest AIC by employing a forward selection procedure. The probability to find the correct model is investigated in a simulation study. Moreover, we heuristically derive the corresponding asymptotic probabilities. An empirical study is also included.

This is a note on the class of semi-linear elliptic equations \[ \Delta u + f(x,u) + g(|x|) x \cdot \nabla u = 0, \] in exterior domains of Euclidean space of dimension \(n \geq 3\). Via theory for linear ordinary differential equations new sufficient conditions for the existence of positive solutions vanishing at infinity are given. In the case when the nonlinearity \(f(x,\cdot)\) has sublinear growth, and the radially symmetric function \(g\) is nonnegative, the authors show existence for a class of equations not covered by the criteria given by the reviewer [Nonlinear Anal., Theory Methods Appl. 64, No.~7 (A), 1608--1620 (2006; Zbl 1101.34022)]. Also other developments of those results are available in the literature. We prove the existence of positive solutions to the scalar equation \(y^{\prime\prime}(x)+F(x,y,y^{\prime})\) \(=0\). Applications to semilinear elliptic equations in exterior domains are considered.

This is a note on the class of semi-linear elliptic equations \[ \Delta u + f(x,u) + g(|x|) x \cdot \nabla u = 0, \] in exterior domains of Euclidean space of dimension \(n \geq 3\). Via theory for linear ordinary differential equations new sufficient conditions for the existence of positive solutions vanishing at infinity are given. In the case when the nonlinearity \(f(x,\cdot)\) has sublinear growth, and the radially symmetric function \(g\) is nonnegative, the authors show existence for a class of equations not covered by the criteria given by the reviewer [Nonlinear Anal., Theory Methods Appl. 64, No.~7 (A), 1608--1620 (2006; Zbl 1101.34022)]. Also other developments of those results are available in the literature. Using the technique of measures of noncompactness and, in particular, a consequence of Sadovskii's fixed point theorem, we prove a theorem about the existence and asymptotic stability of solutions of a functional integral equation. Moreover, in order to illustrate our results, we include one example and compare our results with those obtained in other papers appearing in the literature.

Cohomology classes defined by polylogarithms have played a decisive role in the investigations on special values of \(L\)-functions, most notably in the study of the Tamagawa number conjecture. These extension classes were defined for relative curves by Beilinson and Deligne (unpublished) and for abelian schemes by [\textit{J. Wildeshaus}, Realizations of polylogarithms. Lecture Notes in Mathematics. 1650. Berlin: Springer (1997; Zbl 0877.11001)]. The polylog extension on curves has the advantage of being a one-extension of lisse sheaves, thus it can be itself represented by a lisse sheaf. Whereas the polylog extension on an abelian scheme of relative dimension \(d\) is a \((2d-1)\) extension. However, it is shown in this note that the two notions are closely related: roughly speaking, the polylog extension on an abelian scheme is the push-forward of the polylog on a sub-curve. In the situation of a curve embedded in its Jacobian, the polylog on the Jacobian is the cup product of the polylog on the curve with the fundamental class of the curve (to simplify the exposition, only the étale realization is treated). Beilinson's conjecture on special values of \(L\)-functions generalize the classical analytic class number formula, and it relates integral motivic cohomology groups and Deligne cohomology groups of smooth projective schemes defined over \(\mathbb Q\). The weak version of Beilinson's conjecture involves the existence of a certain subspace of the integral motivic cohomology group of a given scheme, and it requires the construction of explicit elements in motivic cohomology. In the cases where the approach of constructing explicit elements proved to be successful, the underlying geometric objects are special cases of pure or mixed Shimura varieties. It is expected in these cases that the collection of the images under the regulator of these explicit elements is interpolated by a single object called the polylogarithmic extension. This monograph attempts to provide a unified explanation for such constructions. It provides a sheaf-theoretic foundation of the theory of polylogarithms and generalizes the approaches of Beilinson, Deligne and Levin to the context of Shimura varieties. This monograph consists of five parts. The first two parts are mostly expository describing the generic sheaf and the canonical construction on a comprehensive level. Part I discusses the construction and properties of the mixed structure on the completed group ring of the topological fundamental group of a scheme, which is smooth over a number field. Special emphasis is given to the universal property satisfied by this mixed structure. The construction can be performed in a relative setting when there is a morphism \(\pi: X \to Y\) of schemes satisfying a certain regularity condition. In this case the generic sheaf on \(X\) can be defined. In Part II, such a construction is specialized to the case, where the morphism \(\pi\) is the natural projection of a mixed Shimura variety to the underlying pure Shimura variety. The main tool used for this purpose is the identification of the generic sheaf, called the logarithmic sheaf, with the canonical construction of a certain pro-representation. The central part of this book is Part III, which generalizes the definition of polylogarithmic extensions in the context of mixed Shimura varieties and discusses characteristic features of such an extension such as the splitting principle, rigidity, and norm compatibility. The treatment of the special case of the morphism \(\pi: G_{m, \mathbb Q} \to \text{Spec} (\mathbb Q)\) is contained in Part IV, where both the Hodge and \(\ell\)-adic versions are described, recovering results of Beilinson and Deligne. Part V discusses the case of the universal elliptic curve \(\pi: \mathcal E \to M\) over a modular curve \(M\) defined over \(\mathbb Q\). It also discusses the base change to CM-points of \(M\). \{Authors remark: This monograph is based on the thesis with the same title reviewed in Zbl 0824.14018\}

Cohomology classes defined by polylogarithms have played a decisive role in the investigations on special values of \(L\)-functions, most notably in the study of the Tamagawa number conjecture. These extension classes were defined for relative curves by Beilinson and Deligne (unpublished) and for abelian schemes by [\textit{J. Wildeshaus}, Realizations of polylogarithms. Lecture Notes in Mathematics. 1650. Berlin: Springer (1997; Zbl 0877.11001)]. The polylog extension on curves has the advantage of being a one-extension of lisse sheaves, thus it can be itself represented by a lisse sheaf. Whereas the polylog extension on an abelian scheme of relative dimension \(d\) is a \((2d-1)\) extension. However, it is shown in this note that the two notions are closely related: roughly speaking, the polylog extension on an abelian scheme is the push-forward of the polylog on a sub-curve. In the situation of a curve embedded in its Jacobian, the polylog on the Jacobian is the cup product of the polylog on the curve with the fundamental class of the curve (to simplify the exposition, only the étale realization is treated). This book is devoted to an introduction to the deterministic theory of continuous time systems described by ordinary differential equations. The main topics dealt with are the following: operations on systems, quantitative and qualitative methods, state space methods, frequency domain description, linear and nonlinear time-varying systems, Hammerstein, Volterra and Wiener models. The approach and the exposition are of the engineering level, without deeper mathematical developments. Many examples, mostly from electrical engineering, are presented. According to the author, the book is intended for electrical and electronic engineering students.

[For the review of the first ed. see Zbl 1054.74001.] The emphasis of the book is static structural design. The authors develop elastic equations for a variety of physical systems, from a taut rope to plates and shells. For each system they then show how to setup and analyze the problem using the finite element method. A good reminder for the need of such a book is contained in the preface. ``A working knowledge of the fundamentals of the finite element method, and classical structural mechanics is a prerequisite for any sound finite element analysis. Only a well trained engineer wil have the skills to continually examine the computed results.'' The book is well written and would be a valuable addition for the library of a structural engineer. This book is the revised and unessentially extended English edition of the German version [the authors, Statik mit finiten Elementen. (Statics with finite elements). Berlin: Springer (2002; Zbl 0991.74001)]. It is a matter of fact that there exist a number of (more or less) excellent books on the finite element method (FE), especially with respect to structural analysis. However, in practice this method is often and increasingly used formally without an understanding of its limitations and of the real structural behavior. Therefore the authors intend to eliminate these deficiencies and to explain the FE method -- differing from the usual procedure -- preferably by using alternatively influence functions and by taking into account real engineering problems. Emphasis is rather on the discussion of ideas than on systematical and complete derivations. A reader's sufficient knowledge of structural analysis, solid mechanics and mathematics is assumed apparently. Chapter 1 comprises an introduction to the FE method. It is devoted to various aspects of this method, without emphasizing the fact that it is a discretisation method for the approximate solution of continuous problems (and not a matrix method for the exact solution of discontinuous problems). The discretisation is based alternatively on the methods of substitute loads, projection, energy and influence functions respectively. As one-dimensional models there are considered non-extensible cables, elastic bars and beams, and two-dimensional elastic plates without and with holes. Additionally, error estimates and convergence problems as well as adaptive methods are discussed. Chapter 2 represents a short introduction to the boundary element method with hints about its application, and chapter 3 is devoted to the stiffness of beams and frames. Chapter 4 deals with the discretisation of plane elasticity problems, using triangular and rectangular elements and including geometrically nonlinear problems. Further, adaptive mesh refinements, convergence tests and error estimates are considered here, too. Some applications to real constructions and soil mechanics problems are presented, together with meaningful illustrations. Chapter 5 starts with the analysis of Kirchhoff plates on the basis of a displacement model and continues with a hybrid stress model. There follows a discussion of Mindlin plates (called frequently, but incorrectly Reissner-Mindlin plates), and the corresponding shear locking effect is examined. In chapter 6 a short summary of the theory of shells (including membranes) and of the corresponding finite elements is given. This theory, however, is useless for a reader who is not familiar with tensor analysis. Finally, in chapter 7 a summary of mathematical definitions and denotations together with some important formulas including functional analysis, Green's functions and error estimates are given, with the exception of the elements of tensor analysis. This book is neither a compendium on structural analysis nor on the FE method, Rather, it is intended to give the reader a basic understanding of possibilities and limitations of the method with respect to real constructions in civil engineering. In this sense the book gives important hints to the user of FE software. Many examples result from real constructions. However, most of the formulas -- not all of them given correctly -- are presented without any derivation and force the interested reader to consult additionally the original literature which can be found via the extensive list of references.

[For the review of the first ed. see Zbl 1054.74001.] The emphasis of the book is static structural design. The authors develop elastic equations for a variety of physical systems, from a taut rope to plates and shells. For each system they then show how to setup and analyze the problem using the finite element method. A good reminder for the need of such a book is contained in the preface. ``A working knowledge of the fundamentals of the finite element method, and classical structural mechanics is a prerequisite for any sound finite element analysis. Only a well trained engineer wil have the skills to continually examine the computed results.'' The book is well written and would be a valuable addition for the library of a structural engineer. Let \({\mathcal L}(Q,\dot Q)={1\over 2}|\dot Q|^2+ h(Q,\dot Q)\) with \(h\) analytic of small norm. The problem of Arnold's diffusion consists in finding conditions on \(h\) which guarantee the existence of orbits \(Q\) of \({\mathcal L}\) with \(\dot Q\) connecting two arbitrary points of frequency space. Recently, J. N. Mather has found a sufficient condition for Arnold's diffusion; this condition is not read on \(h\) itself, but on the set of all action-minimizing orbits of \({\mathcal L}\). In this paper, we try to characterize those action-minimizing orbits whose mean frequency is close to periodic.

This survey paper is a continuation of an earlier survey by the author [Rend. Semin. Mat. Fis. Milano 68, 59--120 (1998; Zbl 1098.32505)] and describes several important recent developments in the theory of positive currents. Among the themes covered are: extension, products, pullback and regularization of currents, as well as applications in holomorphic dynamics. Together with the earlier survey, this gives a very broad overview of the field. This is a survey paper on the positive currents on complex manifolds. The author is concerned with some important results from the theory of the currents such as currents defined by analytic varieties, holomorphic chains and their boundaries, the closed and positive currents etc. In the first part of the paper the author deals with the closed positive currents on complex manifolds, thought of as natural generalizations of the submanifolds. After a historical introduction, she discusses the Lelong numbers and regularization. Then she deals with the currents and submanifolds, theorems of support, cut-off and extension, currents and holomorphic maps, currents and metrics. In the second part, the author deals with positive, pluriharmonic or plurisubharmonic currents thought of as generalizations of the plurisubharmonic functions as well as of the closed currents. She discusses the following arguments: products and intersections of currents, purisubharmonic currents: extension and support problems, currents and modifications (the Lelong numbers of the plurisubharmonic currents). At the end the author presents an appendix dealing with some introductory notions in current theory.

This survey paper is a continuation of an earlier survey by the author [Rend. Semin. Mat. Fis. Milano 68, 59--120 (1998; Zbl 1098.32505)] and describes several important recent developments in the theory of positive currents. Among the themes covered are: extension, products, pullback and regularization of currents, as well as applications in holomorphic dynamics. Together with the earlier survey, this gives a very broad overview of the field. The AneuRisk65 data [\textit{L. M. Sangalli} et al., ibid. 8, No. 2, 1879--1890, electronic only (2014; Zbl 1305.62376)] are analysed using methodology from statistical shape analysis. The internal carotid arteries are aligned using translation and rotation in three dimensions, together with shifts of the abscissa coordinate. Spline interpolation and weighted Procrustes methods are used to estimate the mean size-and-shapes in each of the six groups. Differences in torsion and curvature of the group means are highlighted, and permutation and bootstrap tests confirm there is weak evidence for differences in shape between the upper aneurysm group compared to the others. Finally shape variability, analysis of mean radii and classification are explored. (Concerns [\textit{L. M. Sangalli} et al., ibid. 8, No. 2, 1879--1890, electronic only (2014; Zbl 1305.62376)].)

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by \textit{G. Roth} and \textit{H. Caswell} [Theor. Popul. Biol. 120, 62--77 (2018; Zbl 1397.91523)] for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogeneous Markov chains and the discrete-time McKendrick-von Foerster equation, authors derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. Authors apply our approach to study a specific time-dependent model of the southern fulmar (\textit{Fulmarus glacialoides}), and obtain insights into how the number of breeding attempts depends on external conditions that vary through time. As an individual moves through its life cycle, it passes through a series of states (age classes, size classes, reproductive states, spatial locations, health stati, etc.) before its eventual death. The occupancy time in a state is the time spent in that state over the individual's life. Depending on the life cycle description, the occupancy times describe different demographic variables, for example, lifetime breeding success, lifetime habitat utilisation, or healthy longevity. Models based on absorbing Markov chains provide a powerful framework for the analysis of occupancy times. Current theory, however, can completely analyse only the occupancy of single states, although the occupancy time in a set of states is often desired. For example, a range of sizes in a size-classified model, an age class in an age \(\times\) stage model, and a group of locations in a spatial stage model are all sets of states. We present a new mathematical approach to absorbing Markov chains that extends the analysis of life histories by providing a comprehensive theory for the occupancy of arbitrary sets of states, and for other demographic variables related to these sets (e.g., reaching time, return time). We apply this approach to a matrix population model of the Southern Fulmar (\textit{Fulmarus glacialoides}). The analysis of this model provides interesting insight into the lifetime number of breeding attempts of this species. Our new approach to absorbing Markov chains and its implementation in matrix oriented software makes the analysis of occupancy times more accessible to population ecologists and directly applicable to any matrix population models.

During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by \textit{G. Roth} and \textit{H. Caswell} [Theor. Popul. Biol. 120, 62--77 (2018; Zbl 1397.91523)] for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogeneous Markov chains and the discrete-time McKendrick-von Foerster equation, authors derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. Authors apply our approach to study a specific time-dependent model of the southern fulmar (\textit{Fulmarus glacialoides}), and obtain insights into how the number of breeding attempts depends on external conditions that vary through time. In his paper in J. Math. Soc. Japan 24, 20-59 (1972; Zbl 0226.14013), the author studied properties of elliptic surfaces and their numerical characters like geometric genus, Betti numbers, and Hodge numbers and determined their Mordell-Weil group. In the present paper he uses this approach to study the elliptic surface \(F_ m\) defined by: \(Y^ 2=X(X+Z_ 1^ m)(X-Z_ 2^ m)\) with \(Z_ 1^ m+Z_ 2^ m=1\) which is called ``Fermat modular surface of \(F\)- level \(m\)''. The author proves that the Mordell-Weil group of \(F_ m\) is isomorphic to \(({\mathbb{Z}}/2{\mathbb{Z}})^ 2\) or \(({\mathbb{Z}}/4{\mathbb{Z}})^ 2\) according as m is odd or even (theorem 4), and he determines the discriminant of its Néron-Severi group (theorem 7). In the last section the author explains how \(F_ m\) can be interpreted as modular elliptic surface (cf. the paper cited above) with respect to a non congruence subgroup of PSl(2,\({\mathbb{Z}})\), if \(m\neq 1,2,4\), or 8.

A commutative ring \(R\) is called almost perfect if \(R/I\) is perfect (that is, it satisfies the DCC on principal ideals) for every non-zero ideal \(I\) of \(R\). The author continues the investigations of local almost perfect domains made by \textit{S. Bazzoni} and \textit{L. Salce} [Colloq. Math. 95, 285--301 (2003; Zbl 1048.13014)]. He considers in detail the Archimedean valuation domains dominating a one-dimensional local domain and in particular an almost perfect domain. A commutative unitary ring \(R\) is called almost perfect if, for every nonzero ideal \(I\) of \(R\), the quotient ring \(R/I\) is a perfect ring [i.e. every \(R/I\)-module has a projective cover; see \textit{H. Bass}, Trans. Am. Math. Soc. 95, 466--488 (1960; Zbl 0094.02201)]. In a recent paper [\textit{S. Bazzoni} and \textit{L. Salce}, J. Lond. Math. Soc. (2) 66, No. 2, 276--294 (2002; Zbl 1009.13003)], the authors investigated the case of integral domains that are almost perfect. Among the results already known, proved in that paper, we mention the following characterizations: (a) an integral domain \(R\) is almost perfect if and only if all flat \(R\)-modules are strongly flat; (b) an integral domain \(R\) is almost perfect if and only if \(R\) is \(h\)-local and, for each maximal ideal \(M\) of \(R\), \(R_{M}\) is almost perfect. In the paper under review, the authors pursue the study of the almost perfect rings, focusing their interest on the ``central case'' of local integral domains, because of the local property proved in (b) and because of the following preliminary result: An almost perfect commutative ring, which is not an integral domain, is necessarily a perfect ring. After a revisitation of the work made by \textit{J. R. Smith} [Pac. J. Math. 30, 233--245 (1969; Zbl 0201.04603)], who also studied the local integral domains that are almost perfect, using a different terminology (that is, local domains with TTN), the authors obtain several new results on these domains and their modules. Particularly interesting is the last section of the paper where the authors describe three different explicit constructions for obtaining examples of (non-Noetherian) almost perfect local domains. In particular, they give an example of an integrally closed local almost perfect integral domain which is not a valuation domain.

A commutative ring \(R\) is called almost perfect if \(R/I\) is perfect (that is, it satisfies the DCC on principal ideals) for every non-zero ideal \(I\) of \(R\). The author continues the investigations of local almost perfect domains made by \textit{S. Bazzoni} and \textit{L. Salce} [Colloq. Math. 95, 285--301 (2003; Zbl 1048.13014)]. He considers in detail the Archimedean valuation domains dominating a one-dimensional local domain and in particular an almost perfect domain. In the oligopoly e-commerce market, the oligarch retailers sell products through traditional channel, while others through both network and traditional channel in order to obtain greater profits. Instead of discussing classic Bertrand game model, which past studies have done, we considered dual-channel retailer who makes price decision through both in network channel and traditional channel. This paper used the bifurcation theory of dynamical system, considering dual-channel retailer who makes delay decision. We performed a numerical simulation on system with different conditions, and some complex phenomenons occured, such as bifurcation and chaos. The results showed that adopting price delay decision in tradition channel would make the system more stable. While, adopting price delay decision in network channel makes the system less stable. When the market is in chaotic state, the using of delay decision would have an effect on the system stability in either traditional or network channels. The system become stable from chaos and would return to chaotic again with the increasing of weight in past period. Some interesting phenomenons happened when dual-channel retailer adopted delay decision in both channels. The superposition of delay decision would make the system more complex. At last, we measured the system's performance by using profit index. We analyzed the profits of different oligarchs when the system is in different states. When the system is in chaos, the total profits of the oligarchs are obviously less than that in a stable state. Adopting delay decision is a way to avoid profit loss when system is in chaotic period, but this requires the retailer has rich operational experience. That is because adopting delayed decision may not always enhance the competitive strength of oligarchs.

Let \(\|\cdot\|\) be a nonnegative function on a space \(A\) of analytic functions on the unit disk \(\mathbb{D}\) satisfying \(\sup_{| z|\leq r}| f(z)|\leq C(r)\cdot\| f\|\) for every \(f\in A\) \((0<r<1)\). A function \(f: \mathbb{D}\to\mathbb{C}\) is said to have the BSE condition if there exists a constant \(K>0\) such that for every \(n\in\mathbb{N}\), \(c_ 1,\dots,c_ n\in\mathbb{C}\) and \(z_ 1,\dots,z_ n\in\mathbb{D}\) the inequality \[ \left|\sum^ n_{j=1} c_ j f(z_ j)\right|\leq K\sup_{h\in A_ 1}\left|\sum^ n_{j=1} c_ j h(z_ j)\right| \] holds \((A_ 1=\{f\in A: \| f\|\leq 1\})\). Associated with \(A\) is the set \(\tilde A=\{f\in C(D)| f\) satisfies BSE\}. The author now shows that, with respect to a suitable norm, \(\tilde A\) is a dual Banach space. The main part of the paper is devoted to the explicit determination of \(\tilde A\) for various classes of analytic functions as \(A(\mathbb{D})\), \(H^ p\), Bloch space, VMOA, Bergman spaces, etc. As an example, we mention that \(\widetilde{A(\mathbb{D})}=H^ \infty\) and \(\widetilde{H^ p}=H^ p\) \((1\leq p\leq \infty)\). The BSE condition has previously been studied by \textit{S. Takahasi} and \textit{O. Hatori} [Proc. Am. Math. Soc. 110, No. 1, 149-158 (1990; Zbl 0722.46025)] in the context of Banach algebras. A class of commutative Banach algebras which satisfy a Bochner- Schönberg-Eberlein-type inequality is introduced. Commutative \(C^*\)- algebras, the disk algebra and the Hardy algebra on the open disk are examples.

Let \(\|\cdot\|\) be a nonnegative function on a space \(A\) of analytic functions on the unit disk \(\mathbb{D}\) satisfying \(\sup_{| z|\leq r}| f(z)|\leq C(r)\cdot\| f\|\) for every \(f\in A\) \((0<r<1)\). A function \(f: \mathbb{D}\to\mathbb{C}\) is said to have the BSE condition if there exists a constant \(K>0\) such that for every \(n\in\mathbb{N}\), \(c_ 1,\dots,c_ n\in\mathbb{C}\) and \(z_ 1,\dots,z_ n\in\mathbb{D}\) the inequality \[ \left|\sum^ n_{j=1} c_ j f(z_ j)\right|\leq K\sup_{h\in A_ 1}\left|\sum^ n_{j=1} c_ j h(z_ j)\right| \] holds \((A_ 1=\{f\in A: \| f\|\leq 1\})\). Associated with \(A\) is the set \(\tilde A=\{f\in C(D)| f\) satisfies BSE\}. The author now shows that, with respect to a suitable norm, \(\tilde A\) is a dual Banach space. The main part of the paper is devoted to the explicit determination of \(\tilde A\) for various classes of analytic functions as \(A(\mathbb{D})\), \(H^ p\), Bloch space, VMOA, Bergman spaces, etc. As an example, we mention that \(\widetilde{A(\mathbb{D})}=H^ \infty\) and \(\widetilde{H^ p}=H^ p\) \((1\leq p\leq \infty)\). The BSE condition has previously been studied by \textit{S. Takahasi} and \textit{O. Hatori} [Proc. Am. Math. Soc. 110, No. 1, 149-158 (1990; Zbl 0722.46025)] in the context of Banach algebras. The authors propose a quasi-likelihood approach analogous to two-way analysis of variance for the estimation of the parameters of generalized linear mixed models with two components of dispersion. They discuss both the asymptotic and small-sample behaviour of their estimators, and illustrate their use with salamander mating data.

The authors prove the existence of a travelling wave solution of the conservation laws equations with shock profile for a nonconvex flux, if its speed is subcharacteritic. The main purpose of the paper is to prove the stability of such a wave and to justify the relaxation schemes introduced in the paper of \textit{T. P. Liu} [Commun. Math. Phys. 108, 153-175 (1987; Zbl 0633.35049)], for a nonconvex flux under the assumption, that the Rankine-Hugoniot and entropy conditions are satisfied. The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman- Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.

The authors prove the existence of a travelling wave solution of the conservation laws equations with shock profile for a nonconvex flux, if its speed is subcharacteritic. The main purpose of the paper is to prove the stability of such a wave and to justify the relaxation schemes introduced in the paper of \textit{T. P. Liu} [Commun. Math. Phys. 108, 153-175 (1987; Zbl 0633.35049)], for a nonconvex flux under the assumption, that the Rankine-Hugoniot and entropy conditions are satisfied. We present an efficient and novel procedure to design two-dimensional (2D) linear-phase IIR filters which requires less hardware resources. A 2D linear-phase FIR filter prototype is first designed using semidefinite programming (SDP). The prototype filter is then decomposed into modular structures via the Schur decomposition method (SDM). Each section is reduced to IIR structures using a novel digital system identification technique called the discrete-time vector fitting (VFz). Examples of image processing applications show that the algorithm exhibits fast convergence and produces low hardware costs and accurate filters.

To solve the linear system \(A\mathbf{u}=\mathbf{f}\) (\(A\in\mathbb{R}^{n\times n}\)) by algebraic multigrid, use is made of a prolongation (or interpolation) operator \(P: \mathbb{R}^n\to\mathbb{R}^{n_c}\) (\(n\) is the dimension of the fine grid, and \(n_c\) of the coarse grid). The propagator \(P\) has to be in balance with the smoother \(M\) to optimize the performance of the resulting iteration matrix \((I-M^TA)(I-PA_cP^TA)(I-M^{-1}A)\) in which \(A_c=P^TAP\) is the projected matrix. In [\textit{R. D. Falgout} and \textit{P. S. Vassilevski}, SIAM J. Numer. Anal. 42, No. 4, 1669--1693 (2004; Zbl 1077.65129)] an ideal interpolation \(P_*\) was defined as one that minimizes a worst case measure for this performance measure. This defines a set \(\mathbb{P}_*\) of such matrices \(P_*\). The result of this paper is to give characterizations of three other sets \(\mathbb{P}_k, k=0,1,2\) such that \(\mathbb{P}_0\subseteq \mathbb{P}_2=\mathbb{P}_*\subseteq \mathbb{P}_1\). The analysis holds under certain conditions (e.g. \(A\) is symmetric positive definite). An explicit expression is given for the unique \(P\in\mathbb{P}_0\), but if other properties are desired like for example sparsity, then the freedom provided by the parameters involved (like the weight matrix used in the definition of the performance criterion and the choice of the coarse grid), then another interpolation operator can be chosen from the larger set. The authors present a theory that generalizes the algebraic multigrid (AMG) framework to address even broader of problems . This paper will provide a guidance in the development of new AMG methods able to handle difficult problems such as Maxwell's equations. Starting by introducing two new measures in applying the relaxation methods for linear systems, a new convergence theory is proposed. Then, a min-max problem for the new measures is formulated and its optimal solution is analyzed. Using these results, necessary conditions for obtaining a uniformly convergent method is proved. The last sections are concerned with the concept of ``compatible relaxation''. The authors present four variants of compatible relaxation, each having its own advantages and disadvantages, and suggest a simple algorithm for using these techniques to choose coarse grids in AMG methods. Two examples illustrating the theoretical results are discussed : an anisotropic diffusion problem with Dirichlet boundary conditions and a non trivial geometric two-grid method for the \(H\)(div) space.

To solve the linear system \(A\mathbf{u}=\mathbf{f}\) (\(A\in\mathbb{R}^{n\times n}\)) by algebraic multigrid, use is made of a prolongation (or interpolation) operator \(P: \mathbb{R}^n\to\mathbb{R}^{n_c}\) (\(n\) is the dimension of the fine grid, and \(n_c\) of the coarse grid). The propagator \(P\) has to be in balance with the smoother \(M\) to optimize the performance of the resulting iteration matrix \((I-M^TA)(I-PA_cP^TA)(I-M^{-1}A)\) in which \(A_c=P^TAP\) is the projected matrix. In [\textit{R. D. Falgout} and \textit{P. S. Vassilevski}, SIAM J. Numer. Anal. 42, No. 4, 1669--1693 (2004; Zbl 1077.65129)] an ideal interpolation \(P_*\) was defined as one that minimizes a worst case measure for this performance measure. This defines a set \(\mathbb{P}_*\) of such matrices \(P_*\). The result of this paper is to give characterizations of three other sets \(\mathbb{P}_k, k=0,1,2\) such that \(\mathbb{P}_0\subseteq \mathbb{P}_2=\mathbb{P}_*\subseteq \mathbb{P}_1\). The analysis holds under certain conditions (e.g. \(A\) is symmetric positive definite). An explicit expression is given for the unique \(P\in\mathbb{P}_0\), but if other properties are desired like for example sparsity, then the freedom provided by the parameters involved (like the weight matrix used in the definition of the performance criterion and the choice of the coarse grid), then another interpolation operator can be chosen from the larger set. This article describes pedagogic issues in setting objective tests in mechanics using Question Mark Perception, coupled with MathML mathematics mark-up and the Scalable Vector Graphics (SVG) syntax for producing diagrams. The content of the questions (for a range of question types such as multi-choice, numerical input and variants such as confidence-based questions) is scripted with random parameters, thereby producing many millions of realizations of the underlying `question style'. This means that the question setter must completely specify the algebraic and pedagogic structure of the question. For some question types, we need to understand and encode the ways in which students make mistakes, offering them as distracters or recognizing their use in numerical inputs (we call this responsive numerical input). We have examined several years' worth of exam scripts to discover what `mal-rules' are used for each question and attempted to characterize them with metadata that makes students' responses recorded in the answer files easier to understand. Results from evaluation experiments are presented; in particular, we are interested in whether the feedback `feeds forward' to affect students' approaches to doing problems in a repeat test or exam, delayed by a variable time period (almost immediately, after 1 week, 1 month or more). To quantify this when examining end of semester exam scripts, we looked at four indicators: using units, identifying vectors, using diagrams and emulating the good layout of the feedback screens in their own written solutions.

This book was written for second year students in technical universities. It is a continuation of the book [\textit{S. Balac} and \textit{F. Sturm}, Algebra and analysis. Course on mathematics for first year students with solved exercises. Collection des Sciences Appliquées de l'INSA de Lyon. (Lausanne): Presses Polytechniques et Universitaires Romandes. (2003; Zbl 1130.00001)]. The book is very well written and organized. Its content is divided in 13 chapters, covering four main parts: sequences and series of (real and complex) functions, bilinear algebra, differential calculus and integral calculus for functions of more variables. Each chapter contains completely solved problems. The computer algebra system Maple is very often used for underlying some notions or results. Ce livre est un cours de Mathématiques dispensé en première année du cycle préparatoire de l'INSA de Lyon dans la filière ASINSA. Il est divisé en 20 chapitres correspondant aux programmes d'Algèbre et d'Analyse. Ces chapitres sont regroupes en cinq grandes parties: (1) Ensembles numériques fondamentaux: le corps des réels, le corps des complexes, suites numériques. (2) Polynômes et fractions rationnelles. (3) Algèbre linéaire: les espaces vectoriels, les applications linéaires, les systèmes d'équations linéaires, réductions des endomorphismes. (4) Calcul différentiel: continuité des fonctions réelles d'une variable réelle, fonctions usuelles, comparaison locale des fonctions, dérivabilité des fonctions réelles d'une variable réelle, déve\-loppements limites. (5) Calcul intégral: l'intégrale de Riemann, l'intégrale généralisée, équations différentielles linéaires. Ce livre pourra être utilisé avec profit par les étudiants des différents cycles préparatoires intègres, les étudiants des DEUG scientifiques, les étudiants en formation continue, etc. Les auteurs se sont attachés à donner des définitions précises et à présenter des raisonnements rigoureux sans toutefois chercher l'exhaustivité. Ainsi, certains résultats énoncés sont admis sans démonstration. D'une manière générale, les démonstrations techniques sont omises au profit des démonstrations pouvant améliorer la compréhension du résultat énoncé, illustrant l'utilisation des notions déjà introduites ou mettant en avant des idées ou méthodes susceptibles d'être réutilisées par la suite. Celles-ci sont soigneusement détaillées et commentées et une attention toute particulière a été apportée à leur rédaction. Par ailleurs, ils ont cherché à motiver les notions introduites et à les illustrer par des exemples et des remarques. Chaque chapitre contient de courts exercices visant à tester la bonne compréhension des notions introduites. Il se termine par des exercices de synthèse qui font appel à la fois aux résultats présentés dans le chapitre concerné et aux notions acquisés dans les chapitres antérieurs. Ces exercices sont souvent issus de dévoirs et d'interrogations écrites et doivent permettre d'assimiler des méthodes de raisonnement et des techniques de calcul. Les exercices sont intégralement corrigés en fin de chaque chapitre.

This book was written for second year students in technical universities. It is a continuation of the book [\textit{S. Balac} and \textit{F. Sturm}, Algebra and analysis. Course on mathematics for first year students with solved exercises. Collection des Sciences Appliquées de l'INSA de Lyon. (Lausanne): Presses Polytechniques et Universitaires Romandes. (2003; Zbl 1130.00001)]. The book is very well written and organized. Its content is divided in 13 chapters, covering four main parts: sequences and series of (real and complex) functions, bilinear algebra, differential calculus and integral calculus for functions of more variables. Each chapter contains completely solved problems. The computer algebra system Maple is very often used for underlying some notions or results. The correctness of the general boundary value problem without initial data for singular parabolic equations is proved.

Let \(M\) and \(M'\) be compact orientable hyperbolic 3-manifolds with totally geodesic boundaries each of which is homeomorphic to a closed connected orientable surface \(\Sigma\). For a pseudo-Anosov homeomorphism \(\varphi:\Sigma \to\Sigma\), let \(N_n\) denote the closed 3-manifold obtained from \(M\) and \(M'\) by attaching \(\partial M\) to \(\partial M'\) by the \(n\)th iteration \(\varphi^n\), that is, \(N_n=M \cup_{\varphi^n}M'\). The Thurston uniformization theorem [\textit{W. P. Thurston}, Bull. Am. Math. Soc., New Ser. 6, 357--379 (1982; Zbl 0496.57005)] implies that \(N_n\) is hyperbolic. The author studies an asymptotic behaviour of the volume of the hyperbolic 3-manifolds \(N_n\). Suppose that \(M\) and \(M'\) are irreducible with incompressible boundaries \(\partial M\) and \(\partial M'\), both of which are acylindrical and homeomorphic to a closed connected surface \(\Sigma\) of genus \(>1\). Using Gromov's simplicial volume \(\|N_n\|\) instead of hyperbolic volume, the author proves that \(\lim_n {\|N_n\|\over n}=\|\Sigma\times I_\varphi \|>0\), where \(\Sigma \times I_\varphi\) denotes the mapping torus of \(\varphi\). In particular, it follows that \(\lim_n\|N_n\|=\infty\). This gives an affirmative answer to a question in [\textit{T. Kuessner}, Efficient fundamental cycles of cusped hyperbolic manifolds, http://arxiv.org/abs/math.GT/0007003]. This paper is the written version of a talk that Thurston gave at the AMS Symposium on the Mathematical Heritage of Henri Poincaré held at Bloomington in April 1980. It is a Research announcement. [Before 1978 the Bulletin of the AMS did not accept Research announcements longer than 100 lines. For this reason, Thurston could not submit there his preprint about surface diffeomorphisms [Th7] which dates back to 1976 and which is written in the same informal style as this one. After a couple of refusals, he stopped trying to publish it and expected until 1988 to do so (see the introduction to [Th7]).] In his ``Analysis Situs'' paper, Poincaré introduces simplicial homology and the fundamental group, with the aim of finding invariants which could distinguish 3-manifolds. He concludes the Vth complement [P] discussing the question: Is any simply connected closed 3-manifold homeomorphic to the 3-sphere? This question, often understood as a conjecture, motivated much of the development of 3-dimensional topology. At the beginning of his paper Thurston proposes, ``fairly confidently'', his Geometrization Conjecture: \textit{The interior of any compact 3-manifold has a canonical decomposition into pieces that carry a geometric structure.} By Kneser and Milnor any orientable compact 3-manifold is homeomorphic to the connected sum of manifolds which are either irreducible or homeomorphic to a sphere bundle over \(S^1\). By Jaco-Shalen and Johannson any orientable irreducible 3-manifold can be decomposed along a disjoint union of embedded tori into pieces that are atoroidal or Seifert-fibered (see in [Bon2] the corresponding statement for non-orientable manifolds). Saying that a manifold carries a \textit{geometric structure} means that it has a complete Riemannian metric which is locally modelled on one of the eight geometries (which are introduced in Section 4 of the paper): the constant curvature ones \({\mathbb H}^3\), \({\mathbb S}^3\), \({\mathbb R} ^3\), the Seifert-fibered ones \(\widetilde{\mathrm{PSL}(2,{\mathbb R})}\), \({\mathbb H} ^2\times {\mathbb R}\), \({\mathbb S}^2\times {\mathbb R}\), \(\mathrm{Nil}\) and \(\mathrm {Sol}\). The pieces which are referred to in the Geometrization Conjecture are the pieces of the Jaco-Shalen-Johannson decomposition; more precisely, all the pieces except those which one needs to keep undecomposed any component of the sphere decomposition which is a torus bundle over \(S^1\) with Anosov monodromy, since those bundles carry a Sol-geometry. This conjecture contains the Poincaré Conjecture as a very special case. But the main difference with the Poincaré Conjecture is that the Geometrization Conjecture proposes a global vision of \textit{all} compact 3-manifolds. Furthermore, the geometric structure on a given 3-manifold when it exists is often unique, up to isometry; consequently, all the Riemannian invariants of the geometric structure on a 3-manifold like the volume, the diameter, the length of the shortest closed geodesic, the Chern-Simons invariant, etc.\ are then topological invariants. The Geometrization Conjecture was proven in 2003 by Grigori Perelman, who followed an analytical approach that had been initiated by Richard Hamilton in [H]. This approach was entirely different from that of Thurston, but there is no doubt that the results announced in his paper contributed to give a very strong evidence for the truth of the Geometrization Conjecture, and therefore for the truth of the Poincaré Conjecture (in the introduction of [Mo3], John Morgan points out this apparent paradox that a conjecture might look less plausible than another one which is much stronger). Among the results announced in this paper which gave strong evidence for the validity of the Geometrization Conjecture, the most important is certainly the Hyperbolization Theorem for Haken manifolds (Theorem 2.5): \textit{Any Haken 3-manifold which is homotopically atoroidal is hyperbolic, except the particular case of the twisted \(I\)-bundle over the Klein bottle}. One corollary is that the Geometrization Conjecture is true for all Haken manifolds, and in particular for any irreducible 3-manifold with non-empty boundary (once accepted the Hyperbolization Theorem, the proof of this corollary reduces to checking that Seifert fibered spaces and \(T^2\)-bundles over the circle are geometric too). Haken manifolds form a large class of 3-manifolds for which the Geometrization Conjecture is true. However, as Thurston writes after the statement of the Hyperbolization Theorem, even if the incompressible surface is an essential tool in the proof of the Hyperbolization Theorem for Haken manifolds, this surface seems to have little to do with the existence of the hyperbolic metric. And then he discusses the manifolds obtained by Dehn surgery on a 3-manifold with boundary \(M\) when the interior of \(M\) carries a finite volume hyperbolic metric. Theorem 2.6, \textit{the Hyperbolic Dehn Surgery Theorem}, states that for all surgery data, except those involving a finite set of slopes on each boundary component, the resulting 3-manifold is hyperbolic. Furthermore, few of the resulting manifolds are Haken: Thurston had shown this when \(M\) is the figure-eight knot complement [Th1], and Hatcher-Thurston had shown this when \(M\) is a \(2\)-bridge knot complement [HT] (a paper which circulated as a preprint as early as 1979). So these manifolds form a new family of hyperbolic manifolds which are not covered by the Hyperbolization Theorem. Thurston explains that he found using a computer program that most Dehn surgeries on certain punctured tori bundles over the circle were hyperbolic and that he constructed then by hand geometric structures for all the other Dehn surgeries. He insists on the beauty of the geometric structures ``when you learn to see them'' that are often revealed by computer pictures. The following section ``Applications'' describes some properties of Haken manifolds and of hyperbolic manifolds. {\parindent=6mm \begin{itemize}\item[--] When \(M\) is Haken and atoroidal, the group of isotopy classes of homeomorphisms of \(M\) is finite and it lifts to a group of homeomorphisms of \(M\). \item[--] The fundamental group of a Haken manifold is \textit{residually finite}. \item[--] Theorem 3.4: the Gromov-Thurston theorem on the behavior of volume under a non-zero degree map between hyperbolic manifolds \(M \to N\): \(\mathrm{vol}\, M\geq \mathrm{degree}\cdot\mathrm{vol} \,N\) with equality only when the covering map is homotopic to a regular cover. \item[--] Theorem 3.5: the Jørgensen-Thurston theorem that says that the set of volumes of hyperbolic manifolds is well-ordered. In particular, there is a finite volume manifold with smallest volume. \item[--] Thurston suggests that the volume and the eta invariant of a hyperbolic 3-manifold should be considered as the real and imaginary parts of a complex number. \end{itemize}} Then Thurston describes concrete approaches ``to make the calculation of hyperbolic structures routine'' and he explains how Robert Riley used computers for constructing hyperbolic structures [R1]. Riley was maybe the first person who tried to ``hyperbolize'' a given topological 3-manifold, namely the complement in \({\mathbb S}^{3}\) of certain knots (see [R2]). The last section addresses two theorems which are directly related to the proof of the hyperbolization of Haken manifolds. Theorem 5.7, \textit{the Double Limit Theorem}, is the main step of the hyperbolization for manifolds which fiber over the circle; it gives a condition for a sequence \((\rho_i)\) of quasi-Fuchsian groups to contain a subsequence which converges up to conjugacy in the space of representations of \(\pi _1 (S)\) in \(\mathrm{PSL(}2,{\mathbb C})\). The condition, formulated in terms of the Ahlfors-Bers coordinates of \(\rho_{i}\) is that the two coordinates diverge to laminations whose reunion fills up the surface. This theorem led to beautiful objects, in particular to \textit{the Cannon-Thurston map} which is a sphere-filling curve \(S^{1}\to \partial {\mathbb H}^{3}\) which conjugates the action of the surface group on \(S^{1}\) as a Fuchsian group to its action on \(\partial {\mathbb H}^{3}\) as the limiting Kleinian group; furthermore this map can be described in geometric terms. Theorem 5.8 is \textit{the Compactness Theorem of the space of discrete and faithful representations into \(\mathrm{PSL(2,}{\mathbb C})\) of the fundamental group of acylindrical 3-manifolds}, one of the key ingredients of the proof in the non-fibered case. The paper contains neither proofs nor hints of proofs. But the Princeton lecture notes [Th1] were circulating widely since the end of the 70's. These notes were not aimed to give a proof of the Hyperbolization Theorem, but they did contain important material which is relevant to the proof (in particular in the notoriously difficult Chapters 8 and 9 which discuss algebraic versus geometric convergence of hyperbolic 3-manifolds). Also Thurston had already lectured about the Hyperbolization Theorem at several conferences (see [Th2] and [Th3]). The first versions of [Th4] (proving the Compactness Theorem 5.8 above) and [Th5] (the Double Limit Theorem 5.7) were circulating since 1980. The preprint [Th6] which contains a refinement of the main result of [Th4] was distributed later, in 1986. In 1982, the most detailed presentation of the proof of the Hyperbolization Theorem in the non-fibered case, written by John Morgan, was published in the Proceedings of the Smith Conjecture conference [Mo2], a conference held in 1979; this presentation follows the proof described in [Th3]. Dennis Sullivan had reported about the proof in the fibered case at the Bourbaki seminar in 1980 [Su]. The Hyperbolization Theorem for Haken manifolds was sometimes named \textit{the Monster Theorem}, because of the length of the proof and the unusual amount of different techniques that were required. It became a challenge to find other approaches. John Hubbard observed a deep relation between the main step of the proof for the non-fibered case -- \textit{the fixed point theorem for the skinning map} -- and a conjecture of Irwin Kra in Teichmüller theory saying that the classical Theta operator acting on holomorphic quadratic differentials is a contraction for the \(L^1\)-norm. This conjecture was solved by Curt McMullen who showed how to apply it to the fixed point problem [McMullen1], [McMullen2] (see also [O2]). A proof of the fibered case is contained in [O1]; it is based on a proof of the Double Limit Theorem, different from Thurston's and uses \({\mathbb R}\)-trees instead of the delicate ``Uniform injectivity of doubly incompressible pleated surfaces'' Theorem (see also [Ka]). Thurston concludes his paper with a list of problems about 3-manifolds and about Kleinian groups which contains many of the most difficult problems of these fields that had already been posed. Any mathematician can appreciate the importance and the impact of this paper through the fact that among these 24 problems, 22 were solved by 2012 (and indeed the two remaining ones, Problem 19 -- on properties of arithmetic hyperbolic manifolds -- is more a research theme than a problem and Problem 23 -- on the rational independence of volumes of hyperbolic manifolds -- leads to difficult conjectures in number theory). Any topologist knows how much the new perspective presented in this paper influenced and inspired research in 3-dimensional topology, in hyperbolic geometry and even much beyond: it contributed in a lot of ways to the developments of geometric group theory, of complex dynamics, to the study of spaces of representations, to the study of the Weil-Petersson geometry of Teichmüller space etc. Describing these influences is not the purpose of this review and I will end by trying to describe each problem and for all but two, by giving references to the papers which brought the solution. As the copious references to be found in Zentralblatt or in MathSciNet testify, this work has inspired a vast literature. For reasons of space, I have been unable to cite many important papers which solved significant intermediate steps. \textbf{The list of problems} I have classified the problems according to the following themes: Geometrization of 3-manifolds and of 3-orbifolds, Topology of 3-manifolds, Kleinian groups, Subgroups of the fundamental groups of 3-manifolds, Computer programs and tabulations, Arithmetic properties of Kleinian groups. It appeared that there were connections between certain of the problems. For the exposition, I followed the chronological order of the solutions within a given theme, rather than the order in which Thurston presented the problems. \textbf{Geometrization of 3-manifolds and of 3-orbifolds: Problems 1, 3.} 1. \textit{The Geometrization Conjecture for \(3\)-manifolds.} This conjecture is proven by Perelman in [P1], [P2], [P3]. The content of these preprints is explained with details by Bruce Kleiner and John Lott in [KL]. John Morgan and Gang Tian give complete proofs of the Poincaré Conjecture and of the geometrization of 3-manifolds with finite fundamental groups in [MT]. A complete proof of the Geometrization Conjecture for arbitrary 3-manifolds, including a simplification of the original argument (but assuming the Hyperbolization Theorem for Haken manifolds) is contained in [BBBMP]. 3. \textit{The Geometrization Conjecture for \(3\)-orbifolds.} Thurston makes the comment that this problem contains Problem 2 on the classification of finite group actions on 3-manifolds. Indeed the language of orbifold gives a way to ``encode'' non-free actions of finite groups (but it does cover much more examples). If \(G\) is a finite group of diffeomorphisms acting on a 3-manifold \(M\), the quotient space \(M/G\) is naturally the underlying space of an orbifold whose singular locus is the image of the set of points in \(M\) which are fixed by some non-trivial element in \(G\) and labeled by the isotropy groups. Showing that the action of \(G\) on \(M\) is geometric is equivalent to showing that this orbifold is geometric. The {Geometrization Conjecture for 3-orbifolds} can be formulated in a way close to the Geometrization Conjecture for 3-manifolds [Bon2]. In a footnote added in proof, Thurston announces that he has proven this conjecture when the singular locus of the orbifold has dimension at least 1: this is the \textit{Orbifold Theorem}. Thurston did not write anything about the Orbifold Theorem comparable to what he wrote about the Hyperbolization Theorem, although he planned to do so in 1986 (cf.\ the Introduction to [Th4]). In lectures at Durham in 1984, he outlined his proof. A complete proof of the Orbifold Theorem (for orientable orbifolds) is given by Michel Boileau, Bernhard Leeb and Joan Porti in [BLP]; previously Boileau and Porti had solved the case when the singular locus is a 1-dimensional submanifold [BP]. The proofs in both papers differ from Thurston's at the delicate point of recognizing Seifert fibered pieces: those pieces are identified after a simplicial volume computation, using Gromov's vanishing theorem. Another reference, closer to Thurston's original proof, is [CHK]. \textbf{Topology of 3-manifolds: Problems 2, 4, 24.} 2. \textit{Is any finite group action on a (geometric) \(3\)-manifold equivalent to an isometric action?} This problem is a broad extension of \textit{the Smith conjecture}: \textit{Any orientation preserving periodic diffeomorphism of \({\mathbb S} ^3\) is conjugated to an orthogonal rotation} solved in [Mo1] (see in particular the history at the end of [Sh]). When the geometry on \(M\) is \(\widetilde{\mathrm{PSL(2,}{\mathbb R})}\), \({\mathbb H} ^2\times {\mathbb R}\), \(\mathrm{Nil}\), \({\mathbb R} ^3\) and \(\mathrm{Sol}\), Problem 2 is solved by Williams Meeks and Peter Scott in [MS] using minimal surfaces and topological techniques. The case of the \({\mathbb S}^2\times {\mathbb R}\) geometry is solved by Meeks and Yau in [MY], except for actions of the alternating group \(A_5\). The case when the geometry is \({\mathbb S}^3\) or \({\mathbb H}^3\) is solved by Jonathan Dinkelbach and Bernhard Leeb in [DL] by making equivariant Perelman's proof; it is important to observe that this more recent result applies to an \textit{arbitrary} finite group action on \({\mathbb S}^{3}\) or \({\mathbb H}^{3}\) whereas those (non-orientable) actions which have only isolated fixed points cannot be covered by the Orbifold theorem. 4. \textit{Develop a global theory of hyperbolic Dehn surgery.} Let \(M\) be a finite volume hyperbolic 3-manifold with \(k\) cusps; choose disjoint horoball neighborhoods of these cusps which are bounded by tori denoted \(T_i\). If for \(i=1, \dots , k\), \(s_i\) is \textit{a slope}, i.e., a non trivial isotopy class of simple closed curves on \(T_i\), \(M(s_1,\dots, s_k)\) denotes the manifold obtained from \(M\) by Dehn filling along \(s=(s_i)\). The \textit{exceptional set} is the set of the \(s\)'s such that \(M(s)\) is not hyperbolic. Ian Agol and Mark Lackenby find independently conditions on the slopes \(s_i\) that imply that \(M(s)\) is irreducible with a \textit{Gromov hyperbolic} fundamental group ([A1], [L]). When \(\partial M\) is connected, Lackenby and Meyerhoff prove that the exceptional set contains at most 10 slopes [LM], as it was conjectured by Cameron Gordon; the figure-eight knot has 10 exceptional slopes ([Th1], \S4) and is still conjectured to be the only knot with this property. A different approach to a quantitative version of the Dehn Surgery Theorem is due to Craig Hodgson and Steve Kerckhoff using deformations of hyperbolic cone-manifolds ([HK1], [HK2]). When \(\partial M\) is connected, other important universal properties of the exceptional set are also the \textit{Cyclic Dehn Surgery Theorem} of Mark Culler, Cameron Gordon, John Luecke, and Peter Shalen [CGLS] which says that there are at most three slopes such that \(M(s)\) has cyclic fundamental group, and the \textit{Finite Dehn Surgery Theorem} of Steve Boyer and Xingru Zhang which says that there are at most six slopes such that \(M(s)\) has a finite fundamental group [BZ]. 24. \textit{Show that most \(3\)-manifolds with Heegaard diagrams of a given genus have hyperbolic structures.} By the Geometrization Theorem, this problem amounts to show that most 3-manifolds with Heegaard diagrams of a given genus are irreducible and atoroidal. These properties can be checked on the curve complex of the Heegaard surface in terms of the distance between the two subcomplexes which are respectively generated by the meridian systems of each splitting [H]. But Thurston had maybe in mind a more concrete description of the hyperbolic metric, closer to the description that comes with the Hyperbolic Dehn Surgery Theorem. In [NS1] Hossein Namazi and Juan Souto consider 3-manifolds obtained by glueing 2 copies of a handlebody by a power \(\phi ^n\) of a pseudo-Anosov diffeomorphism which satisfies some genericity assumption. For all sufficiently high powers \(n\), they construct a negatively curved Riemannian metric on the resulting manifold with curvature arbitrarily close to \(-1\). Also in this case the Geometrization Theorem implies that those manifolds are hyperbolic. However, the advantage of the approach in [NS1] is to show the quasi-isometry type of the hyperbolic metric in terms of \(\phi\) and \(n\). \textbf{Kleinian groups: Problems 5--14.} In the 60's, Lipman Bers constructed an embedding of Teichmüller space as a bounded domain in \({\mathbb C}^n\). He deduced using a topological argument that most points in the frontier of this embedding did correspond to Kleinian groups which are \textit{degenerate}, in the sense that they are not \textit{geometrically finite} (see also [G]). Explicit examples of degenerate groups had been given by Troels Jørgensen in [Jo1] (see also [Jo2], a preprint from around 1975 which describes the fundamental domains for doubly degenerate once punctured torus groups). But before Thurston, no general geometric properties of degenerate Kleinian groups had been established. For proving the Hyperbolization Theorem, Thurston needed to consider those Kleinian groups and he began their study. Problems 5 to 14 address mostly those groups. 5. \textit{Are all Kleinian groups geometrically tame?} Let \(G\) be a finitely generated Kleinian group with infinite covolume that we will also suppose without parabolic elements for this exposition. By a Theorem of Peter Scott, the quotient manifold \(N={\mathbb H} ^3/G\) has \textit{a compact core}, i.e.,, a codimension 0 submanifold \(M\) such that the inclusion \(M\hookrightarrow N\) is a homotopy equivalence. Each component of \(\partial M\) separates the interior of \(M\) from an \textit{end} of \(N\). The groups studied by Thurston in \S 8 and \S 9 of [Th1] have the following two properties: (1) \(M\) is boundary-incompressible (i.e., \(G\) does not split as a free product or as an HNN extension over the trivial group) and (2) \(G\) is a limit of \textit{geometrically finite groups} (i.e., the inclusion homomorphism \(G\to \mathrm{PSL(2},{\mathbb C})\) is a limit of group embeddings \(\rho _i : G\to \mathrm{PSL(2},{\mathbb C})\) such that for all \(i\), the manifold \({\mathbb H}^3/\rho _i (G)\) has a compact core with convex boundary). A consequence of (1) is that for any boundary component \(S\) of \(\partial M\) bounding an end \(E\), the inclusion \(S \hookrightarrow E \) is a homotopy equivalence. Thurston introduced the notion of \textit{pleated surface} (also called \textit{uncrumpled surface} in [Th1], \S8). He showed that when \(G\) satisfies (2), then each end of \(N\) is \textit{geometrically tame}, i.e., either \(E\) is geometrically finite, or there exists a sequence of pleated surfaces \(f_i : S\to E\) homotopic to the inclusion map \(S\hookrightarrow E\) and such that the maps \(f_i\) tend to \(\infty\). The existence of this sequence has two important consequences. First \(N\) is \textit{topologically tame}: it is diffeomorphic to the interior of a compact 3-manifold. Second, \(N\) is \textit{analytically tame}, a property that implies in particular that \(G\) satisfies \textit{the Ahlfors conjecture} which says that the Lebesgue measure of the limit set of any finitely generated Kleinian group is \(0\) or \(1\); even more, analytical tameness implies that the action of \(G\) on its limit set is ergodic when this limit set has full measure. The notion of ``geometric tameness'' which appears in Problem 5 is this one. As indicated before, Thurston had solved this problem for the Kleinian groups which satisfy (1) and (2) in [Th1]. The first breakthrough on Problem 5 is due to Francis Bonahon. He shows in [Bon1] that any finitely generated Kleinian group is geometrically tame when it satisfies property~(1). In fact, his theorem applies to groups \(G\) which satisfy a property weaker than (1), which requires that no parabolic element of \(G\) is conjugated to an element which is contained in a factor of a decomposition of \(G\) as a free product or as an HNN extension over the trivial group. One of the tools that Bonahon introduced in his proof are \textit{the geodesic currents}; these are the transverse measures to the geodesic foliation on the unit tangent bundle of \(S\). Those objects had been much studied before, in particular in the context of Anosov flows, but they were considered by Bonahon as a natural generalization of the notion of measured geodesic laminations. The next important progress on Problem 5 after Bonahon is due to Richard Canary: he shows in his thesis, using a beautiful branched covering trick, that, when \(N={\mathbb H}^3/G\) is the interior of a compact 3-manifold, then it is geometrically tame [Ca1]. In the 70's, Al Marden had conjectured that for any finitely generated Kleinian group \(G\), \({\mathbb H}^3/G\) is \textit{topologically tame}, meaning that it is the interior of a compact manifold [Ma]; this \textit{Marden tameness conjecture} is Problem 9 from Thurston's list. Therefore, if the Marden conjecture is true, it will follow from Canary's theorem that any Kleinian group is geometrically tame. 9. \textit{Are all Kleinian groups topologically tame?} The Marden conjecture was proven in 2004 independently by Ian Agol [A2] and by Danny Calegari and David Gabai [CG] following distinct approaches. Teruhiko Soma gave a simplification of the argument of Calegari-Gabai [Som] making use of the notion of ``disk-busting'' that Agol used. Brian Bowditch exposes a self-contained proof in [Bow1] which simplifies the original approach and avoids in particular the use of the ``end-reductions'' as in the previous references. Canary surveys the history and applications of the tameness theorem in [Ca4]. 10. \textit{The Ahlfors measure \(0\) problem.} Lars Ahlfors showed in [Ah] that if \(G\) is a geometrically finite Kleinian group, then its limit set either has measure 0, or is equal to the whole sphere and he conjectured that the same holds for any finitely generated Kleinian group. Thurston had shown in [Th1] that when a Kleinian group is geometrically tame and indecomposable, then it satisfies the Ahlfors conjecture (and furthermore the action of \(G\) on the boundary of \({\mathbb H}^3\) is ergodic when its limit set has full measure). In [Ca1] Canary shows how to adapt Thurston's argument to the decomposable case. Therefore the Ahlfors conjecture follows from the positive answer to the geometric tameness question (Problem 5) which follows from the truth of the Marden conjecture (Problem 9). 11. \textit{Classify geometrically tame representations of a given group.} This problem is the \textit{Ending Lamination Conjecture}. It comes with 12. \textit{Describe the quasi-isometry type of a given group.} Let \(N={\mathbb H}^3/G\) be an hyperbolic 3-manifold with indecomposable fundamental group and let \(M\hookrightarrow N\) be a compact core. When \(N\) is geometrically tame, Thurston defined \textit{end invariants} which retain geometric information about the ends of \(N\). By Bonahon's theorem, his definition applies to any hyperbolic manifold with indecomposable fundamental group. For simplicity, suppose again that \(G\) has no parabolic elements: then each component of \(\partial M\) faces exactly one end of \(N\). If the end \(E\) facing the component \(S\) is geometrically finite, one classical invariant of \(E\) is the \textit{conformal structure at infinity}, which is an element of the Teichmüller space of \(S\). When the (geometrically tame) end facing \(S\) is not geometrically finite, Thurston defined an \textit{ending lamination}: it is a measured geodesic lamination on \(S\), well-defined up to the transversal measure. Problem 11 asks wether this set of invariants, the elements in Teichmüller space and the ending laminations are sufficient to reconstruct \(N\). When \(N\) has indecomposable fundamental group, the problem can be reduced to the case when \(N\) has the homotopy type of a closed surface. Minsky solves the case when \(N\) has \textit{bounded geometry} (the length of the closed geodesics is bounded from below by a positive constant) in [Mi1]. His proof provides also a \textit{geometric model} for \(N\), i.e., a metric space constructed directly from the end invariants which is quasi-isometric to \(N\) (this is the ``formula'' sought for in Problem 12): in the \textit{doubly degenerate case}, this model is the metric on \(S\times {\mathbb R}\) such the metric on the slice \(S\times \{t\}\) describes a Teichmüller geodesic between the two ending laminations. Understanding the case in the presence of short geodesics required a lot of efforts and the development of new techniques; one can say that much of the study from the geometric view point of the \textit{curve complex of a surface} that was initiated by Howard Masur and Yair Minsky was motivated by this problem. In [Mi2], Minsky shows that the property that \(N\) has bounded geometry in one end can be read on the ending lamination. In [Mi3], he introduces a new model, which depends on the end invariants. This model is a metric on \(S\times {\mathbb R}\) which predicts (in terms of the geodesic in the curve complex joining the end invariants) which should be the short geodesics of \(N\) and where they should be located; Minsky constructs also a map from this model to \(N\) which is Lipschitz on the thick part. In [BCM] Jeff Brock, Canary and Minsky show that this model is biLipschitz equivalent to \(N\); they announce also the same result (the existence of a model) for any finitely generated Kleinian group. Another approach to the same result is also explained in [Bow2]. Therefore, if two hyperbolic manifolds have the same end invariants, they are biLipschitz equivalent and then by Sullivan's No Invariant Line Fields Theorem, they are isometric. 6. \textit{Is every Kleinian group a limit of geometrically finite groups?} Let \( G\) be a finitely generated Kleinian group. The space of representations of \(G\) into \(\mathrm{PSL(}2,{\mathbb C})\) that send each parabolic element to a parabolic element is an affine algebraic set; the faithful representations with discrete image form a closed subset \(\mathcal {DF}(G)\) of this space. The geometrically finite representations without accidental parabolics form an open subset \(\mathcal {GF}(G)\subset \mathcal {DF}(G)\). The question asks if this subset is dense in \( \mathcal {DF}(G)\). When \(G\) is a surface group and for representations contained in a \textit{Bers slice}, this had been conjectured by Bers in [B] and it was solved by Kenneth Bromberg in [Brom] using cone manifolds technics. A positive answer to Thurston's question in full generality is given independently by Hossein Namazi and Juan Souto in [NS2] and by Ken'ichi Ohshika in [Os]; previously Brock and Bromberg solve the case of indecomposable groups without parabolic elements in [BB] (see also [BCM]). The proofs in [NS2] and [Os] depend on the Ending Lamination Theorem (Problem 11). By this theorem, any hyperbolic 3-manifold is determined up to isometry by its topology and by its end invariants. Therefore it suffices to show that for any \(\rho \in \mathcal {DF}(G)\), there is a representation \(\rho _{\infty}\) in the closure of \({\mathcal {GF}(G)}\) which has the same end invariants. The existence of such a \(\rho _{\infty}\) comes from a generalization of the Double Limit Theorem. A different solution to Problem 6, which does not use the full strength of the Ending Lamination Theorem, has been announced by Bromberg and Souto. 8. \textit{Analyse limits of quasi-Fuchsian groups with accidental parabolics.} One important and difficult step of Thurston's original proof of the Hyperbolization of Haken manifolds is the theorem which says that when a sequence of geometrically tame representations into \(\mathrm{PSL(2},{\mathbb C})\) of the fundamental group of a 3-manifold with incompressible boundary which preserves the type of the elements and converges algebraically, then it converges also geometrically (up to possibly extracting a subsequence) if the limit representation has \textit{no accidental parabolics}. This last hypothesis is really necessary since Kerckhoff and Thurston give an example of a sequence of quasi-Fuchsian representations of a punctured torus group which converges algebraically but such the geometric limit contains a \({\mathbb Z}+{\mathbb Z}\) parabolic subgroup [KS]. The geometric limit of an algebraically converging sequence of quasi-Fuchsian groups can even be non finitely generated and for many reasons ([BO], [Brock]). Ohshika and Soma have announced a complete description of the topological type of the limit hyperbolic manifolds, and also a classification up to isometry of those limits in the spirit of the Ending Lamination Theorem [OsS]. 7. \textit{Develop a theory of Schottky groups and their limits.} In [Th1], Thurston studies Kleinian groups which satisfy properties (1) and (2) (cf. the discussion of Problem 5), in particular surface groups which are limits of quasi-Fuchsian groups. For those groups he defines the end invariants in \S 8 and he shows in \S 9 the theorem mentioned in the last paragraph: ``algebraic convergence without accidental parabolics implies geometric convergence''. Problem 7 is to develop a similar study for the Kleinian groups which are limits of \textit{Schottky groups}, i.e., limits of geometrically finite free groups. Brock, Canary and Minsky announce in [BCM] that using Canary's work on Problem 5, the solution of Marden's conjecture (Problem 9) and extending their work on the Ending lamination Theorem in the boundary incompressible case, they have constructed biLipschitz models for the geometry of \({\mathbb H}^3/G\) when \(G\) is an arbitrary finitely generated Kleinian group; they show indeed that each end of \({\mathbb H}^3/G\) is quasi-isometric to an end of a degenerate surface group (with quasi-isometric constants depending on \(G\)). Jørgensen conjectured that for any finitely generated Kleinian group \(G\), a sequence of representations \(\rho _{i}:G\to \mathrm{PSL(2,}{\mathbb C})\) which converges algebraically to a representation \(\rho \) without accidental parabolics converges also geometrically to \(\rho\) up to extracting a subsequence. This is proven by Jim Anderson and Canary in [AC] when the domain of discontinuity of \(\rho\) is non empty. When the domain of discontinuity is empty, then any end of \({\mathbb H}^{3}/\rho (G)\) is degenerate; it follows from the tameness of the algebraic limit and from the Thurston-Canary covering theorem [Ca3] that \(\rho\) has finite index in the limit \(\rho _{\infty}\) of any subsequence which converges geometrically, and therefore \(\rho=\rho _{\infty}\). One could also interpet this problem as Thurston asking for a generalization of the Double Limit Theorem to the context of Schottky groups. He had conjectured a boundedness criterion for a sequence of geometrically finite representations of a free group that he formulated in terms close to those of the Double Limit Theorem but involving laminations in the \textit{Masur domain}. First attempts to prove this conjecture are [O3] and [KS] (see also [Ca5]). In its most general form, when the free group is replaced by an arbitrary finitely generated Kleinian group, the conjecture is solved by Namazi-Souto in [NS2] and by Ohshika in [Os] (cf. also the announcement by Ingkang Kim, Cyril Lecuire and Ohshika of a slightly more general result in [KLO]). 13. \textit{If the Hausdorff dimension of the limit set of a Kleinian group is \(< 2\), is it geometrically finite?} Bishop and Jones give a positive answer to this question using analytical tools in [BJ]. Another solution can be deduced from the geometric tameness of finitely generated Kleinian groups (which was proven after [BJ]). Canary observes in [Ca2] that if \(N\) is a geometrically tame hyperbolic 3-manifold which is not geometrically finite, its Cheeger constant is 0 and therefore \(\lambda _{0}(N)\), the lowest eigenvalue of the Laplacian on \(N\) is \(0\). By a result of Sullivan, for any hyperbolic 3-manifold \(N={\mathbb H}^{3}/G\), \(\lambda _{0}(N)=\delta _{G}(2-\delta _{G})\), when \textit{the critical exponent} \(\delta _{G}\) of \(G\) is \( \geq 1\) or \(\lambda _{0}(N)=1\). Therefore \(\delta _{G}=2\) when \(G\) is tame and not geometrically finite. By another result from [BJ] (true for any non-elementary discrete group of isometries of \( {\mathbb H}^n\), for any dimension) \(\delta _{G}\) equals the Hausdorff dimension of \textit{the radial limit set} of \(G\). Therefore the limit set of \(G\) has Hausdorff dimension~\(2\). 14. \textit{Existence of Cannon-Thurston maps}. Let \(H\) be a finitely generated Kleinian group and let \(M\) be a compact core of \(N={\mathbb H}^3/H\). It follows from the Hyperbolisation Theorem that there exists a convex cocompact Kleinian group \(G\) such that \(M\) is homeomorphic to the quotient by \(G\) of the reunion \({\mathbb H}^3\cup \Omega (G)\) of \({\mathbb H}^3\) and of the domain of discontinuity of \(G\). The group \(G\) is not unique but its limit set \(L(G)\) and the action of \(G\) on it are. They can be defined indeed in purely combinatorial terms, due to the fact that \(G\) is convex cocompact: \(L(G)\) is equivariantly homeomorphic to \textit{the Floyd boundary} of \(G\). Denote by \(\rho\) the isomorphism between \(G\) and \(H\) induced by the homotopy equivalence \(M\hookrightarrow N\). The problem asks if there is a continuous \(G\)-equivariant map from \(L(G)\) to \(L(\rho (G))\); such a map is called \textit{a Cannon-Thurston map} since it had been shown to exist when \(N\) is the cyclic cover of an hyperbolic manifold which fibers over the circle by Cannon and Thurston in [CT] (previously Bill Floyd had constructed the map when \(N\) is geometrically finite in [Fl]). Progress on this problem followed progress on the Ending lamination conjecture. Minsky solves the case when \(N\) has the homotopy type of a surface and has bounded geometry [Mi1]. Mahan Mj solves the case of pared manifolds with incompressible boundary and bounded geometry in [Mj1] (see also [Kl]). More recently, using the Minsky model for surface groups, Mahan Mj solves the general case when \(N\) has the homotopy type of a closed surface [Mj2]. He also announces the solution for an arbitrary hyperbolic 3-manifold. \textbf{Subgroups of the fundamental group of 3-manifolds: {Problems 15--18.}} 15. \textit{Are finitely generated subgroups of a Kleinian group separable?} A group \(G\) is LERF (\textit{locally extended residually finite}) if each finitely generated subgroup \(H\subset G\) is \textit{separable}, i.e., \(H\) equals the intersection of the finite index subgroups of \(G\) which contain \(H\). This algebraic property for a group has important geometric consequences: Peter Scott had shown in [Sc] that closed surface groups are LERF and deduced that any finitely generated subgroup of a surface group \(\pi _{1} (S)\) is the fundamental group of a subsurface in a finite cover of \(S\). Thurston asks the question because of its potential implications to the Geometrization Conjecture. Even if the Geometrization was obtained by quite different routes, the research on the LERF property has led to a vast literature which shows strong and profound connections between geometric group theory and hyperbolic manifolds. In particular, Daniel Wise developed a vast research program to attack this problem which is centered around the notion of \textit{cube complexes} [Wi]. This program is an elaboration of Scott's approach (for proving that surface groups are LERF, Scott exploited the property that the fundamental group of the non-orientable closed surface of Euler characteristic \(-1\) -- which is a quotient of any closed surface -- acts on \({\mathbb H}\) with a right-angled pentagon as fundamental domain). A \textit{cube complex} is a cell complex whose cells are isomorphic to cubes \([-1,1]^{n}\) and such that the attaching maps between cells are isometries. One says that a group \(G\) is \textit{cubulated} when it acts freely and cocompactly on a simply connected \(\mathrm{CAT(0)}\) cube complex. In 2007, Frédéric Haglund and Daniel Wise introduced the notion of \textit{special cube complex} which imposes some restrictions on the behavior of the \textit{hyperplanes} [HW]; they prove that if a compact cube complex \(X\) is special and has a Gromov hyperbolic fundamental group, then all the \textit{quasi-convex subgroups} of \(\pi _{1 } (X)\) are separable. Wise conjectures in [Wi] that any compact cube complex which has a Gromov hyperbolic fundamental group has a finite cover that is \textit{special}. This is precisely the conjecture that Agol solves in [AGM]. The solution of Problems 15 to 18 follows from this. The separability of quasi-convex subgroups (i.e., convex cocompact subgroups) of a cocompact Kleinian group is a direct application of this result since Nicolas Bergeron and Daniel Wise proved in 2009 that any cocompact Kleinian group is cubulated [BW] (subgroups which are not convex cocompact are virtual fibers as a consequence of the tameness and the Thurston-Canary covering theorem). Bergeron and Wise used a method introduced by Michah Sageev who constructed, from any finite collection of quasi-convex subgroups of a group \(G\) (which satisfy certain conditions) an action of \(G\) on a \(\mathrm{CAT(0)}\) cube complex. This theorem of Bergeron and Wise could be proven in such a generality thanks to the recent result of Jeremy Kahn and Vlad Markovic which says that any cocompact Kleinian group contains (many) quasi-Fuchsian subgroups [KM]. 16. \textit{Is every irreducible \(3\)-manifold with infinite fundamental group virtually Haken?} This problem contains the question whether the fundamental group of an irreducible 3-manifold contains a surface group when it is infinite. For a cocompact Kleinian group, this is solved by Kahn and Markovic in [KM]; they use in particular fine properties of the geodesic flow on the unit tangent bundle of hyperbolic manifolds such as the \textit{exponential mixing with respect to the Liouville measure}. By the LERF property of Kleinian groups, any surface subgroup of a cocompact Kleinian group \(G\) provided by this theorem is the fundamental group of a closed surface which is embedded in an appropriate finite cover of \({\mathbb H}^3/G\). This solves Problem 16 for the case of hyperbolic manifolds which was the remaining case (by the Geometrization Theorem and since all the other geometries can be handled directly). 17. \textit{Does every aspherical \(3\)-manifold have a finite cover with positive first Betti number?} 18. \textit{Does every finite volume hyperbolic \(3\)-manifold have a finite cover which fibers over the circle?} In [A3], Agol introduces a new class of groups, the \textit{residually finite rational solvable} (RFRS) groups and shows that this class contains all \textit{right-angled Artin groups}. He observes that if the fundamental group of a 3-manifold \(M\) is not abelian but RFRS, then \(M\) has a first Betti number virtually infinite; he shows also that when \(M\) is irreducible with \(\chi (M)=0\) and \(\pi _{1}(M)\) RFRS, then \(M\) virtually fibers over the circle. In [HW], among the properties of special cube complexes they study, Haglund and Wise show that if a group is cubulated by a special cube complex, then some finite index subgroup of it embeds into a right-angled Artin group. Therefore, any cocompact Kleinian group has a finite index subgroup which is RFRS and therefore Problems 17 and 18 have positive answers. \textbf{Computer programs and tabulations: Problems 20--22.} In his PhD thesis, Jeff Weeks wrote the program \texttt{SnapPea} which computes the hyperbolic structure of a link complement in \({\mathbb S}^3\) when this structure exists (\url{http://www.geometrygames.org/SnapPea/}). This program provides a lot of useful information: it shows the Ford domain, it gives the shape of the cusps and shows the induced tessellation of those, it computes many invariants of the hyperbolic metric like the volume, the length of its shortest closed geodesic etc.\ [We]. It allows to tabulate many families of 3-manifolds like the Callahan-Hildebrand-Weeks census of non compact finite volume hyperbolic 3-manifolds, or the Hodgson-Weeks census of small volume closed hyperbolic 3-manifolds. It shows evidence for several problems in the list; in particular in [DT], Nathan Dunfield and Thurston check the virtual Haken conjecture for all 3-manifolds from the Hodgson-Weeks census (all have indeed positive virtual Betti number). The \textit{Weeks manifold} was observed to have the smallest volume among the known hyperbolic manifolds and was conjectured to be the closed hyperbolic manifold with smallest volume: this conjecture is now solved by David Gabai, Robert Meyerhoff and Peter Milley (see [GMM1], [GMM2]). Furthermore, \texttt{SnapPea} permits to ``feel'' the well ordering of hyperbolic manifolds by the volume as Thurston asks in Problem 22: it gives evidence that this ordering is compatible with the topological complexity given by the \textit{Mom number} [GMM1]. \texttt{SnapPea} is a powerful tool, which in addition to helping the intuition with many problems, suggests also new ones. For instance, its computation of the hyperbolic structure on a link complement exhibits in all cases a triangulation of this complement by \textit{geodesic ideal tetrahedra}; however there are no theoretical proofs yet of the existence of such a triangulation. Other software doing computations with 3-manifolds or with surface diffeomorphisms can be found at the CompuTop.org Software Archive website \url{http://www.math.illinois.edu/~nmd/computop/index.html}. \textbf{Arithmetic properties of Kleinian groups: Problems 19, 23.} 19. \textit{Find topological and geometric properties of quotient spaces of arithmetic subgroups of \(\mathrm{PSL(2},{\mathbb C})\).} This problem is more a theme of research than a problem like the ones above. Certain topological problems from the list were first established for arithmetic hyperbolic 3-manifolds: Lackenby proves the Surface subgroup conjecture for arithmetic manifolds in [L2] and Agol applies his criterion for virtual fibering to Bianchi groups in [A3]. A geometric property of arithmetic hyperbolic manifolds that is often used is that each commensurability class of arithmetic 3-manifolds contains many representatives with a non-trivial isometry group. An example of this is the following property which characterizes the arithmetic manifolds among all hyperbolic finite volume 3-manifolds [CLR]: \textit{Any closed geodesic \(\gamma\) in an arithmetic hyperbolic 3-manifold \(N\) lifts to a geodesic \(\tilde \gamma\) in some finite cover \(N_{\gamma}\) of \(N\) which admits an isometric involution whose fixed point set contains \(\tilde \gamma\)}. One central question, closely related to questions in number theory like the Lehmer conjecture, is the Short Geodesic Conjecture: it asks wether the injectivity radius of arithmetic hyperbolic 3-orbifolds is bounded from below by a global positive number (cf. [MR], \S12). For more information regarding this problem, see the survey by Alan Reid [Reid1]. 23. \textit{Show that the volumes of hyperbolic \(3\)-manifolds are not all rationally related.} Today, one knows very little about arithmetic properties of volumes of hyperbolic 3-manifolds and this problem is far from being solved; one does not even know of one single hyperbolic 3-manifold for which one could decide whether its volume is rational or irrational. However, the algebraic framework for studying arithmetic properties of volumes is now well established. Given a field \(k\subset {\mathbb C}\), its \textit{Bloch group} \(\mathcal B (k)\) is defined as a certain subspace of a certain quotient of the free \({\mathbb Z}\)-module generated by the elements of \(k\setminus \{0,1\}\); there is also a \textit{Bloch regulator map} \(\rho : B ( k)\to {\mathbb C}/{\mathbb Q}\) [NY]. Walter Neumann and Jun Yang assign to any finite volume hyperbolic 3-manifold \(N={\mathbb H}^3/G\) an element \(\beta (N)\in \mathcal B ( k(N))\subset \mathcal B ({\mathbb C})\), where \(k(N)\) is the \textit{invariant trace field of \(N\)} (i.e., the subfield of \({\mathbb C}\) generated by the squares of the traces of the elements of \(G\)). They show that, up to a constant multiple, the volume of \(N\) and its Chern-Simons invariant are respectively, the imaginary part and the real part of \(\rho (\beta (N))\) (this is one realization of Thurston's hint that volume and Chern-Simons invariant should be considered simultaneously has the real and imaginary parts of a complex number (see also [Y])). It is conjectured that when \(k=\overline{{\mathbb Q}}\), the algebraic closure of \({\mathbb Q}\), the imaginary part of the Bloch regulator map is injective. 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Let \(M\) and \(M'\) be compact orientable hyperbolic 3-manifolds with totally geodesic boundaries each of which is homeomorphic to a closed connected orientable surface \(\Sigma\). For a pseudo-Anosov homeomorphism \(\varphi:\Sigma \to\Sigma\), let \(N_n\) denote the closed 3-manifold obtained from \(M\) and \(M'\) by attaching \(\partial M\) to \(\partial M'\) by the \(n\)th iteration \(\varphi^n\), that is, \(N_n=M \cup_{\varphi^n}M'\). The Thurston uniformization theorem [\textit{W. P. Thurston}, Bull. Am. Math. Soc., New Ser. 6, 357--379 (1982; Zbl 0496.57005)] implies that \(N_n\) is hyperbolic. The author studies an asymptotic behaviour of the volume of the hyperbolic 3-manifolds \(N_n\). Suppose that \(M\) and \(M'\) are irreducible with incompressible boundaries \(\partial M\) and \(\partial M'\), both of which are acylindrical and homeomorphic to a closed connected surface \(\Sigma\) of genus \(>1\). Using Gromov's simplicial volume \(\|N_n\|\) instead of hyperbolic volume, the author proves that \(\lim_n {\|N_n\|\over n}=\|\Sigma\times I_\varphi \|>0\), where \(\Sigma \times I_\varphi\) denotes the mapping torus of \(\varphi\). In particular, it follows that \(\lim_n\|N_n\|=\infty\). This gives an affirmative answer to a question in [\textit{T. Kuessner}, Efficient fundamental cycles of cusped hyperbolic manifolds, http://arxiv.org/abs/math.GT/0007003]. Cet exposé présente des résultats qui font suite à l'article de \textit{R. Beals} et \textit{R.Coifman} [Commun. Pure. Appl. Math. 37, 39-90 (1984; Zbl 0519.34022)]. Il s'agit du problème inverse pour un opérateur non nécessairement auto-adjoint tel que \(P=D^ 3-q_ 1(x)D-q_ 0(x), D=-id/dx,\) considéré sur \({\mathbb{R}}\). L'étude du problème spectral \((1)\quad Pu=z^ 3u, z\in {\mathbb{C}}\), conduit à une fonction matricielle continue v(z) définie sur la réunion S des rayons \(S_ k=\{z\in {\mathbb{C}}\backslash 0\); arg z\(=(\pi /6)+k(\pi /3)\}\), \(k=0,1,...,5\), et sur un ensemble borné discret \(B\subset {\mathbb{C}}\backslash S\). La fonction V, appelée de scattering, est déterminée par la comportement asymptotique de fonctions propres de (1) pour \(z\in B\cup S\). L'A. montre que l'application \((q_ 0,q_ 1)\mapsto v\) est injective et que pour un ensemble ouvert d'un espace de Banach convenable de fonctions v, v provient d'un unique opérateur v.

Extending the results of \textit{G. Helzer} [Am. Math. Mon. 107, No. 3, 219--237 (2000; Zbl 1012.83002)], the author defines, in two-dimensional Minkowski space-time, Minkowskian angles and pseudo-angles between two null directions and between a null direction and any non-null direction, respectively, together with their hyperbolic trigonometric functions. Particular care is taken to define every notion, as well as to provide illustrations, by means of several handmade colored figures, of the facts described by the formulas, making this paper readable by undergraduates. The paper starts with several problems formulated by Rick Norwood in 1992, concerning the relativist treatment of a rotational movement. Following the author's opinion, an explanation of the difficulties in solving such problems resides in a poor knowledge of the fundamental mathematical structures intrinsically involved in the special relativity. As a remarkable example in this respect, the author reminds us of the incompatibility of special relativity with the use of classical calculus. Aiming at clarifying the natural framework, and at removing several difficulties encountered in developing the special relativity, the paper contains the following three sections: Planar Curves, Accelerated Points, and Acceleraled Segments.

Extending the results of \textit{G. Helzer} [Am. Math. Mon. 107, No. 3, 219--237 (2000; Zbl 1012.83002)], the author defines, in two-dimensional Minkowski space-time, Minkowskian angles and pseudo-angles between two null directions and between a null direction and any non-null direction, respectively, together with their hyperbolic trigonometric functions. Particular care is taken to define every notion, as well as to provide illustrations, by means of several handmade colored figures, of the facts described by the formulas, making this paper readable by undergraduates. This approximately 100 page thesis contains, in a very readable form, just about all one wants to know about the theory of domains, algebraic or continuous directed complete posets. Starting from the definition of a partial ordering, the author quickly gets to new results in domain theory. The work is divided into four chapters: Basic Concepts, Domains with least elements, Domains without a least element, and Continuous domains. The theme is to classify such domains by finding maximal subcategories with certain properties. The first such theorem is Smyth's result that if D is an algebraic directed complete poset with least element and if the function space [D\(\to D]\) is \(\omega\)-algebraic, then D is bifinite. This theorem is generalized in the thesis in various ways: to the uncountable case, and to the case that the domains have no least element. The last chapter treats retracts of algebraic directed complete posets. The paper is quite well-written and can serve as either an introduction to an interesting subject or as a source of conjectures and new results for the expert.

The authors analyze a Hardy-Sobolev inequality introduced by \textit{M. Badiale} and \textit{G. Tarantello} [Arch. Ration. Mech. Anal. 163, No. 4, 259--293 (2002; Zbl 1010.35041)], and they show that the extremal functions for which the inequality is optimal satisfy certain symmetry properties. We analyze the existence and non-existence of cylindrical solutions for the following problem: \[ -\Delta u(x)= \varphi(r)|u|^{p-2}u \quad\text{in }\mathbb{R}^3, \qquad u(x)>0\quad\text{in }\mathbb{R}^3, \qquad \int_{\mathbb{R}^3} \varphi(r)u^{p-1} dx< +\infty, \] with \(p>1\), which has been proposed as a model for the dynamics of galaxies. We prove a general integral inequality of Sobolev-Hardy type that allows us to use variational methods when the power \(p\) belongs to the interval [4,6]. We find solutions in the range \(4 < p \leq 6\). The value \(p = 4\) seems to have characteristics similar to those of the critical Sobolev exponent \(p = 6\).

The authors analyze a Hardy-Sobolev inequality introduced by \textit{M. Badiale} and \textit{G. Tarantello} [Arch. Ration. Mech. Anal. 163, No. 4, 259--293 (2002; Zbl 1010.35041)], and they show that the extremal functions for which the inequality is optimal satisfy certain symmetry properties. Contributions: --Shreeram S. Abhyankar, Recent progress on the Jacobian conjecture: p.1077 --Manuel Blickle (joint with Raphael Bondu), Local cohomology multiplicities via local etale cohomology: p.1080 -- Holger Brenner (joint with Moty Katzman), On the arithmetic of tight closure and of strong semistability: p.1082 --Ragnar-Olaf Buchweitz (joint with Graham Leuschke and Michel van den Bergh), Noncommutative desingularization of the generic determinant: p.1084 --Aldo Conca, Products of linear spaces, polymatroids and integral posets: p.1087 --Lawrence Ein, Singularities of pairs: p.1090 --David Eisenbud (joint with Frank-Olaf Schreyer), Computing direct images with free resolutions: p.1092 --Joseph Gubeladze, Effective factorization of invertible matrices over monoid rings: p.1093 --R. V. Gurjar, A geometric proof of Boutot's result on singularities of quotients: p.1096 --Robin Hartshorne, Gorenstein liaison of algebraic varieties: p.1097 --Srikanth Iyengar (joint with Luchezar Avramov, Ragnar-Olaf Buchweitz, Claudia Miller), Levels in triangulated categories and perfect complexes over commutative rings: p.1100 --Shulim Kaliman (joint with Nikolai Saveliev), \({\mathbb C}_+\)-actions on contractible threefolds: p.1103 --Kazuhiko Kurano, The singular Riemann-Roch formula and Hilbert-Kunz functions: p.1104 --Graham J. Leuschke (joint with Ragnar-Olaf Buchweitz), Factoring the adjoint and maximal CM modules: p.1107 --Masayoshi Miyanishi, Affine pseudo-planes and affine pseudo-coverings: p.1110 --Tim Römer (joint with Morten Brun, Winfried Bruns), Cohomology of partially ordered sets: p.1113 --Anurag K. Singh (joint with Uli Walther), Remarks on F-pure rings: p.1115 --Bernd Sturmfels (joint with Seth Sullivant), Combinatorial secant varieties: p.1117 --Shunsuke Takagi, A subadditivity formula for multiplier ideals on singular varieties: p.1118

Let \(A\) and \(B\) be positive invertible operators on a complex Hilbert space, and let \(\sigma \) be an operator mean. \textit{D.~Kainuma} and \textit{E. Kamei} [Math. Jap. 28, 627--631 (1983; Zbl 0531.46045)] proved that if \(\phi\) is a \(C^{*}\)-homomorphism, then \(\phi(A\sigma B)=\phi(A)\sigma\phi(B)\) holds for any positive operators \(A\) and \(B\). In this paper, the authors discuss properties of the function \(\phi\) satisfying \( \phi(A\sigma B)=\phi(A)\sigma \phi(B)\). They obtain implication relations among the following assertions: (i) \(\phi(A\sharp B)=\phi(A)\sharp \phi(B)\), (ii) \(\phi(A\nabla_{s,t} B)=\phi(A)\nabla_{s,t} \phi(B)\) for \(s,t>0\) and (iii) \(\phi(A!_{s,t} B)=\phi(A)!_{s,t} \phi(B)\) for \(s,t>0\), where \(\sharp\), \(\nabla_{s,t}\) and \(!_{s,t}\) mean geometric, generalized arithmetic and generalized harmonic means, respectively. Next, the authors obtain properties of a positive increasing function satisfying \(f(x^{-1})=(f(x))^{-1}\) on \([0,\infty)\) via shorted operators. The theory of means of positive operators acting on a Hilbert space H is established by \textit{F. Kubo} and \textit{T. Ando} [Math. Ann. 246, 205-224 (1980; Zbl 0412.47013)]: A binary operation m among positive operators is called a mean if m satisfies the following axioms: (I) \(A\leq C\) and \(B\leq D\) imply Am\(B\leq CmD,\) (II) Transformer inequality: \(C^*(AmB)C\leq C^*ACmC^*BC,\) (III) \(A_ n\downarrow A\), \(B_ n\downarrow B\) imply \(A_ nmB_ n\downarrow AmB,\) (IV) \(1m1=1.\) Ando proved the ''submultiplicativity'' \(\Phi\) (AmB)\(\leq \Phi(A)m\Phi(B)\), for a positive linear map \(\Phi\), invertible positive A and B, and \(m=h\) (the harmonic mean) and g(the geometric mean), i.e. \(AhB=2(A^{- 1}+B^{-1})^{-1}\), \(AgB=A^{\frac{1}{2}}(A^{-frac{1}{2}}BA^{- frac{1}{2}})^{\frac{1}{2}}A^{\frac{1}{2}}\). The main theorem of this note is to give a partial converse of Ando's theorem: Theorem: If \(\Phi\) is a normal unital positive linear map among von Neumann algebras, then the following conditions are equivalent: (i) \(\Phi\) is a \(C^*\)-homomorphism, (ii) \(\Phi(AgB)=\Phi(A)g\Phi(B)\) for A,\(B\geq 0\) (iii) \(\Phi(AhB)=\Phi(A)h\Phi(B)\) for A,\(B\geq 0\), where a \(C^*\)- homomorphism is a positive linear map \(\Phi\) on a \(C^*\)-algebra such that \(\Phi(A^ 2)=\Phi(A)^ 2\) for every selfadjoint A. Moreover, certain allied topics are also discussed.

Let \(A\) and \(B\) be positive invertible operators on a complex Hilbert space, and let \(\sigma \) be an operator mean. \textit{D.~Kainuma} and \textit{E. Kamei} [Math. Jap. 28, 627--631 (1983; Zbl 0531.46045)] proved that if \(\phi\) is a \(C^{*}\)-homomorphism, then \(\phi(A\sigma B)=\phi(A)\sigma\phi(B)\) holds for any positive operators \(A\) and \(B\). In this paper, the authors discuss properties of the function \(\phi\) satisfying \( \phi(A\sigma B)=\phi(A)\sigma \phi(B)\). They obtain implication relations among the following assertions: (i) \(\phi(A\sharp B)=\phi(A)\sharp \phi(B)\), (ii) \(\phi(A\nabla_{s,t} B)=\phi(A)\nabla_{s,t} \phi(B)\) for \(s,t>0\) and (iii) \(\phi(A!_{s,t} B)=\phi(A)!_{s,t} \phi(B)\) for \(s,t>0\), where \(\sharp\), \(\nabla_{s,t}\) and \(!_{s,t}\) mean geometric, generalized arithmetic and generalized harmonic means, respectively. Next, the authors obtain properties of a positive increasing function satisfying \(f(x^{-1})=(f(x))^{-1}\) on \([0,\infty)\) via shorted operators. The book is intended as an introductory textbook at undergraduate level for students of computer science. The central topics are formalisms and methods for representing and manipulating knowledge: Logic, semantic nets, production rules, reasoning. These principles are illustrated by easy to understand examples which mostly are taken from cardiovascular medicine. Most of the principles of expert systems have been worked out in small programs (PROLOG, LISP). Chapter 1 introduces into the problem domain, Chapter 2 is concerned with propositional and first order logic. The basic principles of resolution are discussed as well as the subject of efficiency in the context of automated reasoning: Semantic, SLD resolution, first-order predicate logic. Logic as a reprsentation formalism, and the importance for the construction of logic-based expert systems, will be exemplified. Chapter 3 deals with knowledge representation in production systems. Inference methods with production rules (top-down inference), pattern representation and matching. Chapter 4 deals with semantic net formalism and inheritance as a reasoning mechanism in frame hierarchies. Reasoning with uncertainty is the main topic of Chapter 5: Fundamentals of probability theory and the generalized Dempster-Shafer theory of uncertainty with an adaption for application in production systems. Chapter 6 presents tools for knowledge and inference inspection with concepts for implementing user interfaces. In Chapter 7, OPSS, LOOPS, and CENTAUR as languages used for the development of expert systems are discussed. The appendix yields an overview of the PROLOG and LISP language. The textbook distinguishes from many other books on expert systems. It is suitable for teaching the concepts and techniques of knowledge representation and inference. Primarily, the teaching subject is devoted to a computer science audience, but the book can also be recommended to those students in other scientific fields which are interested in fundamentals and practical aspects of building expert systems.

An appropriate combination of mesh refinement (\(h\)-version) and increase of the shape functions' polynomial degree (\(p\)-version) yield, for linear elliptic boundary value problems, an exponential rate of convergence in the energy norm even if the exact solution is singular. The present paper presents a method which uses an \(h\)-\(p\)-approximation for a domain decomposition with fully overlapping domains, allowing \(C^ 0\)- continuous approximation for geometrically incompatible mesh refinement to be obtained [cf. \textit{J. H. Bramble, R. E. Ewing, R. R. Parashkevov} and \textit{J. E. Pasciak}, SIAM J. Sci. Stat. Comput. 13, No. 1, 397-410 (1992; Zbl 0746.65084)]. The method is applied to a reaction-diffusion problem resulting from semiconductor process simulation. By means of numerical examples it is illustrated that the foregoing approach is well suited for problems with sharp interior layers or shock-like behaviour. A flexible mesh refinement strategy for the approximation of solutions of elliptic boundary value problems in considered. A preconditioning based on overlapping domain decomposition methods is considered. It is shown that the iteration schemes converge to the discrete solution at a rate which is independent of the mesh parameters in the case of two dimensions. The results in three dimensions are somewhat weaker. Numerical experiments are presented.

An appropriate combination of mesh refinement (\(h\)-version) and increase of the shape functions' polynomial degree (\(p\)-version) yield, for linear elliptic boundary value problems, an exponential rate of convergence in the energy norm even if the exact solution is singular. The present paper presents a method which uses an \(h\)-\(p\)-approximation for a domain decomposition with fully overlapping domains, allowing \(C^ 0\)- continuous approximation for geometrically incompatible mesh refinement to be obtained [cf. \textit{J. H. Bramble, R. E. Ewing, R. R. Parashkevov} and \textit{J. E. Pasciak}, SIAM J. Sci. Stat. Comput. 13, No. 1, 397-410 (1992; Zbl 0746.65084)]. The method is applied to a reaction-diffusion problem resulting from semiconductor process simulation. By means of numerical examples it is illustrated that the foregoing approach is well suited for problems with sharp interior layers or shock-like behaviour. In this paper, strong forms of Schur's Lemma and its converse are proved for maps taking non-negative functions to non-negative functions and having formal adjionts. This results are applied to give best constants in a large class of weighted Lebesgue norm inequalities for non-negative integral operators. Since general mesures are used, norm of non-negative matrix operators may be calculated by the same method.

This work extends the analysis of the first part [see the review above]. The conditions which must be satisfied for the existence of the nonlinear driving-point immittance in the nonuniform case are stronger than in the case of the uniform ladder. The author has proved that the nonlinear driving-point immittance can be computed for a nonlinear ladder network. The ladder network which has been under investigation is infinite but uniform. The series resistances and the shunt conductances of the ladder must be continuous functions of current or voltage and monotone in a neighborhood of the origin. An interesting example of a network is presented where the characteristic immittance is linear though the network elements are nonlinear.

This work extends the analysis of the first part [see the review above]. The conditions which must be satisfied for the existence of the nonlinear driving-point immittance in the nonuniform case are stronger than in the case of the uniform ladder. The efficient solution to the coupled system of PDEs governing the mass and the energy balance in deformable porous media requires advanced numerical algorithms. A combination of mixed/Galerkin finite elements and finite volumes along with a staggered method are employed. Fluid flow and heat transfer are addressed iteratively by a fully coupled approach and the medium deformation by an explicitly coupled scheme, at each time step. Such formulation allows for stable numerical solutions, element-wise conservative velocity fields and accurate prediction of sharp temperature convective fronts. The proposed model is experimented with in realistic applications of a deep aquifer fluid injection.

A bounded linear operator \(T\) acting on a Banach space \(X\) satisfies property \((w)\), a variant of Weyl's theorem, provided that the approximate point spectrum of \(T\) is the disjoint union of the Weyl essential approximate point spectrum of \(T\) and the set of all isolated points in the spectrum of \(T\) which are eigenvalues of finite multiplicity [\textit{V. Rakočević}, Mat. Vesn. 37, 423--426 (1985; Zbl 0596.47001)]. The authors use spectra related to the notion of topological uniform descent, introduced in this paper, to give necessary and sufficient conditions for \(T\) to satisfy property \((w)\). As application, they investigate the stability of property \((w)\) under commuting perturbations by power finite rank operators. Let X be an infinite-dimensional complex Banach space and denote the set of bounded linear operators on X by B(X). Let \(\sigma_ a(A)\) and \(\sigma_{ea}(A)\) denote, respectively, the approximate point spectrum and essential approximate point spectrum of an element A of B(X) [Mat. Vesnik 5, 389-391 (1981; Zbl 0504.47004)]. In this note we study the class of operators A in B(X) such that \(\sigma_{ea}(A)=\sigma_ a(A)\setminus \pi_{\infty}(A)\), where \(\pi_{\infty}(A)\) denotes the set of isolated eigenvalues of A with finite multiplicities.

A bounded linear operator \(T\) acting on a Banach space \(X\) satisfies property \((w)\), a variant of Weyl's theorem, provided that the approximate point spectrum of \(T\) is the disjoint union of the Weyl essential approximate point spectrum of \(T\) and the set of all isolated points in the spectrum of \(T\) which are eigenvalues of finite multiplicity [\textit{V. Rakočević}, Mat. Vesn. 37, 423--426 (1985; Zbl 0596.47001)]. The authors use spectra related to the notion of topological uniform descent, introduced in this paper, to give necessary and sufficient conditions for \(T\) to satisfy property \((w)\). As application, they investigate the stability of property \((w)\) under commuting perturbations by power finite rank operators. Problem solving and symbolisation among students who fail in school. Directed at students who fail in the school system, the aim of this research is to analyse the impact of interaction between different kinds of symbolisation in relation to concepts in problem solving. There are two sorts of methods for symbolisation: natural language and schematisation. These symbolic accompaniments of thought are in reality a new type of mediation which can allow students to have the perspective and distance necessary to face cognitive obstacles which up until then have been insurmountable. The experience (1999-2001) has been based on the theoretical relationship between psychology (genetic and cognitive), semiotics and mathematical pedagogy. Within this framework, problems known as '\(4^{th}\) proportional' were presented to 15-year old students of SEGPA. The specificity of these mathematical problems is that they engage students in problem solving. Students worked on them during clinical sessions as well as in the classroom, which encouraged pair work interaction. During the classroom work, the clinicians, helped by the teachers, picked opportunities to use symbolisations to, among other things, provoke a rupture in the thought process, so that students could modify their representations of the problems. Individual follow-up of selected students with contrasting results, allowed the measurement of the evolution of their cognitive profile throughout the experimentation period. (orig.)

Consider an \(n\)-dimensional, conformally flat, minimal, Lagrangian submanifold \(M\) of an \(n\)-dimensional Kähler manifold \(\widetilde M\) of constant holomorphic sectional curvature \(4c\). The paper studies such submanifolds in terms of multiplicities of the eigenvalues of the Schouten tensor, for \(n\geq4\). Since conformally flat Riemannian metrics in dimension \(n\geq4\) have vanishing Weyl tensor, their Schouten tensors are of Codazzi type, and the article under review proves the following two theorems: \textbf{Theorem.} If the Schouten tensor of \(M\) does not admit an eigenvalue of multiplicity one, then it has a single eigenvalue. The submanifold \(M\) is then of constant sectional curvature \(\bar c\) and is either totally geodesic or \(c=1, \bar c=0\), and \(M\) is congruent to an open part of the Lagrangian torus (detailed in the introduction before Theorem 1.1). \textbf{Theorem.} If the Schouten tensor of \(M\) admits exactly one eigenvalue of multiplicity one, then it admits only one eigenvalue of higher multiplicity. The submanifold \(M\) is then locally isometric to a warped product \(I\times M_0\), where \(I\subset\mathbb{R}\) is an interval and \(M_0\) is a manifold with constant sectional curvature. The next three theorems, proven in the paper, are local classification theorems for the following specific situations: \begin{itemize} \item[1.] Assume \(\widetilde M\) is \(\mathbb{C}^n\), \(n\geq4\), and the Schouten tensor admits exactly one eigenvalue of multiplicity one. In this case, three normal forms are obtained, two of which appear to be missing in a previous work by \textit{D. E. Blair} [Can. Math. Bull. 50, No. 3, 321--333 (2007; Zbl 1149.53048)]. \item[2.] When \(\widetilde M\) is \(\mathbb{CP}^n(4)\), \(n\geq4\), and the Schouten tensor admits exactly one eigenvalue of multiplicity one. Three normal forms are obtained. \item[3.] When \(\widetilde M\) is the complex hyperbolic space \(\mathbb{CH}^n(-4)\), \(n\geq4\), and the Schouten tensor admits exactly one eigenvalue of multiplicity one. Four normal forms are obtained. \end{itemize} The mentioned normal forms can be found in Theorems 1.5, 1.6 and 1.7 of the paper, respectively. The author studies the question of conformally flat, minimal and Lagrangian submanifolds in \(\mathbb C^n\) and shows that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one.

Consider an \(n\)-dimensional, conformally flat, minimal, Lagrangian submanifold \(M\) of an \(n\)-dimensional Kähler manifold \(\widetilde M\) of constant holomorphic sectional curvature \(4c\). The paper studies such submanifolds in terms of multiplicities of the eigenvalues of the Schouten tensor, for \(n\geq4\). Since conformally flat Riemannian metrics in dimension \(n\geq4\) have vanishing Weyl tensor, their Schouten tensors are of Codazzi type, and the article under review proves the following two theorems: \textbf{Theorem.} If the Schouten tensor of \(M\) does not admit an eigenvalue of multiplicity one, then it has a single eigenvalue. The submanifold \(M\) is then of constant sectional curvature \(\bar c\) and is either totally geodesic or \(c=1, \bar c=0\), and \(M\) is congruent to an open part of the Lagrangian torus (detailed in the introduction before Theorem 1.1). \textbf{Theorem.} If the Schouten tensor of \(M\) admits exactly one eigenvalue of multiplicity one, then it admits only one eigenvalue of higher multiplicity. The submanifold \(M\) is then locally isometric to a warped product \(I\times M_0\), where \(I\subset\mathbb{R}\) is an interval and \(M_0\) is a manifold with constant sectional curvature. The next three theorems, proven in the paper, are local classification theorems for the following specific situations: \begin{itemize} \item[1.] Assume \(\widetilde M\) is \(\mathbb{C}^n\), \(n\geq4\), and the Schouten tensor admits exactly one eigenvalue of multiplicity one. In this case, three normal forms are obtained, two of which appear to be missing in a previous work by \textit{D. E. Blair} [Can. Math. Bull. 50, No. 3, 321--333 (2007; Zbl 1149.53048)]. \item[2.] When \(\widetilde M\) is \(\mathbb{CP}^n(4)\), \(n\geq4\), and the Schouten tensor admits exactly one eigenvalue of multiplicity one. Three normal forms are obtained. \item[3.] When \(\widetilde M\) is the complex hyperbolic space \(\mathbb{CH}^n(-4)\), \(n\geq4\), and the Schouten tensor admits exactly one eigenvalue of multiplicity one. Four normal forms are obtained. \end{itemize} The mentioned normal forms can be found in Theorems 1.5, 1.6 and 1.7 of the paper, respectively. Bemerkung über die Ausbreitungsgeschwindigkeit der verschiedenen Arten von Erdbebenwellen und Kritik an gewissen Folgerungen aus Messungsergebnissen, die Dichte der durchlaufenen Erdschichten betreffend.

The notions of primitivity and exponent of a square nonnegative matrix \(A\) are classical: \(A\) is primitive provided there is a nonnegative integer \(k\) such that \(A^k\) is entrywise positive and in the case \(A\) is primitive the exponent of \(A\) is the smallest such \(k\). \textit{E. Fornasini} and \textit{M. E. Valcher} [ibid. 263, 275-310 (1997; Zbl 0887.93033)] have extended the notion of primitivity to pairs \((A,B)\) of square nonnegative matrices of the same order. The pair \((A,B)\) is primitive provided there exist nonnegative integers \(h\) and \(k\) such that the sum of all products formed by words consisting of \(hA\)'s and \(kB\)'s is entrywise positive. The paper defines the exponent of a nonnegative matrix pair to be the smallest value of \(h+k\) over all such \(h\) and \(k\). It is then shown that the largest exponent of a primitive pair of \(n\) by \(n\) nonnegative matrices lies in the interval \([(n^3-5n^2)/2\), \((3n^3+2n^2 -2n)/2]\). In addition, the exponent of a pair of nonnegative matrices is related to properties of an associated two-dimensional dynamical system. A notion of a 2D digraph and a 2D strongly connected digraph are introduced. Equivalent descriptions of the irreducibility of matrices are obtained. Dynamical characterizations of irreducible matrix pairs and their characteristic polynomials are presented. Primitivity is introduced as a special case of irreducibility of matrix pairs.

The notions of primitivity and exponent of a square nonnegative matrix \(A\) are classical: \(A\) is primitive provided there is a nonnegative integer \(k\) such that \(A^k\) is entrywise positive and in the case \(A\) is primitive the exponent of \(A\) is the smallest such \(k\). \textit{E. Fornasini} and \textit{M. E. Valcher} [ibid. 263, 275-310 (1997; Zbl 0887.93033)] have extended the notion of primitivity to pairs \((A,B)\) of square nonnegative matrices of the same order. The pair \((A,B)\) is primitive provided there exist nonnegative integers \(h\) and \(k\) such that the sum of all products formed by words consisting of \(hA\)'s and \(kB\)'s is entrywise positive. The paper defines the exponent of a nonnegative matrix pair to be the smallest value of \(h+k\) over all such \(h\) and \(k\). It is then shown that the largest exponent of a primitive pair of \(n\) by \(n\) nonnegative matrices lies in the interval \([(n^3-5n^2)/2\), \((3n^3+2n^2 -2n)/2]\). In addition, the exponent of a pair of nonnegative matrices is related to properties of an associated two-dimensional dynamical system. The notions of essential and inessential maps defined by Granas are well known: the map \(f\) acting on a closure of some open set \(U\) is essential if any map \(j\) homotopic to \(f\) and equal to it on the boundary of \(U\) has a fixed point; otherwise \(f\) is inessential. In this paper the author proves theorems of the following scheme for different classes of multivalued maps: if a map is inessential on some set \(U_1\) and essential on a subset \(U_2\), then it has a fixed point on \(U_1\setminus U_2\). A similar result for a coincidence problem (between a linear map and a multivalued map) is also shown.

The main result of the paper under review is to give a simple and elegant proof of the following formula in Gromov-Witten theory by using virtual localization: \[ \int_{[\overline{M}_{1,0}(\mathbb{P}^r,d)]^{vir}}e(E)=\frac{(-1)^{(r+1)d}(r+1)}{24d}. \] Here \(\overline{M}_{1,0}(\mathbb{P}^r,d)\) is the moduli space of degree \(d\) stable maps from genus 1 curves to \(\mathbb{P}^r\), and \(E\) is a vector bundle on the moduli space whose fiber at the moduli point \([f:C\to \mathbb{P}^r]\) is naturally identified with \(H^1(C,f^*\mathcal{O}_{\mathbb{P}^r}^{\oplus(r+1)})\). This formula is a generalization of the genus 1 multiple cover formula in the case \(r=1\). An independent proof for the formula is given by \textit{Y. Iwao} et al. [J. Reine Angew. Math. 663, 67--90 (2012; Zbl 1260.14068)]. As a corollary of the formula a proof for the preservation of the Gromov-Witten invariants under simple flops for exceptional curve classes is obtained. In the present paper, the authors continue the work initiated in [Ann. Math. (2) 172, No. 1, 243--290 (2010; Zbl 1272.14040)] in order to establish the invariance of Gromov-Witten theory under simple flops. More precisely, given smooth projective manifolds \(X\) and \(X'\) related by a simple flop \(f\), the graph closure induces an identification of the Chow motives of \(X\) and \(X'\). Moreover, their big quantum cohomology rings, or equivalently genus zero Gromov-Witten invariants with \(3\) or more insertions, are isomorphic after analytic continuation of the Käler parameters. In the paper under review the authors study the case of higher genera Gromov-Witten invariants. In this case, the authors show that the natural framework to consider is Gromov-Witten invariants with ancestors rather then with descendants, as for the last the invariance would not be attained. The ancestors are pull-backs of the \(\psi\) classes in \(\overline M_{g,n}\) to the Kontsevich moduli space of stable maps under stabilization and forgetful morphisms. The authors then consider the ancestor potential, which is a formal series on the Novikov variables involving generating functions of Gromov-Witten invariants with ancestors, and show that, as in the genus zero case, a simple flop will maintain the ancestor potential invariant after analytic continuation in the Kähler moduli space. The same statement is shown to be false in the descendant case. Still, invariance up to analytic continuation is still shown to hold for generating functions with both ancestor and descendant insertions of special type with respect to \(f\). The methods of proof include a degeneration analysis, which allows the authors to work in a suitable local model, where Givental's quantization formalism can be applied to deduce the results in higher genera from the author's previous results in genus zero. The authors also observe that, since flops are crepant transformations, the results of the paper under review are consistent with the crepant transformation conjecture and suggest that, in order to study it, it might be more natural to work within the framework of Gromov-Witten theory with ancestors rather than with descendants.

The main result of the paper under review is to give a simple and elegant proof of the following formula in Gromov-Witten theory by using virtual localization: \[ \int_{[\overline{M}_{1,0}(\mathbb{P}^r,d)]^{vir}}e(E)=\frac{(-1)^{(r+1)d}(r+1)}{24d}. \] Here \(\overline{M}_{1,0}(\mathbb{P}^r,d)\) is the moduli space of degree \(d\) stable maps from genus 1 curves to \(\mathbb{P}^r\), and \(E\) is a vector bundle on the moduli space whose fiber at the moduli point \([f:C\to \mathbb{P}^r]\) is naturally identified with \(H^1(C,f^*\mathcal{O}_{\mathbb{P}^r}^{\oplus(r+1)})\). This formula is a generalization of the genus 1 multiple cover formula in the case \(r=1\). An independent proof for the formula is given by \textit{Y. Iwao} et al. [J. Reine Angew. Math. 663, 67--90 (2012; Zbl 1260.14068)]. As a corollary of the formula a proof for the preservation of the Gromov-Witten invariants under simple flops for exceptional curve classes is obtained. Es handelt sich um den für den Gymnasialunterricht jedenfalls interessanten Weg, die Haupteigenschaften der Kegelschnitte aus trigonometrischen Formeln abzuleiten. Zugrunde gelegt wird ein Dreieck \(ABC\), von dem \(A\) und \(B\) als Brennpunkte zweier konfokalen Kegelschnitte angesehen werden, die sich in \(C\) schneiden. Durch die vier Hauptachsen dieser beiden Kegelschnitte werden alle Bestimmungsstücke ausgedrückt. Die Parabel erscheint zuletzt als Grenzfall beider Kurven.

The topics connected to singular perturbations are quite various and most of them arise from physical sciences. So is the definition of a singular perturbation. The author of this textbook starts from problems and solutions: if \(f_\varepsilon:\mathbb R\mapsto X\) is a family of mappings (\(X\) being a metric space and \(\varepsilon\) a small parameter) representing the solution of a mathematical problem, the equations of this problem define a regularly perturbed problem if the convergence of \(f_\varepsilon(x)\) for \(\varepsilon\rightarrow 0\) is uniform in \(x\). If there is a solution which does not converge uniformly, the problem is singularly perturbed. The textbook under review is somehow different among many other but not an exception. Like the classical reference ``Perturbation methods'' of \textit{A. H. Nayfeh} [Reprint of the 1973 original. Wiley Classics Library. New York, NY: Wiley (2000; Zbl 0995.35001)], the present book is also application oriented in the sense that the approach is defined and illustrated through applications. The specific feature of this book is however the ``classroom level'' of the applications: they are structured in ``problems'' which have the length of few pages. For instance, the first chapter, entitled ``What is a singular perturbation'' has 28 pages and displays 7 problems arising from such different fields of application as: radiation reaction, convection-diffusion boundary layer and modulated oscillations. The following 3 chapters deal with standard methods in singular perturbations: Asymptotic expansions, Matched asymptotic expansions, Matched asymptotic expansions in partial differential equations. Within these chapters the ``problems'' are mainly illustrations of the difficulties to overcome and pitfalls to avoid, for instance: Another outrageous example, Asymptotic expansions of integrals -- the usual suspects. The 4th chapter, on matched asymptotic expansions in partial differential equations, is oriented to more complex problems: Chapman-Enskog asymptotics, Projected Lagrangian, Singularly perturbed eigenvalue problems. The 5th chapter, Prandtl boundary layer theory, is dedicated to one of the most important applications since it is the basics for the practical approximate solutions of the Navier Stokes equations. It appears however that the main interest in this book goes to the modulated oscillations -- the last 3 chapters: Modulated oscillations, Modulation theory by transforming variables, Nonlinear resonance. Since signal modulation used to be basic in radio engineering, revisiting its backgrounds with reference to singular perturbations might be fruitful and, with respect to this, reading the book under review can be rewarding. Each chapter is endowed with a guide to bibliography which is helpful to potential readers, in particular, students. In all, this book is a valuable completion to the literature on singular perturbations. It might be the first reference to read but also a good auxiliary in understanding more specialized books or papers. The book is a reprint of the 1973 original edition. This classical monograph was translated into many languages, in particular, in Russian in 1976. Entire generations of experts in the field of asymptotical mathematics, including the author of present review, had used Nayfeh's book as textbook as well as handbook. The book includes the following chapters: introduction (general information concerning asymptotic expansions); straightforward expansions and sources of nonuniformity; the method of strained coordinates; the method of matched and asymptotic expansions; variation of parameters and method of averaging, the method of multiple scales; asymptotic solutions of linear equations (WKB approximation and its generalizations). During the last 29 years a lot of fresh results were obtained, but classical approaches change very slowly. This reader-friendly and good-organised book may be used as introductory course in the field of asymptotic approaches. It can be highly recommended for students, engineers, and experts in various branches of science.

The topics connected to singular perturbations are quite various and most of them arise from physical sciences. So is the definition of a singular perturbation. The author of this textbook starts from problems and solutions: if \(f_\varepsilon:\mathbb R\mapsto X\) is a family of mappings (\(X\) being a metric space and \(\varepsilon\) a small parameter) representing the solution of a mathematical problem, the equations of this problem define a regularly perturbed problem if the convergence of \(f_\varepsilon(x)\) for \(\varepsilon\rightarrow 0\) is uniform in \(x\). If there is a solution which does not converge uniformly, the problem is singularly perturbed. The textbook under review is somehow different among many other but not an exception. Like the classical reference ``Perturbation methods'' of \textit{A. H. Nayfeh} [Reprint of the 1973 original. Wiley Classics Library. New York, NY: Wiley (2000; Zbl 0995.35001)], the present book is also application oriented in the sense that the approach is defined and illustrated through applications. The specific feature of this book is however the ``classroom level'' of the applications: they are structured in ``problems'' which have the length of few pages. For instance, the first chapter, entitled ``What is a singular perturbation'' has 28 pages and displays 7 problems arising from such different fields of application as: radiation reaction, convection-diffusion boundary layer and modulated oscillations. The following 3 chapters deal with standard methods in singular perturbations: Asymptotic expansions, Matched asymptotic expansions, Matched asymptotic expansions in partial differential equations. Within these chapters the ``problems'' are mainly illustrations of the difficulties to overcome and pitfalls to avoid, for instance: Another outrageous example, Asymptotic expansions of integrals -- the usual suspects. The 4th chapter, on matched asymptotic expansions in partial differential equations, is oriented to more complex problems: Chapman-Enskog asymptotics, Projected Lagrangian, Singularly perturbed eigenvalue problems. The 5th chapter, Prandtl boundary layer theory, is dedicated to one of the most important applications since it is the basics for the practical approximate solutions of the Navier Stokes equations. It appears however that the main interest in this book goes to the modulated oscillations -- the last 3 chapters: Modulated oscillations, Modulation theory by transforming variables, Nonlinear resonance. Since signal modulation used to be basic in radio engineering, revisiting its backgrounds with reference to singular perturbations might be fruitful and, with respect to this, reading the book under review can be rewarding. Each chapter is endowed with a guide to bibliography which is helpful to potential readers, in particular, students. In all, this book is a valuable completion to the literature on singular perturbations. It might be the first reference to read but also a good auxiliary in understanding more specialized books or papers. In this paper, the problem of nonzero-sum stochastic differential game between two competing insurance companies is considered, i.e., the relative performance concerns. A certain proportion of reinsurance can be taken out by each insurer to control his own risk. Moreover, each insurer can invest in a risk-free asset and risk asset with the price dramatically following the constant elasticity of variance (CEV) model. Based on the principle of dynamic programming, a general framework regarding Nash equilibrium for nonzero-sum games is established. For the typical case of exponential utilization, we, respectively, give the explicit solutions of the equilibrium strategy as well as the equilibrium function. Some numerical studies are provided at last which assist in obtaining some economic explanations.

This paper is devoted to studying the average values of class numbers of quadratic extensions of an arithmetic ring \(R\) of the form \(\mathbb{F}_ q[t]\). For \(q\) odd such an extension is of the form \(R[\sqrt m]\). The authors consider the case where \(m\) is monic. Let \(L(s,\chi_ m)\) be the corresponding \(L\)-functions. Then the authors first derive a closed formula for \(\sum_{\deg(m)=M;m \text{monic}}L(s,\chi_ m)\) which can be used to determine the average values of \(h_ m\) (the class number of \(R[\sqrt m])\). After this the authors consider the much more difficult problem of determining the average over the set of square-free \(m\). Whereas the treatment of the first type of sum was essentially elementary these sums are treated through the use of the theory of Eisenstein series on the double metaplectic cover of \(GL_ 2(k_{\mathbb{A}})\) where \(k=\mathbb{F}_ q(t)\) and \(k_{\mathbb{A}}\) is the corresponding adele ring. It is analogous to earlier work of \textit{D. Goldfeld} and \textit{J. Hoffstein} [Invent. Math. 80, 185-208 (1985; Zbl 0564.10043)] in the case of \(\mathbb{Q}\) but it can be made more complete in that the analogues of functions which can be proved to be meromorphic in the classical case are rational functions of \(q^{-s}\). Also the problems with places of residual characteristic 2 do not appear. The authors study the asymptotic behaviour of sums of type \(\sum_{0<\pm m<x,\quad m \text{ squarefree}}L(\rho,\chi_ m)\), where \(\rho\in {\mathbb{C}}\), Re(\(\rho)\geq\) and \(L(s,\chi_ m)\) is the Dirichlet L-function attached to (m/\(\cdot)\) if \(m\equiv 1 mod 4\) and (4m/\(\cdot)\) if \(m\equiv 2,3 mod 4\). For fixed \(\epsilon >0\) one has for all \(\rho\) with Re\((\rho)\geq 1\) \[ \sum_{m\text{ as above}}L(\rho,\chi_ m)=c(\rho) x+O(x^{+\epsilon}), \] where c(\(\rho)\) is an explicitly given analytic function (theorem 1). For \(\leq Re(\rho)\leq 1\) there are similar results. The special case \(\rho =\) was previously investigated by \textit{M. Jutila} [Analysis 1, 149-161 (1981; Zbl 0485.10029)]. Sums of different but related type occur in the work of \textit{L. A. Takhtadzhyan} and \textit{A. I. Vinogradov} [Sov. Math., Dokl. 22, 555-559 (1980); translation from Dokl. Akad. Nauk SSSR 254, 1298-1301 (1980; Zbl 0468.10022)]. In theorem 2 the authors describe in detail the analytic behaviour of the Dirichlet series \[ Z_{\pm}(\rho,s)=\sum_{\pm m>1,\quad m\text{ squarefree}}L(\rho,\chi_ n) | m|^{-s} \] for \(Re(s)>\). In their proofs the authors make extensive use of the analytic properties of Eisenstein series of half-integral weight.

This paper is devoted to studying the average values of class numbers of quadratic extensions of an arithmetic ring \(R\) of the form \(\mathbb{F}_ q[t]\). For \(q\) odd such an extension is of the form \(R[\sqrt m]\). The authors consider the case where \(m\) is monic. Let \(L(s,\chi_ m)\) be the corresponding \(L\)-functions. Then the authors first derive a closed formula for \(\sum_{\deg(m)=M;m \text{monic}}L(s,\chi_ m)\) which can be used to determine the average values of \(h_ m\) (the class number of \(R[\sqrt m])\). After this the authors consider the much more difficult problem of determining the average over the set of square-free \(m\). Whereas the treatment of the first type of sum was essentially elementary these sums are treated through the use of the theory of Eisenstein series on the double metaplectic cover of \(GL_ 2(k_{\mathbb{A}})\) where \(k=\mathbb{F}_ q(t)\) and \(k_{\mathbb{A}}\) is the corresponding adele ring. It is analogous to earlier work of \textit{D. Goldfeld} and \textit{J. Hoffstein} [Invent. Math. 80, 185-208 (1985; Zbl 0564.10043)] in the case of \(\mathbb{Q}\) but it can be made more complete in that the analogues of functions which can be proved to be meromorphic in the classical case are rational functions of \(q^{-s}\). Also the problems with places of residual characteristic 2 do not appear. We develop a method to build new 5D \(\mathcal{N} = 1\) gauge models based on Sasaki-Einstein manifolds \(Y^{p, q}\). These models extend the standard 5D ones having a unitary \(\mathrm{SU}(p)_q\) gauge symmetry based on \(Y^{p, q} \). Particular focus is put on the building of a gauge family with symplectic \(\mathrm{SP}(2r, \mathbb{R})\) symmetry. These super QFTs are embedded in M-theory compactified on folded toric Calabi-Yau threefolds \(\hat{X}(Y^{2r, 0})\) constructed from conical \(Y^{2r, 0}\). By using outer-automorphism symmetries of 5D \(\mathcal{N} = 1\) BPS quivers with unitary \(\mathrm{SU}(2r)\) gauge invariance, we also construct BPS quivers with symplectic \(\mathrm{SP}(2r, \mathbb{R})\) gauge symmetry. Other related aspects are discussed.

In view of results proven by G. E. Bredon, T. Chang and T. Skjelbred, for any finite dimensional connected \(\mathbb Z/p\)-CW complex \(X\) (\(p\) prime) such that \(H^*(X)\) is a Poincaré duality algebra, \(H^*(F)\) is also a Poincaré duality algebra, where \(F\subset X^{\mathbb Z/p}\) is a component of the fixed point set. In the paper a new, more concise proof of the above theorem is given. It uses the property that, roughly speaking, the Poincaré duality is preserved under algebraic deformations. This method depends essentially on the monograph [\textit{C. Allday} and \textit{V. Puppe}, Cohomological methods in transformation groups, Camb. Stud. Adv. Math. 32 (1993; Zbl 0799.55001)]. The book under review deals with the theory of compact transformation groups and, more specially, with the study of torus and \(p\)-torus actions by means of Algebraic Topology. At the same time an introduction for the beginner and a reference for the specialist, this book has a dual presentation of results, well adapted at each category of readers. For the beginner there are statements using CW-complexes and ordinary singular cohomology. Specialists will be interested in sections written for more general spaces and using the Alexander-Spanier cohomology. A major part of this text is devoted to recent research on these transformation groups, research to which the two authors made a large contribution. In the thirties, P. A. Smith initialized the use of algebraic-topologic methods in the study of group actions; nowadays, this domain bears the name of ``P. A. Smith Theory''. Other important contributions were made by A. Adem, A. Borel, G. Bredon, W. Browder, G. Carlsson, T. Chang, P. Conner, E. Floyd, D. Gottlieb, S. Halperin, W.-Y. Hsiang, R. Oliver, D. Quillen, T. Skjelbred, J. C. Su and many others. The book takes its place in the series of synthesis works on the study of transformation groups and completes them; let us cite in particular: \textit{G. E. Bredon}, Introduction to compact transformation groups (1972; Zbl 0246.57017); \textit{W.-Y. Hsiang}, Cohomology theory of topological transformation groups, Ergebn. Math. Grenzgeb. 85 (1975; Zbl 0429.57011); \textit{T. tom Dieck}, Transformation groups, de Gruyter Stud. Math. 8 (1987; Zbl 0611.57002)]. The algebraic invariants of a torus action \(G\) on a space \(X\) belong to an algebraic model of the Borel space \(X_ G\); the nature of this model depends on the type of invariants under consideration. Since the space \(X_ G\) is the total space of a fibration with basis the classifying space \(BG\) and fibre the space \(X\), the authors can use the perturbation theory of Hirsch-Brown to construct such a model from the tensor product of the models of the basis and of the fibre. In the case of an \(S^ 1\) action, the algebraic setting is the theory of minimal models of \textit{D. Sullivan} [Publ. Math., Inst. Haut. Étud. Sci. 47(1977), 269-331 (1978; Zbl 0374.57002)]. For actions of \(G = \mathbb{Z}_ p\), the models come from cellular chain complexes and use the homotopy theory of differential modules recalled in Appendix B. Let us now describe in detail the contents of this work, which is divided in 5 Chapters and 2 Appendices. Chapter 1 recalls the definition of a \(G\)-CW-complex \(X\), the construction of the associated Borel fibration \(X_ G\) and the equivariant cohomology of \(X: H^*_ G(X) := H^*(X_ G)\). The major part of this chapter is devoted to an algebraic Borel construction in the case of a \(p\)-torus, with an explicit description of the diagonal and of the induced module and algebra structures. It is also shown how this construction gives the cohomology of fixed point sets using evaluations. The practical use of these models is illustrated by proofs of several results of P. A. Smith. Chapter 2 is a summary of Sullivan's rational homotopy theory in terms of commutative graded differential algebras. This presentation contains most of the notions required for the study of rational spaces by means of algebraic objects: the algebra of PL-forms, algebraic homotopy, minimal models, models of a fibration and formality. Only the realization functor is omitted. Although this chapter does not contain any exercises, numerous examples and bibliographic citations will give the beginner ample opportunity to acquaint himself with the subject. However, the list of references has to be completed by the book of \textit{P. A. Griffiths} and \textit{J. W. Morgan}, Rational homotopy theory and differential forms, Prog. Math. 16 (1981; Zbl 0474.55001)]. Chapter 3 begins with the main result of the theory: the Borel localization theorem for \(G\)-CW-complexes of finite dimension with a finite number of orbit types: Let \(G\) be a compact Lie group and \(k\) be a commutative ring. If \(S\) is a multiplicative subset of the center of \(H^*(BG;k)\), then the inclusion of the fixed point set \(X^ S \hookrightarrow X\) induces an isomorphism between the localizations \(S^{-1}H^*_ G(X;k) \cong S^{-1}H^*_ G(X^ S;k)\). For a compact connected group acting with fixed points, this result is extended to rational equivariant homotopy defined in section 3. Section 5 consists of an evaluation theorem in the case of a torus action, \((S^ 1)^ m\). Many applications to the structure of fixed point sets are given; in particular one has, under suitable hypotheses, \(\dim_ kH^*(X^ G;k) \leq \dim_ k H^*(X;k)\), (Theorem 3.10.4). Note that many sections of this chapter are devoted to the cohomology of Alexander- Spanier. The generality of certain results imply an increased technicality; for instance, the non-localized version of the Hsiang fundamental theorem of fixed points consists of a statement two pages long (Theorem 3.8.7). A part from the models of rational homotopy, the proofs require numerous tools coming from commutative algebra; most of them are recalled in Appendix \(A\). Chapter 4 contains a study of the torus rank, \(\text{rank}_ 0 X\), of a space \(X\), defined as the maximal dimension of a torus acting almost- freely on \(X\). Section 3 recalls the various upper bounds of \(\text{rank}_ 0X\) coming from the Lie algebra of rational homotopy. Section 4 is devoted to the conjecture on the torus rank attributed to S. Halperin: If \(\text{rank}_ 0 X \geq r\) then \(\dim_ \mathbb{Q} H^*(X;\mathbb{Q}) \geq 2^ r\). This conjecture is true for homogeneous spaces and Kähler manifolds; the main part of section 4 consists of its proof in case \(r \leq 3\). In the next section, the exponent of Browder and Gottlieb \(e(X,G)\) is introduced for the action of a finite torus, \((\mathbb{Z}_ p)^ r\); \(e(X,G)\) is defined as the order of the cokernel of \(H^ n_ G(X;\mathbb{Z}) \to H^ n(X;\mathbb{Z})\), when \(X\) verifies \(H^ n(X;\mathbb{Z}) = \mathbb{Z}\) and \(H^ j(X;\mathbb{Z}) = 0\), \(j > n\). If \(X\) is a topological manifold, \(e(X,G)\) is equal to the order of the smallest orbit, [Browder-Gottlieb]. This definition is then extended to the Tate equivariant cohomology. Chapter 4 contains also a proof of a theorem due to Dwyer-Wilkerson which determines \(H^*_ G(X^ K;\mathbb{F}_ p)\) as a function of the structure of modules over the Steenrod algebra of a localization of \(H^*_ G(X;\mathbb{F}_ p)\). Chapter 5 deals with the particular case of \(k\)-PD-spaces, that is to say, spaces \(X\) whose cohomology \(H^*(X;k)\) is a Poincaré algebra, not necessarily in a graded sense. The first result generalizes the case of a compact Lie group acting differentiably on a compact manifold: If \(X\) is a \(G\)-CW-complex of finite dimension which is a \(k\)-PD-space, then all the components of fixed point sets are also \(k\)-PD-spaces, (\(G = S^ 1\) and \(k = \mathbb{Q}\), or \(G\) is a \(p\)-torus and \(k = \mathbb{F}_ p\)). This chapter ends with the definitions of Gysin morphisms and Euler classes in the \(k\)-PD-spaces setting. In particular, they show that there exists a generalization of a Borel formula between the dimensions of a \(G\)-space \(X\), its fixed point set \(X^ G\), and the fixed point sets \(X^ K\) of subtori \(K \subset G\) of codimension one. To conclude, the book contains a lot of precise statements, references and remarks which will be appreciated by the specialist, while the numerous exercises and examples will help the beginner in his first contact with his theory. It should be noted that despite the dual purpose in the presentation both kinds of readers can easily find their way. However, the beginner has to get a good grasp of Borel constructions, Sullivan's homotopy theory and various tools of commutative algebra before he can start to study Chapter 3 and the subsequent ones.

In view of results proven by G. E. Bredon, T. Chang and T. Skjelbred, for any finite dimensional connected \(\mathbb Z/p\)-CW complex \(X\) (\(p\) prime) such that \(H^*(X)\) is a Poincaré duality algebra, \(H^*(F)\) is also a Poincaré duality algebra, where \(F\subset X^{\mathbb Z/p}\) is a component of the fixed point set. In the paper a new, more concise proof of the above theorem is given. It uses the property that, roughly speaking, the Poincaré duality is preserved under algebraic deformations. This method depends essentially on the monograph [\textit{C. Allday} and \textit{V. Puppe}, Cohomological methods in transformation groups, Camb. Stud. Adv. Math. 32 (1993; Zbl 0799.55001)]. Let \(H\) be a Hilbert space, let \(B_0:{\mathcal D}(B_0)\to H\) and \(B:{\mathcal D}(B)\to H\) be closed linear operators, \(\overline{{\mathcal D}(B_0)}=H\), and let the operator \(B\) be subordinate to \(B_0\). Consider the operator \(A=B_0-\varepsilon B\), where \(\varepsilon\in \mathbb{C}\) is a parameter. Its spectral pairs, i.e., the pairs formal from the eigenvalues and corresponding eigenvectors of this operator depend, in general, on \(\varepsilon\). A spectral pair \((\lambda(\varepsilon), y(\varepsilon))\) of the operator \(A\) is analytically dependent on \(\varepsilon\) if for \(|\varepsilon |<\varepsilon_0\), \[ \lambda(\varepsilon)= \sum^\infty_{j=0} \lambda_j\varepsilon^j, \qquad y(\varepsilon)= \sum^\infty_{j=0} x_j\varepsilon^j, \qquad B_0y(\varepsilon)= \sum^\infty_{j=0} (B_0x_j)\varepsilon^j, \tag{1} \] where the numbers \(\lambda_j\) and the vectors \(x_j\) are independent of \(\varepsilon\), the series for \(\lambda(\varepsilon)\) converges in \(\mathbb{C}\), and the series for \(y(\varepsilon)\) and \(B_0y(\varepsilon)\) converges in the norm of \(H\). The series (1) are called the series of perturbation theory. If the spectral pairs \((\lambda(\varepsilon), y(\varepsilon))\) of the operator \(A\) depend analytically on \(\varepsilon\), then the numbers \(\lambda_j\) and the vectors \(x_j\) satisfy the recursion relations \[ (B_0-\lambda_0{\mathcal J})x_0=0,\;|x_0|\neq 0, \qquad (B_0-\lambda_0{\mathcal J})x_p= Bx_{p-1}+ \sum_{j=0}^{p-1} \lambda_{j+1} x_{p-1-j},\;\forall p\in\mathbb{N}, \tag{2} \] where \({\mathcal J}\) is the identity operator. In addition, in this case \((\lambda_0,x_0)= (\lambda(0),y(0))\) is a spectral pair for the operator \(B_0\). \dots The purpose of the present article is to exhibit the spectral pairs of the operator \(A\) that depend analytically on \(\varepsilon\).

This work arises from trying to answer Problem 4 presented in [\textit{B. Sturmfels}, IMA Vol. Math. Appl. 149, 351--363 (2009; Zbl 1158.13300)]: ``General problem: Study the geometry of conditional independence models for multivariate Gaussian random variables.'' Thus, this survey develops the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. Throughout the 30 pages, the authors introduce and classify oriented gaussoids, connect valuated gaussoids to tropical geometry, addresses the realizability problem for gaussoids and oriented gaussoids and so on. The reader can find additional materials on the web \url{www.gaussoids.de}. Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field, with main emphasis on graphical models with hidden variables, maximum likelihood estimation, and multivariate Gaussian distributions.

This work arises from trying to answer Problem 4 presented in [\textit{B. Sturmfels}, IMA Vol. Math. Appl. 149, 351--363 (2009; Zbl 1158.13300)]: ``General problem: Study the geometry of conditional independence models for multivariate Gaussian random variables.'' Thus, this survey develops the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. Throughout the 30 pages, the authors introduce and classify oriented gaussoids, connect valuated gaussoids to tropical geometry, addresses the realizability problem for gaussoids and oriented gaussoids and so on. The reader can find additional materials on the web \url{www.gaussoids.de}. We discuss the problem of designing a telecommunication network with the survivability requirement that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. Furthermore, the rings should be connected at certain transit nodes. The traffic between rings may pass through other rings. Each ring is associated with a certain fixed cost. We describe the problem, modeled as a linear integer programming problem, and a heuristic solution method, based on column generation and Lagrangean relaxation.

\textit{M.\,V.\thinspace Solodov} and \textit{B.\,F.\thinspace Svaiter} [Math.\ Program.\ 87, No.\,1(A), 189--202 (2000; Zbl 0971.90062)] proved the strong convergence of the proximal point algorithm for finding zeroes of a maximal monotone operator in a Hilbert space, and their convergence results are stated in terms of the metric projection. The present paper extends the latter result to a maximal monotone operator defined in the dual of a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. This latter result is stated in terms of the concept of \textit{sunny generalized nonexpansive retraction}, which generalizes the concept of metric projection. This paper is concerned with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by \textit{O. Güler} [SIAM J. Control Optim. 29, 403--419 (1991; Zbl 0737.90047)] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in the authors paper [J. Convex Anal. 6, 59--70 (1999; Zbl 0961.90128)]. Strong convergence is foreed by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two uniknowns.

\textit{M.\,V.\thinspace Solodov} and \textit{B.\,F.\thinspace Svaiter} [Math.\ Program.\ 87, No.\,1(A), 189--202 (2000; Zbl 0971.90062)] proved the strong convergence of the proximal point algorithm for finding zeroes of a maximal monotone operator in a Hilbert space, and their convergence results are stated in terms of the metric projection. The present paper extends the latter result to a maximal monotone operator defined in the dual of a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. This latter result is stated in terms of the concept of \textit{sunny generalized nonexpansive retraction}, which generalizes the concept of metric projection. Es wird eine Methode vorgestellt, Unterricht auf eine eher ungewöhnliche Weise vorzubereiten, zu ergänzen oder zu begleiten, um dadurch wertvolle Denkprozesse bzw. Rückmeldungen für den weiteren Unterricht zu ermöglichen. Es handelt sich darum, mit Schülern einen Projekttag durchzuführen, an dem sie einmal anders ihrem Fach, ihren Mitschülern, ihrem Kursleiter und sich selbst begegnen können. Der Projekttag wird näher beschrieben. Einzelne Ergebnisse und einige methodische Schritte werden vor- bzw. dargestellt. (orig.)

By a Mealy automaton the authors mean a quadruple \((A,\Sigma,\delta,\rho)\) where \(A\) and \(\Sigma\) are finite non-empty states (called, respectively, the state set and the alphabet) and \(\delta\colon A\times\Sigma\to A\) and \(\rho\colon A\times\Sigma\to\Sigma\) are totally defined functions. If \(\mathcal A=(A,\Sigma,\delta,\rho)\) is a Mealy automaton, then each state \(x\in A\) defines the function \(\rho_x\colon\Sigma\to\Sigma\) by \(\rho_x(i)=\rho(x,i)\) and each letter \(i\in\Sigma\) defines the function \(\delta_i\colon A\to A\) by \(\delta_i(x)=\delta(x,i)\). The function \(\rho_x\) extends to \(\Sigma^*\) in a natural way: if \(i\in\Sigma\) and \(u\in\Sigma^*\), then \(\rho_x(iu)=\rho_x(i)\rho_{\delta_i(x)}(u)\). The semigroup of all mappings from \(\Sigma^*\) to \(\Sigma^*\) generated by all the functions \(\rho_x\) with \(x\in A\) is called the semigroup of \(\mathcal A\) and is denoted by \(\langle\mathcal A\rangle_+\). If all functions \(\rho_x\colon\Sigma\to\Sigma\) happen to be bijective, the group of all mappings from \(\Sigma^*\) to \(\Sigma^*\) generated by the extensions of \(\rho_x\)'s with \(x\in A\) to \(\Sigma^*\) is called the group of \(\mathcal A\) and is denoted by \(\langle\mathcal A\rangle\). The authors study the following finiteness problem: given a Mealy automaton, determine whether its (semi)group is finite. If \(\mathcal A=(A,\Sigma,\delta,\rho)\) is a Mealy automaton, its dual \(\mathfrak d(\mathcal A)\) is the Mealy automaton obtained by switching the roles of \(A\) and \(\Sigma\). \textit{D. Savchuk} and \textit{Y. Vorobets} [J. Algebra 336, No. 1, 53-66 (2011; Zbl 1239.20032)] have shown that the semigroup \(\langle\mathcal A\rangle_+\) is finite if and only if so is the semigroup \(\langle\mathfrak d(\mathcal A)\rangle_+\). The authors observe that the minimization of a Mealy automaton does not change its semigroup but the minimization of an automaton with a minimal dual can make the dual automaton non-minimal. Therefore they suggest alternatingly minimize and dualize a given automaton until both the resulting automaton and its dual become minimal. If this process leads to the automaton with one state over a one-letter alphabet, then the semigroup of the initial automaton is finite. This sufficient condition for finiteness is not necessary but its implementation has allowed the authors to establish finiteness for a number of automaton semigroups whose analysis was not possible with previously known finiteness conditions. The helix graph of a Mealy automaton \(\mathcal A=(A,\Sigma,\delta,\rho)\) is the directed graph with nodes \(A\times\Sigma\) and arcs \((x,i)\to(\delta_i(x),\rho_x(i))\) for all \((x,i)\). The authors show that if all functions \(\rho_x\) and \(\delta_i\) are bijective and the group \(\langle\mathcal A\rangle\) is finite, then the helix graph of \(\mathcal A\) must be a union of disjoint cycles. This necessary condition for finiteness is not sufficient but again its implementation has proved to be superior over previously known finiteness conditions. We construct a family of automata with \(n\) states, \(n\geqslant 4\), acting on a rooted binary tree that generate the free products of cyclic groups of order 2.

By a Mealy automaton the authors mean a quadruple \((A,\Sigma,\delta,\rho)\) where \(A\) and \(\Sigma\) are finite non-empty states (called, respectively, the state set and the alphabet) and \(\delta\colon A\times\Sigma\to A\) and \(\rho\colon A\times\Sigma\to\Sigma\) are totally defined functions. If \(\mathcal A=(A,\Sigma,\delta,\rho)\) is a Mealy automaton, then each state \(x\in A\) defines the function \(\rho_x\colon\Sigma\to\Sigma\) by \(\rho_x(i)=\rho(x,i)\) and each letter \(i\in\Sigma\) defines the function \(\delta_i\colon A\to A\) by \(\delta_i(x)=\delta(x,i)\). The function \(\rho_x\) extends to \(\Sigma^*\) in a natural way: if \(i\in\Sigma\) and \(u\in\Sigma^*\), then \(\rho_x(iu)=\rho_x(i)\rho_{\delta_i(x)}(u)\). The semigroup of all mappings from \(\Sigma^*\) to \(\Sigma^*\) generated by all the functions \(\rho_x\) with \(x\in A\) is called the semigroup of \(\mathcal A\) and is denoted by \(\langle\mathcal A\rangle_+\). If all functions \(\rho_x\colon\Sigma\to\Sigma\) happen to be bijective, the group of all mappings from \(\Sigma^*\) to \(\Sigma^*\) generated by the extensions of \(\rho_x\)'s with \(x\in A\) to \(\Sigma^*\) is called the group of \(\mathcal A\) and is denoted by \(\langle\mathcal A\rangle\). The authors study the following finiteness problem: given a Mealy automaton, determine whether its (semi)group is finite. If \(\mathcal A=(A,\Sigma,\delta,\rho)\) is a Mealy automaton, its dual \(\mathfrak d(\mathcal A)\) is the Mealy automaton obtained by switching the roles of \(A\) and \(\Sigma\). \textit{D. Savchuk} and \textit{Y. Vorobets} [J. Algebra 336, No. 1, 53-66 (2011; Zbl 1239.20032)] have shown that the semigroup \(\langle\mathcal A\rangle_+\) is finite if and only if so is the semigroup \(\langle\mathfrak d(\mathcal A)\rangle_+\). The authors observe that the minimization of a Mealy automaton does not change its semigroup but the minimization of an automaton with a minimal dual can make the dual automaton non-minimal. Therefore they suggest alternatingly minimize and dualize a given automaton until both the resulting automaton and its dual become minimal. If this process leads to the automaton with one state over a one-letter alphabet, then the semigroup of the initial automaton is finite. This sufficient condition for finiteness is not necessary but its implementation has allowed the authors to establish finiteness for a number of automaton semigroups whose analysis was not possible with previously known finiteness conditions. The helix graph of a Mealy automaton \(\mathcal A=(A,\Sigma,\delta,\rho)\) is the directed graph with nodes \(A\times\Sigma\) and arcs \((x,i)\to(\delta_i(x),\rho_x(i))\) for all \((x,i)\). The authors show that if all functions \(\rho_x\) and \(\delta_i\) are bijective and the group \(\langle\mathcal A\rangle\) is finite, then the helix graph of \(\mathcal A\) must be a union of disjoint cycles. This necessary condition for finiteness is not sufficient but again its implementation has proved to be superior over previously known finiteness conditions. We begin with a brief review of the so-called WKB approximation, its deficiencies in higher order and attempts by several authors to remedy them. It is then shown that these deficiencies do not appear in the phase-integral approximation generated from an a priori unspecified base function, which was originally devised by the present authors in 1974 and is presented here in a way which clarifies the role of the ``small'' parameter in the differential equation. The advantage of this approximation versus the WKB approximation in higher order is also discussed. In a discussion of relations between solutions of the Schrödinger equation and the \(q\)-equation, the Ermakov-Lewis invariant is considered. In the concluding section we mention other items which, beside the phase-integral approximation in question, constitute the phase-integral method.

The author shows that every orientation-preserving periodic diffeomorphism \(f\) on a closed orientable 3-manifold \(M\) has a surgery description, that is, there is an integral framed link \(L\) in \(S^3\) which is invariant by a standard rotation \(\varphi \) around a trivial knot, such that \(M\) is obtained by surgery on \(L\) and that \(f\) is conjugate to the periodic differomorphism induced by \(\varphi\). This theorem gives a generalization of the results of \textit{J. H. Przytycki} and \textit{M. V. Sokolov} [Math. Proc. Camb. Philos. Soc. 131, No. 2, 295-307 (2001; Zbl 0985.57013)] for periodic maps of prime periods. The author also gives visualizations of isometries of the complements of 2-component hyperbolic links with \(\leq 9\) crossings which do not extend to periodic maps of \(S^3\). A (framed) link \(L\) in the standard 3-sphere \(S^3\) is said to be \(p\)-periodic if there is a \(\mathbb{Z}_p\)-action on \(S^3\), with a circle as a fixed point set, which maps \(L\) onto itself, and such that \(L\) is disjoint from the fixed point set. A 3-manifold \(M\) is called \(p\)-periodic if it admits an orientation preserving action of the cyclic group \(\mathbb{Z}_p\) with a circle as a fixed point set, and the action is free outside the circle. The authors prove that \(p\)-periodic 3-manifolds (where \(p\) is a fixed prime integer, \(p\geq 2)\) are obtained by surgeries on \(p\)-periodic links. As a consequence, they prove that for any \(p\)-periodic 3-manifold \(M\), where \(p\) is an odd prime integer, \(p\geq 3\), the first homology group of \(M\) with \(\mathbb{Z}_p\)-coefficients is different from \(\mathbb{Z}_p\). Finally, a similar criterion of 2-periodicity for rational homology 3-spheres completes this nice paper.

The author shows that every orientation-preserving periodic diffeomorphism \(f\) on a closed orientable 3-manifold \(M\) has a surgery description, that is, there is an integral framed link \(L\) in \(S^3\) which is invariant by a standard rotation \(\varphi \) around a trivial knot, such that \(M\) is obtained by surgery on \(L\) and that \(f\) is conjugate to the periodic differomorphism induced by \(\varphi\). This theorem gives a generalization of the results of \textit{J. H. Przytycki} and \textit{M. V. Sokolov} [Math. Proc. Camb. Philos. Soc. 131, No. 2, 295-307 (2001; Zbl 0985.57013)] for periodic maps of prime periods. The author also gives visualizations of isometries of the complements of 2-component hyperbolic links with \(\leq 9\) crossings which do not extend to periodic maps of \(S^3\). The elastic longitudinal stresses associated with the flow of jets of dilute polymer solutions from a short nozzle and their effect on the stability of the free jet are investigated theoretically and experimentally. The results obtained make it possible to take a fresh look at the ways in which a polymer additive affects the stability of high-velocity capillary jets.

The scalar differential delay equation \[ \dot x(t)=-\mu x(t)+f(x(t-1)), \tag{1} \] where \(\mu>0, f: \mathbb{R}\rightarrow \mathbb{R}\) is continuous and strictly increasing with \(f(0)=0\) and \(|f(x)|<\mu |x|\) for all \(x\) outside a bounded neighborhood of \(0\), possesses a global attractor \({\mathcal A}\). The global attractor \({\mathcal A}\) is a nonempty compact set of the phase space \({\mathcal C}:=C([-1,0], \mathbb{R})\) which is invariant under the semiflow defined by equation (1) and which attracts all bounded sets of \(\mathcal C\). In an earlier work by the authors and \textit{J. Wu} [Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback. Fields Institute Monograph Series. 11. Providence, RI: AMS (1999; Zbl 1004.34002)], under additional assumptions on equation (1), the existence of an invariant three-dimensional set \(\overline{W}\) was established and its fine structure studied. The assumptions imposed in the paper under review, that are somewhat stronger and also imply the existence of \(\overline{W}\), are (H1) \(f(0)=0, f^\prime(x)>0\) for all \(x\in \mathbb{R}\), \(f(x)-\mu x\) has exactly one zero \(\xi^-\) in \((-\infty,0)\) and exactly one zero \(\xi^+\) in \((0,\infty),\) \(f^\prime(\xi^-)<\mu, f^\prime(\xi^+)<\mu\), (H2) \(f^\prime(0)>\mu/\cos\theta\) for \(\theta\in (3/2\pi, 2\pi)\) with \(\theta=-\mu\tan\theta\). They guarantee the existence of exactly two more constant solutions \(\xi_-\) and \(\xi_+\) to equation (1) both stable and hyperbolic, and of a set \(S\) that is a separatrix of nonoscillatory behavior. The invariant set \(\overline{W}\) may be visualized as a smooth solid spindle which is split by the invariant disk \(\overline{W}\cap S\) into the basins of attraction towards the tips \(\xi^-\) and \(\xi^+\). In the present paper, the authors indicate additional to (H1) and (H2) conditions under which the invariant set \(\overline{W}\) is a global attractor, that is \(\overline{W}={\mathcal A}\). The conditions are (H3) \(f^\prime(0)<\mu/\cos\theta\) for \(\theta\in (7\pi/2, 4\pi)\) with \(\theta=-\mu\tan\theta\), (H4) \(f(-x)=-f(x)\) for all \(x\in \mathbb{R}\), and the function \(x f^\prime (x)/f(x)\) is strictly decreasing in \((0,\infty)\). The assumptions (H2) and (H3) guarantee that the linearization of equation (1) about \(0\) has precisely three eigenvalues with positive real parts. The assumption (H4) then allows one to prove that there exists exactly one rapidly oscillating periodic solution which forms, together with the equilibria \(0\), \(\xi^-\) and \(\xi^+\), a part of the global attractor. The authors study the scalar delay differential equation \[ \dot x(t)= -\mu x(t)+ f(x(t- 1)),\tag{\(*\)} \] where \(\mu\geq 0\) and \(f(x)\) is an increasing bounded \(C^1\) function with \(f(0)= 0\). A set of additional assumptions is imposed such that (1) Equation \((*)\) has two more equilibria (positive and negative) both stable and attracting; (2) The corresponding linearization at zero \[ \dot y(t)= -\mu y(t)+ f'(0) y(t-1) \] has at least a three-dimensional unstable subspace. An invariant subset of the original nonlinear equation is described in great detail (which takes the whole space of 165 pages of the paper). The authors provide the following brief description of a main part of their results: ``Under natural and mild additional conditions the leading 3-dimensional local unstable manifold at the stationary point \(0\) extends in forward time to a smooth solid spindle with singularities at its tip, that are further stationary points both stable and attractive. An invariant smooth disk of solution curves winding from \(0\) towards a bordering unstable periodic orbit splits the spindle into invariant halves each of which is attracted to one of its tips.'' The book also contains a set of eight appendices which represent an independent interest by themselves.

The scalar differential delay equation \[ \dot x(t)=-\mu x(t)+f(x(t-1)), \tag{1} \] where \(\mu>0, f: \mathbb{R}\rightarrow \mathbb{R}\) is continuous and strictly increasing with \(f(0)=0\) and \(|f(x)|<\mu |x|\) for all \(x\) outside a bounded neighborhood of \(0\), possesses a global attractor \({\mathcal A}\). The global attractor \({\mathcal A}\) is a nonempty compact set of the phase space \({\mathcal C}:=C([-1,0], \mathbb{R})\) which is invariant under the semiflow defined by equation (1) and which attracts all bounded sets of \(\mathcal C\). In an earlier work by the authors and \textit{J. Wu} [Shape, smoothness and invariant stratification of an attracting set for delayed monotone positive feedback. Fields Institute Monograph Series. 11. Providence, RI: AMS (1999; Zbl 1004.34002)], under additional assumptions on equation (1), the existence of an invariant three-dimensional set \(\overline{W}\) was established and its fine structure studied. The assumptions imposed in the paper under review, that are somewhat stronger and also imply the existence of \(\overline{W}\), are (H1) \(f(0)=0, f^\prime(x)>0\) for all \(x\in \mathbb{R}\), \(f(x)-\mu x\) has exactly one zero \(\xi^-\) in \((-\infty,0)\) and exactly one zero \(\xi^+\) in \((0,\infty),\) \(f^\prime(\xi^-)<\mu, f^\prime(\xi^+)<\mu\), (H2) \(f^\prime(0)>\mu/\cos\theta\) for \(\theta\in (3/2\pi, 2\pi)\) with \(\theta=-\mu\tan\theta\). They guarantee the existence of exactly two more constant solutions \(\xi_-\) and \(\xi_+\) to equation (1) both stable and hyperbolic, and of a set \(S\) that is a separatrix of nonoscillatory behavior. The invariant set \(\overline{W}\) may be visualized as a smooth solid spindle which is split by the invariant disk \(\overline{W}\cap S\) into the basins of attraction towards the tips \(\xi^-\) and \(\xi^+\). In the present paper, the authors indicate additional to (H1) and (H2) conditions under which the invariant set \(\overline{W}\) is a global attractor, that is \(\overline{W}={\mathcal A}\). The conditions are (H3) \(f^\prime(0)<\mu/\cos\theta\) for \(\theta\in (7\pi/2, 4\pi)\) with \(\theta=-\mu\tan\theta\), (H4) \(f(-x)=-f(x)\) for all \(x\in \mathbb{R}\), and the function \(x f^\prime (x)/f(x)\) is strictly decreasing in \((0,\infty)\). The assumptions (H2) and (H3) guarantee that the linearization of equation (1) about \(0\) has precisely three eigenvalues with positive real parts. The assumption (H4) then allows one to prove that there exists exactly one rapidly oscillating periodic solution which forms, together with the equilibria \(0\), \(\xi^-\) and \(\xi^+\), a part of the global attractor. [For the entire collection see Zbl 0664.00012.] Conditions are found which guarantee existence of a solution of the equation \[ u_{xx}+a(x,y,t)u_{xy}+b(x,y,t)u_{yy}+c(x,y,t)u_ x+d(x,y,t)u_ y+e(x,y,t)u=f(x,y,t) \] not depending on the parameter t.

A graph \(G\) is equimatchable if every matching in \(G\) extends to a maximum matching in \(G\). In [Discrete Math. 266, No.1--3, 263--274 (2003; Zbl 1022.05065)], \textit{K.~Kawarabayashi}, \textit{M. D. Plummer} and \textit{A. Saito} showed that there are only a finite number of 3-connected equimatchable planar graphs. The present paper shows that in a surface (orientable or non-orientable) of any fixed genus there are only a finite number of 3-connected equimatchable graphs having a minimal imbedding of representativity at least three. (The latter hypothesis can be dropped, for non-bipartite graphs.) The proof uses the Gallai-Edmonds decomposition theorem for matchings. In this paper, equimatchability in several well-known families of graphs in studied. In particular, equimatchable cubic graphs and the equimatchable 3-connected planar (3-polytopal) graphs are determined. The authors have used extensively the Gallai-Edmonds decomposition theory for graphs in terms of their maximum matchings to obtain these results. It is shown that there are precisely 23 3-connected planar graphs which are equimatchable and that there are only 2 cubic equimatchable graphs.

A graph \(G\) is equimatchable if every matching in \(G\) extends to a maximum matching in \(G\). In [Discrete Math. 266, No.1--3, 263--274 (2003; Zbl 1022.05065)], \textit{K.~Kawarabayashi}, \textit{M. D. Plummer} and \textit{A. Saito} showed that there are only a finite number of 3-connected equimatchable planar graphs. The present paper shows that in a surface (orientable or non-orientable) of any fixed genus there are only a finite number of 3-connected equimatchable graphs having a minimal imbedding of representativity at least three. (The latter hypothesis can be dropped, for non-bipartite graphs.) The proof uses the Gallai-Edmonds decomposition theorem for matchings. The boundary of the Siegel disc of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not yet been proved. The present work considers a cylinder renormalization -- a novel type of renormalization for holomorphic maps with a Siegel disc which is better suited for a hyperbolicity proof. A key element of a cylinder renormalization of a holomorphic map is a conformal isomorphism of a dynamical quotient of a subset of \(\mathbb{C}\) to a bi-infinite cylinder \(\mathbb{C}/\mathbb{Z}\). A construction of this conformal isomorphism is an implicit procedure which can be performed using the measurable Riemann mapping theorem. We present a constructive proof of the measurable Riemann mapping theorem and obtain rigorous bounds on a numerical approximation of the desired conformal isomorphism. Such control of the uniformizing conformal coordinate is of key importance for a rigorous computer-assisted study of cylinder renormalization.

Blow-up of radial solutions of the exponential reaction-diffusion equation is studied. The blow-up profile of solutions with incomplete blow-up is found using semigroup estimates. This improves a previous result of \textit{M. Fila} and \textit{A. Pulkkinen} [Tohoku Math. J. (2) 60, No. 3, 303--328 (2008; Zbl 1158.35056)] which was established under the assumption of radial monotonicity. We study the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term. Our main aim is to show that solutions which can be continued beyond blow-up possess a nonconstant selfsimilar blow-up profile. For some particular solutions we determine this profile precisely.

Blow-up of radial solutions of the exponential reaction-diffusion equation is studied. The blow-up profile of solutions with incomplete blow-up is found using semigroup estimates. This improves a previous result of \textit{M. Fila} and \textit{A. Pulkkinen} [Tohoku Math. J. (2) 60, No. 3, 303--328 (2008; Zbl 1158.35056)] which was established under the assumption of radial monotonicity. This paper deals with the problem of determining the economic operating policy for inventory items which are manufactured jointly but packaged individually. A modification is proposed to Silver's algorithm, which is generally believed to be the simplest heuristic algorithm for determining the economic operating policy for jointly replenished items.

This carefully written and highly readable paper is the last of a long series by author and \textit{G. Stuck}, originally motivated by a paper of \textit{N. Kowalsky} [Ann. Math. (2), 144, No. 3, 611-640 (1996; Zbl 0871.53048)]. The author characterizes in it the connected Lie groups \(G\) with simply connected nilradical which admit an orbit nonproper action by isometries of some connected Lorentz manifold. Here, orbit nonproper means that for some point \(x\), the evaluation map \(g\mapsto g(x)\) is nonproper. Such an action exists if and only if \(G\) satisfies one of the following conditions: the adjoint action \(G\to \text{GL} ({\mathfrak g})\) is nonproper, or \({\mathfrak g}\) has a direct summand \({\mathfrak so} (n,i)\), \(i\leq 2\leq n\), or \({\mathfrak g}\) has a nonzero abelian ideal \({\mathfrak a}\) admitting a positive definite or Minkowski quadratic form such that the adjoint action of \({\mathfrak g}\) on \({\mathfrak a}\) is conformal with respect to this form. There is another, equivalent formulation of the third condition which can be checked directly by looking at the adjoint representation of a Levi factor on the center of the nilradical. The proof of the theorem depends on a detailed analysis of the structure of \({\mathfrak so}(n,1)\) and of the representations of \({\mathfrak sl}_2\) and of noncompact simple or reductive groups. The author classifies those noncompact simple \(\mathbb{R}\)-algebraic groups \(G\) which act non-properly on a Lorentzian manifold \(M\) and preserve its metric. The main achievement over earlier work by other authors is that in the present paper \(M\) may be noncompact. The author proves that \(G\) must be locally isomorphic to \(\text{SL}(2,\mathbb{R})\) if all \(G\)-stabilizers are discrete. If \(G\) has arbitrary stabilizers, \(G\) can also be locally isomorphic to \(\text{SO}(n,1)\) or \(\text{SO}(n,2)\) for some \(n\). Part of her results can be generalized to pseudo-Riemannian manifolds.

This carefully written and highly readable paper is the last of a long series by author and \textit{G. Stuck}, originally motivated by a paper of \textit{N. Kowalsky} [Ann. Math. (2), 144, No. 3, 611-640 (1996; Zbl 0871.53048)]. The author characterizes in it the connected Lie groups \(G\) with simply connected nilradical which admit an orbit nonproper action by isometries of some connected Lorentz manifold. Here, orbit nonproper means that for some point \(x\), the evaluation map \(g\mapsto g(x)\) is nonproper. Such an action exists if and only if \(G\) satisfies one of the following conditions: the adjoint action \(G\to \text{GL} ({\mathfrak g})\) is nonproper, or \({\mathfrak g}\) has a direct summand \({\mathfrak so} (n,i)\), \(i\leq 2\leq n\), or \({\mathfrak g}\) has a nonzero abelian ideal \({\mathfrak a}\) admitting a positive definite or Minkowski quadratic form such that the adjoint action of \({\mathfrak g}\) on \({\mathfrak a}\) is conformal with respect to this form. There is another, equivalent formulation of the third condition which can be checked directly by looking at the adjoint representation of a Levi factor on the center of the nilradical. The proof of the theorem depends on a detailed analysis of the structure of \({\mathfrak so}(n,1)\) and of the representations of \({\mathfrak sl}_2\) and of noncompact simple or reductive groups. [For the entire collection see Zbl 0633.00025.] In this paper short and elegant proofs of \(\Sigma_{\omega +1}\)- completeness of index sets \(L=\{x:\exists n(W_ x^{(n)}\equiv_ T \emptyset^{(n)})\}\), \(H=\{x:\exists n(W_ x^{(n)}\equiv_ T \emptyset^{(n+1)})\}\), \(\Pi_{\omega +1}\)-completeness of index sets \(S=\{<x,y>:\forall n(W_ x^{(n)}\nleq_ T W_ y^{(n)}\&W_ y^{(n)}\nleq_ T W_ x^{(n)})\}\) and \(Int=\{x:\forall n(\emptyset^{(n)}<_ T W_ x^{(n)}<_ T \emptyset^{(n+1)})\}\), are given. The basic result of this paper is the following: Theorem. For every \(n\geq 0\), if f is a function partial recursive in \(\emptyset^{(n+2)}\), then there is a total recursive function g such that (\(\forall x\in dom f)(W^{(n)}_{g(x)}\equiv_ m W^{(n)}_{f(x)})\). Here for \(A\subseteq \omega\), \(A^{(0)}=A\). If f is a function partial recursive in \(\emptyset^{(\omega)}\), then there is a total recursive function g such that for all \(x\in dom f\exists n(W^{(n)}_{f(x)}\equiv_ m W^{(n)}_{g(x)})\).

Suppose that \(A\) is an indecomposable cellular algebra such that the spectrum of its Cartan matrix is of the form \(\{n,1,1,\dots,1\}\). The authors show that the number of non-isomorphic simple modules of \(A\) lies in a certain set [see \textit{J. J. Graham} and \textit{G. I. Lehrer}, Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029) for the definition of cellular algebras]. The authors also give a partial answer to the question as to whether, for each number \(m\) in this set, there is such an algebra with \(m\) non-isomorphic simple modules. The proof involves the application of some number-theoretic results in terms of sums of squares and the consideration of special sets of partitions. The authors define a class of associative algebras (``cellular'') by means of multiplicative properties of a basis, show that they have cell representations whose structure depends on certain invariant bilinear forms, and then obtain a general description of their irreducible representations and block theory as well as criteria for semisimplicity. These concepts are used to discuss the Brauer centraliser algebras, the Ariki-Koike algebras and the Temperley-Lieb and Jones algebras.

Suppose that \(A\) is an indecomposable cellular algebra such that the spectrum of its Cartan matrix is of the form \(\{n,1,1,\dots,1\}\). The authors show that the number of non-isomorphic simple modules of \(A\) lies in a certain set [see \textit{J. J. Graham} and \textit{G. I. Lehrer}, Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029) for the definition of cellular algebras]. The authors also give a partial answer to the question as to whether, for each number \(m\) in this set, there is such an algebra with \(m\) non-isomorphic simple modules. The proof involves the application of some number-theoretic results in terms of sums of squares and the consideration of special sets of partitions. A module \(M\) is called square-free if it contains no non-zero isomorphic submodules \(A\) and \(B\) with \(A \cap B =0\). Dually, \(M\) is called dual-square-free if \(M\) has no proper submodules \(A\) and \(B\) with \(M =A +B\) and \(M/A \cong M/B\). In this paper we show that if \(M = \oplus_{i \in I}M_i\), then \(M\) is square-free iff each \(M_i\) is square-free and \(M_j\) and \(\oplus_{j \neq i \in I}M_i\) are orthogonal. Dually, if \(M = \oplus_{i =1}^nM_i\), then \(M\) is dual-square-free iff each \(M_i\) is dual-square-free, \(1 \leqslant i \leqslant n\), and \(M_j\) and \(\oplus_{i \neq j}^nM_i\) are factor-orthogonal. Moreover, in the infinite case, we show that if \(M = \oplus_{i \in I}S_i\) is a direct sum of non-isomorphic simple modules, then \(M\) is a dual-square-free. In particular, if \(M =A \oplus B\) where \(A\) is dual-square-free and \(B = \oplus_{i \in I}S_i\) is a direct sum of non-isomorphic simple modules, then \(M\) is dual-square-free iff \(A\) and \(B\) are factor-orthogonal; this extends an earlier result by the authors in [Commun. Algebra 47, No. 7, 2954--2966 (2019; Zbl 1470.16005), Proposition 2.8].

For asymptotically bounded holomorphic functions defined in a polysector in \({\mathbb C}^n\), it is proved that the existence of a strong asymptotic expansion in Majima's sense along a single multidirection towards the vertex implies a global asymptotic expansion on the whole polysector. A similar result is also obtained for Gevrey's strong asymptotic expansions. These generalize the results of \textit{A. Fruchard} and \textit{C. Zhang} [Ann. Fac. Sci. Toulouse, Math. (6) 8, No. 1, 91--115 (1999; Zbl 1157.30322)] for asymptotic expansions in one variable. The authors consider analytic functions \(f(z)\) bounded on a sector \(\alpha\leq \arg\,z\leq \beta, 0<|z|<\rho\), with vertex at the origin of the complex \(z\)-plane. They show that if \(f(z)\) has an asymptotic expansion as \(|z|\rightarrow 0\) along the ray \(\arg\,z=\theta_0\) of the sector, then the asymptotics remain valid on the whole sector. The type of expansion considered is a Gevrey expansion of order \(k\). This is defined as follows: if \(f(z)\) has an asymptotic expansion \(\widehat{f}_N(z)\) along the ray \(\arg\,z=\theta_0\), where \(N\) is a positive integer, then \(|f(z)-\widehat{f}_N(z)|<C_N|z|^N (|z|\rightarrow 0)\), where \(C_N\) is a constant independent of \(z\). If \(C_N\) has the form \(C_N=CA^N\Gamma(1+Nk^{-1})\), with \(A\) and \(C\) positive constants independent of \(N\), then \(f(z)\) possesses \(\widehat{f}_N(z)\) as an asymptotic expansion of Gevrey type of order \(k\) in the direction \(\theta_0\). In the particular case \(k=1\), this corresponds to the familiar ``factorial divided by a power'' (when \(z\) is replaced by \(1/z\) as the asymptotic variable) employed in Dingle's 1973 book. When \(k>1\) the growth in the coefficients is slower than \(N!\). If, in addition, the constant \(A\) in the above bound has the form \(A=1/R(\theta)+\delta\) for some \(\delta>0\), then it is established how the quantity \(R(\theta)\) varies across the sector \(\alpha\leq\arg\,z\leq\beta\) for different \(k\). Exponentially small expansions are also considered and the behaviour of the exponent \(R(\theta)\) in the order estimate \(f(z)=O(\exp {-R(\theta)|z|})\), valid along the ray \(\theta_0\) in the sector, is obtained throughout the sector \(\alpha\leq\arg\,z\leq\beta\). A \(q\)-analog of these results is also discussed.

For asymptotically bounded holomorphic functions defined in a polysector in \({\mathbb C}^n\), it is proved that the existence of a strong asymptotic expansion in Majima's sense along a single multidirection towards the vertex implies a global asymptotic expansion on the whole polysector. A similar result is also obtained for Gevrey's strong asymptotic expansions. These generalize the results of \textit{A. Fruchard} and \textit{C. Zhang} [Ann. Fac. Sci. Toulouse, Math. (6) 8, No. 1, 91--115 (1999; Zbl 1157.30322)] for asymptotic expansions in one variable. For the review of the English original (1983), cf. Zbl 0525.01017.

The starting point of the author's consideration is the generalized version of the electro-diffusion equations which model the transport of two sorts of charged ``species'' in a device under external forces (such as the applied bias \(V\)): (1) \(\varepsilon\psi''(x)= q_ 1 n_ 1(x)+ q_ 2 n_ 2(x)- N(x)\), \(n_ 1'(x)= \alpha_ 1 q_ 1 n_ 1(x)\psi'(x)+ q_ 1 J_ 1(x)\), \(n_ 2'(x)= \alpha_ 2 q_ 2 n_ 2(x)\psi'(x)+ q_ 2 J_ 2(x)\), \(J_ 1'(x)= 0\), \(J_ 2'(x)=0\), \(x\in \Omega\), with boundary conditions \(\psi/\Gamma=\psi_ D(V)\), \(n_ i/\Gamma= n^ i_ D\) for \(i=1,2\). Here \(\Omega=(a,b)\), \(\Omega\) bounded, \(\Gamma=\{a,b\}\), \(\psi_ D(V)= \{\psi_ a,\psi_ b\}\), \(n_ D^ i=\{n^ i_ a,n^ i_ b\}\) and \(\psi_ a-\psi_ b\) depends linearly on the parameter \(V\). The physical meanings of the unknowns in (1) are: \(\psi\) the electrostatic potential, \(n_ i\) the density of the \(i\)th species and \(J_ i\) the current density of the \(i\)th species. The parameter \(\varepsilon\) is assumed to be a small and strictly positive number, the numbers \(q_ i\) can be positive, negative or zero, while the \(\alpha_ i\) are strictly positive numbers. The function \(N(x)\) models the scaled doping profile, if it refers to the semiconductor device or biological membrane. Many results concerning the existence of the solution for the systems like this were obtained earlier, but on the other hand, there is a lack of results on the analysis of uniqueness or multiplicity of the solutions, especially for the case of arbitrary applied bias \(V\) and different forms of doping profile \(N(x)\). The paper under review is devoted to just this case. Let be \(X= H^ 2(\Omega)\times (H^ 1(\Omega))^ 2\), \(Y= (L^ 2(\Omega))^ 2\), \(W=(\psi,n_ 1,n_ 2,J_ 1,J_ 2)\). If one defines a mapping: \({\mathcal F}: \mathbb{R}\times X\times \mathbb{R}^ 2\mapsto Y\times \mathbb{R}^ 6\) by \({\mathcal F}(V,W)= (\varepsilon\psi''-(q_ 1 n_ 1+ q_ 2 n_ 2- N)\), \(n_ 1'-\alpha_ 1 q_ 1 n_ 1\psi'-q_ 1 J_ 1\), \(n_ 2'-\alpha_ 2 q_ 2 n_ 2\psi'- q_ 2J_ 2\), \(\psi(a)- \psi_ a\), \(\psi(b)- \psi_ b\), \(n_ 1(a)- n^ 1_ a\), \(n_ 1(b)-n^ 1_ b\), \(n_ 2(a)- n^ 2_ a\), \(n_ 2(b)-n^ 2_ b)\), then the nonlinear system (1) can be written in the form (2) \({\mathcal F}(V,W)= 0\), where \(\mathcal F\) is a continuously Fréchet differentiable map. Then the author comes from (2) to the consideration of the linearized homogeneous equation (3) \({\mathcal F}(V,W)=0\) for which he is using his decoupling method [C. R. Acad. Sci., Paris, Ser. I 311, 589-592 (1990; Zbl 0717.34006)]. He establishes an upper bound of the dimension of the kernel of the linearized operator (3). Then he gives a uniqueness theorem for the system (1) under different hypotheses on \(N(x)\), in particular, he proves the global uniqueness theorem in case of vanishing doping profile and local uniqueness in case of piecewise-constant doping profiles and sufficiently strong bias \(V\). The nonlinear system of coupled equations \[ \epsilon \psi ''(x)=\sum^{M}_{k=1}q_ kn_ k(x)-N(x),\quad 0<x<1, \] \[ (n'_ k(x)-q_ kn_ k(x)\psi '(x))'=0,\quad 1\leq k\leq M, \] \[ \psi (0)=\psi_ 0,\quad \psi (1)=\psi_ 1,\quad n_ k(0)=n^ 0_ k,\quad n_ k(1)=n^ 1_ k,\quad 1\leq k\leq M \] describes the electro- diffusional phenomena occurring in biological membranes and semiconductor devices. This note is concerned with uniqueness of solutions for this system. We prove uniqueness for bipolar membranes with a vanishing fixed charged density and for unipolar membranes. These results are the first ones, as far as we know, valid without restriction on the applied voltages. Our method is based on the maximum principle.

The starting point of the author's consideration is the generalized version of the electro-diffusion equations which model the transport of two sorts of charged ``species'' in a device under external forces (such as the applied bias \(V\)): (1) \(\varepsilon\psi''(x)= q_ 1 n_ 1(x)+ q_ 2 n_ 2(x)- N(x)\), \(n_ 1'(x)= \alpha_ 1 q_ 1 n_ 1(x)\psi'(x)+ q_ 1 J_ 1(x)\), \(n_ 2'(x)= \alpha_ 2 q_ 2 n_ 2(x)\psi'(x)+ q_ 2 J_ 2(x)\), \(J_ 1'(x)= 0\), \(J_ 2'(x)=0\), \(x\in \Omega\), with boundary conditions \(\psi/\Gamma=\psi_ D(V)\), \(n_ i/\Gamma= n^ i_ D\) for \(i=1,2\). Here \(\Omega=(a,b)\), \(\Omega\) bounded, \(\Gamma=\{a,b\}\), \(\psi_ D(V)= \{\psi_ a,\psi_ b\}\), \(n_ D^ i=\{n^ i_ a,n^ i_ b\}\) and \(\psi_ a-\psi_ b\) depends linearly on the parameter \(V\). The physical meanings of the unknowns in (1) are: \(\psi\) the electrostatic potential, \(n_ i\) the density of the \(i\)th species and \(J_ i\) the current density of the \(i\)th species. The parameter \(\varepsilon\) is assumed to be a small and strictly positive number, the numbers \(q_ i\) can be positive, negative or zero, while the \(\alpha_ i\) are strictly positive numbers. The function \(N(x)\) models the scaled doping profile, if it refers to the semiconductor device or biological membrane. Many results concerning the existence of the solution for the systems like this were obtained earlier, but on the other hand, there is a lack of results on the analysis of uniqueness or multiplicity of the solutions, especially for the case of arbitrary applied bias \(V\) and different forms of doping profile \(N(x)\). The paper under review is devoted to just this case. Let be \(X= H^ 2(\Omega)\times (H^ 1(\Omega))^ 2\), \(Y= (L^ 2(\Omega))^ 2\), \(W=(\psi,n_ 1,n_ 2,J_ 1,J_ 2)\). If one defines a mapping: \({\mathcal F}: \mathbb{R}\times X\times \mathbb{R}^ 2\mapsto Y\times \mathbb{R}^ 6\) by \({\mathcal F}(V,W)= (\varepsilon\psi''-(q_ 1 n_ 1+ q_ 2 n_ 2- N)\), \(n_ 1'-\alpha_ 1 q_ 1 n_ 1\psi'-q_ 1 J_ 1\), \(n_ 2'-\alpha_ 2 q_ 2 n_ 2\psi'- q_ 2J_ 2\), \(\psi(a)- \psi_ a\), \(\psi(b)- \psi_ b\), \(n_ 1(a)- n^ 1_ a\), \(n_ 1(b)-n^ 1_ b\), \(n_ 2(a)- n^ 2_ a\), \(n_ 2(b)-n^ 2_ b)\), then the nonlinear system (1) can be written in the form (2) \({\mathcal F}(V,W)= 0\), where \(\mathcal F\) is a continuously Fréchet differentiable map. Then the author comes from (2) to the consideration of the linearized homogeneous equation (3) \({\mathcal F}(V,W)=0\) for which he is using his decoupling method [C. R. Acad. Sci., Paris, Ser. I 311, 589-592 (1990; Zbl 0717.34006)]. He establishes an upper bound of the dimension of the kernel of the linearized operator (3). Then he gives a uniqueness theorem for the system (1) under different hypotheses on \(N(x)\), in particular, he proves the global uniqueness theorem in case of vanishing doping profile and local uniqueness in case of piecewise-constant doping profiles and sufficiently strong bias \(V\). This book studies forecasting of short time series. A time series is defined to be short if the number of observations is less than the minimum required to achieve a given accuracy of forecasts. This is compensated for by the use of expert information where experts give opinions about future values of the time series in the form of inequalities (these may be between future values at different horizons, such as ``the series will be monotone over the next 5 time units''). The models considered are trend plus noise where the trend is linear in the unknown parameters while the time functions may be arbitrary. The basic idea is that the expert information is of higher priority than the fit. The problem is formulated as a minimization problem where the loss function, representing the fit, is minimized over a subset of the parameter space consistent with the expert information. The various problems considered in the book are best seen from its table of contents (abbreviated here): Chapter~1: Short time series and their description; Chapter~2: Forecasting under consistent expert judgements; Chapter~3: Forecasting under inconsistent (contradictory) expert judgements; Chapter~4: Model selection; Chapter~5: Combining models and predictions. In each section, there is an illustrative example and a summary in algorithmic form of the estimation/model selection method considered in it.

In this paper the notion of \(H^0\)-convergence for diffusion system in perforated domains with Neumann boundary condition [\textit{M. Briane}, \textit{A. Damlamian}, and \textit{P. Donato}, in Cioranescu, D. (ed.) et al., Nonlinear partial differential equations and their applications. College de France seminar. Vol. XIII. Lectures held at the J. L. Lions seminar on applied mathematics, Paris, France, 1994-1996. Harlow: Longman. Pitman Res. Notes Math. Ser. 391, 62-100 (1998)] is extended to the linear elasticity system in perforated domains with a homogeneous traction condition. The authors prove that the main properties of the original \(H^0\)-convergence are valid. The locality and compactness results are established, and a family of correctors is constructed. The considerations are partly based on the results of \textit{F. Murat} and \textit{L. Tartar} [in Cherkaev, Andrej (ed.) et al., Topics in the mathematical modelling of composite materials. Boston, MA: Birkhaeuser. Prog. Nonlinear Differ. Equ. Appl. 31, 1-8 (1997; Zbl 0913.35020)]. Let \(\Omega\) be an open bounded set of \(\mathbb{R}^N\) (\(N=2\) or 3 in general). A problem is to find a subset \(\Omega_1\) of \(\Omega\) such that if \[ a(x)=\begin{cases} \alpha, &\text{ on \(\Omega_1\)}\\ \beta, &\text{ on \(\Omega\setminus\Omega_1\),}\end{cases} \] and \(u\) is the solution of the Dirichlet problem \[ -\sum_i {\partial \over {\partial x_i}}\Biggl(a(x)\;{{\partial u}\over {\partial x_i}}\Biggr)= f\quad\text{in } \Omega, \qquad u=0\quad\text{on }\partial \Omega, \] the functional \(J(a)=\int_\Omega F(x,u(x))dx\) reaches its minimum. If regularity properties are assumed on \(\Omega\), one can prove the existence of an optimal solution. If no condition is imposed, then it may happen that there is no optimal solution. To study the above problem, the authors consider a minimizing sequence \(a_n(x)\) such that \(J(a_n(x))\) decreases to the infimum of \(J(a)\). They give some necessary conditions for \(a_{ij}(x)\), such that the solution of the minimization problem can be found, where \(a_{ij}(x)\) are the coefficents of the Dirichlet problem. The authors also give a corresponding physical interpretation for their problem.

In this paper the notion of \(H^0\)-convergence for diffusion system in perforated domains with Neumann boundary condition [\textit{M. Briane}, \textit{A. Damlamian}, and \textit{P. Donato}, in Cioranescu, D. (ed.) et al., Nonlinear partial differential equations and their applications. College de France seminar. Vol. XIII. Lectures held at the J. L. Lions seminar on applied mathematics, Paris, France, 1994-1996. Harlow: Longman. Pitman Res. Notes Math. Ser. 391, 62-100 (1998)] is extended to the linear elasticity system in perforated domains with a homogeneous traction condition. The authors prove that the main properties of the original \(H^0\)-convergence are valid. The locality and compactness results are established, and a family of correctors is constructed. The considerations are partly based on the results of \textit{F. Murat} and \textit{L. Tartar} [in Cherkaev, Andrej (ed.) et al., Topics in the mathematical modelling of composite materials. Boston, MA: Birkhaeuser. Prog. Nonlinear Differ. Equ. Appl. 31, 1-8 (1997; Zbl 0913.35020)]. Die für die Realschule bearbeitete kleine Schrift enthält in gedrängter Uebersicht die Sätze und Regeln der sieben elementaren algebraischen Operationen. Eine wissenschaftliche systematische Entwickelung des Zahlbegriffs war nicht beabsichtigt.

The authors obtain a new sufficient condition for the global attractivity of solutions of the delay difference equation \(x_{n+1}=x_nf(\,x_n,x_{n-1})\), \(n=0,1,\dots \) As an application, they consider the equation \[ x_{n+1}= (a+bx_n)/(A+x_{n-1}), \quad n=0,1,\dots, \tag \(*\) \] \(a,b,A\in (0,\infty )\) and obtain that \(a\leq A(2A-b)\) is a sufficient condition for global attractivity of the positive equilibrium point \(x^{*}\) of \((*)\). I am not sure that this condition adds any new region for the global attractivity comparing with the regions obtained by \textit{V. L. Kocic, G. Ladas} and \textit{I. W. Rodrigues} [J. Math. Anal. Appl. 173, No. 1, 127--157 (1993; Zbl 0777.39002)] and mentioned in this paper. The present study is related to the global asymptotic behavior, the oscillatory character and the periodic nature of all solutions of the rational recursive sequences; \(x_{n+1}=[a+\sum^{k-1}_{i=0}b_ ix_{n-i}]/x_{n-k}\), and \(x_{n+1}=(a+bx_ n)/(A+x_{n-k})\), \(n=0,1,2,\dots\), which arises due to an open problem [problem \# 1343, Math., Mag. 63, No. 2, 125 (1990)].

The authors obtain a new sufficient condition for the global attractivity of solutions of the delay difference equation \(x_{n+1}=x_nf(\,x_n,x_{n-1})\), \(n=0,1,\dots \) As an application, they consider the equation \[ x_{n+1}= (a+bx_n)/(A+x_{n-1}), \quad n=0,1,\dots, \tag \(*\) \] \(a,b,A\in (0,\infty )\) and obtain that \(a\leq A(2A-b)\) is a sufficient condition for global attractivity of the positive equilibrium point \(x^{*}\) of \((*)\). I am not sure that this condition adds any new region for the global attractivity comparing with the regions obtained by \textit{V. L. Kocic, G. Ladas} and \textit{I. W. Rodrigues} [J. Math. Anal. Appl. 173, No. 1, 127--157 (1993; Zbl 0777.39002)] and mentioned in this paper. Our aim in this paper is to prove that the Cohen-Macaulay property is inherited by tensor products of \(k\)-algebras. To this purpose, the first section investigates the grade of three specific types of ideals that play a primordial role within the ideal structure of the tensor product of two \(k\)-algebras. This allows us, in the second section, to establish the main theorem, that is, for \(k\)-algebras \(A\) and \(B\) such that \(A\otimes_k B\) is Noetherian, \(A\otimes_kB\) is a Cohen-Macaulay ring if and only if so are \(A\) and \(B\).

The class of QB-rings was introduced by \textit{G. K. Pedersen} and the authors [in J. Algebra 230, No. 2, 608-655 (2000; Zbl 0963.16008)], as a generalization of both rings with (K-theoretic) stable rank one and purely infinite simple rings. The present paper investigates QB-rings \(R\) which satisfy separative cancellation for finitely generated projective modules, meaning that \(A\oplus A\cong A\oplus B\cong B\oplus B\) implies \(A\cong B\). It is not known whether all QB-rings satisfy this condition. The authors establish it here for a class of QB-rings whose quasi-invertible elements have suitable decompositions relative to central idempotents, a class that includes all prime QB-rings. It is proved that over a separative QB-ring \(R\), all quasi-invertible square matrices can be diagonalized via row and column operations. This is exploited to develop results about lifting units and index theory modulo ideals. In particular, the authors prove that the natural map \(\text{GL}_1(R)\to K_1(R)\) is surjective; more generally, they derive a relative version that holds for any QB-ideal in a separative ring. Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the \(QB\)-rings. These constitute a considerable enlargement of the class of rings with stable rank one (\(B\)-rings) and include examples like the ring of endomorphisms of a vector space \(V\) over some field \(\mathbb{F}\), and the ring of all row- and column-finite matrices over \(\mathbb{F}\). We show that the category of \(QB\)-rings is stable under the formation of corners, ideals, and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of \(QB\)-rings to be a \(QB\)-ring, and show that extensions of \(B\)-rings often lead to \(QB\)-rings. Specializing to the category of exchange rings we characterize the subset of exchange \(QB\)-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e., a quasi-invertible element. Finally we show that the \(C^*\)-algebras that are \(QB\)-rings are exactly the extremally rich \(C^*\)-algebras studied by L. G. Brown and the second author.

The class of QB-rings was introduced by \textit{G. K. Pedersen} and the authors [in J. Algebra 230, No. 2, 608-655 (2000; Zbl 0963.16008)], as a generalization of both rings with (K-theoretic) stable rank one and purely infinite simple rings. The present paper investigates QB-rings \(R\) which satisfy separative cancellation for finitely generated projective modules, meaning that \(A\oplus A\cong A\oplus B\cong B\oplus B\) implies \(A\cong B\). It is not known whether all QB-rings satisfy this condition. The authors establish it here for a class of QB-rings whose quasi-invertible elements have suitable decompositions relative to central idempotents, a class that includes all prime QB-rings. It is proved that over a separative QB-ring \(R\), all quasi-invertible square matrices can be diagonalized via row and column operations. This is exploited to develop results about lifting units and index theory modulo ideals. In particular, the authors prove that the natural map \(\text{GL}_1(R)\to K_1(R)\) is surjective; more generally, they derive a relative version that holds for any QB-ideal in a separative ring. Mechanism design is considered in the context of fair allocations of indivisible goods with monetary compensation, by focusing on problems where agents' declarations on allocated goods can be verified before payments are performed. A setting is considered where verification might be subject to errors, so that payments have to be awarded under the presumption of innocence, as incorrect declared values do not necessarily mean manipulation attempts by the agents. Within this setting, a mechanism is designed that is shown to be truthful, efficient, and budget-balanced. Moreover, agents' utilities are fairly determined by the Shapley value of suitable coalitional games, and enjoy highly desirable properties such as equal treatment of equals, envy-freeness, and a stronger one called individual-optimality. In particular, the latter property guarantees that, for every agent, her/his utility is the maximum possible one over any alternative optimal allocation. The computational complexity of the proposed mechanism is also studied. It turns out that it is \(\#\)P-complete so that, to deal with applications with many agents involved, two polynomial-time randomized variants are also proposed: one that is still truthful and efficient, and which is approximately budget-balanced with high probability, and another one that is truthful in expectation, while still budget-balanced and efficient.

Group-groupoids, which are known in the literature as 2-groups, are internal groupoids in the category of groups and are categorically equivalent to crossed modules of groups. In the reference [the reviewer and \textit{T. Şahan}, Georgian Math. J. 26, No. 3, 437--447 (2019; Zbl 1451.22001)], using this equivalence, the notions of actions and coverings of group-groupoids was described. In this paper, as an extension of these equivalences, the author defines liftings of crossed semimodules and proves a categorical equivalence between liftings of crossed semimodules and actions and coverings of Schreier internal categories in the category of monoids. Further, in the paper an equivalence between crossed semimodules and simplicial monoids whose Moore complex has length one is given The aim of this paper is to define the notion of lifting via a group morphism for a crossed module and give some properties of this type of liftings. Further, we obtain a criterion for a crossed module to have a lifting crossed module. We also prove that the category of the lifting crossed modules of a certain crossed module is equivalent to the category of group-groupoid actions on groups, where the group-groupoid corresponds to the crossed module.

Group-groupoids, which are known in the literature as 2-groups, are internal groupoids in the category of groups and are categorically equivalent to crossed modules of groups. In the reference [the reviewer and \textit{T. Şahan}, Georgian Math. J. 26, No. 3, 437--447 (2019; Zbl 1451.22001)], using this equivalence, the notions of actions and coverings of group-groupoids was described. In this paper, as an extension of these equivalences, the author defines liftings of crossed semimodules and proves a categorical equivalence between liftings of crossed semimodules and actions and coverings of Schreier internal categories in the category of monoids. Further, in the paper an equivalence between crossed semimodules and simplicial monoids whose Moore complex has length one is given For ordinals \(\alpha\), \(\beta\), \(\gamma_0\), \(\gamma_1\), \(\delta_0\), and \(\delta_1\), the balanced polarized partition relation \[ \left(\begin{matrix} \alpha\\ \beta \end{matrix}\right) \rightarrow \left(\begin{matrix} \gamma_0 & \gamma_1\\ \delta_0 & \delta_1 \end{matrix}\right)^{1,1} \] holds if for any partition of \(\beta\times\alpha=K_0\cup K_1\) into two classes, either there are \(D_0\in [\beta]^{\delta_0}\) and \(C_0\in [\alpha]^{\gamma_0}\) with \(D_0\times C_0\subseteq K_0\), or there are \(D_1\in [\beta]^{\delta_1}\) and \(C_1\in [\alpha]^{\gamma_1}\) with \(D_1\times C_1\subseteq K_1\). In the paper under review it is shown that if \(\text{cf}\kappa=\omega\) and \(\lambda=2^{<\kappa}\), then \[ \left(\begin{matrix} \lambda^+\\ \lambda \end{matrix}\right) \rightarrow \left(\begin{matrix} \lambda^+ & \alpha\\ \lambda & \kappa \end{matrix}\right)^{1,1} \] for all \(\alpha<\omega_1\).

There are several examples of applications of Mathematical Logic to Diophantine Geometry. The most celebrated one seems to be the Matijasevic theorem saying that Hilbert's Tenth Problem is not decidable, i.e. there is no general effective way to decide whether a polynomial with integral coefficients -- and arbitrarily many variables -- has integral roots. Hrushovski's recent proof of the Mordell-Lang Conjecture for function fields in any characteristic provides a new, deep, more direct and qualitative level of interaction. Let us recall briefly the statement of Hrushovski's result. Let \(k < K\) be algebraically closed fields, \(A\) be a semiabelian variety defined over \(k\). Call a closed subvariety \(S\) of \(A\) special if there are a connected algebraic subgroup \(A_1\) of \(A\), a semiabelian variety \(A_0\) defined over \(k\), a subvariety \(S_0\) of \(A_0\), also defined over \(k\), and a surjective homomorphism of algebraic groups \(h\) from \(A_1\) to \(A_0\) such that \(S = h^{-1} (S_0) + c\) for some \(c\) in \(A\). Hrushoski shows that, if \(X\) is a closed subvariety of \(A\) and \(\Gamma\) is a finite rank subgroup of \(A\), then there are finitely many special subvarieties \(S_1,\ldots,S_k\) of \(A\) such that \(X \cap \Gamma \subseteq S_1 \cup \ldots \cup S_k \subseteq X\). The novelty of Hrushovski's approach is its large resort to modern Model Theory, involving some typical model theoretic concepts and tools, like orthogonality, modular groups, Zariski sets, as well as the model theoretic analysis of differentially closed and separably closed fields (actually the use of differential algebra was inspired by some previous contributions of A. Buium). The paper under review is an illuminating, essential and well written exposition of Hrushovski's work. First a short and clear historical introduction to the Mordell-Lang and the Manin-Mumford Conjecture is provided. Then the author gives a quick, but comprehensive report of the model theoretic background. Finally proofs are briefly sketched; here, the role of Model Theory is well underlined and clearly explained. Also, it is emphasized how Hrushovski's approach can even provide much better effective bounds for the Manin-Mumford Conjecture over number fields. The Mordell-Lang conjecture is a hypothesis formulated by Lang and extending the Mordell conjecture on rational points of curves, as well as the Manin-Mumford conjecture on torsion points of Abelian varieties. In characteristic 0, the Mordell-Lang conjecture was proved in a series of papers by Raynaud, Faltings and Vojta. In positive characteristic, only partial cases had been solved before this paper. Hrushovski presents here a uniform proof valid in any characteristic and using a model theoretic approach. Here is the statement of the main theorem. One deals with an algebraically closed field \(k\) and a field extension \(K/k\). Let \(S\) be a semi-Abelian variety over \(K\) and \(X\) be a subvariety of \(S\). Take a subgroup \(\Gamma\) of \(S\) such that \({\mathbb Q}_p \otimes \Gamma\) is finitely generated as a \({\mathbb Q}_p\)-module (here \({\mathbb Q}_p\) denotes \({\mathbb Q}\) if \(p=0\), and \(\{ m/n \in {\mathbb Q} : n\) prime to \(p \}\) otherwise). Suppose that \(X \cap \Gamma\) is Zariski dense in \(X\). Then there are a semi-Abelian variety \(S_0\) defined over \(k\), a subvariety \(X_0\) of \(S_0\) also defined over \(k\), and a rational homomorphism \(h\) from a group subvariety of \(S\) into \(S_0\) such that \(X\) is a translate of \(h^{-1}(X_0)\). As a consequence, when \(S\) has \(K/k\) trace 0, then \(X \cap \Gamma\) is a finite union of cosets of subgroups of \(\Gamma\). The core of the Hrushovski approach does not depend on the characteristic \(p\). But, just to explain the idea, let us treat before the case \(p=0\). The argument in this case uses differential algebra and was inspired by some recent papers of Buium. Without loss of generality, one can assume that \(K\) is algebraically closed and has infinite transcendence degree over \(k\). Accordingly, one can equip \(K\) with a derivation \(D\) making \(K\) a differential field, and even a differentially closed field (so an \(\omega\)-stable structure) with constant field \(k\). Model theory assigns an ordinal dimension to every non-empty set definable in \((K, D)\) (its Morley rank); in particular, the Morley rank agrees with the usual dimension for varieties. \(X\) is definable and its Morley rank is an integer. By using the Manin homomorphism in \((K, D)\), one sees that, with no loss of generality, \(\Gamma\) is definable, too, and its Morley rank is again an integer. Let us apply a dichotomy theorem of Hrushovski and Zil'ber to certain groups related to \(X\) and \(\Gamma\). It comes out that such a group is either a module over a suitable local ring, with no additional structure, or an algebraic group over an algebraically closed field. A theorem of Sokolovic says that the only algebraically closed field definable in \((K, D)\) is -- up to definable isomorphism -- the constant field \(k\). So the previous dichotomy just leads to the subvarieties mentioned in the main theorem: group subvarieties, or subvarieties defined over \(k\). And actually the analysis of the two cases accomplishes the proof. What happens when \(p>0\)? Here the plan is similar, but needs some changes. Most notably, differential algebra can be avoided now. Separably closed fields \(F\) of characteristic \(p\) such that \(|F : F^p|\) is finite and \(>1\) are enough. Their theory is not \(\omega\)-stable, but only stable, so Morley rank cannot be used in this setting. However some suitable modifications let the previous machinery work; for instance, the role of the Manin homomorphism is played now by the projection modulo \(\bigcap_n p^n S\) (which is not a definable set, but is \(\infty\)-definable). The paper is well written and well worth reading for several reasons. As recalled before, it gives the first proof of the conjecture (in the general form stated above). Moreover it provides a very brilliant and ingenious application of Model Theory to a question of Diophantine Geometry, using some deep model theoretic tools (like differentially and separably closed fields, or the Hrushovski-Zil'ber Dichotomy Theorem) to obtain the positive solution. So the paper is a significant step within the increasing connection between Model Theory and Algebraic Geometry.

There are several examples of applications of Mathematical Logic to Diophantine Geometry. The most celebrated one seems to be the Matijasevic theorem saying that Hilbert's Tenth Problem is not decidable, i.e. there is no general effective way to decide whether a polynomial with integral coefficients -- and arbitrarily many variables -- has integral roots. Hrushovski's recent proof of the Mordell-Lang Conjecture for function fields in any characteristic provides a new, deep, more direct and qualitative level of interaction. Let us recall briefly the statement of Hrushovski's result. Let \(k < K\) be algebraically closed fields, \(A\) be a semiabelian variety defined over \(k\). Call a closed subvariety \(S\) of \(A\) special if there are a connected algebraic subgroup \(A_1\) of \(A\), a semiabelian variety \(A_0\) defined over \(k\), a subvariety \(S_0\) of \(A_0\), also defined over \(k\), and a surjective homomorphism of algebraic groups \(h\) from \(A_1\) to \(A_0\) such that \(S = h^{-1} (S_0) + c\) for some \(c\) in \(A\). Hrushoski shows that, if \(X\) is a closed subvariety of \(A\) and \(\Gamma\) is a finite rank subgroup of \(A\), then there are finitely many special subvarieties \(S_1,\ldots,S_k\) of \(A\) such that \(X \cap \Gamma \subseteq S_1 \cup \ldots \cup S_k \subseteq X\). The novelty of Hrushovski's approach is its large resort to modern Model Theory, involving some typical model theoretic concepts and tools, like orthogonality, modular groups, Zariski sets, as well as the model theoretic analysis of differentially closed and separably closed fields (actually the use of differential algebra was inspired by some previous contributions of A. Buium). The paper under review is an illuminating, essential and well written exposition of Hrushovski's work. First a short and clear historical introduction to the Mordell-Lang and the Manin-Mumford Conjecture is provided. Then the author gives a quick, but comprehensive report of the model theoretic background. Finally proofs are briefly sketched; here, the role of Model Theory is well underlined and clearly explained. Also, it is emphasized how Hrushovski's approach can even provide much better effective bounds for the Manin-Mumford Conjecture over number fields. The paper under review studies an M/G/1 queueing system with server vacation, i.e., upon a service completion, the server takes a vacation with probability \(p\) or continues to serve customers (if any) with complementary probability \(1-p\). The vacation time is assumed to be exponentially distributed. The paper derives transform for transient (time-dependent) state probabilities.

Semilocal convergence of Newton-like methods \(x_{k+1} = x_k - A(x_k)^\#F(x_k)\), \(k \geq 0\) is discussed for solving the nonlinear equation \(\Gamma F(x)=0\). Here \(F\) is a twice \(F\)-differentiable nonlinear operator between Banach spaces and \(\Gamma\) is a bounded linear operator, furthermore, \(A(x)\) is a bounded linear operator approximating \(F'(x)\) and \(A(x)^\#\) a bounded outer inverse of \(A(x)\), i.e., \(A(x)^\# A(x) A(x)^\# = A(x)^\#\). It is an extension of the work by \textit{M. Z. Nashed} and \textit{X. Chen} [Numer. Math. 66, No. 2, 235-257 (1993; Zbl 0797.65047)]. Convergence theorems of (semilocal) Kantorovich-type and Mysovskij-type are proved under hypotheses on the second \(F\)-derivatives of \(F(x)\). For singular \(F\)-differentiable nonlinear operators between Banach spaces convergence theorems of (semilocal) Kantorovich-type are proved for Newton-like methods \(x_{k+1} = x_ k - A(x_ k)^ \#F(x_ k)\), \(k \geq 0\). Here \(A(x)\) is a bounded linear operator approximating \(F'(x)\) and \(A(x)^ \#\) a bounded outer inverse of \(A(x)\); that is, \(A(x)^ \# A(x) A(x)^ \# = A(x)^ \#\). The results reduce precisely to the well- known Kantorovich and Mysovskii theorems for the case of invertible derivatives. The analysis utilizes Banach-type lemmas and perturbation bounds for outer inverses which show that (for bounded linear operators) the set of outer inverses contains elements that behave similar to bounded linear inverses. Several examples compare the results with related known results and also addresses some computational issues. Various interesting remarks relevant to this setting conclude the paper.

Semilocal convergence of Newton-like methods \(x_{k+1} = x_k - A(x_k)^\#F(x_k)\), \(k \geq 0\) is discussed for solving the nonlinear equation \(\Gamma F(x)=0\). Here \(F\) is a twice \(F\)-differentiable nonlinear operator between Banach spaces and \(\Gamma\) is a bounded linear operator, furthermore, \(A(x)\) is a bounded linear operator approximating \(F'(x)\) and \(A(x)^\#\) a bounded outer inverse of \(A(x)\), i.e., \(A(x)^\# A(x) A(x)^\# = A(x)^\#\). It is an extension of the work by \textit{M. Z. Nashed} and \textit{X. Chen} [Numer. Math. 66, No. 2, 235-257 (1993; Zbl 0797.65047)]. Convergence theorems of (semilocal) Kantorovich-type and Mysovskij-type are proved under hypotheses on the second \(F\)-derivatives of \(F(x)\). We give some characterizations of \(n\) dimensional \((n\geq 2)\) hyperbolic cylinder, spherical cylinder or Euclidean cylinder in a Lorentzian space form. We show that the hyperbolic cylinder, spherical cylinder or Euclidean cylinder is the only complete space-like hypersurface in an \((n+1)\) dimensional Lorentzian space form \(M^{n+1}_1(c)\) with non-zero constant mean curvature and two distinct principal curvatures one of which is simple, if the norm square of the second fundamental form of \(M^n\) satisfies some pinching conditions, respectively.

\textit{G. A. Elliott} and \textit{D. Kucerovsky} [Pac. J. Math. 198, 385--409 (2001; Zbl 1058.46041)] showed that for an essential unital extension \[ 0 \longrightarrow B\longrightarrow C \longrightarrow A \longrightarrow 0 \] of a stable separable C*-algebra \(B\) and a unital separable C*-algebra \(A\) and a weakly nuclear splitting morphism \(s: A \to C\) the following conditions are equivalent: i) The extension is absorbing in the nuclear sense. ii) The extension has the purely large property on a subalgebra (that contains a stable full in \(B\) subalgebra) in the multiplier algebra. iii) The extension algebra has the approximation property. In the paper the author weakens the fullness condition in section 3, thereby weakening the absortion condition, in proving an analytic form of the Weyl-von Neumann theorem (Theorem 19) for extensions which are not full: Let \(B\) be a separable and stable C*-algebra, \(C\) a unital, factorizable and separable subalgebra of the multiplier algebra \(\mathcal M(B)\) containing \(B\). Let \(\Phi: C \to \mathcal M(B)\) be a completely positive unital map, dominated by the identity. There exists a sequence \((v_n)\) such that, for each \(c\in C\), \(\Phi(c) -v_ncv_n^*\) is in \(B\) and goes to zero in norm as \(n \to \infty\). In Theorem 34 and Theorem 37 he clarifie some conditions on purely large criteria for full extensions to be absorbing. The present paper is devoted to a generalization of so called Weyl-von Neumann theorems of Voiculescu, Kasparov, Kirchberg and Lin, which in turn extends a result of \textit{L. G. Brown}, \textit{R. G. Douglas} and \textit{P. A. Fillmore} [Ann. Math. (2) 105, 265--324 (1977; Zbl 0376.46036)]. Namely, an intrinsic characterization is obtained of those extensions of one separable \(C^*\)-algebra by another which are absorbing in a certain natural nuclear sense. If the first algebra (which is assumed to be a stable ideal) or the quotient is nuclear, then this condition reduces to that considered by earlier authors.

\textit{G. A. Elliott} and \textit{D. Kucerovsky} [Pac. J. Math. 198, 385--409 (2001; Zbl 1058.46041)] showed that for an essential unital extension \[ 0 \longrightarrow B\longrightarrow C \longrightarrow A \longrightarrow 0 \] of a stable separable C*-algebra \(B\) and a unital separable C*-algebra \(A\) and a weakly nuclear splitting morphism \(s: A \to C\) the following conditions are equivalent: i) The extension is absorbing in the nuclear sense. ii) The extension has the purely large property on a subalgebra (that contains a stable full in \(B\) subalgebra) in the multiplier algebra. iii) The extension algebra has the approximation property. In the paper the author weakens the fullness condition in section 3, thereby weakening the absortion condition, in proving an analytic form of the Weyl-von Neumann theorem (Theorem 19) for extensions which are not full: Let \(B\) be a separable and stable C*-algebra, \(C\) a unital, factorizable and separable subalgebra of the multiplier algebra \(\mathcal M(B)\) containing \(B\). Let \(\Phi: C \to \mathcal M(B)\) be a completely positive unital map, dominated by the identity. There exists a sequence \((v_n)\) such that, for each \(c\in C\), \(\Phi(c) -v_ncv_n^*\) is in \(B\) and goes to zero in norm as \(n \to \infty\). In Theorem 34 and Theorem 37 he clarifie some conditions on purely large criteria for full extensions to be absorbing. A susceptible infected susceptible (SIS) model with infected prey population and combined harvesting has been analyzed. It has been observed that harvesting reduces the level of equilibrium in one of the positions. In the absence of harvesting it reduces to carrying capacity, extinction. All equilibrium positions lie on the boundary of feasible domain.

In the first part [Algebra Anal. 13, No. 5, 16--43 (2001; Zbl 1016.30004)], a new approach to deriving inequalities for algebraic polynomials was suggested. This is based on constructing, for a given polynomial, a conformal and univalent mapping of a certain class, which is followed by applying certain results of the geometric theory of functions of a complex variable to the class in question. Several methods for constructing conformal mappings were considered, and some inequalities were obtained, which, in terms of the theory of functions, can be treated as covering and distortion theorems for algebraic polynomials. The present paper supplements the previous paper. An analog of the Schwarz boundary lemma is proved for non-univalent regular mappings of subsets of the unit disk onto a disk. Based on this result, certain strengthened inequalities of Bernstein type for algebraic polynomials are obtained. The generalized Mendeleev problem is discussed. Two-sided bounds for the module of the derivative of a polynomial with critical points on an interval are established. Bounds for the coefficients of polynomials under certain constraints are provided. The reviewed paper contains a new method of proving different types of inequalities for polynomials. The starting point are well known inequalities, like distortion theorems or Grunsky type inequalities for holomorphic and univalent functions \(f(z)\) in the unit disk \(\mathbb D=\{z:|z|<1\}\) and such that \(f(0)=0\) and \(|f(z)|<1\), \(z\in\mathbb D\). The author has constructed such univalent mappings via certain polynomials with prescribed properties like positions of their zeros or critical points. Applications of the mentioned inequalities for polynomials lead to the resulting inequalities for polynomials containing covering results and interesting generalizations of Markov and Baernstein inequalities. One of the main result is the following Theorem 4.1 Let \(P(z)=c_0+c_1z+c_2z^2+\dots+c_nz^n\), and let \(k\) be a nonnegative integer. Then for every point \(z\) on the circle \(|z|=1\) we have \[ |zP'(z)-kP(z)|\leq\begin{cases} (n-k-1)M(P)+\sqrt{|c_n|M(P)}&\text{if }k<\frac n2,\\ (\frac n2-1)M(P)+\sqrt{\|c_n|-|c_0\|M(P)}&\text{if } k=\frac n2,\\ (k-1)M(P)+\sqrt{|c_0|M(P)}&\text{if }k>\frac n2,\end{cases} \] where \(M(P)=\max\{|P(z)|:|z|=1\}\). Equality is attained in the first and in the second case, e.g., for \(P(z)=c_nz^n\), and in third case for \(P(z)=c_0\).

In the first part [Algebra Anal. 13, No. 5, 16--43 (2001; Zbl 1016.30004)], a new approach to deriving inequalities for algebraic polynomials was suggested. This is based on constructing, for a given polynomial, a conformal and univalent mapping of a certain class, which is followed by applying certain results of the geometric theory of functions of a complex variable to the class in question. Several methods for constructing conformal mappings were considered, and some inequalities were obtained, which, in terms of the theory of functions, can be treated as covering and distortion theorems for algebraic polynomials. The present paper supplements the previous paper. An analog of the Schwarz boundary lemma is proved for non-univalent regular mappings of subsets of the unit disk onto a disk. Based on this result, certain strengthened inequalities of Bernstein type for algebraic polynomials are obtained. The generalized Mendeleev problem is discussed. Two-sided bounds for the module of the derivative of a polynomial with critical points on an interval are established. Bounds for the coefficients of polynomials under certain constraints are provided. See the review in Zbl 0708.30035.

The author studies the geometry of complex Finsler submanifolds with respect to the Rund connection [for similar results using the Chern-Finsler connection see \textit{G. Munteanu}, Period. Math. Hung. 55, No.~1, 97--112 (2007; Zbl 1164.53021)]. In particular, he shows that the holomorphic curvature of the Rund connection of a complex Finsler submanifold coincides with the holomorphic curvature of the connection induced by the ambient Rund connection, and that it is not greater than the holomorphic curvature computed in the ambient manifold. Furthermore, he also obtains a characterization of totally geodesic submanifolds of complex Finsler manifolds of constant holomorphic curvature. The proofs are mainly based on the Gauss, Codazzi and Ricci equations for the submanifold. It should be remarked that the definition of holomorphic curvature used by the author is slightly different from the definition of holomorphic curvature used for the Chern-Finsler connection; however, he also gives a formula relating this alternative holomorphic curvature (always with respect to the Rund connection) computed in the ambient manifold and computed in the submanifold. Using the Chern-Finsler connection and its induced tangent connection, the author deduces the Gauss, Codazzi and Ricci equations for a holomorphic submanifold of a complex Finsler manifold. He investigates the relationship between the holomorphic curvature of the Chern-Finsler connection and its induced tangent connection. A concept of complex totally geodesic immersed submanifold is introduced and studied. Consequences on holomorphic submanifolds of some classical complex Finsler manifolds are discussed.

The author studies the geometry of complex Finsler submanifolds with respect to the Rund connection [for similar results using the Chern-Finsler connection see \textit{G. Munteanu}, Period. Math. Hung. 55, No.~1, 97--112 (2007; Zbl 1164.53021)]. In particular, he shows that the holomorphic curvature of the Rund connection of a complex Finsler submanifold coincides with the holomorphic curvature of the connection induced by the ambient Rund connection, and that it is not greater than the holomorphic curvature computed in the ambient manifold. Furthermore, he also obtains a characterization of totally geodesic submanifolds of complex Finsler manifolds of constant holomorphic curvature. The proofs are mainly based on the Gauss, Codazzi and Ricci equations for the submanifold. It should be remarked that the definition of holomorphic curvature used by the author is slightly different from the definition of holomorphic curvature used for the Chern-Finsler connection; however, he also gives a formula relating this alternative holomorphic curvature (always with respect to the Rund connection) computed in the ambient manifold and computed in the submanifold. [For the entire collection see Zbl 0753.00030.] We show that the question of checking whether there are ground instances of a term \(t\) which are not instances of the terms \(t1,\dots,tn\) modulo sets of associativity and commutativity axioms is decidable. This questions belongs to the well-known class of complement problems. Its solution provides a formal basis for automating the process of learning from examples, verifying the sufficient-completeness property of algebraic specifications, designing associative-commutative compilation algorithms, finding solutions of systems of equations and disequations in associative and commutative theories, etc.

Let \(\mathbb G\) be a semisimple simply connected algebraic group defined over \(\mathbb R\), \(G\) the group of its real points and \(\mathfrak g\) the Lie algebra of \(G\). The author studies the fundamental group \(\pi_1(G)\) and nontrivial covering groups of \(G\) (such a cover is not realizable as a linear Lie group). Let \(\theta\) denote the Cartan involution of \(\mathbb G\) corresponding to \(G\). The root system \(\Delta\) of \(\mathbb G\) with respect to a \(\theta\)-stable Cartan subgroup is considered. Using the natural action of \(\theta\) on \(\Delta\), the author gives uniform proofs of several basic facts concerning \(\pi_1(G)\) and nonlinear covers of \(G\). Some of these have been already proved by different authors using the classification. The first result claims that \(\pi_1(G)\neq 1\) if and only if there exists a long root \(\alpha\in\Delta\), such that \(\theta(\alpha) = \alpha\) and \(\theta(X_{\alpha}) = -X_{\alpha}\), where \(X_{\alpha}\in\mathfrak g(\mathbb C)\) is a root vector of \(\alpha\). The relation with a result of \textit{G. Prasad} [Adv. Math. 181, 160--164 (2004; Zbl 1037.22040)] is explained. Another result (proved in the ``if'' direction by B. Kostant) claims that \(\pi_1(G)\neq 1\) if and only if the minimal nilpotent orbit in \(\mathfrak g(\mathbb C)\) is defined over \(\mathbb R\). Now suppose that \(\mathfrak g\) is simple. It is proved that \(\pi_1(G)\) is trivial or is isomorphic to \(\mathbb Z\) or to \(\mathbb Z/2\mathbb Z\). A list of all the groups with \(\pi_1(G) = 1\) is given, and a method to determine \(\pi_1(G)\) from the Kac diagram of \(G\) is described. Let \(F\) be a local field different from the field of complex numbers. Let \(G\) be an absolute (almost) simple, simple connected algebraic group over the field \(F\). \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 84, 35-89 (1996; Zbl 0930.20043)] has constructed a topological central extension \[ (*) 1 \rightarrow \mu(F) \rightarrow G(F)\tilde \rightarrow G(F) \rightarrow 1. \] of \(G(F),\) where \(\mu(F)\) is a finite cyclic group of roots of identity contained in \(F\). \textit{G. Prasad} and \textit{A.S. Rapinchuk} [Publ. Math., Inst. Hautes Étud. Sci. 84, 91-187 (1996; Zbl 0941.22019)] proved that, if \(G\) is \(F\)-isotropic, then extension (*) is a universal topological central extension of \(G(F)\). This proof used global arguments. In the present paper the author gives a purely ``local'' proof of this result.

Let \(\mathbb G\) be a semisimple simply connected algebraic group defined over \(\mathbb R\), \(G\) the group of its real points and \(\mathfrak g\) the Lie algebra of \(G\). The author studies the fundamental group \(\pi_1(G)\) and nontrivial covering groups of \(G\) (such a cover is not realizable as a linear Lie group). Let \(\theta\) denote the Cartan involution of \(\mathbb G\) corresponding to \(G\). The root system \(\Delta\) of \(\mathbb G\) with respect to a \(\theta\)-stable Cartan subgroup is considered. Using the natural action of \(\theta\) on \(\Delta\), the author gives uniform proofs of several basic facts concerning \(\pi_1(G)\) and nonlinear covers of \(G\). Some of these have been already proved by different authors using the classification. The first result claims that \(\pi_1(G)\neq 1\) if and only if there exists a long root \(\alpha\in\Delta\), such that \(\theta(\alpha) = \alpha\) and \(\theta(X_{\alpha}) = -X_{\alpha}\), where \(X_{\alpha}\in\mathfrak g(\mathbb C)\) is a root vector of \(\alpha\). The relation with a result of \textit{G. Prasad} [Adv. Math. 181, 160--164 (2004; Zbl 1037.22040)] is explained. Another result (proved in the ``if'' direction by B. Kostant) claims that \(\pi_1(G)\neq 1\) if and only if the minimal nilpotent orbit in \(\mathfrak g(\mathbb C)\) is defined over \(\mathbb R\). Now suppose that \(\mathfrak g\) is simple. It is proved that \(\pi_1(G)\) is trivial or is isomorphic to \(\mathbb Z\) or to \(\mathbb Z/2\mathbb Z\). A list of all the groups with \(\pi_1(G) = 1\) is given, and a method to determine \(\pi_1(G)\) from the Kac diagram of \(G\) is described. This article derives the probability density function of the sum of a normal random variable and a (sphered) Student's \(t\)-distribution on odd degrees of freedom greater than or equal to three. Apart from its intrinsic interest applications of this result include Bayesian wavelet shrinkage, Bayesian posterior density derivations, calculations in the theoretical analysis of projection indices and computation of certain moments.