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The author continues his investigation of finite p'-nilpotent groups [Part I, ibid. 10, 135-146 (1987; Zbl 0614.20009)]. (A group is said to be p'-nilpotent if it has a normal Sylow p-subgroup with nilpotent complement; so it is p'-nilpotent if and only if it is q-nilpotent for all primes \(q\neq p.)\) The main theorem of the paper characterizes the nonsolvable simple groups in which each proper subgroup is q'-nilpotent for some prime q. (For example, each proper subgroup of \(A_ 5\) is either 2'-nilpotent, 3'-nilpotent or 5'-nilpotent.) The proof uses Thompson's classification of the minimal simple groups and Dickson's list of all the subgroups of \(PSL(2,p^ n)\). A p'-nilpotent group is an extension of a p-group by a nilpotent group (which can, of course, be assumed to be a p'-group). The author observes that the p-Frattini subgroup of a group is p'-nilpotent and that the p'- nilpotent groups form a saturated formation. He generalizes some work of M. Torres on \(\Phi^*(G)\), the product of the p-Frattini subgroups of the group G for all the prime factors of the order of G, showing, for example, that the Fitting length of G is less than 3 if and only if \(G'\subseteq \Phi^*(G)\). He shows also that if the group G contains three p'-nilpotent subgroups with pairwise relatively prime indices then G is p'-nilpotent. In the final section of the paper the author investigates the \({\mathcal F}\)-hypercenter of certain solvable groups for the formation \({\mathcal F}\) of p'-nilpotent groups. Only finite groups are treated in the paper.

The author continues his investigation of finite p'-nilpotent groups [Part I, ibid. 10, 135-146 (1987; Zbl 0614.20009)]. (A group is said to be p'-nilpotent if it has a normal Sylow p-subgroup with nilpotent complement; so it is p'-nilpotent if and only if it is q-nilpotent for all primes \(q\neq p.)\) The main theorem of the paper characterizes the nonsolvable simple groups in which each proper subgroup is q'-nilpotent for some prime q. (For example, each proper subgroup of \(A_ 5\) is either 2'-nilpotent, 3'-nilpotent or 5'-nilpotent.) The proof uses Thompson's classification of the minimal simple groups and Dickson's list of all the subgroups of \(PSL(2,p^ n)\). Negotiation is an interaction allowing agents to resolve conflicts and reach agreements over shared concerns. The shared concerns of negotiation may be issues such as price or response time. Negotiation has been studied in different fields including management, social sciences, decision and game theory, artificial intelligence and intelligent agents. Most of the literature devoted to negotiation considers the decision-making during the negotiation in terms of choosing appropriate negotiation strategies. However, there are some decisions that have to be made before the negotiation starts. One of such decisions is the selection of negotiation partners that typically is made by the agent's user. The selection of negotiation partners is crucial because negotiation may be a very time-consuming activity and failed encounters can waste time and resources. The selection problem may be solved by predicting the negotiation capability of each potential partner and choosing the required number of partners with the highest chance of success in a potential negotiation. The selection mechanism is very important because of the practicality and efficiency of multi-agent system interactions.

Recently, by means of the Weierstrass-type representation for harmonic maps from a Riemann surface into symmetric spaces discovered by Dorfmeister, Pedit and Wu, Dorfmeister and Haak have constructed constant mean curvature surfaces by applying the Sym-Bobenko formula to the loop-group-valued maps given by integrating the meromorphic potentials. On the other hand, Kenmotsu's work on the Weierstrass representation for immersions with prescribed mean curvature from a simply connected Riemann surface into Euclidean 3-space was generalized by \textit{K. Akutagawa} and \textit{S. Nishikawa} to Minkowski 3-space [Tôhoku Math. J., II. Ser. 42, 67-82 (1990; Zbl 0679.53002)]. Motivated by these results, the author of this paper considers two aspects. The first is to establish a natural correspondence between the following two spaces: the space of conformal spacelike immersions with constant mean curvature from a simply connected Riemann surface \(\Sigma\) into Minkowski 3-space, and that of nowhere anti-holomorphic harmonic maps from \(\Sigma\) into the Poincaré half plane, regarded as the Riemannian symmetric space \(SL(2, \mathbb{R})/SO(2)\). The second is to prove the Lorentzian version of the Sym-Bobenko formula and apply it to construct spacelike immersions with constant mean curvature into Minkowski 3-space. The main idea in this paper is to define an \(sl(2, \mathbb{R})\)-valued 1-form \(\Lambda^f\) on a Riemann surface \(\Sigma\) associated to a smooth map \(f:\Sigma\to SL(2, \mathbb{R})/SO(2)\) and show that the harmonicity of \(f\) is equivalent to the \(d\)-closedness of \(\Lambda^f\). For an oriented spacelike surface M in Minkowski 3-space \(L^ 3\), the Gauss map G is defined to be a mapping of M into the unit pseudosphere \({\mathbb{H}}\) in \(L^ 3\) assigning to each point p of M the timelike unit normal vector at p translated parallelly to the origin. In this paper, the authors prove a representation formula for spacelike surfaces with prescribed mean curvature in terms of their Gauss maps. More precisely, the following are proved. (1) Arbitrary oriented spacelike surfaces in \(L^ 3\) satisfy a system of first order partial differential equations involving the mean curvature function H and the Gauss map G. (2) The complete integrability condition for this system yields a system of second order partial differential equations identifying the gradient of H and the tension field of G, which simply means that the Gauss map G should be a harmonic mapping if the mean curvature H is constant. (3) Conversely, given a nowhere holomorphic smooth mapping G of a simply connected Riemann surface M into the pseudosphere \({\mathbb{H}}\) satisfying the complete integrability condition for some nonvanishing smooth function H on M, one can construct explicitly a spacelike immersion of M into \(L^ 3\) such that the mean curvature of M is H and the Gauss map of M is given by G. These constitute a Lorentzian counterpart of the Weierstrass-Enneper-Kenmotsu representation formula for surfaces in Euclidean 3-space.

Recently, by means of the Weierstrass-type representation for harmonic maps from a Riemann surface into symmetric spaces discovered by Dorfmeister, Pedit and Wu, Dorfmeister and Haak have constructed constant mean curvature surfaces by applying the Sym-Bobenko formula to the loop-group-valued maps given by integrating the meromorphic potentials. On the other hand, Kenmotsu's work on the Weierstrass representation for immersions with prescribed mean curvature from a simply connected Riemann surface into Euclidean 3-space was generalized by \textit{K. Akutagawa} and \textit{S. Nishikawa} to Minkowski 3-space [Tôhoku Math. J., II. Ser. 42, 67-82 (1990; Zbl 0679.53002)]. Motivated by these results, the author of this paper considers two aspects. The first is to establish a natural correspondence between the following two spaces: the space of conformal spacelike immersions with constant mean curvature from a simply connected Riemann surface \(\Sigma\) into Minkowski 3-space, and that of nowhere anti-holomorphic harmonic maps from \(\Sigma\) into the Poincaré half plane, regarded as the Riemannian symmetric space \(SL(2, \mathbb{R})/SO(2)\). The second is to prove the Lorentzian version of the Sym-Bobenko formula and apply it to construct spacelike immersions with constant mean curvature into Minkowski 3-space. The main idea in this paper is to define an \(sl(2, \mathbb{R})\)-valued 1-form \(\Lambda^f\) on a Riemann surface \(\Sigma\) associated to a smooth map \(f:\Sigma\to SL(2, \mathbb{R})/SO(2)\) and show that the harmonicity of \(f\) is equivalent to the \(d\)-closedness of \(\Lambda^f\). In mining operations, the time delay between grade estimations and decision-making based on those estimations can be substantial. This may lead to the scheduling of stopes mining that is based on information which is seriously out of date and, consequently, results in a substantial mined resources and reserves bias. To mitigate this gap between the grade estimation of an orebody and its exploitation, this paper proposes a method to quickly update resources and reserves that are integrated into the concept of real-time mining. The current standard for grade data collection in underground mines relies on a conventional chemical lab analysis of sparse drill hole or chip/face samples. The proposed methodology for the continuous and swift updating of mine resources and reserves requires a constant and rapid stream of measurements at the stopes. Consequently, this work proposes using portable X-ray fluorescence (XRF) devices to carry out the fast and abundant monitoring of ore grades. However, these fast data are highly uncertain; hence, the objective of this proposed method is to use the total data measurements and integrate their uncertainty into the resources modeling. The first step in the proposed methodology consists of creating a joint distribution function between ``uncertain'' XRF and the corresponding ``hard'' measurements based on empirical historical data. Then, the uncertainty of the XRF measurements is derived from these joint distributions through the conditional distribution of the real values applied to the known XRF measurement. The second step involves updating the reserves by integrating these uncertain XRF data, which are quantified by conditional distributions, into the grade characterization models. To achieve this, a stochastic simulation with point distributions is applied. An actual case study of a copper sulfide deposit illustrates the proposed methodology.

It is known that a nonlinear state feedback can be designed for the input torques of a rigid robot arm so to exactly linearize its closed-loop dynamics. When the initial conditions are not matched with the desired trajectory, the resulting linear dynamics of the error will be driven by external acceleration inputs. An optimal control problem can then be formulated by minimizing a quadratic function of the state error and of either the acceleration or the torque control effort. The weighting of the torque considered in this paper transforms the optimal control problem back into the nonlinear setting. The authors show that the optimal control law can be determined in closed form through the solution of a Riccati matrix equation which is of the differential type if the time horizon is finite and of the algebraic type if the horizon is infinite. Similar results for the infinite horizon were obtained also by \textit{R. Johansson} [IEEE Trans. Automatic Control 35, No. 11, 1197-1208 (1990; Zbl 0721.49032)]. This paper presents algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid body motion are found by solving an algebraic matrix equation. The system stability is investigated according to Lyapunov function theory and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution results in natural design parameters in the form of square weighting matrices as known from linear quadratic optimal control. The proposed optimal control is useful both for motion control, trajectory planning, and motion analysis.

It is known that a nonlinear state feedback can be designed for the input torques of a rigid robot arm so to exactly linearize its closed-loop dynamics. When the initial conditions are not matched with the desired trajectory, the resulting linear dynamics of the error will be driven by external acceleration inputs. An optimal control problem can then be formulated by minimizing a quadratic function of the state error and of either the acceleration or the torque control effort. The weighting of the torque considered in this paper transforms the optimal control problem back into the nonlinear setting. The authors show that the optimal control law can be determined in closed form through the solution of a Riccati matrix equation which is of the differential type if the time horizon is finite and of the algebraic type if the horizon is infinite. Similar results for the infinite horizon were obtained also by \textit{R. Johansson} [IEEE Trans. Automatic Control 35, No. 11, 1197-1208 (1990; Zbl 0721.49032)]. It is a key challenge to exploit the label coupling relationship in multi-label classification (MLC) problems. Most previous works focused on label pairwise relations, in which generally only global statistical information is used to analyze the coupled label relationship. In this work, firstly Bayesian and hypothesis testing methods are applied to predict the label set size of testing samples within their \(k\) nearest neighbor samples, which combines global and local statistical information, and then a priori algorithm is used to mine the label coupling relationship among multiple labels rather than pairwise labels, which can exploit the label coupling relations more accurately and comprehensively. The experimental results on text, biology and audio datasets show that, compared with the state-of-the-art algorithm, the proposed algorithm can obtain better performance on 5 common criteria.

The authors study the phase transition (PT), intended as the lack of analyticity of the pressure function, in the set up of ergodic theory. They consider as dynamical system the full shift, which is uniformly hyperbolic, and look for explicit potentials which are suitable to produce a PT. The kind of PT they consider is the ``freezing PT'', and they show that such a transition does happen for the Fibonacci quasi-crystal. This is a continuation of a preceding work by the authors [Commun. Math. Phys. 321, No. 1, 209--247 (2013; Zbl 1271.82017)], in which a similar result was proved for the Thue-Morse quasi-crystal. We examine the thermodynamic formalism for a class of renormalizable dynamical systems which, in the symbolic space, is generated by the Thue-Morse substitution, and in complex dynamics, by the Feigenbaum-Coullet-Tresser map. The basic question addressed is whether fixed points \(V\) of a renormalization operator \({\mathcal{R}}\) acting on the space of potentials are such that the pressure function \({\beta \mapsto \mathcal{P}(-\beta V)}\) exhibits phase transitions. This extends the work by \textit{A. Baraviera} et al. [Stoch. Dyn. 12, No. 4, Paper No. 1250005 (2012; Zbl 1268.37057)] on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attractor of renormalization is a Cantor set (rather than a single fixed point), which admits various classes of fixed points of \({\mathcal{R}}\), some of which do and some of which do not exhibit phase transitions. In particular, we show it is possible to reach, as a ground state, a quasi-crystal before temperature zero by freezing a dynamical system.

The authors study the phase transition (PT), intended as the lack of analyticity of the pressure function, in the set up of ergodic theory. They consider as dynamical system the full shift, which is uniformly hyperbolic, and look for explicit potentials which are suitable to produce a PT. The kind of PT they consider is the ``freezing PT'', and they show that such a transition does happen for the Fibonacci quasi-crystal. This is a continuation of a preceding work by the authors [Commun. Math. Phys. 321, No. 1, 209--247 (2013; Zbl 1271.82017)], in which a similar result was proved for the Thue-Morse quasi-crystal. For a given set of identical capacitated bins, a set of weighted items, and a set of precedences among such items, we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied. The problem, denoted as the bin packing problem with precedence constraints (BPP-P), has a very intriguing combinatorial structure and models many assembly and scheduling issues. According to our knowledge, the BPP-P has received little attention in the literature, and in this paper we address it for the first time with exact solution methods. In particular, we develop reduction criteria, a large set of lower bounds, a variable neighborhood search upper bounding technique, and a branch-and-bound algorithm. We show the effectiveness of the proposed algorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques.

An algorithm for proving geometric theorems of ruler-constructible type in two- and three- dimensional projective geometry is presented on the basis of some results on Cayley algebra and bracket algebra. In essence, the idea is to combine Cayley expansion and Cramer's rules with Cayley factorization and simplification techniques to get shorter proofs. For a geometric entity or constraint, there are often several different Cayley expressions to represent it which, in addition, usually have many ways to be expanded into bracket polynomials. Furthermore, the choice of an adequate representation can lead to extremely simple proofs whereas an unlucky choice can lead to extremely complicated computations. The central idea to overcome the difficulty of multiple representation, eliminations and expansions is to use what the authors call `breefs', that is, bracket-oriented representation, elimination and expansion for factored and shortest results. In this paper, the Cayley expansions of some typical Cayley expressions are given and a series of theorems are established on factored and shortest expansions; several Cayley factorization techniques are developed, which are also used in a companion paper on theorem proving in conic geometry [J. Symb. Comput. 36, No. 5, 763--809 (2003; Zbl 1047.03011), reviewed below] and from which a powerful simplification technique, called contraction, has been derived. Several elimination rules for theorem proving are obtained which, together with the Cayley expansion and simplification techniques, form the basis of the short-proof generation algorithm. This paper is a continuation of Part I [J. Symb. Comput. 36, No. 5, 717--762 (2003; Zbl 1047.03010), reviewed above] in which the focus is on proving theorems in plane conic geometry; this field is usually harder than incidence geometry from the standpoint of proof mechanization. The idea of bracket computation stated in Part I is developed now for plane conic geometry, the main algorithm being a conic points selection which selects, to a given construction, a suitable sequence of representative conic points for each bracket or wedge product containing the construction. The study of the particular construction of six-point-on-conic leads to the idea of considering its algebraic representations as computation rules. As a result, three additional simplification techniques are presented in this paper; namely, conic transformation, pseudoconic transformation and conic contraction. The additional concept of rational Cayley factorization is included in this paper; it allows the occurrence of bracket monomials in the denominator (hence the name). Technically, it is just a combination of the Cayley factorization techniques from Part I and the conic combination technique; from the computational side, it allows the authors to significantly simplify bracket computation in conic geometry. An automated theorem proving algorithm is presented at the end of the paper together with some sample applications of some theorems on free conic points, tangents and poles related to tangents.

An algorithm for proving geometric theorems of ruler-constructible type in two- and three- dimensional projective geometry is presented on the basis of some results on Cayley algebra and bracket algebra. In essence, the idea is to combine Cayley expansion and Cramer's rules with Cayley factorization and simplification techniques to get shorter proofs. For a geometric entity or constraint, there are often several different Cayley expressions to represent it which, in addition, usually have many ways to be expanded into bracket polynomials. Furthermore, the choice of an adequate representation can lead to extremely simple proofs whereas an unlucky choice can lead to extremely complicated computations. The central idea to overcome the difficulty of multiple representation, eliminations and expansions is to use what the authors call `breefs', that is, bracket-oriented representation, elimination and expansion for factored and shortest results. In this paper, the Cayley expansions of some typical Cayley expressions are given and a series of theorems are established on factored and shortest expansions; several Cayley factorization techniques are developed, which are also used in a companion paper on theorem proving in conic geometry [J. Symb. Comput. 36, No. 5, 763--809 (2003; Zbl 1047.03011), reviewed below] and from which a powerful simplification technique, called contraction, has been derived. Several elimination rules for theorem proving are obtained which, together with the Cayley expansion and simplification techniques, form the basis of the short-proof generation algorithm. We proposed a back force model for simulating dislocations cutting into a \(\gamma^{\prime}\) precipitate, from the physical viewpoint of work for making or recoveringan antiphase boundary (APB). The first dislocation, or a leading partial of a superdislocation, is acted upon by a back force whose magnitude is equal to the APB energy. The second dislocation, or a trailing partial of a superdislocation, is attracted by the APB with a force of the same magnitude. The model is encoded in a 3D discrete dislocation dynamics (DDD) code and demonstrates that a superdislocation nucleates after two dislocations pile up at the interface and that the width of dislocations is naturally balanced by the APB energy and repulsion of dislocations. The APB energy adopted here is calculated by ab initio analysis on the basis of the density functional theory (DFT). Then we applied our DDD simulations to more complicated cases, namely, dislocations near the edges of a cuboidal precipitate and those at the \(\gamma /\gamma^{\prime}\) interface covered by an interfacial dislocation network. The former simulation shows that dislocations penetrate into a \(\gamma^{\prime}\) precipitate as a superdislocation from the edge of the cube, when running around the cube to form Orowan loops. The latter reveals that dislocations become wavy at the interface due to the stress field of the dislocation network, then cut into the \(\gamma^{\prime}\) precipitate through the interspace of the network. Our model proposed here can be applied to study the dependence of the cutting resistance on the spacing of dislocations in the interfacial dislocation network.

In [\textit{Yu. V. Obnosov}, Russ. Math. 48, No. 7, 50--59 (2004; Zbl 1100.30034); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 48, No. 7, 53--62 (2004)], a two-phase medium with one branch of a hyperbola as an interface was investigated. The present paper is an immediate continuation of ibidem in the case of two hyperbolic inclusions. Using classical methods of complex analysis, the authors derive an explicit solution of the conjugation problem of flows at the hyperbolic border between two dissimilar heterogeneous media. The investigation comprises both cases involving the real and the complex coefficients of the corresponding boundary condition. The considerations of the paper are accompanied by five illustrative figures. The author considers a model of the theory of heterogeneous media. He gives a solution of the problem of hyperbolic inclusion.

In [\textit{Yu. V. Obnosov}, Russ. Math. 48, No. 7, 50--59 (2004; Zbl 1100.30034); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 48, No. 7, 53--62 (2004)], a two-phase medium with one branch of a hyperbola as an interface was investigated. The present paper is an immediate continuation of ibidem in the case of two hyperbolic inclusions. Using classical methods of complex analysis, the authors derive an explicit solution of the conjugation problem of flows at the hyperbolic border between two dissimilar heterogeneous media. The investigation comprises both cases involving the real and the complex coefficients of the corresponding boundary condition. The considerations of the paper are accompanied by five illustrative figures. Let \(\mathsf{T}[1,n]\) be a text of length \(n\) and \(\mathsf{T}[i,n]\) be the suffix starting at position \(i\). Also, for any two strings \(X\) and \(Y\), let \(\mathsf{LCP}(X,Y)\) denote their longest common prefix. The range-LCP of \(\mathsf {T}\) w.r.t. a range \([\alpha,\beta]\), where \(1\leq\alpha<\beta\leq n\) is \[ {{{\mathsf{rlcp}}(\alpha,\beta)=\max \{|{\mathsf{LCP}}({\mathsf{T}}[i,n], T[j,n])| \mid i\neq j} \text{ and } {i,j \in [\alpha,\beta]\}}}. \] \textit{A. Amir} et al. [Lect. Notes Comput. Sci. 7074, 683--692 (2011; Zbl 1350.68298)] introduced the indexing version of this problem, where the task is to build a data structure over \(\mathsf{T}\), so that \(\mathsf{rlcp}(\alpha,\beta)\) for any query range \([\alpha,\beta]\) can be reported efficiently. They proposed an \(O(n\log^{1+\epsilon}n)\) space structure with query time \(O(\log\log n)\), and a linear space (i.e., \(O(n)\) words) structure with query time \(O(\delta\log\log n)\), where \(\delta=\beta-\alpha+1\) is the length of the input range and \(\epsilon > 0\) is an arbitrarily small constant. Later, \textit{M. Patil} et al. [Lect. Notes Comput. Sci. 8214, 263--270 (2013; Zbl 1442.68040)] proposed another linear space structure with an improved query time of \(O(\sqrt{\delta}\log^{\epsilon}\delta)\). This poses an interesting question, whether it is possible to answer \(\mathsf{rlcp}(\cdot,\cdot)\) queries in poly-logarithmic time using a linear space data structure. In this paper, we settle this question by presenting an \(O(n)\) space data structure with query time \(O(\log^{1+\epsilon}n)\) and construction time \(O(n\log n)\).

The both notions of Maslov class and symplectic or Marsden-Weinstein reduction are important in symplectic geometry. The author, in this article, shows that Maslov classes behave well under the reduction procedure, in some good situations. Nice reference for this subject is the standard text book by the author [Symplectic geometry and secondary characteristic classes, Prog. Math. 72 (1987; Zbl 0629.53002)]. The aim of this book is to provide a self-consistent treatment of the Maslov class, which is a fundamental invariant in the asymptotic analysis of partial differential equations of quantum physics. The characteristic classes which generalize the Maslov class are discussed and their properties established. These classes are computed in some important cases, namely for Lagrange and Legendre submanifolds of cotangent bundles. The general method of computation used by the author shows that the Maslov classes depend on a generalized second fundamental form, and on the curvature of the Lagrange (Legendre) submanifold. Because the framework for the secondary characteristic classes is the symplectic geometry, some notions as general and natural symplectic vector bundles that appear on symplectic manifolds and their submanifolds, or on contact manifolds, are presented in detail in the first part of the book. The book is of interest to researchers and graduate students in differential geometry, differential topology, mathematical physics and quantum physics.

The both notions of Maslov class and symplectic or Marsden-Weinstein reduction are important in symplectic geometry. The author, in this article, shows that Maslov classes behave well under the reduction procedure, in some good situations. Nice reference for this subject is the standard text book by the author [Symplectic geometry and secondary characteristic classes, Prog. Math. 72 (1987; Zbl 0629.53002)]. It is shown that the minimal order of a Cayley graph with outdegree \(r\geq 3\) and diameter \(d\geq 5\) is at least \((d+1)(1+[r/2])\).

Regular neighbourhoods (of embedded or immersed submanifolds) are one of the basic tools in manifold theory. The present paper gives a version of this notion for groups, with the main intent to produce an analogue for arbitrary finitely presented groups of the Jaco-Shalen-Johannson decomposition of Haken 3-manifolds (along tori and annuli into simple or hyperbolic 3-manifolds and Seifert fiber spaces), resp. of the corresponding decomposition of their fundamental groups into graphs of groups. The characteristic submanifold of a Haken 3-manifold \(M\) consists of the Seifert fiber pieces of the decomposition and can be thought of as regular neighbourhood of all the essential (embedded or immersed) tori and annuli in \(M\). So one is looking for a decomposition of a group \(G\) as a graph of groups, with Abelian edge groups and a subset of characteristic vertex groups which contains conjugates of certain essential subgroups of the group (such as free groups of rank two). There are various previous approaches to such a decomposition which, however, either do not specialize to the JSJ-decomposition in the case of fundamental groups of Haken 3-manifolds, or have the strong uniqueness properties of the JSJ-decomposition only for certain classes of groups (e.g. word hyperbolic groups) (see the introduction of the present paper for a careful description of the various previous approaches to such a decomposition). ``We find a canonical decomposition for (almost) finitely presented groups which essentially specialize to the classical Jaco-Shalen-Johannson decomposition when restricted to the fundamental groups of Haken 3-manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood of a family of almost invariant subsets of a group. An almost invariant subset is an analogue of an immersion.'' ``Here is an introduction to our ideas. As mentioned above, our choice of the crucial property of the classical JSJ-decomposition is the Enclosing Property for immersions. This property implies that the characteristic submanifold \(V(M)\) of an orientable Haken 3-manifold \(M\) contains a representative of every homotopy class of an essential annulus or torus in \(M\). In this paper, we introduce a natural algebraic analogue of enclosing.'' ``Let \(\sigma\) be a splitting of \(G\). We say that \(\sigma\) is enclosed by a vertex \(v\) of a graph of groups structure \(\Gamma\) of \(G\), if there is a graph of groups structure \(\Gamma_\sigma\) of \(G\), with an edge \(e\) which determines the splitting \(\sigma\), such that collapsing the edge \(e\) yields \(\Gamma\), and \(v\) is the image of \(e\). We emphasize that the condition that \(\sigma\) is enclosed by a vertex \(v\) is in general stronger than the condition that the edge group of \(\sigma\) is conjugate into the vertex group of \(v\). This is particularly clear if \(\sigma\) is a free product decomposition of \(G\), as then the edge group of \(\sigma\) is trivial.'' Previous results of the authors' on algebraic analogues of the fact that curves on surfaces with intersection number zero can be homotoped to be disjoint suggest that an appropriate algebraic analogue of an immersion is an almost invariant subset of \(G\) ([Geom. Topol. 4, 179-218 (2000; Zbl 0983.20024)], contained as an appendix in the present paper). Thinking of the characteristic submanifold as a regular neighbourhood of all the essential annuli and tori in \(M\), ``the key new idea of the paper is an algebraic version of regular neighbourhood theory. We describe an algebraic regular neighbourhood of a family of almost invariant subsets of \(G\). This is a graph of groups structure for \(G\), with the property that certain vertices enclose the given almost invariant sets. As splittings have almost invariant sets naturally associated, this also yields an idea of an algebraic regular neighbourhood of a family of splittings.'' In a previous paper [Geom. Topol. 2, 11-29 (1998; Zbl 0897.20029)], the first author defined the intersection number of two splittings (as an amalgamated free product or HNN extension) of any group \(G\) over any subgroups \(H\) and \(K\). In the special case when \(G\) is the fundamental group of a compact surface \(F\) and these splittings arise from embedded arcs or circles on \(F\), the algebraic intersection number of the splittings equals the topological intersection numbers of the corresponding 1-manifolds. The analogous statement holds when \(G\) is the fundamental group of a compact 3-manifold and these splittings arise from \(\pi_1\)-injective embedded surfaces. ``The first main result of the paper is a generalisation to the algebraic setting of the fact that two simple arcs or closed curves on a surface have intersection number zero if and only if they can be isotoped apart.'' Here the algebraic analogue of isotopy is defined in terms of compatibility of splittings which roughly means that the group \(G\) can be expressed as the fundamental group of a graph of groups realizing simultaneously the given splittings. Now the first main theorem says that a collection of \(n\) splittings over finitely generated subgroups is compatible up to isotopy if and only if each pair of splittings has intersection number zero; also, uniqueness of the underlying graph and of its edge and vertex groups is obtained. Next an algebraic analogue of non-embedded arcs or circles on surfaces resp. of immersed \(\pi_1\)-injective surfaces in 3-manifolds is discussed which is defined in terms of almost invariant subsets of the quotient \(H\setminus G\), for a subgroup \(H\) of \(G\). ``Our second main result is an algebraic analogue of the fact that a singular curve on a surface or a singular surface in a 3-manifold which has self-intersection number zero can be homotoped to cover an embedding. It asserts that if \(H\setminus G\) has an almost invariant subset with self-intersection number zero, then \(G\) has a splitting over a subgroup commensurable with \(H\).'' ``In a separate paper, we use the ideas about intersection numbers of splittings to study the JSJ decomposition of Haken 3-manifolds. The problem is to recognize which splittings of the fundamental group of such a manifold arise from the JSJ decomposition. It turns out that a class of splittings which we call canonical can be defined using intersection numbers, and we use this to show that the JSJ decomposition for Haken 3-manifolds depends only on the fundamental group. This leads to an algebraic proof of Johannson's Deformation Theorem. It seems very likely that similar ideas apply to \textit{Z. Sela}'s JSJ decomposition of hyperbolic groups [Geom. Funct. Anal. 7, No. 3, 561-593 (1997; Zbl 0884.20025)] and thus provide a common thread to the two types of JSJ decompositions. Thus, the use of intersection numbers seems to provide a tool in the study of diverse topics in group theory, and this paper together with the paper of Scott mentioned above provides some of the foundational material''.

Regular neighbourhoods (of embedded or immersed submanifolds) are one of the basic tools in manifold theory. The present paper gives a version of this notion for groups, with the main intent to produce an analogue for arbitrary finitely presented groups of the Jaco-Shalen-Johannson decomposition of Haken 3-manifolds (along tori and annuli into simple or hyperbolic 3-manifolds and Seifert fiber spaces), resp. of the corresponding decomposition of their fundamental groups into graphs of groups. The characteristic submanifold of a Haken 3-manifold \(M\) consists of the Seifert fiber pieces of the decomposition and can be thought of as regular neighbourhood of all the essential (embedded or immersed) tori and annuli in \(M\). So one is looking for a decomposition of a group \(G\) as a graph of groups, with Abelian edge groups and a subset of characteristic vertex groups which contains conjugates of certain essential subgroups of the group (such as free groups of rank two). There are various previous approaches to such a decomposition which, however, either do not specialize to the JSJ-decomposition in the case of fundamental groups of Haken 3-manifolds, or have the strong uniqueness properties of the JSJ-decomposition only for certain classes of groups (e.g. word hyperbolic groups) (see the introduction of the present paper for a careful description of the various previous approaches to such a decomposition). ``We find a canonical decomposition for (almost) finitely presented groups which essentially specialize to the classical Jaco-Shalen-Johannson decomposition when restricted to the fundamental groups of Haken 3-manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood of a family of almost invariant subsets of a group. An almost invariant subset is an analogue of an immersion.'' ``Here is an introduction to our ideas. As mentioned above, our choice of the crucial property of the classical JSJ-decomposition is the Enclosing Property for immersions. This property implies that the characteristic submanifold \(V(M)\) of an orientable Haken 3-manifold \(M\) contains a representative of every homotopy class of an essential annulus or torus in \(M\). In this paper, we introduce a natural algebraic analogue of enclosing.'' ``Let \(\sigma\) be a splitting of \(G\). We say that \(\sigma\) is enclosed by a vertex \(v\) of a graph of groups structure \(\Gamma\) of \(G\), if there is a graph of groups structure \(\Gamma_\sigma\) of \(G\), with an edge \(e\) which determines the splitting \(\sigma\), such that collapsing the edge \(e\) yields \(\Gamma\), and \(v\) is the image of \(e\). We emphasize that the condition that \(\sigma\) is enclosed by a vertex \(v\) is in general stronger than the condition that the edge group of \(\sigma\) is conjugate into the vertex group of \(v\). This is particularly clear if \(\sigma\) is a free product decomposition of \(G\), as then the edge group of \(\sigma\) is trivial.'' Previous results of the authors' on algebraic analogues of the fact that curves on surfaces with intersection number zero can be homotoped to be disjoint suggest that an appropriate algebraic analogue of an immersion is an almost invariant subset of \(G\) ([Geom. Topol. 4, 179-218 (2000; Zbl 0983.20024)], contained as an appendix in the present paper). Thinking of the characteristic submanifold as a regular neighbourhood of all the essential annuli and tori in \(M\), ``the key new idea of the paper is an algebraic version of regular neighbourhood theory. We describe an algebraic regular neighbourhood of a family of almost invariant subsets of \(G\). This is a graph of groups structure for \(G\), with the property that certain vertices enclose the given almost invariant sets. As splittings have almost invariant sets naturally associated, this also yields an idea of an algebraic regular neighbourhood of a family of splittings.'' Routing is an important problem in networks. We look at routing in the presence of line segment constraints (i.e., obstacles that our edges are not allowed to cross). Let \(P\) be a set of \(n\) vertices in the plane and let \(S\) be a set of line segments between the vertices in \(P\), with no two line segments intersecting properly. We present the first 1-local \(O(1)\)-memory routing algorithm on the visibility graph of \(P\) with respect to a set of constraints \(S\) (i.e., it never looks beyond the direct neighbours of the current location and does not need to store more than \(O(1)\)-information to reach the target). We also show that when routing on any triangulation \(T\) of \(P\) such that \(S\subseteq T\), no \(o(n)\)-competitive routing algorithm exists when only considering the triangles intersected by the line segment from the source to the target (a technique commonly used in the unconstrained setting). Finally, we provide an \(O(n)\)-competitive 1-local \(O(1)\)-memory routing algorithm on any such \(T\), which is optimal in the worst case, given the lower bound.

Let \(G=\exp \mathfrak g\) be a connected simply connected nilpotent Lie group and let \(w\) be a continuous symmetric weight on \(G\). In [Stud. Math. 160, No. 3, 205--229 (2004; Zbl 1047.43006)] the first author of the present paper and \textit{D. Alexander} proved that closed two-sided ideals of the weighted group algebra \(L^{1}_{w}\) whose hull is \({\chi_{l}}\) (where \(\chi_{l}\), \(l\in \mathfrak g^*\), is the unitary character of \(G\) defined by \(\chi_{l}(\exp X)=e^{-il(X)} \text{ for any } X\in \mathfrak g)\) correspond to finite dimensional, translation invariant subspaces of complex polynomials dominated by \(w\). The paper under review can be viewed as a continuation of that work. It consists in replacing the character \(\chi_{l}\) by a unitary and irreducible representation \(\pi_{l}\) associated to a flat orbit \(G.l=l+\mathfrak g(l)^{\perp}\) or, more generally, for a set of the form \(\{\pi_{l'}: l'\in l+ \mathfrak n^{\perp}\}\) where \(\mathfrak n\) is a non-trivial ideal contained in \(\mathfrak g (l)\) and \(\mathfrak g (l)\) is the stabilizer of the linear form \(l\). Their method consists in considering the set \(P_{w| N}(N)\) of all polynomials on \(N=\exp \mathfrak n\) bounded by a multiple of the weight \(w| N\), and in associating to a \(G\)-invariant translation invariant subspace \(W\) of \(P_{w| N}(N)\) an induced representation \(\pi_{W}\). The map \(W \mapsto \ker \pi_{W} \) appears to be a bijection between \(P_{w| N}(N)\) and the closed two-sided \(L^{\infty}(G/N)\) invariant \(I\) ideals of \(L^{1}_{w}(G)\) such that \(I\) admits \(\{\pi_{l'}\in \widehat{G}: l'\in l+ \mathfrak n^{\perp}\}\) as a hull. In the case of flat orbits, the hull is replaced by the single point \(\{\pi_{l}\}\). The authors deal with simply connected nilpotent Lie groups \(G.\) They first study the behavior of weights on \(G.\) Then they discuss synthesis problems. The next topic is polynomials and group algebras. Finally, for a subalgebra \(A\) of \(L^1(G)\) containing the Schwartz algebra \(S(G)\) as a dense subset, all closed two-sided ideals of \(A\) whose hull reduces to one point which is a character are characterized. Several important examples are considered.

Let \(G=\exp \mathfrak g\) be a connected simply connected nilpotent Lie group and let \(w\) be a continuous symmetric weight on \(G\). In [Stud. Math. 160, No. 3, 205--229 (2004; Zbl 1047.43006)] the first author of the present paper and \textit{D. Alexander} proved that closed two-sided ideals of the weighted group algebra \(L^{1}_{w}\) whose hull is \({\chi_{l}}\) (where \(\chi_{l}\), \(l\in \mathfrak g^*\), is the unitary character of \(G\) defined by \(\chi_{l}(\exp X)=e^{-il(X)} \text{ for any } X\in \mathfrak g)\) correspond to finite dimensional, translation invariant subspaces of complex polynomials dominated by \(w\). The paper under review can be viewed as a continuation of that work. It consists in replacing the character \(\chi_{l}\) by a unitary and irreducible representation \(\pi_{l}\) associated to a flat orbit \(G.l=l+\mathfrak g(l)^{\perp}\) or, more generally, for a set of the form \(\{\pi_{l'}: l'\in l+ \mathfrak n^{\perp}\}\) where \(\mathfrak n\) is a non-trivial ideal contained in \(\mathfrak g (l)\) and \(\mathfrak g (l)\) is the stabilizer of the linear form \(l\). Their method consists in considering the set \(P_{w| N}(N)\) of all polynomials on \(N=\exp \mathfrak n\) bounded by a multiple of the weight \(w| N\), and in associating to a \(G\)-invariant translation invariant subspace \(W\) of \(P_{w| N}(N)\) an induced representation \(\pi_{W}\). The map \(W \mapsto \ker \pi_{W} \) appears to be a bijection between \(P_{w| N}(N)\) and the closed two-sided \(L^{\infty}(G/N)\) invariant \(I\) ideals of \(L^{1}_{w}(G)\) such that \(I\) admits \(\{\pi_{l'}\in \widehat{G}: l'\in l+ \mathfrak n^{\perp}\}\) as a hull. In the case of flat orbits, the hull is replaced by the single point \(\{\pi_{l}\}\). This study outlines and discusses some aspects of the concept of induction in mathematics. In particular, our study examines the role of the concept of induction in mathematics textbooks and in the teaching of mathematics. Thus on the one hand we discuss a typology of various types of reasoning proposed in a particular mathematics textbook for 16 year-old students. On the other hand, we examine students' and teachers' conceptions about induction.

The physicist \textit{J.-M. Richard} observed, in connection with a problem from ballistics, that the maximal perimeter of a parallelogram inscribed in a given ellipse can be attained by a parallelogram with one vertex at any prescribed point of the ellipse, and he gave a direct computational proof in [Eur. J. Phys. 25, No. 6, 835--844 (2004; Zbl 1162.70320)]. In the paper under review, the authors give two simple proofs, one geometric and the other algebraic, together with a slight generalization, and they describe connection with billiards. A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a `safe' domain. In the simple cases of a constant force, harmonic potential and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic section which can be derived either by simple calculus or by geometrical considerations. The case of harmonic forces reveals a subtle property of the maximal sum of distances within an ellipse.

The physicist \textit{J.-M. Richard} observed, in connection with a problem from ballistics, that the maximal perimeter of a parallelogram inscribed in a given ellipse can be attained by a parallelogram with one vertex at any prescribed point of the ellipse, and he gave a direct computational proof in [Eur. J. Phys. 25, No. 6, 835--844 (2004; Zbl 1162.70320)]. In the paper under review, the authors give two simple proofs, one geometric and the other algebraic, together with a slight generalization, and they describe connection with billiards. It has been well established that the variable structure and sliding mode control design methodologies are appropriate for robust control. This control design framework has many attractive features including the ability to counteract the effect of uncertainties and disturbances which are inevitable in most practical systems. Sliding mode control is a particular class of variable structure control which was first introduced by Emel'yanov and his co-workers. The sliding mode control paradigm has now become a mature technique for the design of robust controllers for a wide class of systems including nonlinear, uncertain and time-delayed systems. This book is a collection of plenary and invited talks delivered at the 12th IEEE International Workshop on Variable Structure System held at the Indian Institute of Technology, Mumbai, India in January 2012. The workshop organisers, together with the IEEE CSS Technical Committee on Variable Structure and Sliding Mode Control, invited leading international researchers to present plenary and invited talks at VSS 2012 in order to articulate the current state of the art both in terms of theory and practice in the discipline. After the workshop, these researchers were invited to develop book chapters for this edited collection in order to reflect the latest results and open research questions in the area.

Let \(M=\sharp_n (S^2\times S^1)\), this is a compact orientable 3-manifold whose fundamental group is the free group \(F_n\). A 2-sphere in \(M\) is essential if it does not bound a 3-ball. Let \(\Sigma\) be a maximal sphere system for \(M\), that is, \(\Sigma\) is a collection of disjointly embedded essential spheres in \(M\) such that every complementary component \(P\) of \(\Sigma\) in \(M\) is a 3-punctured 3-sphere. Let \(T\) be a torus embedded in \(M\); it is essential if the image of \(\pi_1(T)\) in \(\pi_1(M)\) is nontrivial. It is said that \(T\) is in normal form with respect to \(\Sigma\), if their intersection is transverse and for each \(P\), the intersection of \(T\) with \(P\) consists of essential disks in \(P\), cylinders joining two different components of \(\partial P\), or pants joining the 3 components of \(\partial P\). The main result of the paper shows that if \(T\) is an essential torus in \(M\), then it is homotopic to a torus in normal form, and the homotopy process does not increase the number of curves of intersection with any sphere of \(\Sigma\). It is also shown that if \(T\) and \(T'\) are two homotopic normal tori, then they are normally homotopic, that is, there is a homotopy between them that does not increase the number of intersections with any sphere of \(\Sigma\). It follows that if \(T\) is in normal form with respect to \(\Sigma\), then the intersection number of \(T\) with any sphere \(S\) in \(\Sigma\) is minimal among the representatives of the homotopy class of \(T\) in each \(P\). These results generalize previous results of \textit{A. E. Hatcher} [Comment. Math. Helv. 70, No. 1, 39--62 (1995; Zbl 0836.57003)] for normal forms of spheres. These results have potential applications in the study of Out(\(F_n\)), for it is known that there is a \(1-1\) correspondence between the homotopy classes of essential imbedded tori in \(M\) and the equivalence classes of \(\mathbb{Z}\)-splittings of \(F_n\), namely, if the torus is separating it corresponds to an amalgamated free product, and if the torus is non-separating it corresponds to an HNN extension of \(F_n\) [\textit{M. Clay, F. Gültepe} and \textit{K. Rafi}, ``Essential tori and the Dehn twists of the free group'', in preparation]. Let \(F_n\) be the free group of rank \(n\). An imbedding of \(\text{Aut}(F_n)\) into \(\text{Aut} (F_{n + 1})\) is defined by extending an automorphism of \(F_n\) to an automorphism of \(F_{n + 1}\) fixing the \((n + 1)\)st basis element. The main result of this paper is that the corresponding map \(H_i(\text{Aut} (F_n)) \to H_i (\text{Aut} (F_{n + 1}))\) is an isomorphism for \(n > i^2/4 + 2i - 1\), and the map \(H_i (\text{Aut} (F_n)) \to H_i (\text{Out} (F_n))\) is an isomorphism for \(n > i^2 + 5i/2\). Similar stability results are known for various groups, including symmetric groups, many linear groups, and mapping class groups of 2-manifolds. \(\text{Out} (F_n)\) is closely related to the mapping class group of the 3-manifold \(M_{n,1}\) which is a connected sum of \(n\) copies of \(S^2 \times S^1\), with one open ball removed. Precisely, this mapping class group contains a finite abelian normal subgroup (generated by homeomorphisms supported in neighborhoods of imbedded 2-spheres), whose quotient is \(\text{Aut} (F_n)\). The author's approach is analogous to Harer's proof of homological stability for mapping class groups of 2-manifolds, as refined by Ivanov. In the 2-dimensional version simplicial complexes are constructed whose \(k\)-simplices correspond to isotopy classes of \(k + 1\) disjoint simple arcs or closed curves. These complexes are often contractible, or if not have high connectivity, and the mapping class group acts on them in an obvious way. Analysis of this action using standard (but nontrivial) homological methods yields the desired information. In the 3-dimensional version, the author considers families of disjoint imbedded 2-spheres in \(M_{n,1}\) (none of which bounds a ball, is parallel to the boundary sphere of \(M_{n,1}\) and no two of which are isotopic). The simplicial complex \(S_{n,1}\) obtained from such families is proven to be contractible when \(n > 0\) (actually this is proven for the more general case \(S_{n,s}\) when the manifold has \(s \geq 0\) punctures). One of the key ideas is to move systems of 2- spheres into a normal form with respect to a decomposition of \(M_{n,s}\) into twice-punctured 3-balls. Also of interest is the identification of the Culler-Vogtmann space of actions of \(F_n\) on \(R\)-trees with a subcomplex of \(S_{n,0}\). The contraction of \(S_{n,0}\) given by the author's argument preserves this subcomplex, giving an alternative proof of its contractibility. This is used to deduce the virtual cohomological dimensions of \(\text{Aut} (F_n)\) and \(\text{Out} (F_n)\), first determined by Culler and Vogtmann.

Let \(M=\sharp_n (S^2\times S^1)\), this is a compact orientable 3-manifold whose fundamental group is the free group \(F_n\). A 2-sphere in \(M\) is essential if it does not bound a 3-ball. Let \(\Sigma\) be a maximal sphere system for \(M\), that is, \(\Sigma\) is a collection of disjointly embedded essential spheres in \(M\) such that every complementary component \(P\) of \(\Sigma\) in \(M\) is a 3-punctured 3-sphere. Let \(T\) be a torus embedded in \(M\); it is essential if the image of \(\pi_1(T)\) in \(\pi_1(M)\) is nontrivial. It is said that \(T\) is in normal form with respect to \(\Sigma\), if their intersection is transverse and for each \(P\), the intersection of \(T\) with \(P\) consists of essential disks in \(P\), cylinders joining two different components of \(\partial P\), or pants joining the 3 components of \(\partial P\). The main result of the paper shows that if \(T\) is an essential torus in \(M\), then it is homotopic to a torus in normal form, and the homotopy process does not increase the number of curves of intersection with any sphere of \(\Sigma\). It is also shown that if \(T\) and \(T'\) are two homotopic normal tori, then they are normally homotopic, that is, there is a homotopy between them that does not increase the number of intersections with any sphere of \(\Sigma\). It follows that if \(T\) is in normal form with respect to \(\Sigma\), then the intersection number of \(T\) with any sphere \(S\) in \(\Sigma\) is minimal among the representatives of the homotopy class of \(T\) in each \(P\). These results generalize previous results of \textit{A. E. Hatcher} [Comment. Math. Helv. 70, No. 1, 39--62 (1995; Zbl 0836.57003)] for normal forms of spheres. These results have potential applications in the study of Out(\(F_n\)), for it is known that there is a \(1-1\) correspondence between the homotopy classes of essential imbedded tori in \(M\) and the equivalence classes of \(\mathbb{Z}\)-splittings of \(F_n\), namely, if the torus is separating it corresponds to an amalgamated free product, and if the torus is non-separating it corresponds to an HNN extension of \(F_n\) [\textit{M. Clay, F. Gültepe} and \textit{K. Rafi}, ``Essential tori and the Dehn twists of the free group'', in preparation]. In this paper, a quantum private comparison protocol is proposed based on \(\chi \)-type state. According to the protocol, two parties can determine the equality of their information with the assistant of a semi-honest third party. Due to utilizing quantum superdense coding, this protocol provides a high efficiency and capacity. Moreover, its security is also discussed.

This paper considers a large-scale fractal, which is an invariant set generated by a family of expansive maps, named a reverse iterated function system, on a discrete metric space. It is interesting that such a large-scale fractal just corresponds to a small-scale one, and a dimension equality between them is exactly given. In addition, this paper provides a proposition and three examples to answer some questions asked by \textit{R. S. Strichartz} [Can. J. Math. 50, No. 3, 638--657 (1998; Zbl 0913.28005)]. The paper studies the fractal structures at large scales in two ways: reverse iterated function systems (RIFS) and fractal blowups. A reverse iterated function system is defined to be a set of expansive maps \(\{T_1,T_2,\dots, T_m\}\) on a discrete metric space. A set \(F\) is called an invariant set of \(\{T_1,T_2,\dots, T_m\}\) if \(F= \sum^m_{j=1} T_jF\) and a measure \(\mu\) an invariant measure if \(\mu\) is the solution of \(\mu= \sum^m_{j= 1} p_j\mu\circ T^{-1}_j\) for positive weights \(p_j\). The structures and basic properties of such invariant sets and measures are investigated. It is proved that invariant sets can be described as unions of forward orbits of fixed points of iterated maps from the RIFS. A blowup \({\mathcal F}\) of a self-similar set \(F\) in \(\mathbb{R}^n\) is defined to be the union of an increasing sequence of sets, each similar to \(F\). A general construction of blowups is given and the blowups of the Cantor set, the Sierpiński gasket, and the von Koch curve are described.

This paper considers a large-scale fractal, which is an invariant set generated by a family of expansive maps, named a reverse iterated function system, on a discrete metric space. It is interesting that such a large-scale fractal just corresponds to a small-scale one, and a dimension equality between them is exactly given. In addition, this paper provides a proposition and three examples to answer some questions asked by \textit{R. S. Strichartz} [Can. J. Math. 50, No. 3, 638--657 (1998; Zbl 0913.28005)]. The refined de Sitter (dS) conjecture provides two consistency conditions for an effective theory potential of a quantum gravity theory. Any inflationary model can be checked by these conditions and minimal gauge inflation is not an exception. We develop a generic method to analyze a monotonically growing potential with an inflection point on the way to the plateau near the top such as the potential in minimal gauge inflation model and the Higgs inflation. Taking the latest observational data into account, we find the fully consistent parameter space where the model resides in the Landscape rather than in the Swampland.

In [\textit{T. Oikhberg} and \textit{H. P. Rosenthal}, Rocky Mt. J. Math. 37, No. 2, 597--608 (2007; Zbl 1138.46006)] it was proved that if \(X\) is a real linear space equipped with a metric \(d\) that satisfies the following three conditions: (i) the metric \(d\) is translation invariant, that is, \(d(x+z,y+z)=d(x,y)\) for any \(x\), \(y\), \(z\in X\), (ii) for every \(x\in X\) the map from \([0,1]\) to \(X\) defined by \(t\mapsto tx\) is continuous, (iii) each one-dimensional affine subspace of \(X\) is isometric to \(\mathbb{R}\), then there exists a norm \(\left\| \, \cdot \, \right\|\) on \(X\) such that \(d(x,y)=\left\| x-y\right\| \), where \(x,y\in X\). In the paper under review the following extension of the preceding result is stated (see Theorem 1.2): Let \(X\) be a real linear space equipped with a metric \(d\) satisfying: (i) the metric \(d\) is translation invariant, that is \(d(x+z,y+z)=d(x,y)\) for any \(x\), \(y\), \(z\in X\), (ii') for every \(x\in X\) the set \(\left\{ tx:t\in [ 0,1]\right\} \) is bounded in \((X,d)\), (iii') for all \(x\), \(y\in X\) their algebraic midpoint (i.e., \((x+y)/2\)) is a metric midpoint (i.e., \(d(x,y)=2d(x,(x+y)/2)=2d(y,(x+y)/2)\)), then \(\left\| x\right\| :=d(x,0)\), \(x\in X\), is a norm on \(X\) and \( d(x,y)=\left\| x-y\right\| \), \(x\), \(y\), \(z\in X\). The proof of the theorem of Oikhberg and Rosenthal is based on the use of the Mazur-Ulam theorem and the proof of the above Theorem 1.2 is shorter and self-contained. Furthermore, in Proposition 1.1, the authors show that (iii) implies (iii'), but that the converse is not true. The authors prove that a metric \(\rho\) on a linear space \(X\) is induced by a norm provided that \(\rho\) is translation invariant, real continuous (i.e., for any \(x\in X\), the map \([0,1]\to X:t\mapsto tx\) is continuous), every one-dimensional affine subspace of \(X\) is isometric to the field of scalars, and (in the complex case) \(\rho(x,y)=\rho(ix,iy)\). An example is presented showing that the assumptions of this statement cannot be significantly weakened. The exposition is fairly self-contained; only the Mazur--Ulam theorem about isometries in linear normed spaces is used.

In [\textit{T. Oikhberg} and \textit{H. P. Rosenthal}, Rocky Mt. J. Math. 37, No. 2, 597--608 (2007; Zbl 1138.46006)] it was proved that if \(X\) is a real linear space equipped with a metric \(d\) that satisfies the following three conditions: (i) the metric \(d\) is translation invariant, that is, \(d(x+z,y+z)=d(x,y)\) for any \(x\), \(y\), \(z\in X\), (ii) for every \(x\in X\) the map from \([0,1]\) to \(X\) defined by \(t\mapsto tx\) is continuous, (iii) each one-dimensional affine subspace of \(X\) is isometric to \(\mathbb{R}\), then there exists a norm \(\left\| \, \cdot \, \right\|\) on \(X\) such that \(d(x,y)=\left\| x-y\right\| \), where \(x,y\in X\). In the paper under review the following extension of the preceding result is stated (see Theorem 1.2): Let \(X\) be a real linear space equipped with a metric \(d\) satisfying: (i) the metric \(d\) is translation invariant, that is \(d(x+z,y+z)=d(x,y)\) for any \(x\), \(y\), \(z\in X\), (ii') for every \(x\in X\) the set \(\left\{ tx:t\in [ 0,1]\right\} \) is bounded in \((X,d)\), (iii') for all \(x\), \(y\in X\) their algebraic midpoint (i.e., \((x+y)/2\)) is a metric midpoint (i.e., \(d(x,y)=2d(x,(x+y)/2)=2d(y,(x+y)/2)\)), then \(\left\| x\right\| :=d(x,0)\), \(x\in X\), is a norm on \(X\) and \( d(x,y)=\left\| x-y\right\| \), \(x\), \(y\), \(z\in X\). The proof of the theorem of Oikhberg and Rosenthal is based on the use of the Mazur-Ulam theorem and the proof of the above Theorem 1.2 is shorter and self-contained. Furthermore, in Proposition 1.1, the authors show that (iii) implies (iii'), but that the converse is not true. In this paper, in order to predict a customer churn in the telecommunication sector, we have analysed several published articles that had used machine learning (ML) techniques. Significant predictive performance had been seen by utilising deep learning techniques. However, we have seen a tremendous lack of empirically derived heuristic information where we had to influence the hyperparameters consequently. Here, we had demonstrated three experimental findings, where a Relu activation function was embedded and utilised successfully in the hidden layers of the deep network. We can also see that the output layer had the service ability of a sigmoid function, in which we had seen a significant performance of the neural network model and obviously it was improved. Furthermore, we had also seen that the model's performance was noticed to be even better, but it was only considered better though when the batch size in the model was taken less than the test dataset's size, respectively. In terms of accuracy, the RemsProp optimizer beat out the other algorithms such as stochastic gradient descent (SGD). RemsProp was seen even better from the Adadelta algorithm, the Adam algorithm, the AdaGrad algorithm, and AdaMax algorithm as well.

This paper extends the results derived in the author's previous paper with \textit{J. E. Yukich} [Ann. Appl. Probab. 13, No. 1, 277--303 (2003; Zbl 1029.60008)]; the extension includes almost sure convergence, convergence of measures, non-translation invariance, and marked point process. ``Given \(n\) independent random marked \(d\)-vectors (points) \(X_i\) distributed with a common density, define the measure \(\nu_n = \sum_i \xi_i\) , where \(\xi_i\) is a measure (not necessarily a point measure) which stabilizes; this means that \(\xi_i\) is determined by the (suitably rescaled) set of points near \(X_i\). For bounded test functions \(f\) on \(R^d\), we give weak and strong laws of large numbers for \(\nu_n(f)\). The general results are applied to demonstrate that an unknown set \(A\) in \(d\)-space can be consistently estimated, given data on which of the points \(X_i\) lie in \(A\), by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.'' The authors use a coupling argument to establish a general weak law of large numbers for functionals of binomial point processes in \(d\)-dimensional Euclidean space. The limit depends explicitly on the density, possibly non-uniform, of the process. When the point set forms the vertex set of a graph, functionals of interest include total edge length with arbitrary weighting, number of edges and number of components. The general result is applied to various graphs. Weak laws of large numbers are also obtained for functionals of marked point processes, including statistics of Boolean models. The methods were used previously to obtain a strong law of large numbers for functionals on uniform point sets; see e.g. the authors [Ann. Appl. Probab. 12, 272-301 (2002; Zbl 1018.60023)].

This paper extends the results derived in the author's previous paper with \textit{J. E. Yukich} [Ann. Appl. Probab. 13, No. 1, 277--303 (2003; Zbl 1029.60008)]; the extension includes almost sure convergence, convergence of measures, non-translation invariance, and marked point process. ``Given \(n\) independent random marked \(d\)-vectors (points) \(X_i\) distributed with a common density, define the measure \(\nu_n = \sum_i \xi_i\) , where \(\xi_i\) is a measure (not necessarily a point measure) which stabilizes; this means that \(\xi_i\) is determined by the (suitably rescaled) set of points near \(X_i\). For bounded test functions \(f\) on \(R^d\), we give weak and strong laws of large numbers for \(\nu_n(f)\). The general results are applied to demonstrate that an unknown set \(A\) in \(d\)-space can be consistently estimated, given data on which of the points \(X_i\) lie in \(A\), by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.'' The author studies the manifold \(N^n_m\) of nondegenerate \(m\)-pairs of the real projective space \(\mathbb{R} P(n)\), i.e., the manifold whose points are pairs \((A,B)\) of an \(m\)-plane \(A\) and an \(n-m-1\) plane \(B\) not intersecting in \(\mathbb{R} P(n)\). In particular, the author constructs a hyperbolic Kählerian metric \(g\) on \(N^n_m\) which is semi-Riemannian of signature \(((m+ 1)(n-m),(m+ 1)(n-m))\) and a Kähler complex structure \(J\) on \(N^n_m\) such that \(g(JX, JY)= -g(X,Y)\) for all \(X,Y\in TM\) and the form \(\Omega(X, Y)= g(X, JY)\) is symplectic i.e., \(d\Omega= 0\). It is proved that \((N^n_m,g)\) is an Einstein manifold and that \(N^n_0\) has constant holomorphic sectional curvature. At the end, the author proves that \((N^n_m,\Omega)\) is symplectomorphic to the cotangent bundle of the Grassman manifold \(G_{m,n}\) of \(m\)-dimensional subspaces of \(\mathbb{R} P(n)\) with the standard symplectic structure of the cotangent bundle.

D.~Mumford, V.~Drinfeld, L.~Gerritzen et al. have developed \(p\)-adic uniformization of curves. This uniformization gives rise to \(p\)-adic analysis and a \(p\)-adic Teichmüller theory (authors: L.~Gerritzen, F.~Herrlich et al.). The present book has nothing in common with the above and is a sequel to an earlier paper [see \textit{S. Mochizuki}, Publ. Res. Inst. Math. Sci. 32, No. 6, 957-1157 (1996; Zbl 0879.14009)]. The long and very well written introduction of the book (with many references to the above paper) attempts to motivate and clarify its rather technical contents. Here we present a small part of that introduction. A curve \(X\) of type \((g,r)\) is some \(Y\setminus S\) with \(Y\) smooth, absolutely irreducible, projective curve of genus \(g\) and \(S\) a finite set of cardinality \(r\). \(X\) is hyperbolic if \(2g-2+r>0\). If \(\mathbb{C}\) is the base field, then one has the classical uniformization by the upper half plane \({\mathbb{H}}\rightarrow X\). One wants to capture this by a structure \((P\rightarrow X,\nabla _P,\sigma)\), called the canonical indigenous bundle on \(X\), which is algebraic in nature. An indigenous bundle means that \(P\rightarrow X\) is a \({\mathbb{P}}^1\)-bundle, \(\nabla _P\) is a connection and \(\sigma\) is a section of \(P\rightarrow X\) such that \(\nabla _P\sigma\) is an isomorphism between the tangent bundles \(\tau _{X/{\mathbb{C}}}\rightarrow \sigma ^*\tau _{P/X}\). The canonical indigenous bundle is obtained from the trivial bundle \({\mathbb{H}}\times {\mathbb{P}}^1\) on \(\mathbb{H}\), provided with the trivial connection \(\nabla\) and section \({\mathbb{H}}\rightarrow {\mathbb{H}}\times {\mathbb{P}}^1\), given by \(h\mapsto (h,h)\), by diving out the action of the topological fundamental group \(\pi _1\) of \(X\). This fundamental group is the automorphism group of \({\mathbb{H}}\rightarrow X\) and moreover a subgroup of \(\text{PSL}_2({\mathbb{R}})\subset \text{PGL}_2({\mathbb{C}})\). This action of \(\pi _1\) is given by \(\lambda (h,q)=(\lambda h,\lambda q)\). The canonical indigenous bundle is characterized among all indigenous bundles on \(X\) by: invariance under \(\text{Fr}_{\infty}\), i.e., complex conjugation, and by the condition that the homomorphism \(\pi _1\rightarrow \text{PGL}_2({\mathbb{C}})\) is injective and has as image a quasi-Fuchs group. One can summarize this by saying that the canonical indigenous bundle can be determined and moreover encodes the uniformization \({\mathbb{H}}\rightarrow X\). Let \(M_{g,r}\) denote the moduli space of the \(r\)-pointed curves of genus \(g\) and \(S_{g,r}\) the moduli space of these curves provided with an indigenous bundle. The natural morphism \(S_{g,r}\rightarrow M_{g,r}\) has a section \(s_H\) which provides each \(X\) with its canonical indigenous bundle. This section is known to be real analytic. Complex Teichmüller theory comes in the picture if one uses the universal covering of the space \(M_{g,r}\). In the \(p\)-adic theory, a hyperbolic curve \(X\) over some \(p\)-adic field is given. The goal is to find among the indigenous bundles \((P\rightarrow X,\nabla _P,\sigma)\) on \(X\) a ``canonical'' one. A first condition is that this bundle is invariant under the Frobenius \(\text{Fr}_p\). The second one is a certain \({\mathbb{Z}}_p\)-integral structure, already present in the paper cited above. In the book under review all \({\mathbb{Z}}_p\)-integral structures are studied as well as the nature of the ``canonical'' section of \(S_{g,r}\rightarrow M_{g,r}\). More terms and subjects appearing in the introduction are Schwarz torsor, canonical Frobenius lifting, canonical Galois representation, intrinsic Hodge theory, anabelian conjecture, nilcurves, deformation theory, crystals, Lubin-Tate, stable bundles, rigidity, Kodaira-Spencer theory. This reviewer understands only a small part of the book. Especially, possible applications of this \(p\)-adic Teichmüller theory, have probably escaped his attention. For elliptic curves in characteristic \(p\) there is a difference between ordinary curves (the generic case) and supersingular curves. The paper under review gives a generalization to genus \(g\geq 2\), different from the obvious look at the Jacobian. A curve is called ordinary if it admits a certain type of rank two bundle with filtration and Frobenius, or a certain family of \(p\)-divisible groups of dimension one and height two. The moduli space of ordinary curves is étale over the usual moduli space of stable curves, and covers the most degenerate curves (the Mumford curves). Its lift to a formal \(p\)-adic scheme (étale over the moduli space) admits a canonical Frobenius-lift which is also ``ordinary''. In turn this defines the notion of a canonical lift from characteristic \(p\) to characteristic zero, over any base-ring with Frobenius-lift. Finally an ordinary curve contains a dense open set of ordinary points, and on the canonical lift of this curve Frobenius also lifts on these ordinary points. There are also logarithmic variants, and results about local moduli problems. The proofs consist mostly of ingenious calculations of obstruction-classes.

D.~Mumford, V.~Drinfeld, L.~Gerritzen et al. have developed \(p\)-adic uniformization of curves. This uniformization gives rise to \(p\)-adic analysis and a \(p\)-adic Teichmüller theory (authors: L.~Gerritzen, F.~Herrlich et al.). The present book has nothing in common with the above and is a sequel to an earlier paper [see \textit{S. Mochizuki}, Publ. Res. Inst. Math. Sci. 32, No. 6, 957-1157 (1996; Zbl 0879.14009)]. The long and very well written introduction of the book (with many references to the above paper) attempts to motivate and clarify its rather technical contents. Here we present a small part of that introduction. A curve \(X\) of type \((g,r)\) is some \(Y\setminus S\) with \(Y\) smooth, absolutely irreducible, projective curve of genus \(g\) and \(S\) a finite set of cardinality \(r\). \(X\) is hyperbolic if \(2g-2+r>0\). If \(\mathbb{C}\) is the base field, then one has the classical uniformization by the upper half plane \({\mathbb{H}}\rightarrow X\). One wants to capture this by a structure \((P\rightarrow X,\nabla _P,\sigma)\), called the canonical indigenous bundle on \(X\), which is algebraic in nature. An indigenous bundle means that \(P\rightarrow X\) is a \({\mathbb{P}}^1\)-bundle, \(\nabla _P\) is a connection and \(\sigma\) is a section of \(P\rightarrow X\) such that \(\nabla _P\sigma\) is an isomorphism between the tangent bundles \(\tau _{X/{\mathbb{C}}}\rightarrow \sigma ^*\tau _{P/X}\). The canonical indigenous bundle is obtained from the trivial bundle \({\mathbb{H}}\times {\mathbb{P}}^1\) on \(\mathbb{H}\), provided with the trivial connection \(\nabla\) and section \({\mathbb{H}}\rightarrow {\mathbb{H}}\times {\mathbb{P}}^1\), given by \(h\mapsto (h,h)\), by diving out the action of the topological fundamental group \(\pi _1\) of \(X\). This fundamental group is the automorphism group of \({\mathbb{H}}\rightarrow X\) and moreover a subgroup of \(\text{PSL}_2({\mathbb{R}})\subset \text{PGL}_2({\mathbb{C}})\). This action of \(\pi _1\) is given by \(\lambda (h,q)=(\lambda h,\lambda q)\). The canonical indigenous bundle is characterized among all indigenous bundles on \(X\) by: invariance under \(\text{Fr}_{\infty}\), i.e., complex conjugation, and by the condition that the homomorphism \(\pi _1\rightarrow \text{PGL}_2({\mathbb{C}})\) is injective and has as image a quasi-Fuchs group. One can summarize this by saying that the canonical indigenous bundle can be determined and moreover encodes the uniformization \({\mathbb{H}}\rightarrow X\). Let \(M_{g,r}\) denote the moduli space of the \(r\)-pointed curves of genus \(g\) and \(S_{g,r}\) the moduli space of these curves provided with an indigenous bundle. The natural morphism \(S_{g,r}\rightarrow M_{g,r}\) has a section \(s_H\) which provides each \(X\) with its canonical indigenous bundle. This section is known to be real analytic. Complex Teichmüller theory comes in the picture if one uses the universal covering of the space \(M_{g,r}\). In the \(p\)-adic theory, a hyperbolic curve \(X\) over some \(p\)-adic field is given. The goal is to find among the indigenous bundles \((P\rightarrow X,\nabla _P,\sigma)\) on \(X\) a ``canonical'' one. A first condition is that this bundle is invariant under the Frobenius \(\text{Fr}_p\). The second one is a certain \({\mathbb{Z}}_p\)-integral structure, already present in the paper cited above. In the book under review all \({\mathbb{Z}}_p\)-integral structures are studied as well as the nature of the ``canonical'' section of \(S_{g,r}\rightarrow M_{g,r}\). More terms and subjects appearing in the introduction are Schwarz torsor, canonical Frobenius lifting, canonical Galois representation, intrinsic Hodge theory, anabelian conjecture, nilcurves, deformation theory, crystals, Lubin-Tate, stable bundles, rigidity, Kodaira-Spencer theory. This reviewer understands only a small part of the book. Especially, possible applications of this \(p\)-adic Teichmüller theory, have probably escaped his attention. Learning to make good choices is a key skill for adults. This chapter discusses how curricula for teaching numeracy skills can be expanded into an important domain that often is ignored by adult (and K-12) educators: the practice of decision making. The chapter suggests ways in which often-neglected concepts and skills related to probability and statistics can be developed in the context of curricula and case studies in decision making.

The aim of this paper is an improvement of the research work of \textit{M. T. Darvishi} and \textit{P. Hessari} [Appl. Math. Comput. 176, No.~1, 128--133 (2006; Zbl 1101.65033)]. Improvements are presented for bounds of the spectral radius of \(l_{\omega ,r}\), which is the iterative matrix of the generalized accelerated overrelaxation (GAOR) method. The convergence of the GAOR method for diagonally dominant coefficient matrices is investigated and new sufficient conditions are obtained. Two numerical examples are used for the explanation and the establishment of the proposed improvements. The convergence of a generalized accelerated overrelaxation (AOR) method to solve the linear system \(Hy= f\), namely: \(y^{k+1}= \ell_{\omega,r}y^{(k)}+\omega k\), is investigated. Sufficient convergence conditions for \(\omega\) and \(r\) are obtained, for \(H\) being a weak diagonally dominant matrix.

The aim of this paper is an improvement of the research work of \textit{M. T. Darvishi} and \textit{P. Hessari} [Appl. Math. Comput. 176, No.~1, 128--133 (2006; Zbl 1101.65033)]. Improvements are presented for bounds of the spectral radius of \(l_{\omega ,r}\), which is the iterative matrix of the generalized accelerated overrelaxation (GAOR) method. The convergence of the GAOR method for diagonally dominant coefficient matrices is investigated and new sufficient conditions are obtained. Two numerical examples are used for the explanation and the establishment of the proposed improvements. A probabilistic framework for cleaning the data collected by Radio-Frequency IDentification (RFID) tracking systems is introduced. What has to be cleaned is the set of trajectories that are the possible interpretations of the readings: a trajectory in this set is a sequence whose generic element is a location covered by the reader(s) that made the detection at the corresponding time point. The cleaning is guided by integrity constraints and consists of discarding the inconsistent trajectories and assigning to the others a suitable probability of being the actual one. The probabilities are evaluated by adopting \textit{probabilistic conditioning} that logically consists of the following steps. First, the trajectories are assigned \textit{a priori} probabilities that rely on the independence assumption between the time points. Then, these probabilities are revised according to the spatio-temporal correlations encoded by the constraints. This is done by conditioning the \textit{a priori} probability of each trajectory to the event that the constraints are satisfied: this means taking the ratio of this \textit{a priori} probability to the sum of the \textit{a priori} probabilities of all the consistent trajectories. Instead of performing these steps by materializing all the trajectories and their \textit{a priori} probabilities (which is infeasible, owing to the typically huge number of trajectories), our approach exploits a data structure \textit{called conditioned trajectory graph (ct-graph)} that compactly represents the trajectories and their conditioned probabilities, and an algorithm for efficiently constructing the ct-graph, which progressively builds it while avoiding the construction of components encoding inconsistent trajectories.

Let f be a transcendental entire function of finite order which has the power series representation \(f(z)=\sum^{\infty}_{n=0}a_ nz^{\lambda_ n}\) where \(a_ n\neq 0\), \(n=0,1,2,... \). Let g be an arbitrary meromorphic function for which \(T(r,g)=o(T(r,f))\), (r\(\to \infty)\), where T is the Nevanlinna characteristic function. If \(d_ n\) is the highest common factor of all the \(\lambda_{m+1}-\lambda_ m\) for \(m\geq n\) and \(d_ n\) approaches infinity with increasing n, then for every g we have \[ \delta_ s(g(z),f)=1-\liminf_{r\to \infty}(N(r,1/(f-g(z))/(T(r,f)))=0. \] The theorem extends the author's earlier result [Kodai Math. J. 11, 32-37 (1988; reviewed above)]. A similar theorem is given for f of infinite order. Estimates for \(\delta_ s(g(z),f)\) are also given for f whose representation is such that the complement of \(\{\lambda_ n\}\) with respect to the positive integers contains an arithmetic progression. Let F be a transcendental entire function of finite order which has the power series representation \(f(z)=\sum^{\infty}_{n=0}a_ nz^{\lambda_ n}\) where \(a_ n\neq 0\), \(n=0,1,2,... \). Let g be an arbitrary entire function for which \(T(r,g)=o(T(r,g))\), (r\(\to \infty)\), where T is the Nevanlinna characteristic function. If \(d_ n\) is the highest common factor of all the \(\lambda_{m+1}-\lambda_ m\) for \(m\geq n\) and \(d_ n\) approaches infinity with increasing n, then for every g \[ \delta_ s(g(z),f)=1-\liminf_{r\to \infty}(N(r,1/(f- g(z)))/T(r,f))=0. \] The theorem extends an earlier result of \textit{W. K. Haymann} [J. Lond. Math. Soc., II. Ser. 28, 327-338 (1983; Zbl 0488.30023)] where g(z)\(\equiv c\) with c a complex number. Results also exist for f of infinite order.

Let f be a transcendental entire function of finite order which has the power series representation \(f(z)=\sum^{\infty}_{n=0}a_ nz^{\lambda_ n}\) where \(a_ n\neq 0\), \(n=0,1,2,... \). Let g be an arbitrary meromorphic function for which \(T(r,g)=o(T(r,f))\), (r\(\to \infty)\), where T is the Nevanlinna characteristic function. If \(d_ n\) is the highest common factor of all the \(\lambda_{m+1}-\lambda_ m\) for \(m\geq n\) and \(d_ n\) approaches infinity with increasing n, then for every g we have \[ \delta_ s(g(z),f)=1-\liminf_{r\to \infty}(N(r,1/(f-g(z))/(T(r,f)))=0. \] The theorem extends the author's earlier result [Kodai Math. J. 11, 32-37 (1988; reviewed above)]. A similar theorem is given for f of infinite order. Estimates for \(\delta_ s(g(z),f)\) are also given for f whose representation is such that the complement of \(\{\lambda_ n\}\) with respect to the positive integers contains an arithmetic progression. In this expository paper, the basic mathematical concepts needed for the formalization of the notion of `cellular automaton' are presented along with a generalization which uses probabilistic aspects in the local interactions. This generalization (stochastic cellular automata) provides a more realistic modeling of natural phenomena. The generation of complex processes from simple rules leads to the consideration of statistical peculiarities in the evolution of cellular automata. Two procedures are briefly described here. A simple illustration of the deterministic model generating divisibility patterns of certain polynomials and a simple illustration of the stochastic model simulating random walks are exhibited.

This is the author's lecture on the occasion of being awarded the International Dobrushin Prize for 2011. The prize was presented on July 25, 2011, at the International Mathematical Conference in honor of the 50th anniversary of the Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences. The author starts with a short memory of his teacher R. L. Dobrushin [1929 -- 1995]. Then he reviews joint work with \textit{J. S. Athreya} [J. Mod. Dyn. 3, No. 3, 359--378 (2009; Zbl 1184.37007)]. Here he gives a. o. the proof of Theorem 2.2 (loc. cit.) in the case \(n\geq 3\). We prove analogs of the logarithm laws of Sullivan and Kleinbock--Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(\(n, \mathbb R\))/SL\((n, \mathbb Z\)). The key lemma for our results says the measure of the set of unimodular lattices in \(\mathbb R^n\) that does not intersect a `large' volume subset of \(\mathbb R^n\) is `small'. This can be considered as a `random' analog of the classical Minkowski Theorem in the geometry of numbers.

This is the author's lecture on the occasion of being awarded the International Dobrushin Prize for 2011. The prize was presented on July 25, 2011, at the International Mathematical Conference in honor of the 50th anniversary of the Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences. The author starts with a short memory of his teacher R. L. Dobrushin [1929 -- 1995]. Then he reviews joint work with \textit{J. S. Athreya} [J. Mod. Dyn. 3, No. 3, 359--378 (2009; Zbl 1184.37007)]. Here he gives a. o. the proof of Theorem 2.2 (loc. cit.) in the case \(n\geq 3\). Properties of the phase change of the two-level pairing model are investigated in the semi-classical treatment by using the variational approach with a mixed-mode coherent state. In the classical limit, \(\hbar\to 0\), a sharp phase transition appears, and the two phases exist in the region where the force strength is larger than a certain critical value. However, it is shown that, in the semi-classical treatment, the above-mentioned behavior of the phase change disappears in both analytical and numerical treatments. This leads to a new understanding of the properties of the phase change in this model.

For a random subset \(\Pi\) of the real line, Rényi showed that \(\Pi\) is a homogeneous Poisson process with rate \(\lambda>0\) provided \(N(A)= \text{card} [\Pi\cap A]\) is Poisson with mean \(\lambda\ell (A)\) for every finite union \(A\) of intervals where \(\ell(A)\) denotes the Lebesgue measure of \(A\). In his recent monograph on ``Poisson processes'' (1993; Zbl 0771.60001), \textit{J. F. C. Kingman} suggested an interesting characteristic functional approach to Rényi's result. Note that a random subset \(\Pi\) of \(\mathbb{R}^d\) is a Poisson process with (non-atomic) mean measure \(\mu\) if and only if the characteristic functional of the process \(\Pi\) is of the form \[ E(e^{-\Sigma_f})= \exp \left(-\int (f-e^{-f(x)}) \mu(dx) \right) \tag{1} \] for every measurable step function \(f:\mathbb{R}^d\to [0,\infty)\) where \(\Sigma_f= \sum_{x\in\Pi} f(x)\). Then Rényi's result may be stated via the characteristic functional as follows: Let \(\mu\) be a non-atomic measure in \(\mathbb{R}^d\), finite on bounded sets. For a random subset \(\Pi\) of \(\mathbb{R}^d\), if (1) holds for all non-negative two-valued step functions (with one of the two values being 0), then (1) holds for all non-negative step functions. Kingman provided a heuristic justification for the theorem by arguing that two-valued step functions can approximate general step functions from the point of view of integration theory. A direct proof is given in this paper, where it is shown that for a general step function \(f\), \(\exp (-\Sigma_f)\) can be approximated in a probabilistic sense by a linear combination of \(\exp (-\Sigma_{f_i})\) where each \(f_i\) is a two-valued step function. For ease of exposition, the author considers only three-valued step functions \(f(x)= aI_A(x)+ bI_B(x)\) where \(a>0\), \(b>0\), and \(A\) and \(B\) are disjoint, each being a finite union of rectangles. The general case can be treated similarly. This monograph is an introductory account of the modern theory of Poisson processes and its many applications. The titles of the different chapters give a rough description of its contents: 1. Stochastic models for random sets of points. 2. Poisson processes in general spaces. 3. Sums over Poisson processes. 4. Poisson processes on the line. 5. Marked Poisson processes. 6. Cox processes. 7. Stochastic geometry. 8. Completely random measures. 9. The Poisson-Dirichlet distribution. After an introductory chapter containing, amongst other things, various fundamental properties of the usual (\(\mathbb{N}\)-valued) Poisson random variables the author introduces in the second chapter the general definition of a Poisson process on a measurable state space \(S\) (supposed to satisfy a fairly weak hypothesis, verified if \(S=\mathbb{R}^ d\) or any other separable metric space): the process is defined by one ``random variable'' \(\Pi\), defined on some probability space \(\Omega\), whose values \(\Pi(\omega)\), \(\omega\in\Omega\), are denumerable subsets of \(S\) such that for each measurable subset \(A\subset S\), the quantity \(\#\{\Pi(\omega)\cap A\}=N(A)(\omega)\) \((\#\) representing ``cardinality'') is a usual \(\overline\mathbb{N}\)-valued random variable; two hypotheses are then imposed on the stochastic process \(\{N(A)\}_{A\subset S}\) (\(A\) measurable) obtained from the ``count functions'' \(N(A)\): (i) \(N(A_ 1),\dots,N(A_ n)\) are independent if \(A_ 1,\dots,A_ n\) are disjoint; (ii) \(N(A)\) has the Poisson distribution with parameter \(\mu(A)\in[0,\infty]\). The basic properties of such Poisson processes are then given in Chapters 2, 3, 4, notably, a clear discussion of Campbell's theorem (concerning the random variable \(\sum_{X\in\Pi}f(X)\), \(f\) being a real-valued function on the state space \(S)\) and a complete proof of a version of an elegant theorem of Rényi which asserts that if \(\mathbb{P}\text{rob}(N(A)=0)\) is of the form \(\exp(-\mu(A))\) (for a large class of sets \(A\) and \(\mu\) nonatomic), then the \(N(A)\)'s are automatically independent. The first four chapters form about half the book; in the last four chapters processes are treated which are either more general than Poisson processes or else specialisations obtained for various applications as for instance in stochastic geometry, ecology, traffic problems, or queueing theory (already introduced in Chapter 4). The presentation everywhere is rigorous without being fuzzy about measure theoretical details; this would make the monograph suitable for many readers, who are either not interested or not trained in measure theoretical subtelities. In the last four chapters, the author introduces a variety of topics without going into all the proofs or details; however, he does provide references which would permit readers to pursue matters further. All in all, the monograph is a useful addition to the literature both for various beginners as well as for lecturers in the theory of stochastic processes who would find in it a rich array of topics presented clearly.

For a random subset \(\Pi\) of the real line, Rényi showed that \(\Pi\) is a homogeneous Poisson process with rate \(\lambda>0\) provided \(N(A)= \text{card} [\Pi\cap A]\) is Poisson with mean \(\lambda\ell (A)\) for every finite union \(A\) of intervals where \(\ell(A)\) denotes the Lebesgue measure of \(A\). In his recent monograph on ``Poisson processes'' (1993; Zbl 0771.60001), \textit{J. F. C. Kingman} suggested an interesting characteristic functional approach to Rényi's result. Note that a random subset \(\Pi\) of \(\mathbb{R}^d\) is a Poisson process with (non-atomic) mean measure \(\mu\) if and only if the characteristic functional of the process \(\Pi\) is of the form \[ E(e^{-\Sigma_f})= \exp \left(-\int (f-e^{-f(x)}) \mu(dx) \right) \tag{1} \] for every measurable step function \(f:\mathbb{R}^d\to [0,\infty)\) where \(\Sigma_f= \sum_{x\in\Pi} f(x)\). Then Rényi's result may be stated via the characteristic functional as follows: Let \(\mu\) be a non-atomic measure in \(\mathbb{R}^d\), finite on bounded sets. For a random subset \(\Pi\) of \(\mathbb{R}^d\), if (1) holds for all non-negative two-valued step functions (with one of the two values being 0), then (1) holds for all non-negative step functions. Kingman provided a heuristic justification for the theorem by arguing that two-valued step functions can approximate general step functions from the point of view of integration theory. A direct proof is given in this paper, where it is shown that for a general step function \(f\), \(\exp (-\Sigma_f)\) can be approximated in a probabilistic sense by a linear combination of \(\exp (-\Sigma_{f_i})\) where each \(f_i\) is a two-valued step function. For ease of exposition, the author considers only three-valued step functions \(f(x)= aI_A(x)+ bI_B(x)\) where \(a>0\), \(b>0\), and \(A\) and \(B\) are disjoint, each being a finite union of rectangles. The general case can be treated similarly. [For the entire collection see Zbl 0741.68016.] The authors propose a geometric method of reasoning about phase portraits, collections of all solution curves of ordinary differential equations in the phase space. The method is designed: (1) to collect geometric features of solution curves using varieties of qualitative techniques, (2) to infer the topology of the phase portrait from geometric clues, and (3) to reason about the global behaviour by analysing the topology of the phase portrait. These problems are solved as follows: (a) the problem of ambiguity is reduced because the geometry and topology of phase portraits are determined on the basis of qualitative information, while (b) the global analysis based on geometric and topological analysis of phase portraits is theoretically supported by dynamical systems theory. The method is incorporated in the program PSX2NL which works on ordinary differential equations defined on a two-dimensional phase space, but produces symbolic descriptions of total behaviour. The authors define a flow grammar which specifies all possible flow patterns of solution curves in the phase space. The role of the flow grammar in PSX2NL is to reason about complex patterns in a uniform and systematic manner, and to switch to an approximate top-down algorithm when complete geometric clues are not available (due to the difficulty of mathematical problems encountered). Comparisons to other similar works and further research directions are discussed.

The author presents improved constructions for generalized van der Corput sequences by means of linear digit scramblings. This also yields a new lower bound for the extreme discrepancy of such sequences. Furthermore, the corresponding properties for generalized Hammersley point sets are established, see also \textit{P. Kritzer, G. Larcher} and \textit{F. Pillichshammer} [Ann. Mat. Pura Appl. (4) 186, No. 2, 229--250 (2007; Zbl 1150.11026)]. If \(\omega=\{{\mathbf x}_n=(x_n,y_n)\), \(0\leq n\leq 2^m-1\}\) is a Hammersley net (\(x_n=n/2^m\), \(y_n=\sum_{i=0}^{m-1} a_i/2^{i+1}\), \(n=\sum_{i=0}^{m-1}a_i2^i\)), then a \(\sigma\)-shifted Hammersley net with \(\sigma=(\sigma_1,\dots,\sigma_m)^{m-1}\in{\mathbb Z}_2^m\) is the net \(\omega_\sigma=\{{\mathbf x}_n=(x_n,y_n)\), \(0\leq n\leq 2^m-1\}\) with \(y_n=\sum_{i=1}^m (a_i\oplus \sigma_{i+1})/2^i\), where \(\oplus\) is addition modulo 2. The shifted Hammersley nets are introduced because the digital shifts can improve the distribution properties of the classical Hammersley nets. The authors focus their investigation on the leading term in the estimations of the star discrepancy \(D^*\). For instance, while the leading term of the upper estimate of \(2^mD^*_{2^m}(\omega)\) is \(m/3\), there is a shifted Hammersley net \(\omega_\sigma\) with \(2^mD^*_{2^m}(\omega_\sigma)\leq m/6+O(1)\). A detailed investigation of the star discrepancy of shifted Hammersley nets contained in the paper shows that for the star discrepancy of any \(\sigma\)-shifted Hammersley net \(\omega_\sigma\) we have \(2^mD^*_{2^m}(\omega)\geq m/6-4/9\) and that the value of \(2^mD^*_{2^m}(\omega)\) is influenced by the number of ones in \(\sigma\) and by the number of changes from ones to zeros in \(\sigma\). The estimates of the star discrepancy of shifted Hammerley nets depend on certain sums of distances to the nearest integer, which are also studied in the paper. Finally, some interesting results concerning the star discrepancy of shifted van der Corput sequences are proved in the last section of the paper.

The author presents improved constructions for generalized van der Corput sequences by means of linear digit scramblings. This also yields a new lower bound for the extreme discrepancy of such sequences. Furthermore, the corresponding properties for generalized Hammersley point sets are established, see also \textit{P. Kritzer, G. Larcher} and \textit{F. Pillichshammer} [Ann. Mat. Pura Appl. (4) 186, No. 2, 229--250 (2007; Zbl 1150.11026)]. We address the non-Gaussianity (nG) of states obtained by weakly perturbing a Gaussian state and investigate the relationships with quantum estimation. For classical perturbations, i.e. perturbations to eigenvalues, we found that the nG of the perturbed state may be written as the quantum Fisher information (QFI) distance minus a term depending on the infinitesimal energy change, i.e. it provides a lower bound to statistical distinguishability. Upon moving on isoenergetic surfaces in a neighbourhood of a Gaussian state, nG thus coincides with a proper distance in the Hilbert space and exactly quantifies the statistical distinguishability of the perturbations. On the other hand, for perturbations leaving the covariance matrix unperturbed, we show that nG provides an upper bound to the QFI. Our results show that the geometry of non-Gaussian states in the neighbourhood of a Gaussian state is definitely not trivial and cannot be subsumed by a differential structure. Nevertheless, the analysis of perturbations to a Gaussian state reveals that nG may be a resource for quantum estimation. The nG of specific families of perturbed Gaussian states is analysed in some detail with the aim of finding the maximally non-Gaussian state obtainable from a given Gaussian one.

Let \(A_n=(a_{ij})\) be an \(n\times n\) \((0,1)\)-matrix, whose entries are independent random variables, \(P(a_{ij}=1)=p\), \(P(a_{ij}=0)=q=1-p\). Denote by \(E_n\) the event that \(A_n\) contains a 0-set of \(n\) elements (i.e. \(A_n\) contains \(n\) zeros in such a way that none of a pair of them lies in the same row and column). The author proves that \(\lim_{n\to \infty}P(E_n)=1\) if \(p=q=\tfrac{1}{2}\). The result (for arbitrary \(p\) and \(q=1-p)\) follows from a theorem of \textit{P. Erdős} and \textit{A. Rényi} [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 8, 455--461 (1964; Zbl 0133.26003)] on permanents. Let \(P(n,N(n))\) denote the probability that a random \(n\) by \(n\) matrix with \(N(n)\) 1's and \(n^2-N(n)\) 0's has a positive permanent. The authors show that if \(N(n)=n\log n+cn+o(n)\), where \(c\) is an arbitrary constant, then \(\lim_{n \to \infty} P(n,N(n)) = \exp(-2e^{-c})\).

Let \(A_n=(a_{ij})\) be an \(n\times n\) \((0,1)\)-matrix, whose entries are independent random variables, \(P(a_{ij}=1)=p\), \(P(a_{ij}=0)=q=1-p\). Denote by \(E_n\) the event that \(A_n\) contains a 0-set of \(n\) elements (i.e. \(A_n\) contains \(n\) zeros in such a way that none of a pair of them lies in the same row and column). The author proves that \(\lim_{n\to \infty}P(E_n)=1\) if \(p=q=\tfrac{1}{2}\). The result (for arbitrary \(p\) and \(q=1-p)\) follows from a theorem of \textit{P. Erdős} and \textit{A. Rényi} [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 8, 455--461 (1964; Zbl 0133.26003)] on permanents. Definiert man die Elastizitätsgrenze als diejenige spezifische Höchstbelastung, bei der die bleibenden Formänderungen den Wert Null soeben überschreiten, so ergibt sich der Übelstand, daß der kritische Punkt sich mit der Verfeinerung des jeweiligen Meßverfahrens verschiebt. Die Bedürfnisse der technischen Praxis verlangen aber eine Begriffsbestimmung der kritischen Grenzspannung, die ein einfaches und der willkürlichen Deutung entzogenes Meßverfahren zuläßt. Hierzu eignet sich nach dem Verf. der Punkt, bei dem die mit wachsender Formänderung auftretende Abkühlung des Materials in eine Erwärmung übergeht. Dieser Punkt ist, wenn man sich mit dem Verf. das Material aus einer rein elastischen und einer zähflüssigen Phase zusammengesetzt denkt, gemäß der \textit{Clausius-Clapeyron}schen Gleichung durch den Vorzeichenwechsel der Größe \(\beta'-\varepsilon'\) charakterisiert, wo \(\beta', \varepsilon'\) die spezifischen Volumina der beiden Phasen sind; er stimmt, wie die Versuche ergeben haben, mit der Streckgrenze überein. Die Temperatur wurde mit Hülfe von Thermoelementen gemessen. Bei weicherem Flußstahl setzte der Streckvorgang ziemlich rasch ein; ein wiederholt belasteter Messingrundstab hatte seine Streckgrenze stets bei derjenigen Höchstbelastung, der er beim vorhergehenden Versuch ausgesetzt war.

Let \(E\) be a Banach space, \(U\subset E\) an open set and \(S:U\rightarrow E\) a \(C^1\)-map. The authors consider the discrete dynamical system (DS) \(\{S^n\}_{n=1}^{\infty}\) generated by \(S\), extending the theory of exponential attractors from such DS in Hilbert space [\textit{A. Eden, C. Foias, B. Nicolaenko} and \textit{R. Temam}, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics 37, Chichester: Wiley, Paris: Masson (1994; Zbl 0842.58056)] on Banach spaces. The following requirements are postulated: 1. the semiflow is \(C^1\) in some absorbing ball, and 2. the linearized semiflow at every point inside the absorbing ball is splitting into the sum of a compact operator plus a contraction. The book under review is devoted to a new direction of the theory of attractors for dissipative evolution equations. Exponential attractors (EA) are the objects intermediate between global attractors and inertial manifolds. The introduction contains the history of this subject. The following four chapters give the notion of EAs and their construction for dissipative evolution equations of the first order, approximations of EAs, in particular Galerkin approximations. The chapter 5 contains the applications to Kuramoto-Sivashinsky, Kuramoto-Sivashinsky-Spiegel equations, 2D and 3D Navier-Stokes equations, Burgers equations and Chaffee-Infante reaction-diffusion equations. In the chapter 6 this theory is developed for second order evolution equations with damping. The applications to sine-Gordon, Klein-Gordon type equations, to systems of sine-Gordon equations are given. The chapter 7 is devoted to EAs of optimal Hausdorff dimension and optimal outer Lyapunov dimension. The concluding chapters 9-10 contain a review about inertial manifolds and their comparison with EAs, Mañé's projections theory and inertially equivalent dynamical systems. Necessary additional material about Mañé's theorem for Hilbert spaces, estimate of the topological entropy and mathematical background of fractal sets are carried out into appendices A, B and C. Thus the purpose of this work is to develop and present the theory of exponential attractors for dissipative evolution equations of infinite dimension. This book can be considered for a graduate course. It is very interesting because of pointing out various connections between fluid mechanics, partial differential equations and dynamical systems.

Let \(E\) be a Banach space, \(U\subset E\) an open set and \(S:U\rightarrow E\) a \(C^1\)-map. The authors consider the discrete dynamical system (DS) \(\{S^n\}_{n=1}^{\infty}\) generated by \(S\), extending the theory of exponential attractors from such DS in Hilbert space [\textit{A. Eden, C. Foias, B. Nicolaenko} and \textit{R. Temam}, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics 37, Chichester: Wiley, Paris: Masson (1994; Zbl 0842.58056)] on Banach spaces. The following requirements are postulated: 1. the semiflow is \(C^1\) in some absorbing ball, and 2. the linearized semiflow at every point inside the absorbing ball is splitting into the sum of a compact operator plus a contraction. The article presents how children aged 4 to 6 assimilate geometrical figures in the preschool period. They become acquainted with geometrical forms through observing nature and their surroundings. They gain mathematical knowledge through various activities.

It is proved that the function \(\omega(\sigma(p+1))\), where \(\sigma\) is the sum of divisors and \(\omega\) is the number of prime divisors, has a normal limiting distribution with centering \({1\over 2}(\log\log x)^ 2\) and norming \(\sqrt{1/3} (\log\log x)^{3/2}\). A key tool is an approximation of this function by an additive one in the following form: for almost all primes \(P\leq x\) we have \[ \omega(\sigma(P+1))=\sum_{p\mid P+1}\omega(p+1)+O(\log\log x(\log\log\log\log x)^ 2). \] The author obtains a Gaussian limit law for the function \(n\mapsto\omega(\varphi(\varphi(n)))\), more exactly he proves \[ \lim{1\over x}\cdot\#\left\{n\leq x,{\omega(\varphi(\varphi(n)))-{1\over 6}(\log\log n)^ 3\over{1\over\sqrt{10}}\cdot(\log\log n)^{5/2}}\right\}=\Phi(y). \] The author states that by his method he is able to give corresponding results for functions \(f(g(n))\) instead of \(\omega(\varphi(\varphi(n)))\), where \(f=\omega\) or \(f=\Omega\), and \(g=\sigma\circ\varphi\) or \(g=\varphi\circ\sigma\) or \(g=\sigma\circ\sigma\) or \(g=\varphi\circ\varphi\). These results extend former investigations of \textit{P. Erdős} and \textit{C. Pomerance} [Rocky Mt. J. Math. 15, 343-352 (1985; Zbl 0617.10037)], \textit{M. Ram Murty} and \textit{V. Kumar Murty} [Duke Math. J. 51, 57-76 (1984; Zbl 0537.10026); Indian J. Pure Appl. Math. 15, 1090-1101 (1984; Zbl 0557.10033)], who treated the function \(\omega\circ\varphi\), and of the author himself [Ann. Univ. Sci. Budap. Rolando Eötvös Sect. Math. 34, 217-225 (1991); see the preceding review], who showed that \[ {\omega(\sigma(p+1))-{1\over 2}(\log\log p)^ 2\over (1/\sqrt 3)\cdot(\log\log p)^{3/2}} \] is distributed according to the Gaussian law. The proof uses the idea of approximating \(\omega(\varphi(\varphi(n)))\) by a sum of two strongly additive functions, and relies heavily on (standard) tools like the Bombieri-Vinogradov theorem, sieve estimates, the Turán-Kubilius inequality, and the Erdős-Kac theorem.

It is proved that the function \(\omega(\sigma(p+1))\), where \(\sigma\) is the sum of divisors and \(\omega\) is the number of prime divisors, has a normal limiting distribution with centering \({1\over 2}(\log\log x)^ 2\) and norming \(\sqrt{1/3} (\log\log x)^{3/2}\). A key tool is an approximation of this function by an additive one in the following form: for almost all primes \(P\leq x\) we have \[ \omega(\sigma(P+1))=\sum_{p\mid P+1}\omega(p+1)+O(\log\log x(\log\log\log\log x)^ 2). \] Averaging the local characteristics with respect to time transforms a semimartingale with independent increments into a Lévy process. It is shown that such a Lévy process is ``closer'' to any Lévy process than the original process, where the ``greater closeness'' of processes is understood as the greater closeness of all \(f\)-divergences of their distributions, which also admits an equivalent formulation in terms of the comparison of the corresponding binary statistical experiments. Moreover, we prove a criterion for the equivalence of binary experiments that consists of the distribution of a semimartingale to independent increments and the distribution of a Lévy process.

This is the second edition of the book. When compared to the first edition [Zbl 0903.94001], the authors have introduced some minor changes. They have reorganized Chapter 2 -- ``Information and entropy'' -- putting emphasis on Shannon's works, e.g. Shannon's quantification of information. Chapter 3 is also slightly modified. The authors have added a new part on the capacity of \(n\)-any symmetric channels. In turn, in Chapter 4, a new Section on ``The information rate of a code'' has been appended. The book covers most of the elementary problems related to both information theory and data compression. The authors give the basic definitions and theorems from both areas -- explaining their meaning through practical exercises. The reader can find fundamentals of the theory of probability, information theory -- including definitions of information, entropy, channel models and channel capacity bounds, coding theory -- with Shannon's theorem and the Kraft-McMillan inequality, lossless data compression schemes -- consisting of Huffman's, Fano's and Shannon's encoding schemes, as well as adaptive (e.g. Knuth's and Gallager's methods) and dictionary methods. The book also gives certain, however very limited, information on transformation methods and image compression algorithms (JPEG). More advanced pieces of information are attached in the form of appendices. The appendices also include notes on and solutions to some exercises. All of this makes the book a useful compendium of fundamental knowledge on information theory and data compression principles. The book deals with foundations of information theory and the most important compression techniques. In the introduction to further discussion the authors formulate and prove the basic theorems on information, entropy and the capacity of binary channels. They also give a brief overview of prefix-condition codes and describe the Kraft-McMillan inequality as well as optimal and quasi-optimal encoding schemes, like Huffman's and Fano's algorithms. Furtheron they analyse lossless compression methods obtained by using probability models to drive a coder. The authors discuss ordinary and higher-order arithmetic coding schemes, giving both encoding and decoding procedures, as well as calculating compression ratios. Next, the authors proceed to an analysis of adaptive -- Huffman and arithmetic methods describing the Knuth and Gallager methods, and interval and recency rank encoding. In the next part the authors describe dictionary methods, that utilise lists (dictionaries) of phrases and pointers used to replace source fragments. Special interest is put on adaptive dictionary methods, described by Ziv and Lempel. The last part of the book concentrates on different transform methods used for image compression. In all of these methods, contrary to the earlier lossless methods, the partial loss of information is allowed. The authors explain the principles of JPEG and modern wavelet-based compression methods.

This is the second edition of the book. When compared to the first edition [Zbl 0903.94001], the authors have introduced some minor changes. They have reorganized Chapter 2 -- ``Information and entropy'' -- putting emphasis on Shannon's works, e.g. Shannon's quantification of information. Chapter 3 is also slightly modified. The authors have added a new part on the capacity of \(n\)-any symmetric channels. In turn, in Chapter 4, a new Section on ``The information rate of a code'' has been appended. The book covers most of the elementary problems related to both information theory and data compression. The authors give the basic definitions and theorems from both areas -- explaining their meaning through practical exercises. The reader can find fundamentals of the theory of probability, information theory -- including definitions of information, entropy, channel models and channel capacity bounds, coding theory -- with Shannon's theorem and the Kraft-McMillan inequality, lossless data compression schemes -- consisting of Huffman's, Fano's and Shannon's encoding schemes, as well as adaptive (e.g. Knuth's and Gallager's methods) and dictionary methods. The book also gives certain, however very limited, information on transformation methods and image compression algorithms (JPEG). More advanced pieces of information are attached in the form of appendices. The appendices also include notes on and solutions to some exercises. All of this makes the book a useful compendium of fundamental knowledge on information theory and data compression principles. We obtain sharp estimates for the module of functions in the classes of normalized locally quasiconformal authomorphisms of the unit disk with given majorants of M. A. Lavrent'ev's characteristic. The estimates are analogs of Schwarz's lemma and A. Mori's theorem and they imply the classical growth theorems for quasiconformal authomorphisms of the disk. In the classes we also prove sharp estimates of the conformal radius and the radius of covering disk. The main results are obtained by methods of extremal lengths and symmetrization.

This nice book pursues a tradition of Soviet books on the subject and tries to fill a gap in the literature. It collects several results concerning spaces which are partially ordered by some cones and operators in ordered Banach spaces. Perhaps positivity is not always well known and used in full; the present book indicates several applications to (mainly linear) problems arising e.g. in system theory, mechanics and mathematics (spectral theory, numerical problems, forced oscillations in nonlinear systems, problems of stability and so on). Among these results, organized in the four chapters of the book, some were previously available only in research journals and/or in Russian; the list of references contains more than 200 entries. The title of the four chapters are the following: basic notions; applications to spectral properties; applications to iteration procedures; other applications. Several exercises (of different difficulty, often containing facts of independent interest) are spread throughout the book. A selection of the material contained in the book can be used to teach a university course, to the advantage of students and teachers. For the review of the original see Zbl 0578.47030. We begin with the presentation of the contents of this book. It gives a rather good idea of what this book is about. Chapter I. General notions. {\S} 1 Cones and order, {\S} 2 Positive linear functionals and operators, {\S} 3 Smooth points in cones, {\S} 4 Normal cones, {\S} 5 Cones with convergent monotone sequence, {\S} 6 Supremums and infinimus, {\S} 7 Cones of rank k. Chapter II. Applications to spectral properties analysis. {\S} 8 Spectral radius, {\S} 9 Eigenfunctions, {\S} 10 Focussing operators, {\S} 11 Leading eigenvalues, {\S} 12 Spectral gap, {\S} 13 Peripheral spectrum, {\S} 14 Invariant subspaces. Chapter III Applications to iterative procedures analysis. {\S} 15 Simple iterations, {\S} 16 Estimations of spectral radius, {\S} 17 Iterative procedures with discrepancy proportional corrections, {\S} 18 Transform actions of equations, {\S} 19 Iterative agregation, {\S} 20 A posteriori error estimations in the positive eigenfunction problem, {\S} 21 Sequences of positive operators. Chapter IV. Some other applications. {\S} 22 Absolutely positive systems, {\S} 23 Impulse-frequency characteristic of linear element, {\S} 24 Frequency-positive linear elements, {\S} 25 Theorems on positive invortibility, {\S} 26 Forced periodic oscillations in nonlinear systems, {\S} 27 Harmonic balance method, {\S} 28 Positive solutions to nonlinear problem, {\S} 29 Problems with parameters, {\S} 30 Stability and absolute stability criteria. As we see, the unifying theoretical material from ordered Banach spaces is collected in the first chapter and in the following three chapters the authors present a wide variety of applications. The book is well-written and the material presented is well-organized. One of the byproducts of that are very short proofs of many theorems. Some of these proofs simplify substantially the old ones. As far as the reviewer knows it is the first book where such diverse applications are gathered up in such a closed form. Throughout the book many exercises are dispersed which on the one hand supply additional information and on the other hand allow one to check up one's level of understanding. Finally we note that in Chapter I the authors (in accordance with the demands of applications presented) deal mostly with the ordered Banach spaces which are not Banach lattices and this distinguishes this chapter from the majority of well-known books on Banach lattices.

This nice book pursues a tradition of Soviet books on the subject and tries to fill a gap in the literature. It collects several results concerning spaces which are partially ordered by some cones and operators in ordered Banach spaces. Perhaps positivity is not always well known and used in full; the present book indicates several applications to (mainly linear) problems arising e.g. in system theory, mechanics and mathematics (spectral theory, numerical problems, forced oscillations in nonlinear systems, problems of stability and so on). Among these results, organized in the four chapters of the book, some were previously available only in research journals and/or in Russian; the list of references contains more than 200 entries. The title of the four chapters are the following: basic notions; applications to spectral properties; applications to iteration procedures; other applications. Several exercises (of different difficulty, often containing facts of independent interest) are spread throughout the book. A selection of the material contained in the book can be used to teach a university course, to the advantage of students and teachers. For the review of the original see Zbl 0578.47030. Each abelian subgroup of the fundamental group of a compact and locally simply connected \(d\)-dimensional length space with no conjugate points is isomorphic to \(\mathbb{Z}^k\) for some \(0 \leq k \leq d\). It follows from this and previously known results that each solvable subgroup of the fundamental group is a Bieberbach group. In the Riemannian setting, this may be proved using a novel property of the asymptotic norm of each abelian subgroup.

It is well-known that equivalence classes of central extensions of a group \(G\) by an abelian group \(A\) are in one-to-one correspondence with homotopy classes of maps of the Eilenberg-MacLane spaces \(K(G,1)\) and \(K(A,2)\). The authors follow \textit{O. Schreier}'s result [Monatsh. Math. 34, 165-180 (1926; JFM 52.0113.04)] to classify equivalence classes of torsors over a category \({\mathcal B}\) under a categorical group \(({\mathbb G},\otimes)\). Then the one-to-one correspondence between classes of those torsors and homotopy classes of maps of classifying spaces \(B({\mathcal B})\) and \(B({\mathbb G},\otimes)\) leads to the topological significance the main result shows. Sind \({\mathfrak A}\) und \({\mathfrak B}\) zwei vorgegebene abstrakte Gruppen, so heißt \textit{Erweiterung} der Gruppe \({\mathfrak A}\) mit Hilfe von \({\mathfrak B}\) eine Gruppe \({\mathfrak C}\), die \({\mathfrak A}\) als Normalteiler enthält mit einer zu \({\mathfrak B}\) einstufig-isomorphen Faktorgruppe. Verf. untersucht die Konstitution aller dieser Gruppen. Die allgemeine Untersuchung führt zu dem Satze I, der die notwendigen und hinreichenden Bedingungen für die Elemente \(A^B, A_{B', B''}\) angibt, damit die Relationen \[ A\overline{B}=\overline{B}A^B, \quad \overline{B'}\overline{B''}= \overline{B'B''}A_{B, B''} \] zusammen mit den Relationen der Gruppe \({\mathfrak A}\) eine Erweiterung von \({\mathfrak B}\) mit Hilfe von \({\mathfrak B}\) definieren. Da diese Bedingungen nicht unabhängig voneinander sind, eine Reduktion der Bedingungen aber nicht erzielt werden kann ohne genauere Strukturkenntnis von \({\mathfrak B}\), so werden weiterhin spezielle Fälle untersucht, die sich aus Spezialisierungen der Gruppe \({\mathfrak B}\) ergeben. \S\ 2 des ersten Teiles setzt voraus, daß \({\mathfrak B}\) ein direktes Produkt endlich oder unendlich vieler Gruppen sei; die Umformung des Satzes I führt auf den Satz II der Arbeit. Die Erweiterung einer Gruppe mit Hilfe \textit{Abel}scher Gruppen ist der Inhalt des dritten und letzten Paragraphen des ersten Teiles. Der zweite Teil der Arbeit bringt nun einige Anwendungen, insbesondere des dritten Paragraphen. Nach einer kurzen Vorbereitung über Kongruenzen zwischen Matrizen wird das Kriterium für die Existenz von Erweiterungen einer Gruppe mit Hilfe von \textit{Abel}schen. Gruppen auf den Fall spezialisiert, daß auch die zu erweiternde Gruppe \({\mathfrak A}\) eine Abelsche Gruppe endlicher Ordnung ist. Für diesen Fall kann ein Kriterium dafür angegeben werden, daß \({\mathfrak A}\) der Kommutator der Erweiterung ist. Alle diese speziellen Resultate lassen sich nun auf Strukturuntersuchungen für \(p\)-Gruppen (d. h. Gruppen von Primzahlpotenzordnung) anwenden, wenn noch einige Sätze über die Automorphismengruppe einer \textit{Abel}schen \(p\)-Gruppe und über den Kommutator einer \(p\)-Gruppe hinzugezogen werden. \S\ 5 beschäftigt sich mit der Frage der Existenz gewisser Typen von metabelschen \(p\)-Gruppen, der letzte Paragraph mit der Aufstellung aller Gruppen der Ordnungen \(p^3, p^4, p^5\). Diese Untersuchungen bringen zwar keine neuen Resultate, sind aber doch durch, ihre Übertragungsmöglichkeit auf höhere Fälle von großer Bedeutung.

It is well-known that equivalence classes of central extensions of a group \(G\) by an abelian group \(A\) are in one-to-one correspondence with homotopy classes of maps of the Eilenberg-MacLane spaces \(K(G,1)\) and \(K(A,2)\). The authors follow \textit{O. Schreier}'s result [Monatsh. Math. 34, 165-180 (1926; JFM 52.0113.04)] to classify equivalence classes of torsors over a category \({\mathcal B}\) under a categorical group \(({\mathbb G},\otimes)\). Then the one-to-one correspondence between classes of those torsors and homotopy classes of maps of classifying spaces \(B({\mathcal B})\) and \(B({\mathbb G},\otimes)\) leads to the topological significance the main result shows. We consider the mKdV-sine Gordon equation \[ u_{xt}+ A\left(\tfrac 32 u^2_xu_{xx}+u_{xxx}\right) =B\sin u, \] \((A,B\) are real constants) and find new multi-soliton solutions by Hirota's direct method.

Let \(A\) be a unital \(C^*\)-algebra, and let \(Lg(A)\) be the set of \(n\)-tuples of elements of \(A\) which generate \(A\) as a left ideal. The topological stable rank of \(A\) is the least \(n\) such that \(Lg_n(A)\) is dense in \(A^n\) if such an \(n\) exists and \(\infty\) otherwise. The Bass stable rank of \(A\), denoted by \(sr(A)\) is the least integer \(n\geq 1\) such that if \((a_1,\dots, a_{n+1})\in Lg_{n+1}(A)\) then there exists \(b_1,\dots, b_n\in A\) such that \(a_1+b_1 a_{n+1}, a_2 b_2 a_{n+1},\dots, a_n+ b_n a_{n+1})\) is in \(Lg_n(A)\). It is known that \(sr(A)\) coincides with the topological stable rank. \textit{R. H. Herman} and \textit{L. N. Vaserstein} [Invent. Math. 77, 553--556 (1984; Zbl 0559.46025)]. The connected stable rank of \(A\) (\(csr(A)\)) is the least integer \(n\geq 1\) such that \(GL^0(m,A)\) acts transitively on \(Lg_n(A)\) by left multiplication for all \(m\leq n\). The author proves a number of results involving these concepts, including: (1) \(crs(A\otimes{\mathcal K})\leq 2\) for any \(C^*\)-algebra \(A\); (2) \(csr(C(X))< [(m+1)/2]+ 1\) if \(X\) is a compact space of dimension \(m\); (3) \(csr(M_m(A))\leq \{(csr(A)- 1)/m\}+1\), where \(\{x\}\) denotes the least integer greater than \(x\); (4) if \(A\) is an \(AF-C^*\)-algebra and \(B\) a \(C^*\)-algebra then \(sr(A\otimes B)\leq sr(B)\), \(csr(A\otimes B)\leq csr(B)\), with equality if \(A\) is commutative; (5) if \(csr(A)= 1\) then \(csr(A\otimes{\mathcal K})= 1\). For any \(C^*\)-algebra A with 1 it is proved that its stable range (introduced by H. Bass for any ring A) equals its topological stable range (introduced by M. A. Rieffel for topological rings A). This extends an old result of Vaserstein on commutative \(C^*\)-algebras. The example of the disc algebra shows that the result cannot be extended to commutative Banach algebras.

Let \(A\) be a unital \(C^*\)-algebra, and let \(Lg(A)\) be the set of \(n\)-tuples of elements of \(A\) which generate \(A\) as a left ideal. The topological stable rank of \(A\) is the least \(n\) such that \(Lg_n(A)\) is dense in \(A^n\) if such an \(n\) exists and \(\infty\) otherwise. The Bass stable rank of \(A\), denoted by \(sr(A)\) is the least integer \(n\geq 1\) such that if \((a_1,\dots, a_{n+1})\in Lg_{n+1}(A)\) then there exists \(b_1,\dots, b_n\in A\) such that \(a_1+b_1 a_{n+1}, a_2 b_2 a_{n+1},\dots, a_n+ b_n a_{n+1})\) is in \(Lg_n(A)\). It is known that \(sr(A)\) coincides with the topological stable rank. \textit{R. H. Herman} and \textit{L. N. Vaserstein} [Invent. Math. 77, 553--556 (1984; Zbl 0559.46025)]. The connected stable rank of \(A\) (\(csr(A)\)) is the least integer \(n\geq 1\) such that \(GL^0(m,A)\) acts transitively on \(Lg_n(A)\) by left multiplication for all \(m\leq n\). The author proves a number of results involving these concepts, including: (1) \(crs(A\otimes{\mathcal K})\leq 2\) for any \(C^*\)-algebra \(A\); (2) \(csr(C(X))< [(m+1)/2]+ 1\) if \(X\) is a compact space of dimension \(m\); (3) \(csr(M_m(A))\leq \{(csr(A)- 1)/m\}+1\), where \(\{x\}\) denotes the least integer greater than \(x\); (4) if \(A\) is an \(AF-C^*\)-algebra and \(B\) a \(C^*\)-algebra then \(sr(A\otimes B)\leq sr(B)\), \(csr(A\otimes B)\leq csr(B)\), with equality if \(A\) is commutative; (5) if \(csr(A)= 1\) then \(csr(A\otimes{\mathcal K})= 1\). Magnetorheological elastomers are materials with a composite microstructure that consists of an elastomeric matrix and magnetizable inclusions. Because of the magnetic inclusions, magnetorheological elastomers are able to change their properties under magnetic field. Thereby, their effective behavior strongly depends on the microstructure. This calls for homogenization strategies to characterize their macroscopic response. However, for arbitrary macroscopic bodies, this is a non-trivial task. The main difficulty stems from the fact that a magnetic body interacts with its surrounding and thus perturbs the magnetic field it is subjected to. In a multiscale simulation, this interaction has to be accounted for through a physically sound prescription of magnetic boundary conditions. Thus, the goal of this contribution is to establish a two-scale homogenization framework that allows for both (i) the incorporation of the microstructure into the macroscopic simulation and (ii) the application of experimentally motivated boundary conditions on arbitrary macroscopic bodies. We show the capabilities of the approach in several numerical studies, in which we analyze the effective behavior of different specimens. Depending on their microstructure, we observe a contraction or extension of the specimens and find magnetically induced stiffening or weakening. All numerical predictions are in good qualitative agreement with experimental measurements.

The author uses the direct method to extend a result of \textit{Z. Gajda} [Int. J. Math. Math. Sci. 14, No. 3, 431--434 (1991; Zbl 0739.39013)], concerning the stability of the Cauchy functional equation \(f(x+y)=f(x)+f(y)\) to Fréchet's first polynomial equation. For more information and recent results on the stability of functional equations see the survey of \textit{J. Brzdek}, \textit{W. Fechner}, \textit{M.S. Moslehian} and \textit{J. Sikorska} [``Recent developments of the conditional stability of the homomorphism equation'', Banach J. Math. Anal. 9, No. 3, 278--327 (2015)]. Let \(E_ 1\), \(E_ 2\) be real normed spaces with \(E_ 2\) complete, and let \(p\), \(\varepsilon\) be real numbers with \(\varepsilon\geq 0\). When \(f: E_ 1\to E_ 2\) satisfies the inequality \(\| f(x+y)-f(x)- f(y)\|\leq\varepsilon(\| x\|^ p+\| y\|^ p)\) for all \(x,y\in E\), it was shown by \textit{T. M. Rassias} [Proc. Amer. Math. Soc. 72, 299-300 (1978; Zbl 0398.47040)] that there exists a unique additive mapping \(T: E_ 1\to E_ 2\) such that \(\| f(x)- T(x)\|\leq\delta\| x\|^ p\) for all \(x\in E_ 1\), providing that \(p<1\), where \(\delta=2\varepsilon/(2-2^ p)\). The relationship between \(f\) and \(T\) was given by the formula \(T(x)=\lim_{n\to\infty}2^{-n}f(2^ nx)\). Rassias also proved that if the mapping from \(\mathbb{R}\) to \(E_ 2\) given by \(t\to f(tx)\) is continuous for each fixed \(x\in E\), then \(T\) is linear. In the present paper the author extends these results to the case \(p>1\), but now the additive mapping \(T\) is given by \(T(x)=\lim_{n\to\infty}2^ nf(2^{-n}x)\), and the corresponding value of \(\delta\) is \(\delta=2\varepsilon/(2^ p-2)\). The author also gives a counterexample to show that the theorem is false for the case \(p=1\), and any choice of \(\delta>0\) when \(\varepsilon>0\).

The author uses the direct method to extend a result of \textit{Z. Gajda} [Int. J. Math. Math. Sci. 14, No. 3, 431--434 (1991; Zbl 0739.39013)], concerning the stability of the Cauchy functional equation \(f(x+y)=f(x)+f(y)\) to Fréchet's first polynomial equation. For more information and recent results on the stability of functional equations see the survey of \textit{J. Brzdek}, \textit{W. Fechner}, \textit{M.S. Moslehian} and \textit{J. Sikorska} [``Recent developments of the conditional stability of the homomorphism equation'', Banach J. Math. Anal. 9, No. 3, 278--327 (2015)]. Exact constants in Jackson-Stechkin type inequalities are found for the smoothness characteristics \(\Lambda_m (f)\), \(m\in\mathbb{N}\), determined by averaging the norm of finite differences of \(m\)th order of functions \(f \in L_2\). A solution is given of the extremal problem of finding the supremum for best joint polynomial approximations of functions and their successive derivatives on some classes of functions from \(L_2\) whose averaged norms of finite differences are bounded above by 1.

Authors' abstract: A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely \(\mathbf F\mathbf F^\dagger\) and \(\mathbf F^\dagger\mathbf F\), determined by a complex square matrix \(\mathbf F\) and its Moore-Penrose inverse \(\mathbf F^\dagger\). Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by \textit{J. Benítez} and \textit{V. Rakočević} [Appl. Math. Comput. 217, No.~7, 3493--3503 (2010; Zbl 1213.15003)]. Further characteristics of \(\mathbf F\mathbf F^\dagger\) and \(\mathbf F^\dagger\mathbf F\), with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with \(\mathbf F\mathbf F^\dagger\) and \(\mathbf F^\dagger\mathbf F\), the paper develops a general approach whose applicability extends far beyond the characteristics provided therein. The authors investigate the class of square matrices \(A\) such that \(AA^{\dagger } - A^{\dagger }A\) is nonsingular, where \(A^{\dagger }\) stands for the Moore-Penrose inverse of \(A\). Among several characterizations they prove that for a matrix \(A\) of order \(n\), the difference \(AA^{\dagger } - A^{\dagger }A\) is nonsingular if and only if \(\mathcal R(A) \oplus \mathcal R(A^*) = \mathbb C_{n,1}\), where \(\mathcal R(\cdot )\) denotes the range space. Moreover, they investigate matrices \(A\) such that \(\mathcal R(A)^\perp = \mathcal R(A^*)\).

Authors' abstract: A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely \(\mathbf F\mathbf F^\dagger\) and \(\mathbf F^\dagger\mathbf F\), determined by a complex square matrix \(\mathbf F\) and its Moore-Penrose inverse \(\mathbf F^\dagger\). Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by \textit{J. Benítez} and \textit{V. Rakočević} [Appl. Math. Comput. 217, No.~7, 3493--3503 (2010; Zbl 1213.15003)]. Further characteristics of \(\mathbf F\mathbf F^\dagger\) and \(\mathbf F^\dagger\mathbf F\), with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with \(\mathbf F\mathbf F^\dagger\) and \(\mathbf F^\dagger\mathbf F\), the paper develops a general approach whose applicability extends far beyond the characteristics provided therein. In the recent progress, the well-known JC (Jacobian conjecture) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all second-order homogeneous differential operators \(\Lambda\) and \(\Lambda\)-nilpotent polynomials \(P\) (the polynomials \(P(z)\) satisfying \(\Lambda^m P^m=0\) for all \(m\geq 1\)). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on \(\Lambda\)-nilpotent polynomials and the associated VC for second-order homogeneous differential operators \(\Lambda\). This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of \(\Lambda\)-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous \(\Lambda\)-nilpotent polynomials for second-d order homogeneous full rank differential operators \(\Lambda\) with constant coefficients.

Numerical methods for integral equations of Volterra type have been the subject of many investigations over the last decades. The book under review, written by one of the leading experts in this area, is on the one hand focused on a seemingly narrow part of this subject, namely methods of collocation type. On the other hand, it shows an enormous broadness because it covers not only the usual simple problems (such as equations with continuous or even smooth kernels) but also equations with weakly singular kernels, various forms of delay equations, integro-differential equations, integral-algebraic equations and equations with singular perturbations. All these items are discussed in a thorough and very detailed fashion, including a review of the analytical aspects that are relevant for the numerical work, thus turning the monograph into a highly valuable resource for any researcher in the area. The style of the book is very similar to that of the older book written by the author and \textit{P. J. van der Houwen} on a similar subject [The numerical solution of Volterra equations (North-Holland, Amsterdam) (1986; Zbl 0611.65092)]. Each section contains notes and exercises indicating possibilities for further research activities. The list of references is about 80 pages long and seems to be very complete. The authors which are known for their own contributions to the field have remarkably well succeeded in producing this comprehensive and up-to-date monograph. Its main aim is the presentation of discretization methods for integral equations of the form \[ \theta y(t)=g(t)+\int^{t}_{0}k(t,s,y(s))ds,\quad t\in I, \] where \(I=[0,T]\) or [0,\(\infty)\) and \(\theta =1\) (second kind equation) or \(\theta =0\) (first kind equation). As special cases one has a linear Volterra integral equation (k(t,s)y(s)) instead of k(t,s,y(s)), Volterra integral equations with weakly singular kernels \((t-s)^{-\alpha}k(t,s,y(s))\) with \(0<\alpha <1\) instead of k(t,s,y(s)), and Volterra integral equations with convolution kernels. And there are generalizations: systems of Volterra integral equations, systems of Volterra integro- differential equations. The analogies of analytical theory and numerical methods to those of the initial value problem for an ordinary differential equation is immediately obvious if \(\theta =1\) and g(t)\(\equiv y(0)\) are taken. In Chapter 1 the basic theory of the types of equations mentioned above is developed (existence, uniqueness, asymptotic behavior) and useful Gronwall inequalities (continuous and discrete) are given. Chapter 2 is a survey on numerical quadrature and on linear multistep methods for ordinary differential equations. These methods are generalized in Chapter 3 to Volterra linear multistep methods. Chapter 4 is devoted to Runge- Kutta-type methods for Volterra equations, and in Chapter 5 collocation methods for Volterra equations with regular kernels are treated, and questions of superconvergence are discussed. Chapter 6 is on Volterra equations with weakly singular kernels (Abel-type integral equations) and contains the theory of fractional convolution quadratures recently (1983, 1985) introduced by Ch. Lubich, descriptions of collocation methods, product integration methods, and a short discussion of the problem of ill-posedness of first-kind Abel integral equations. In Chapter 7, Numerical stability, the various notions of stability of solution of a system of ordinary differential equations \[ y'(t)=f(t,y(t)),\quad t_ 0\leq t<\infty \] are generalized to Volterra integral equations (and systems of such equations) where now the sensitivity to perturbations of the initial function g(t) is important (in contrast to sensitivity to perturbations of an initial value). A numerical method should imitate qualitatively the asymptotic behavior (as \(t\to \infty)\) of the true solution. These problems are thoroughly treated for linear multistep methods and for Runge-Kutta-type methods, and analogously to the case of ordinary differential equations ''test equations'' are used, the basic one being \(\theta y(t)=g(t)+\xi \int^{t}_{0}y(s)ds.\) A more complicated one is the ''convolution test equation''. Chapter 8 gives a survey on available software and on test examples. Finally, there are 51 pages of references. The book is essentially self-contained, and in places where proofs are omitted easily accessible references are given. Results of many illustrative numerical case studies are displayed, and each chapter ends with ''Notes'' giving many hints to the original literature where topics have been treated and where more details and special results can be found. The book will be very useful for both the beginner who wants to be introduced and for the researcher who needs to be informed on all the aspects of the subject and on the latest developments.

Numerical methods for integral equations of Volterra type have been the subject of many investigations over the last decades. The book under review, written by one of the leading experts in this area, is on the one hand focused on a seemingly narrow part of this subject, namely methods of collocation type. On the other hand, it shows an enormous broadness because it covers not only the usual simple problems (such as equations with continuous or even smooth kernels) but also equations with weakly singular kernels, various forms of delay equations, integro-differential equations, integral-algebraic equations and equations with singular perturbations. All these items are discussed in a thorough and very detailed fashion, including a review of the analytical aspects that are relevant for the numerical work, thus turning the monograph into a highly valuable resource for any researcher in the area. The style of the book is very similar to that of the older book written by the author and \textit{P. J. van der Houwen} on a similar subject [The numerical solution of Volterra equations (North-Holland, Amsterdam) (1986; Zbl 0611.65092)]. Each section contains notes and exercises indicating possibilities for further research activities. The list of references is about 80 pages long and seems to be very complete. The authors consider the full Stokes system \[ -\Delta v+\nabla p=f', \quad -\nabla \cdot v=f_4 \text{ in }\Omega,\quad v=g\text{ on }\partial \Omega\tag{1} \] in an unbounded doinain \(\Omega\subset\mathbb{R}^3\) which has cylindrical outlets at inifinity. They prove a representation formula and a priori estimates for solutions of (1) when the data \((f,g)\) belong to \(H^{l,q}\)-spaces. These results are generalized to function spaces with polynomial weights. The general procedure is to study the system first in a straight cylinder \(\omega\times\mathbb{R}\) by applying the Fourier transform in the third variable and then to extend the results to the more complicated geometry.

There exist many monographs which deal with applications of either Lie group theory or differential geometry to physics, but rarely with both subjects simultaneously. Now we have one that brings these two vital strands of mathematical physics together. It seems that Lam benefitted from a previous collaboration with \textit{S. S. Chern} and \textit{W. H. Chen} which resulted in the text ``Lectures on differential geometry'' published in 1999 by World Scientific (Zbl 0940.53001). The present text is more elementary in that it addresses an audience of advanced undergraduates and beginning graduates. Its major part focuses on Lie groups and algebras, their representations as they occur in physics, but also on the role of finite groups such as the dihedral group in the study fo the benzene molecule. There are chapters on Clifford and exterior algebras as well as the Hodge operator. Then the reader is guided to differential forms and exterior differentiation, i.e. algebraic structures built on differential manifolds. He is then led to consider fiber bundles, especially vector and principal bundles, and finally, algebraic objects (characteristic classes) constructed from analytical data (curvatures, connections). Generally, obfuscating proofs are omitted, leaving room for chapters on Yang-Mills equations, magnetic monopoles, Chern-Simon forms on the physical side, and Riemannian geometry, complex manifolds, Atiyah-Singer index theorem on the mathematical side are included. Based on a lecture series delivered in Peking University in 1980 by a renowned leader in differential geometry, S. S. Chern, the present book is addressed mainly to advanced undergraduate and beginning graduate students in mathematics and physics. The first lecture contains some basic notions about differentiable manifolds and ends with Frobenius' theorem. Multilinear algebra is used in the exterior differential calculus, where the method of exterior differentiation due to E. Cartan is systematically developed. The general theory of connections on vector bundles is introduced. In a lecture devoted to Riemannian geometry, a relation between local and global properties is established by the Gauss-Bonnet theorem. Some basic notions on Lie groups are given and the method of moving frames is used to show that the first and second fundamental forms constitute a complete system of invariants for hypersurfaces in \(\mathbb{R}^{m+1}\). In a lecture on complex manifolds, the Hermitian and Kählerian manifolds play a central role. The main lecture is devoted to Finsler geometry, since according to Prof. Chern's opinion, time has come for the subject of Finsler geometry to take a more prominent position in university curricula of basic differential geometry. Some historical notes and some applications of differential geometry in theoretical physics are given at the end. The material of this book (with several examples and remarks throughout) provides a solid and comprehensive background for more advanced and specialized studies.

There exist many monographs which deal with applications of either Lie group theory or differential geometry to physics, but rarely with both subjects simultaneously. Now we have one that brings these two vital strands of mathematical physics together. It seems that Lam benefitted from a previous collaboration with \textit{S. S. Chern} and \textit{W. H. Chen} which resulted in the text ``Lectures on differential geometry'' published in 1999 by World Scientific (Zbl 0940.53001). The present text is more elementary in that it addresses an audience of advanced undergraduates and beginning graduates. Its major part focuses on Lie groups and algebras, their representations as they occur in physics, but also on the role of finite groups such as the dihedral group in the study fo the benzene molecule. There are chapters on Clifford and exterior algebras as well as the Hodge operator. Then the reader is guided to differential forms and exterior differentiation, i.e. algebraic structures built on differential manifolds. He is then led to consider fiber bundles, especially vector and principal bundles, and finally, algebraic objects (characteristic classes) constructed from analytical data (curvatures, connections). Generally, obfuscating proofs are omitted, leaving room for chapters on Yang-Mills equations, magnetic monopoles, Chern-Simon forms on the physical side, and Riemannian geometry, complex manifolds, Atiyah-Singer index theorem on the mathematical side are included. When the progress of female bark beetles is followed in a small arena as they are attracted to a source emitting male pheromones, the variate that is of interest to biologists is the heading angle, i.e., the angle between the direction toward the source and the direction of forward motion. In the present paper we model the resulting angular time series variate using autoregressive-type models with von Mises random errors. Estimates of parameters for four data sets were obtained by maximizing a von Mises likelihood function. Plots of residuals developed after fitting first order models indicated some interesting behavioral differences between beetles in control groups and those in bioassays with strong pheromone attractants.

The paper under review is a continuation of the analysis developed in [\textit{D. D. Holm} et al., Q. Appl. Math. 67, No. 4, 661-685 (2009; Zbl 1186.68413)]. There the authors formulated metamorphosis, that is, diffeomorphic pattern matching using distances on a space of images or shapes, in an abstract Lagrangian setting and derived the general evolution equations. The key point is that the metamorphosis equations contain the equations for a perfect complex fluid. A main question was left open in the case of matching point sets. An important case because of the extended use of landmarks to match images, for example. The authors proposed in the cited paper that whenever the template and the target were sums of Dirac measures, the geodesic evolution would remain a measure at each time, and this measure could be expressed as a sum of Dirac measures plus an absolutely continuous part. All this is rigorously proved in the paper under review. Nevertheless, beside Dirac measures, the setting needs to be augmented with distributions of Calderón-Zygmund type to obtain the right results. These additional singularities complicate the numerical approximation of the solutions, but this drawback is solved thanks to a Eulerian numerical method where the solutions are approximated using smooth densities and adapting then the discretization used for metamorphosis of smooth \(L^2\) densities. Finally, numerical experiments illustrate the method. In the pattern matching approach to imaging science, the process of ``metamorphosis'' is template matching with dynamical templates [Found. Comput. Math. 5, No.~2, 173--198 (2005; Zbl 1099.68116)]. Here, we recast the metamorphosis equations of that paper into the Euler-Poincaré variational framework of \textit{D. D. Holm, J. E. Marsden} and \textit{T. S. Ratiu} [Adv. Math. 137, No.~1, 1--81 (1998; Zbl 0951.37020)] and show that the metamorphosis equations contain the equations for a perfect complex fluid. This result connects the ideas underlying the process of metamorphosis in image matching to the physical concept of an order parameter in the theory of complex fluids. After developing the general theory, we reinterpret various examples, including point set, image and density metamorphosis. We finally discuss the issue of matching measures with metamorphosis, for which we provide existence theorems for the initial and boundary value problems.

The paper under review is a continuation of the analysis developed in [\textit{D. D. Holm} et al., Q. Appl. Math. 67, No. 4, 661-685 (2009; Zbl 1186.68413)]. There the authors formulated metamorphosis, that is, diffeomorphic pattern matching using distances on a space of images or shapes, in an abstract Lagrangian setting and derived the general evolution equations. The key point is that the metamorphosis equations contain the equations for a perfect complex fluid. A main question was left open in the case of matching point sets. An important case because of the extended use of landmarks to match images, for example. The authors proposed in the cited paper that whenever the template and the target were sums of Dirac measures, the geodesic evolution would remain a measure at each time, and this measure could be expressed as a sum of Dirac measures plus an absolutely continuous part. All this is rigorously proved in the paper under review. Nevertheless, beside Dirac measures, the setting needs to be augmented with distributions of Calderón-Zygmund type to obtain the right results. These additional singularities complicate the numerical approximation of the solutions, but this drawback is solved thanks to a Eulerian numerical method where the solutions are approximated using smooth densities and adapting then the discretization used for metamorphosis of smooth \(L^2\) densities. Finally, numerical experiments illustrate the method. This paper considers testing for linear regression models when covariates are measured with additive error and some additional linear restrictions on the coefficients are available. We propose a test statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotic standard chi-squared distribution. Finally, some simulations are given to examine the performance of our procedure and the results are satisfactory.

In the present paper, the authors study the limiting behaviour of Riemann sums constructed according to Simpson's rule for a fractional Brownian motion. More precisely, let \(B\) be a fractional Brownian motion, that is, a centered Gaussian process with covariance function given by \[ \mathbb{E}[B_s B_t] = \frac{1}{2} \Big( s^{2H} + t^{2H} - |t-s|^{2H} \Big) \quad \text{for all \(s,t \geq 0\),} \] where \(H \in (0,1)\) is the Hurst parameter. Furthermore, let \(f : \mathbb{R} \to \mathbb{R}\) be a smooth function, and consider the Riemann sums constructed according to Simpson's rule with uniform partition: \[ S_n^S(t) := \sum_{j=0}^{\lfloor nt \rfloor - 1} \frac{1}{6} \Big( f' \Big( B_{\frac{j}{n}} \Big) + 4 f' \Big( \Big( B_{\frac{j}{n}} + B_{\frac{j+1}{n}} \Big) / 2 \Big) + f' \Big( B_{\frac{j+1}{n}} \Big) \Big) \Big( B_{\frac{j+1}{n}} - B_{\frac{j}{n}} \Big). \] The authors show that for \(H = 1/10\) this sequence of sums converges weakly to a random variable. More precisely, conditioned on the path \(\{ B_s: s \leq t \}\) one has \[ S_n^S(t) \to f(B_t) - f(0) + \frac{\beta}{2880} \int_0^t f^{(5)}(B_s)\, \mathrm{d}W_s \quad \text{weakly,} \] where \(W\) is a standard Brownian motion, independent of \(B\), and \(\beta \in \mathbb{R}\) is a constant, which is defined in the article. From this result, the authors derive the change-of-variable formula \[ f(B_t) = f(0) + \int_0^t f'(B_s)\, \mathrm{d}^S B_s - \frac{\beta}{2880} \int_0^t f^{(5)}(B_s)\, \mathrm{d}W_s \quad \text{in distribution}, \] where the stochastic integral with differential \(\text{d}^S B_s\) is understood as limit of the Simpson rule sums. The result of the authors contributes to a paper by \textit{M. Gradinaru} et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 4, 781--806 (2005; Zbl 1083.60045)], where it was shown that for \(H > 1/10\) the sequence of sums converges in probability, but generally does not converge in probability for \(H \leq 1/10\). Suppose \(X\) is a continuous process and \(g: {\mathbb R}\to{\mathbb R}\) is a locally bounded Borel function. Given a positive integer \(m\) and a probability measure \(\nu\) on \([0,1]\), the authors introduce \(m\)-order \(\nu\)-integrals as \[ \int_0^tg(X_u)\,d^{\nu,m}X_u:= \lim_{\varepsilon\downarrow 0} \text{ prob}\, {1\over{\varepsilon}}\int_0^t du\, (X_{u+\varepsilon}-X_u)^m\int_0^1 g(X_u+\alpha(X_{u+\varepsilon}-X_u))\,\nu(d\alpha). \] When \(\nu\) is symmetric, the corresponding integral is an extension of symmetric integrals of Stratonovich type. If \(f\in C^{2n}({\mathbb R})\), \(\nu\) is symmetric, and if \(X\) is a continuous process having a \((2n)\)-variation (i.e. \([X,X,\dots,X]\)), then the Itô formula holds with some extra terms consisting of higher order \(\delta_{1/2}\)-integrals. In the case of the fractional Brownian motion \(B^H\), \(0<H<1\), \(m\)-order \(\nu\)-integral vanishes for all odd indices \(m>1/2H\) and any symmetric \(\nu\). If \(\nu\) is any symmetric probability measure, then the Itô-Stratonovich formula holds for \(H>1/6\), but fails to hold for \(H\leq 1/6\). However, if \(H\leq 1/6\), an Itô formula is still valid provided we proceed through a different regularization of the symmetric integral which involves particular symmetric probability measures.

In the present paper, the authors study the limiting behaviour of Riemann sums constructed according to Simpson's rule for a fractional Brownian motion. More precisely, let \(B\) be a fractional Brownian motion, that is, a centered Gaussian process with covariance function given by \[ \mathbb{E}[B_s B_t] = \frac{1}{2} \Big( s^{2H} + t^{2H} - |t-s|^{2H} \Big) \quad \text{for all \(s,t \geq 0\),} \] where \(H \in (0,1)\) is the Hurst parameter. Furthermore, let \(f : \mathbb{R} \to \mathbb{R}\) be a smooth function, and consider the Riemann sums constructed according to Simpson's rule with uniform partition: \[ S_n^S(t) := \sum_{j=0}^{\lfloor nt \rfloor - 1} \frac{1}{6} \Big( f' \Big( B_{\frac{j}{n}} \Big) + 4 f' \Big( \Big( B_{\frac{j}{n}} + B_{\frac{j+1}{n}} \Big) / 2 \Big) + f' \Big( B_{\frac{j+1}{n}} \Big) \Big) \Big( B_{\frac{j+1}{n}} - B_{\frac{j}{n}} \Big). \] The authors show that for \(H = 1/10\) this sequence of sums converges weakly to a random variable. More precisely, conditioned on the path \(\{ B_s: s \leq t \}\) one has \[ S_n^S(t) \to f(B_t) - f(0) + \frac{\beta}{2880} \int_0^t f^{(5)}(B_s)\, \mathrm{d}W_s \quad \text{weakly,} \] where \(W\) is a standard Brownian motion, independent of \(B\), and \(\beta \in \mathbb{R}\) is a constant, which is defined in the article. From this result, the authors derive the change-of-variable formula \[ f(B_t) = f(0) + \int_0^t f'(B_s)\, \mathrm{d}^S B_s - \frac{\beta}{2880} \int_0^t f^{(5)}(B_s)\, \mathrm{d}W_s \quad \text{in distribution}, \] where the stochastic integral with differential \(\text{d}^S B_s\) is understood as limit of the Simpson rule sums. The result of the authors contributes to a paper by \textit{M. Gradinaru} et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 4, 781--806 (2005; Zbl 1083.60045)], where it was shown that for \(H > 1/10\) the sequence of sums converges in probability, but generally does not converge in probability for \(H \leq 1/10\). An extremal curve of the simplest variational problem is a continuously differentiable function. Hilbert's differentiability theorem provides a sufficient condition for the existence of the second derivative of an extremal curve. It is desirable to have a simple example in which the condition of Hilbert's theorem is violated and an extremal curve is not twice differentiable. In this paper, a cubic variational problem with the following properties is analyzed. The functional of the problem is bounded neither above nor below. There exists an extremal curve for this problem which is obtained by sewing together two different extremal curves and not twice differentiable at the sewing point. Despite this unfavorable situation, an attempt to apply the method of steepest descent (in the form proposed by V.F. Dem'yanov) to this problem is made. It turns out that the method converges to a stationary curve provided that a suitable step size rule is chosen.

Given highest weights \(\lambda\), \(\mu\) and \(\nu\) for a finite-dimensional complex semisimple Lie algebra, let \(C_{\lambda \mu}^{\nu}\) be the multiplicity of the irreducible representation \(V_{\nu}\) in the tensor product of \(V_{\lambda}\) and \(V_{\mu}\), that is, \(V_{\lambda} \otimes V_{\mu}=\bigoplus_{\nu} C_{\lambda \mu}^{\nu} V_{\nu}\). In general, the numbers \(C_{\lambda \mu}^{\nu}\) are called Clebsch-Gordan coefficients and, in the specific case of type-\(A\) Lie algebras, they are called Littlewood-Richardson coefficients. De Loera and McAllister show in this paper that when the rank of the Lie algebra is assumed fixed, then one can compute the Clebsch-Gordan coefficients \(C_{\lambda \mu}^{\nu}\) in time polynomial in the input size of the defining weights. Moreover, in the type-\(A\) case, deciding whether \(C_{\lambda \mu}^{\nu} \neq 0\) can be done in polynomial time even when the rank is not fixed. The experiments carried out by the authors show that this algorithm, based on counting lattice points in polytopes, is superior in practice to the standard techniques for computing these coefficients when the weights have large entries but small rank. On the basis of abundant experimental evidence, De Loera and McAllister also propose two conjectured generalizations of the saturation theorem of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] that establish that, given highest weights \(\lambda\), \(\mu\) and \(\nu\) for \(\text{sl}_r({\mathbb C})\), and given an integer \(n>0\), the Littelwood-Richardson coefficient \(C_{\lambda \mu}^{\nu}\) satisfies that \(C_{\lambda \mu}^{\nu} \neq 0\) if, and only if, \(C_{n\lambda , n\mu}^{n\nu} \neq 0\). This is a remarkable paper which is devoted to the final solution of the following old and fundamental combinatorial problem: For given dominant weights \(\lambda,\mu,\nu\) and the corresponding irreducible \(\text{GL} _n({\mathbb{C}})\)-modules \(V_{\lambda},V_{\mu},V_{\nu}\), does the tensor product \(V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}\) contain a \(\text{GL} _n({\mathbb{C}})\)-invariant vector? Without essential lost of generality, replacing \(\nu\) with its dual \(\nu^{\ast}\), another standard formulation of the problem is for which polynomial modules \(V_{\lambda},V_{\mu},V_{\nu^{\ast}}\) is the coefficient \(c_{\lambda,\mu}^{\nu^{\ast}}\) in the Littlewood-Richardson rule \(V_{\lambda}\otimes V_{\mu}\cong \sum c_{\lambda,\mu}^{\nu^{\ast}}V_{\nu^{\ast}}\) is different from 0? The answer to the problem under consideration is known asymptotically. \textit{A. A. Klyachko} [Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010)] gave a finite set of linear inequalities derived from the Schubert calculus and \(\lambda,\mu,\nu\) satisfying this system if and only if there exists a positive integer \(N\) such that \(V_{N\lambda}\otimes V_{N\mu}\otimes V_{N\nu}\) contains a \(\text{GL} _n({\mathbb{C}})\)-invariant vector. Since the set of triples \((\lambda,\mu,\nu)\) corresponding to a \(\text{GL} _n({\mathbb{C}})\)-invariant vector form an additive monoid, the complete solution of the problem for the description of such triples depends on the saturation of the monoid and this is the main result of the paper under review: If \((V_{N\lambda}\otimes V_{N\mu}\otimes V_{N\nu}) ^{\text{GL} _n({\mathbb{C}})}>0\) for some \(N>0\), then \((V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}) ^{\text{GL} _n({\mathbb{C}})}>0\). The main tool for the proof is the Berenstein-Zelevinsky polytope associated to the triple \((\lambda,\mu,\nu)\) in which the number of lattice points is the corresponding Littlewood-Richardson coefficient. The authors give a new description using their honeycomb model. As an immediate consequence the authors obtain an affirmative answer to the Horn conjecture from 1962 which gives a recursive system of inequalities reducing the problem to lower-dimensional Littlewood-Richardson questions. The considered problem is also related to many other classical problems. It turns out that they all have the same answer and are consequences of the result by Klyachko and the saturation conjecture. In particular, the authors give a new proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture for \(\text{GL} _n({\mathbb{C}})\) which states that if \(w\lambda+v\nu\) is in the positive Weyl chamber, then \(V_{w\lambda+v\nu}\) is a constituent of \(V_{\lambda}\otimes V_{\nu}\) (and is known to be true for all Lie groups). Another application is to Hermitian matrices solving a problem with more than 100 years of history: If \(\lambda,\mu,\nu\) are sequences of \(n\) descending real numbers, do there exist Hermitian matrices \(A\) and \(B\) with eigenvalues \(\lambda\) and \(\mu\), respectively, such that \(A+B\) has eigenvalue \(\nu\)?

Given highest weights \(\lambda\), \(\mu\) and \(\nu\) for a finite-dimensional complex semisimple Lie algebra, let \(C_{\lambda \mu}^{\nu}\) be the multiplicity of the irreducible representation \(V_{\nu}\) in the tensor product of \(V_{\lambda}\) and \(V_{\mu}\), that is, \(V_{\lambda} \otimes V_{\mu}=\bigoplus_{\nu} C_{\lambda \mu}^{\nu} V_{\nu}\). In general, the numbers \(C_{\lambda \mu}^{\nu}\) are called Clebsch-Gordan coefficients and, in the specific case of type-\(A\) Lie algebras, they are called Littlewood-Richardson coefficients. De Loera and McAllister show in this paper that when the rank of the Lie algebra is assumed fixed, then one can compute the Clebsch-Gordan coefficients \(C_{\lambda \mu}^{\nu}\) in time polynomial in the input size of the defining weights. Moreover, in the type-\(A\) case, deciding whether \(C_{\lambda \mu}^{\nu} \neq 0\) can be done in polynomial time even when the rank is not fixed. The experiments carried out by the authors show that this algorithm, based on counting lattice points in polytopes, is superior in practice to the standard techniques for computing these coefficients when the weights have large entries but small rank. On the basis of abundant experimental evidence, De Loera and McAllister also propose two conjectured generalizations of the saturation theorem of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] that establish that, given highest weights \(\lambda\), \(\mu\) and \(\nu\) for \(\text{sl}_r({\mathbb C})\), and given an integer \(n>0\), the Littelwood-Richardson coefficient \(C_{\lambda \mu}^{\nu}\) satisfies that \(C_{\lambda \mu}^{\nu} \neq 0\) if, and only if, \(C_{n\lambda , n\mu}^{n\nu} \neq 0\). Item response theory (IRT) is analysed within the framework of dichotomous response items. Rasch models have some good properties among IRT models. We are particularly interested in the random effect model. We present a linear model applied to the latent variable (random effect) inside the Rasch model with binary preditors. Within this framework, we analyse treatment-group comparisons in a diabetes study. The group's comparison test is a LR test.

Let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(P\). Then \(B\) determines a fusion system \(\mathcal F\) on \(P\). Following work of \textit{B. Külshammer} and \textit{L. Puig} [Invent. Math. 102, No. 1, 17-71 (1990; Zbl 0739.20003)], one can attach an element \(\alpha_Q\in H^2(\Aut_{\mathcal F}(Q),k^\times)\) to every \(\mathcal F\)-centric subgroup \(Q\) of \(P\). The ``gluing problem'' asks whether these elements can be ``glued'' uniquely to an element \(\alpha\in H^2(\mathcal F^c,k^\times)\) where \(\mathcal F^c\) denotes the full subcategory of \(\mathcal F\) whose objects are the \(\mathcal F\)-centric subgroups of \(P\). In the paper under review, the author considers the cases where \(B\) is either tame (so that \(p=2\) and \(P\) is dihedral, semidihedral or quaternion) or the principal \(p\)-block of \(\text{PSL}_3(p)\) where \(p\neq 2\) (so that \(P\) is extraspecial of order \(p^3\) and exponent \(p\)). In these cases he proves the existence of \(\alpha\) but observes that \(\alpha\) is not unique when \(p\equiv 1\pmod 3\) and \(B\) is the principal \(p\)-block of \(\text{PSL}_3(p)\). Let \(O\) be a discrete valuation ring whose residue field \(k\) is algebraically closed of characteristic \(p\). Let \(G\) be a finite group, and let \(H\) be a normal subgroup of \(G\). \textit{M. Broué} and the second author [Invent. Math. 56, 117-128 (1980; Zbl 0425.20008)] introduced the notion of nilpotent blocks of \(H\), and the second author [Invent. Math. 93, 77-116 (1988; Zbl 0646.20010)] essentially completed the analysis of them by determining the structure of their so-called source algebras. The purpose of the present paper is to study the source algebras of blocks of \(G\) that lie over nilpotent blocks of \(H\). The main result is that the structure of such a block is completely determined by two \(O\)-algebras, one a twisted group algebra, and the other the endomorphism ring of a capped endo-permutation module for a certain \(p\)-subgroup of \(G\). The details are too lengthy and involved to be explained here.

Let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(P\). Then \(B\) determines a fusion system \(\mathcal F\) on \(P\). Following work of \textit{B. Külshammer} and \textit{L. Puig} [Invent. Math. 102, No. 1, 17-71 (1990; Zbl 0739.20003)], one can attach an element \(\alpha_Q\in H^2(\Aut_{\mathcal F}(Q),k^\times)\) to every \(\mathcal F\)-centric subgroup \(Q\) of \(P\). The ``gluing problem'' asks whether these elements can be ``glued'' uniquely to an element \(\alpha\in H^2(\mathcal F^c,k^\times)\) where \(\mathcal F^c\) denotes the full subcategory of \(\mathcal F\) whose objects are the \(\mathcal F\)-centric subgroups of \(P\). In the paper under review, the author considers the cases where \(B\) is either tame (so that \(p=2\) and \(P\) is dihedral, semidihedral or quaternion) or the principal \(p\)-block of \(\text{PSL}_3(p)\) where \(p\neq 2\) (so that \(P\) is extraspecial of order \(p^3\) and exponent \(p\)). In these cases he proves the existence of \(\alpha\) but observes that \(\alpha\) is not unique when \(p\equiv 1\pmod 3\) and \(B\) is the principal \(p\)-block of \(\text{PSL}_3(p)\). The articles of this volume will be reviewed individually. The preceding symposium has been reviewed (see Zbl 1069.68012).

Catalan's conjecture states that the only consecutive integers that are both powers are 8 and 9. Here is a very brief summary of what is known. For more information one may also consult the very fine book of the author [Catalan's conjecture. Are 8 and 9 the only consecutive powers? Academic Press, New York (1994; Zbl 0824.11010)] on this subject. It is enough to consider \(x^p- y^q=1\), with \(p\) and \(q\) odd primes. By using linear forms in logarithms and other techniques it is known that (assuming \(p<q\)), \(10^5<p<3.31* 10^{12}\), \(10^6<q< 4.13*10^{17}\). The present article is a very fine informal lecture on Catalan's conjecture which explains why the problem is challenging and natural and also describes the methods, both algebraic and analytical, used in attacking the problem. It gives a good overview of the problem for those seeing it for the first time. With the solution of Fermat's last theorem, the second most notorious unsolved problem in diophantine equations is Catalan's conjecture: Does the equation \(x^ p- y^ q =1\) have any solutions in integers \(x\), \(y\) and primes \(p\), \(q\) other than \((3,2,2,3)\)? Since Catalan proposed this in 1844 many famous mathematicians have worked on the problem. The problem may be attacked from two directions. The first is to find an upper bound for \(p\) and \(q\). In 1976, \textit{R. Tijdeman} and \textit{M. Langevin} [Acta Arith. 29, 197-209 (1976; Zbl 0316.10008), respectively Unpublished Manuscript, 1976], in a celebrated paper, established explicit upper bounds for \(x\), \(y\), \(p\), \(q\) by using Baker's theory of linear forms in logarithms of algebraic numbers. Since then successive improvements in Baker's constants have yielded successive improvements in these bounds. Currently, the best known bounds for \(p\) and \(q\) to date are \(10^{18}\) and \(10^{13}\) respectively. These were found by \textit{Tim O'Neil}, a PhD student in mathematics at BGSU (submitted), who improved some results of Bennet et al. on linear forms in logarithms of three algebraic numbers (submitted) to obtain them. The problem is also being attacked from below. Here one must resort to algebraic number theory to make any headway. The use of ideals and class numbers of the relevant fields is essential in this regard. Using this approach \textit{K. Inkeri} [Acta Arith. 9, 285-290 (1964; Zbl 0127.271); J. Number Theory 34, 142-152 (1990; Zbl 0699.10029)] was able to rule out many \(p\) and \(q\) for which Catalan's equation admits a nontrivial solution. He also gave criteria for those \(p\) and \(q\) for which a solution can exist. However, these lead to questions about the class numbers of cyclotomic fields, which can be very large and difficult to compute. \textit{M. Mignotte} (to appear) improved the latter criterion by showing that one only needs to compute the class number of the maximal 2 subfield of the cyclotomic field. Again, this leads to a difficult criterion: The class numbers of these fields may be split into two factors, the first factor and the relative class number. The first factor is, in general, very difficult to compute. However, \textit{W. Schwarz} (to appear) circumvented this problem by showing that it suffices just to compute the relative class number of the relevant fields. Though it has been possible to compute these relative class numbers for the last 50 years, all known results involved obtaining two large numbers and cancelling out their common factors. This made implementation of the algorithm on a computer quite difficult. The theory was developed by Borevich and Shafarevich in Russia and Hasse in Germany. Recently, in a fine Masters thesis at Bowling Green State University, \textit{Rob Clother} was able to merge their ideas into a single formula, thus taking advantage of both viewpoints. Further, he used some ideas of Kummer from the 1850's to eliminate the computation of all large numbers in his formula. This led to an algorithm for computation of class numbers of Mignotte's fields which was much faster than any achieved to date. Finally, \textit{M. Mignotte} (to appear) has given methods for dealing with those cases of Catalan's problem for which all these criteria fail. In view of all this, it is safe to say that the problem will (hopefully) be completely solved quite soon. Ribenboim's book is not only a book about Catalan's problem, but also an excellent one on diophantine equations and related topics. Thus, after some preliminary sections on basic facts needed in the proofs, we find others devoted to the Pythagorean equation, continued fractions, results of Størmer on Pell's equation, representation of integers by binary cubic forms, \textit{J. H. E. Cohn}'s [J. Lond. Math. Soc. 42, 750-752 (1967; Zbl 0154.29701)] results on binary quartic diophantine equations, powerful numbers and many other topics, some of which are still open. However, we also find a complete treatment of all classical results on Catalan's equation and their proofs. In the last part of the book the author proves an effective version of Tijdeman's result on an upper bound for \(p\) and \(q\). As pointed out above, this is not the best upper bound known, but it does give the essential ideas of Tijdeman's work. We also find a derivation of \textit{S. Hyyrö}'s [Ann. Univ. Turku, Ser. A I 79 (1964; Zbl 0127.01904)] algorithm to determine all eventual solutions \((x, y)\). Unfortunately, the known upper bounds on \(x\) and \(y\) are too large for this algorithm to be useful. It would require a major improvement in known lower bounds for linear forms in two logarithms for this. All in all, this is a very fine book on an exciting research topic -- indeed, it is the first to be devoted to this topic. What makes it particularly noteworthy is that many useful results on diophantine equations from many journals are gathered together in one place for the first time. It is certainly worthy of translation into many languages.

Catalan's conjecture states that the only consecutive integers that are both powers are 8 and 9. Here is a very brief summary of what is known. For more information one may also consult the very fine book of the author [Catalan's conjecture. Are 8 and 9 the only consecutive powers? Academic Press, New York (1994; Zbl 0824.11010)] on this subject. It is enough to consider \(x^p- y^q=1\), with \(p\) and \(q\) odd primes. By using linear forms in logarithms and other techniques it is known that (assuming \(p<q\)), \(10^5<p<3.31* 10^{12}\), \(10^6<q< 4.13*10^{17}\). The present article is a very fine informal lecture on Catalan's conjecture which explains why the problem is challenging and natural and also describes the methods, both algebraic and analytical, used in attacking the problem. It gives a good overview of the problem for those seeing it for the first time. When a bi-stable oscillator undergoes a supercritical Hopf bifurcation due to a galloping instability, intra-well limit cycle oscillations of small amplitude are born. The amplitude of these oscillations grows as the flow speed is increased to a critical speed at which the dynamic trajectories escape the potential well. The goal of this paper is to obtain a simple yet accurate analytical expression to approximate the escape speed. To this end, three different analytical approaches are implemented: (i) the method of harmonic balance, (ii) the method of multiple scales using harmonic and elliptic basis functions, and (iii) the Melnikov criterion. All methods yielded an identical expression for the escape speed with only one key difference which lies in the value of a constant that changes among the different methods. A comparison between the approximate analytical solutions and a numerical integration of the equation of motion demonstrated that the escape speed obtained via the multiple scales method using the elliptic functions and the Melnikov criterion are in excellent agreement with the numerical simulations. On the other hand, the first-order harmonic balance technique and the multiple scales using harmonic functions provide analytical estimates that significantly underestimate the actual escape speed. Using the Melnikov criterion, the influence of parametric and additive noise on the escape speed was also studied.

Let \(R\) be a ring and let \(M\) be a nonsingular left \(R\)-module such that \(B = \text{End}_ R(M)\) is a left nonsingular ring. It is proved that there exists a lattice isomorphism between the lattice \(C(M)\) of all complement (i.e. closed) submodules of \(_ R M\) and the lattice \(C(B)\) of all complement left ideals of \(B\). In particular, this means that \(_ R M\) is a CS-module (i.e. every complement submodule is a direct summand) if and only if \(_ B B\) is a CS-module. In particular, this is the case if \(M\) is retractable, i.e. \(\text{Hom}_ R(M,N) \neq 0\) for every non- zero submodule \(N\) of \(M\). This last result is a theorem of \textit{S. M. Khuri} [Bull. Aust. Math. Soc. 43, 63-71 (1991; Zbl 0719.16004)]. Let M be a left module over an associative ring R with identity. M is said to be retractable (resp. e-retractable) if \(Hom_ R(M,U)\neq 0\) for every nonzero submodule (resp. nonzero complement submodule) U of M. M is said to be nondegenerate if Tm\(\neq 0\) for every nonzero \(m\in M\), where T is the trace of M in R. M is said to be a CS module if every complement submodule of M is a direct summand in M. Finally, a ring A is a left CS ring if \({}_ AA\) is a CS module. Some connections between nondegenerate, retractable, and e-retractable modules are given. It is also proved that if M is nonsingular and retractable, then \(End_ R(M)\) is a left CS ring if and only if M is a CS module.

Let \(R\) be a ring and let \(M\) be a nonsingular left \(R\)-module such that \(B = \text{End}_ R(M)\) is a left nonsingular ring. It is proved that there exists a lattice isomorphism between the lattice \(C(M)\) of all complement (i.e. closed) submodules of \(_ R M\) and the lattice \(C(B)\) of all complement left ideals of \(B\). In particular, this means that \(_ R M\) is a CS-module (i.e. every complement submodule is a direct summand) if and only if \(_ B B\) is a CS-module. In particular, this is the case if \(M\) is retractable, i.e. \(\text{Hom}_ R(M,N) \neq 0\) for every non- zero submodule \(N\) of \(M\). This last result is a theorem of \textit{S. M. Khuri} [Bull. Aust. Math. Soc. 43, 63-71 (1991; Zbl 0719.16004)]. Let \(P\) be a ``property''; that is, a class of topological spaces. Then \(P\) is localizable if whenever \(x\in X\in P\), \(x\) has arbitrarily small neighborhoods (not necessarily open) in \(X\) that has property \(P\). It is shown that, for localized and certain ``nice'' properties \(P\), by a standard ``kernel'' inductive procedure, one can characterize those spaces that are expressible as the finite union of subspaces each having the property \(P\) locally, and determine the least such number. Besides, it is shown that this least number, minus one, behaves ``logarithmically'': it is additive for product spaces. Also, specific examples of such properties are given, and some open questions are raised.

Continuing own investigations [\textit{K. Gürlebeck} and \textit{U. Kähler}, J. Anal. Anwend. 15, No. 2, 283-297 (1996; Zbl 0864.30039)] the author deals with the properties of several operators in Clifford analysis. Mainly he investigates the Teodorescu transform and the Cauchy type operator, the latter in a domain and on the boundary, a generalized \(\pi\)-operator is defined and some of its properties are proved. A main result gives connections between the invertibility of the \(\pi\)-operator and some orthogonal decompositions of \(L_2\). Several relations between the various types of operators are given. The complex Pompeiu or \(T\)-operator can be generalized to the Clifford setting without any difficulties. It is very useful in the Clifford just as in the complex case. The shifting of the \(\Pi\)-operator from the complex plane to the Clifford algebra \(C\) was used sometimes, e.g. \textit{W. Sprößig} [Dissertation TH Karl-Marx-Stadt (1979)]. The authors analyze very carefully the properties of this Clifford \(\Pi\)-operator, defining it by \(\Pi f=\overline DTf\), where \(D\) is the usual Cauchy-Riemann operator in \(C\), \(Tf\) defined as usually by the domain integral over the Cauchy kernel multiplied by the function \(f\). \(\Pi\) is a strongly singular integral operator, its mapping and invertibility properties are examined, an application to the solution of the Clifford-Beltrami equation is given.

Continuing own investigations [\textit{K. Gürlebeck} and \textit{U. Kähler}, J. Anal. Anwend. 15, No. 2, 283-297 (1996; Zbl 0864.30039)] the author deals with the properties of several operators in Clifford analysis. Mainly he investigates the Teodorescu transform and the Cauchy type operator, the latter in a domain and on the boundary, a generalized \(\pi\)-operator is defined and some of its properties are proved. A main result gives connections between the invertibility of the \(\pi\)-operator and some orthogonal decompositions of \(L_2\). Several relations between the various types of operators are given. We consider the wreath product of two permutation groups \(G\leq\) Sym \(\Gamma \) and \(H\leq\) Sym \(\Delta \) as a permutation group acting on the set \(\Pi \) of functions from \(\Delta \) to \(\Gamma \). Such groups play an important role in the O'Nan-Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let \(X\) be a subgroup of Sym \(\Gamma \) \(\wr\) Sym \(\Delta \). Our main result is that, in a suitable conjugate of \(X\), the subgroup of Sym \( \Gamma \) induced by a stabiliser of a coordinate \(\delta \in \Delta \) only depends on the orbit of \(\delta \) under the induced action of \(X\) on \(\Delta \). Hence, if \(X\) is transitive on \(\Delta \), then \(X\) can be embedded into the wreath product of the permutation group induced by the stabiliser \(X_{\delta }\) on \(\Gamma \) and the permutation group induced by \(X\) on \(\Delta \). We use this result to describe the case where \(X\) is intransitive on \(\Delta \) and offer an application to error-correcting codes in Hamming graphs.

For the determination of the diffusion coefficient, in the inverse problem on medicine's substances diffusion from polymer films, a universal method does not exist. Therefore, the solution is usually found sorting out a series of direct problems on some algorithm with the minimization of a chosen criterion for the deviation of designed data from experimental ones. In this resolving process, it is required to solve the direct problems with high accuracy. With this aim, the authors use the Galerkin method with discontinuous basic functions [\textit{B. Cockburn}, Lect. Notes Math. 1697, 151--268 (1998; Zbl 0927.65120)]. We study the Runge-Kutta discontinuous Galerkin method for numerically solving nonlinear hyperbolic systems and its extension for convection-dominated problems, the so-called local discontinuous Galerkin method. Examples of problems to which these methods can be applied are the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semi-conductor device simulation. The main features that make the methods under consideration attractive are their formal high-order accuracy, their nonlinear stability, their high parallelizability, their ability to handle complicated geometries, and their ability to capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The purpose of these notes is to provide a short introduction to the devising and analysis of these discontinuous Galerkin methods.

For the determination of the diffusion coefficient, in the inverse problem on medicine's substances diffusion from polymer films, a universal method does not exist. Therefore, the solution is usually found sorting out a series of direct problems on some algorithm with the minimization of a chosen criterion for the deviation of designed data from experimental ones. In this resolving process, it is required to solve the direct problems with high accuracy. With this aim, the authors use the Galerkin method with discontinuous basic functions [\textit{B. Cockburn}, Lect. Notes Math. 1697, 151--268 (1998; Zbl 0927.65120)]. For a given matrix \(H\) which has \(d\) singular values larger than \(\varepsilon \), an expression for all rank-\(d\) approximants \(\widehat H\) such that \((H - \widehat H)\) has 2-norm less than \(\varepsilon\) is derived. These approximants have minimal rank, and the set includes the usual `truncated singular value decomposition' low-rank approximation. The main step in the procedure is a generalized Schur algorithm, which requires only \(O (1/2 m^ 2n)\) operations (for an \(m \times n\) matrix \(H)\). The column span of the approximant is computed in this step, and updating and downdating of this space is straightforward. The algorithm is amenable to parallel implementation.

Let \(X\) be a real smooth linear normed space, that is, there exists the limit \[ n'(x;y):=\lim_{t\rightarrow 0}\frac{\| x+ty\|^2-\| x\|^2}{2t} \] for all \(x,y\in X\). Using a Euclidean property of heights of a triangle in [\textit{T. Precupanu} and \textit{I. Ionică}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 55, No.~1, 35--47 (2009; Zbl 1199.46066)], it is established that, in a smooth real linear normed space \(X\) with \(\dim X\geq 3\), the norm is Hilbertian if and only if \[ n'(y;x)[n'(x;y)n'(x-y;x)-\| x \|^2 n'(x-y;y)]=n'(x;y)[\| y\|^2 n' (x-y;x)-n'(y;x)n'(x-y;y)] \] for all \(x,y\in X\). In the present paper, the author shows that this result is also true if \(\dim X=2\). To this end, the author uses the fact that \(n'(x;\cdot)\) is a linear functional for any fixed \(x\in X\) and that there exist two real functions \(A,B\) in two variables and homogeneous of degree one such that \(n'(x;y)=A(x_1,x_2)y_1+B(x_1,x_2)y_2\) and \(\| x \|^2=A(x_1,x_2)x_1 + B(x_1,x_2)x_2\) for all \(x=(x_1,x_2)\) and \(y=(y_1,y_2)\). Finally, the differentiability of the norm is exploited. The paper under review studies some geometric properties of inner product spaces using norm derivatives. The authors characterize Hilbertian norms via symmetry of norm derivatives as well as the problem of triangle heights with respect to Birkhoff's orthogonality.

Let \(X\) be a real smooth linear normed space, that is, there exists the limit \[ n'(x;y):=\lim_{t\rightarrow 0}\frac{\| x+ty\|^2-\| x\|^2}{2t} \] for all \(x,y\in X\). Using a Euclidean property of heights of a triangle in [\textit{T. Precupanu} and \textit{I. Ionică}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 55, No.~1, 35--47 (2009; Zbl 1199.46066)], it is established that, in a smooth real linear normed space \(X\) with \(\dim X\geq 3\), the norm is Hilbertian if and only if \[ n'(y;x)[n'(x;y)n'(x-y;x)-\| x \|^2 n'(x-y;y)]=n'(x;y)[\| y\|^2 n' (x-y;x)-n'(y;x)n'(x-y;y)] \] for all \(x,y\in X\). In the present paper, the author shows that this result is also true if \(\dim X=2\). To this end, the author uses the fact that \(n'(x;\cdot)\) is a linear functional for any fixed \(x\in X\) and that there exist two real functions \(A,B\) in two variables and homogeneous of degree one such that \(n'(x;y)=A(x_1,x_2)y_1+B(x_1,x_2)y_2\) and \(\| x \|^2=A(x_1,x_2)x_1 + B(x_1,x_2)x_2\) for all \(x=(x_1,x_2)\) and \(y=(y_1,y_2)\). Finally, the differentiability of the norm is exploited. The paper deals with controlling systems of the form \[ \begin{aligned} \dot x_i & =x_{i+1},\;i=1,\dots,n-1\\ \dot x_n & =f(x_1,\dots,x_n) +g(x_1,\dots, x_n)u(t),\;y=x_1 ,\end{aligned} \] in particular with stabilizing the equilibria. Asymptotic stability is analyzed by a suitable Lyapunov function constructed using the back-stepping method. A third-order example is given together with simulation results.

\textit{A. Gyárfás} [Fruit salad, Electron. J. Comb. 4, Research paper R8 (1997; Zbl 0885.05103); printed version J. Comb. 4, 65-72 (1997)] conjectured that a graph in which each path spans a \(3\)-chromatic subgraph is \(k\)-colorable, for a constant \(k\) (possibly \(k= 4\)). The authors prove that such a graph \(G\) is colorable with \(3\cdot\lfloor\lg_c|V(G)|\rfloor\) colors for a suitable constant \(c = 8/7\). Paul Erdős liked fruit salad. I mixed this one for him from ingredients obtained while working on some of his problems. He was pleased by it and carried it to several places to offer to others as well. It is very sad that I have to add to the manuscript: dedicated to his memory.

\textit{A. Gyárfás} [Fruit salad, Electron. J. Comb. 4, Research paper R8 (1997; Zbl 0885.05103); printed version J. Comb. 4, 65-72 (1997)] conjectured that a graph in which each path spans a \(3\)-chromatic subgraph is \(k\)-colorable, for a constant \(k\) (possibly \(k= 4\)). The authors prove that such a graph \(G\) is colorable with \(3\cdot\lfloor\lg_c|V(G)|\rfloor\) colors for a suitable constant \(c = 8/7\). Let \(K\) be a compact subset of \([0,2\pi]\), and consider a triangular array \(X\) of interpolation points in \(K\). The author characterizes those arrays \(X\) for which a \(p>0\) exists such that, for every continuous function \(f\) on \(K\) (with \(f\) being \(2\pi\)-periodic if \(K=[0,2\pi])\), the trigonometric Lagrange interpolants converge to \(f\) in (weighted) \(L_p(K)\). A similar characterization is given for the convergence of polynomial interpolants in the Hardy space \(H_p\) for some \(p>0\) and for all \(f\) belonging to the disk algebra.

\textit{S. V. Konyagin} [Math. Notes 44, No. 6, 910--920 (1988; Zbl 0688.42003); translation from Mat. Zametki 44, No. 6, 770--784 (1988)] proved that a trigonometric series cannot converge to \(\infty\) on a set of positive measure. Note that Haar and Walsh series have the same property.\par In this paper, an orthonormal polynomial spline system \(\{f_n\}_{n=-k+2}^{\infty}\) of order \(k\) with respect to a dyadic partition of \([0,\,1]\) is considered. For \(k=2\), the orthonormal linear spline system is the Franklin system. Let \(\sigma_m(x)\) be the \((2^m + k +1)\)th partial sum of the Ciesielski series \[\sum_{n=-k+2}^{infty} a_n\, f_n(x)\,.\] Then the authors show that the measure of \(\{x \in [0,\,1] : \, \lim_{m\to \infty} \sigma_m(x) = \infty\}\) is zero. The author proves the following main theorem: for any trigonometric series: \(\sum^{\infty}_{n=0}c_ n \cos (nx+\theta_ n)\), let \(S_ n(x)=\sum^{n}_{m=0}c_ m \cos (mx+\theta_ m)\) and \(\bar S(x)= \limsup_{n\to \infty}S_ n(x)=F(x)\), \(\bar S(x)= \limsup_{n\to \infty}S_ n(x)=g(x)\), then \(mes\{x\in [-\pi,\pi]:\) \(-\infty <\underline S(x)\leq S(x)=+\infty \}=0.\) In particular \(mes\{x\in [-\pi,\pi]:\) \(\lim S(x)=\infty \}=0.\) This theorem is closely related to the work of \textit{D. E. Menchoff} [C. R. Acad. Sci. URSS, Nov. Ser. 26, 214-216 (1940; Zbl 0023.02901)]. By the end of this paper the author poses four interesting problems related to the above theorem.

\textit{S. V. Konyagin} [Math. Notes 44, No. 6, 910--920 (1988; Zbl 0688.42003); translation from Mat. Zametki 44, No. 6, 770--784 (1988)] proved that a trigonometric series cannot converge to \(\infty\) on a set of positive measure. Note that Haar and Walsh series have the same property.\par In this paper, an orthonormal polynomial spline system \(\{f_n\}_{n=-k+2}^{\infty}\) of order \(k\) with respect to a dyadic partition of \([0,\,1]\) is considered. For \(k=2\), the orthonormal linear spline system is the Franklin system. Let \(\sigma_m(x)\) be the \((2^m + k +1)\)th partial sum of the Ciesielski series \[\sum_{n=-k+2}^{infty} a_n\, f_n(x)\,.\] Then the authors show that the measure of \(\{x \in [0,\,1] : \, \lim_{m\to \infty} \sigma_m(x) = \infty\}\) is zero. In this paper, we consider coupled wave equation of Kirchhoff type in a noncylindrical domain. This work is devoted to prove the existence and uniqueness of global solutions and decay for the energy of solutions.

The main result of the author's previous work [ibid. 31, No. 4, 1105-1114 (1991; Zbl 0798.22006)] is generalized to the case of more than countably infinite index set \(A\) of nonabelian \(G_ \alpha\), by using direct integral of non-separable Hilbert spaces. In particular, let \(G = \prod_{\alpha \in A}' G_ \alpha\) be the restricted direct product of finite groups \(G_ \alpha\). Let \(d\) be the cardinality of \(A^{non} = \{\alpha \in A \mid G_ \alpha \text{ is non-commutative}\}\). Then there exist \(2^ d\) mutually disjoint irreducible decompositions of the left regular representation \(\ell\) of \(G\) of type \[ \{\ell, L^ 2(G)\} = {^{\{\Phi_ h\}}} \int^ \oplus_ \Xi \{{^{\{F_ h(\xi)\}}}\int^ \oplus_{K(\xi)} \{\rho_{\xi,k}^{{\mathcal O}(\xi)}, {\mathbf V}(\xi,{\mathcal O} (\xi);k)\} d \nu_ \xi(k)\} d\mu (\xi). \] It is not hard to see that the restricted direct product \(G = \prod_{\alpha \in A}' G_ \alpha\) of a countably infinite family of finite groups \(G_ \alpha\), \(\alpha \in A\), with discrete topology, in general is not of type I. Its representations have therefore not a unique decomposition into irreducible components. The author studies decompositions of the left regular representation \(L\). He decomposes firstly \(L\) into factor representations. Each component of the central decomposition of \(L\) is an infinite tensor product of an irreducible representation \(\xi_ \alpha\) of \(G_ \alpha\) in a Hilbert space of finite dimension \(d(\xi_ \alpha)\). The reference vector of this infinite tensor product is a sequence of identity operators in the algebra \(B(\xi_ \alpha) = \text{End }V(\xi_ \alpha)\). Thus the factors are of type II\(_ 1\) if \(\dim B(\xi_ \alpha) \geq 2\) for infinitely many \(\alpha \in A\) or of type I otherwise. The second step of decomposing these factors into direct integrals of irreducible representations essentially depends on choice of orthogonal basis of \(V(\xi_ \alpha)\). The author constructs continuously many, mutually disjoint irreducible decompositions of the left regular representations \(L\) into irreducible components.

The main result of the author's previous work [ibid. 31, No. 4, 1105-1114 (1991; Zbl 0798.22006)] is generalized to the case of more than countably infinite index set \(A\) of nonabelian \(G_ \alpha\), by using direct integral of non-separable Hilbert spaces. In particular, let \(G = \prod_{\alpha \in A}' G_ \alpha\) be the restricted direct product of finite groups \(G_ \alpha\). Let \(d\) be the cardinality of \(A^{non} = \{\alpha \in A \mid G_ \alpha \text{ is non-commutative}\}\). Then there exist \(2^ d\) mutually disjoint irreducible decompositions of the left regular representation \(\ell\) of \(G\) of type \[ \{\ell, L^ 2(G)\} = {^{\{\Phi_ h\}}} \int^ \oplus_ \Xi \{{^{\{F_ h(\xi)\}}}\int^ \oplus_{K(\xi)} \{\rho_{\xi,k}^{{\mathcal O}(\xi)}, {\mathbf V}(\xi,{\mathcal O} (\xi);k)\} d \nu_ \xi(k)\} d\mu (\xi). \] We study the existence of equilibria and approximate equilibria avoiding any assumption of convexity both for the domain and for the bifunction. Our approach is based on the concept of cyclic monotonicity for bifunctions. First, we exploit this notion to obtain an Ekeland's variational principle for bifunctions which leads to the existence of approximate solutions of the so-called Minty equilibrium problem. Then, we prove the existence of equilibria in compact and noncompact settings. We introduce a new notion as a key tool for deriving a Minty's lemma avoiding the use of convexity.

``In this paper we propose a new public-key encryption scheme that is based on \textit{M. O. Rabin}'s trapdoor one-way permutation [Digital signatures and public key functions as intractable as factorization. Technical Report MIT/LCS/TR-212, Massachusetts Institute of Technology (January 1979)]. We can prove that the security of our scheme against adaptive chosen-ciphertext attacks (CCA security) is equivalent to the factoring assumption. Furthermore, the scheme is practical as its encryption performs only roughly two, and its decryption roughly one modular exponentiation. To the best of our knowledge, this is the first scheme that simultaneously enjoys those two properties.'' The highly readable introduction contains, moreover, more interesting information, a.o. on the history of the problem, random oracle schemes, details of the authors' construction, details of proof, and the efficiency. The result of this paper were announced by the first two authors in [Advances in cryptology -- EUROCRYPT 2009. 28th annual international conference on the theory and applications of cryptographic techniques. Lect. Notes Comput. Sci. 5479, 313--332 (2009; Zbl 1239.94052)]. We propose a practical public-key encryption scheme whose security against chosen-ciphertext attacks can be reduced in the standard model to the assumption that factoring is intractable.

``In this paper we propose a new public-key encryption scheme that is based on \textit{M. O. Rabin}'s trapdoor one-way permutation [Digital signatures and public key functions as intractable as factorization. Technical Report MIT/LCS/TR-212, Massachusetts Institute of Technology (January 1979)]. We can prove that the security of our scheme against adaptive chosen-ciphertext attacks (CCA security) is equivalent to the factoring assumption. Furthermore, the scheme is practical as its encryption performs only roughly two, and its decryption roughly one modular exponentiation. To the best of our knowledge, this is the first scheme that simultaneously enjoys those two properties.'' The highly readable introduction contains, moreover, more interesting information, a.o. on the history of the problem, random oracle schemes, details of the authors' construction, details of proof, and the efficiency. The result of this paper were announced by the first two authors in [Advances in cryptology -- EUROCRYPT 2009. 28th annual international conference on the theory and applications of cryptographic techniques. Lect. Notes Comput. Sci. 5479, 313--332 (2009; Zbl 1239.94052)]. The operad associated to \(n\)-ary algebras (algebras with \(n\)-ary partially associative multiplication) \(p\text{Ass}_0^n\) for the first time were studied by \textit{A. V. Gnedbaye} [Contemp. Math. 202, 83--113 (1997; Zbl 0880.17003)]. The main result of the author is the proof of non-Koszulity of this operad where \(n = 3\). For this the dual operad of \(p\text{Ass}_0^3\) is defined, which extends the definition of Ginzburg and Kapranov given in the case of binary operations. The author gives examples of this construction. It is proved, that the dual operad of the \(p\text{Ass}_0^3\) is the operad of totally associative algebras with the operation of degree 1. From this fact the main result is deduced.

In his youth, Paul Adrien Maurice Dirac (1902--1985), who won the Nobel Price in physics in 1933 (together with Erwin Schrödinger), happened to be somewhat of a solitary figure, with one exception: his classmate Herbert Charles Wiltshire (1902--1968) at his young years, at school and later at Bristol University. As to this all, view [\textit{G. Farmelo}, The strangest man. The hidden life of Paul Dirac, mystic of the atom. New York, NY: Basic Books; London: Faber and Faber (2009; Zbl 1230.01028)]. The aim of the paper under review is to describe the life and whereabouts of Wiltshire, more or less contrary those things of Dirac. Wiltshire made a career all over the world in electrical engineering (for instance, in Argentina, in England, in the U.S.A., and in particular after World War 2 at Cranfield, the department of aircraft economics and production). Later again, Wiltshire made visits to several universities in the U.S.A., such as Columbia University, Harvard, Cornell, M.I.T., etc. The author has given a fine overview of Wiltshire's life and work, with nice details. The (later in time) restoration of the contacts between Wiltshire and Dirac is also mentioned in the paper. The reader should take knowledge of the bibliography. Let me finish with quoting paraphrased parts of the author's conclusion: ``Paul Dirac left a legacy in the foundations of Quantum Mechanics, but his classmate took a different route. Wiltshire left behind a flourishing Work Study school with an international reputation at Cranfield. It has been unique in British higher education as an institution concerned only with postgraduate work, but one that had no recognized university status. It received a Royal Charter in 1969 (one year after the death of Wiltshire). Its name was changed to `The Cranfield Institute of Technology' and nowadays it is the Cranfield University.'' Of all the people who had visited [Bohr's] institute, Dirac was `the strangest man'.'' This assessment by Niels Bohr casts light on one of the most influential characters of 20th-century physics (and mathematics), Paul A. M. Dirac. But it gives only a partial idea of how Dirac appeared to his contemporaries. The opinion of the theoretician John Slater, as described in the book under review, helps to delineate the figure of Dirac: ``In Slater's view, there are two types of theoretical physicist. The first consists of people like himself, `the prosaic, pragmatic, matter-of-fact sort, who [\(\dots\)] tries to write or speak in the most comprehensible manner possible'. The second was `the magical, or hand-waving type, who like a magician, waves his hand as if he were drawing a rabbit out of a hat, and who is not satisfied unless he can mystify his readers or hearers'. For Slater and many others, Dirac was a magician.'' Why this opinion? In the beautiful book by Graham Farmelo, this is explained in a very detailed way, although the account is never boring. The biographical notes begin with a picture of Dirac as a student in engineering that reveals the roots of subsequent anecdotes: ``True to form, Dirac strode ahead in mathematics and was a `student who got all the answers exactly right, but who had not the faintest idea of how to deal with apparatus'. Not only was he maladroit, his mind was on other things: he spent much of his time in the physics library, reflecting on the fundamentals of science.'' Or, in the words of a first-hand witness: ``there came to these lectures one whose shoe-laces I was not worthy to unloose. This was Dirac, then a very young student, whose budding genius had been recognized by the department of engineering and was in the process of being fostered by the department of mathematics.'' Among his teachers in Cambridge who realized quite soon his precocious genius, we find Ralph H. Fowler, who ``gauged the ability of his new student by asking him to tackle a nontrivial but tractable problem: to find a theoretical description of the breaking up of the molecules of gas in a closed tube whose temperature gradually changes from one end to the other. Some five months later, when Dirac found the solution, he wanted to file it away and forget it, a suggestion that dismayed Fowler: `if you're not going to write your work up, you might as well shut up shop!' Dirac succumbed and forced himself to learn the art of writing academic articles. Words did not come easily to him, but he gradually developed the style for which he was to become famous, a style characterised by directness, confident reasoning, powerful mathematics, and plain English. Dirac's curriculum in mathematics ranged from projective geometry to Grassmann algebra--two topics that revealed their influence in later achievements--but, according to Farmelo, it was the theory of relativity that had impressed Dirac since his youth. And, when later in his life he heard the news from Princeton that Einstein had died, his tears clearly testified to Dirac's estimation for such a piece of science and its discoverer: ``It was for a hero, not a friend, that Dirac shed those tears. During those first hours of grief, he may have recalled his student days in Bristol when he first became acquainted with relativity theory, which inspired him to be a theoretician. What mattered most to Dirac were Einstein's science, his individualism, his indifference to orthodoxy and the ability he demonstrated later in life to ignore his critics' catcalls, muted only by timidity and cowardice. Dirac's most famous contribution in mathematics was the invention of the delta function, a mathematical construction that made no sense within conventional mathematics at that time. ``Dirac knew but did not care that pure mathematicians would regard the function as preposterous as it did not behave according to the usual rules of mathematical logic. He conceded that the function was not `proper' but added blithely that one can use it `as though it were a proper function for practically all purposes in quantum mechanics without getting incorrect results'. It was not until the late 1940s that mathematicians accepted the function as a concept of unimpeachable respectability. Nevertheless, Dirac astonished physicists as well with his unconventional skills in mathematics, and a delicious anecdote reported by Farmelo illustrates this well. ``Once, he gave a devastating performance in a game that had been introduced in Göttingen in 1929. The challenge was to express any whole number using the number 2 precisely four times, and using only well-known mathematical symbols. The first few numbers are easy: \(1=(2+2)/(2+2)\), \(2=(2/2)+(2/2)\), \(3=(2\times2)-(2/2)\), \(4=2+2+2-2\). Soon, the game becomes much more difficult, even for Göttingen's finest mathematical minds. They spent hundreds of hours playing the game with ever-higher numbers---until Dirac found a simple and general formula enabling any number to be expressed using four 2s, entirely within the rules. He had rendered the game pointless. Dirac's simple but powerful mathematical logic was revealed to physicists when he introduced the system of bra and ket symbols, which enabled the formulae of quantum mechanics to be written with a special neatness and concision. The subject of quantum mechanics was, indeed, the first where Dirac gave a major contribution in physics, and not only for its symbology. From the present book, we learn that he greatly benefitted from his early mathematical studies: ``[Projective geometry] was most useful for research, but I did not mention it in my published work [\(\dots\)] because I felt that most physicists were not familiar with it. When I had obtained a particular result, I translated it into an analytic form and put down the argument in terms of equations.'' Dirac's synthesis of quantum mechanics can be admired in his famous book The principles of quantum mechanics, which, according to Farmelo, ``had been written with no regard to his readers' intellectual shortcomings, without the slightest sign of emotion, with not a single leavening metaphor or simile. For Dirac, the quantum world was not like anything else people experience, so it would have been misleading to include comparisons with everyday behaviour.'' Among the admirers of such a book we find--unexpectedly--Einstein, who regarded it as ```the most logically perfect presentation of quantum theory'[\(\dots\)] [He] often took it on vacation for leisure reading and, when he came across a difficult quantum problem, would mutter to himself, `Where is my Dirac?''' Dirac's most famous contribution to physics remains, however, the Dirac equation for quantum relativistic spin \(1/2\) particles. Although no significant letter or record of conversations with anyone exists accounting for the genesis of that equation, Farmelo is nonetheless able to give an illuminating anecdote about it: Dirac ``broke his silence only before he set off to Bristol for the Christmas vacation when he bumped into his friend Charles Darwin, a grandson of the great naturalist and one of Britain's leading theoretical physicists. On Boxing Day, in a long letter to Bohr, Darwin wrote: `[Dirac] has now got a completely new system of equations for the electron which does the spin right in all cases and seems to be the thing.' The name of Dirac is, however, also associated to many other ``things'', ranging from the introduction in physics of two of its best-known technical terms, namely fermion and boson (denoting particles following the Fermi-Dirac or the Bose-Einstein statistics, respectively), to his study of the gravitational field's energy, which indicated that ``it is delivered in separate quanta, which he called gravitons, a long-neglected term first introduced a quarter of a century before in the journal Under the Banner of Marxism. After Dirac reintroduced the name, it stuck. These particles will be much harder to detect than photons, he pointed out, but experimenters should lose no time in beginning the hunt for them.'' The introduction of the concept of gravitational quanta followed another well-known contribution by Dirac, that is, quantum electrodynamics, which came from his efforts to find a quantum version of Maxwell's classical electromagnetism. Here the problem was mainly that of describing properly the processes of creation and annihilation of particles (electrons, positrons and photons). Dirac ``associated each creation with a mathematical object, a creation operator, which is closely related to but quite distinct from another object associated with annihilation, an annihilation operator''. Later in his life, he was also involved in solving one of the pathologies of quantum electrodynamics---that related to negative-energy photons---by using the technical device of an indefinite metric. Quite interesting, and unexpected for many readers (including the reviewer), is the fact that during the Second World War Dirac himself also contributed significantly to a practical project regarding nuclear isotope separation. ``He tried to find a general theory of all processes that might separate isotopic mixtures [\(\dots\)], aiming to deduce the conditions that would most effectively separate them. To solve the problem, he had to use all his talents: the mathematician's analytical skills, the theoretician's penchant for generalisation and the engineer's insistence on producing useful results. Intriguing as well was Dirac's repugnance for the theory of renormalization, aimed to cure the appearance of infinities in quantum gauge theories. ``Despite the success of the technique, Dirac abominated it, partly because he could see no way of visualising its mathematics but mainly because he felt that the process of renormalisation was artificial, an inelegant way of sweeping the fundamental problems of [the] theory under the carpet. In his opinion, a fundamental theory of nature must be beautiful, whereas renormalisation seemed to Dirac's taste to be as devoid of beauty as the dissonances of Arnold Schönberg.'' In order to give a viable alternative, Dirac produced a new theory where point-like particles were replaced by tiny, one-dimensional things that he called strings: applied to quantum electrodynamics, his theory gave the same results as the renormalized version. ``Dirac had found what he was seeking: `a model in which a bare electron is inconceivable, because the end of a piece of string is inconceivable without the string'. But it was only the germ of an idea, not a complete new theory. Several of his students examined it but soon set it aside, as Dirac did soon afterwards. Years later, it would transpire that he had once again been ahead of his time.'' Dirac's aversion for the theory of renormalization was, however, not just a whim of his own, but rather a manifestation of what he firmly believed in, that is, the principle of mathematical beauty. ``The success of relativity and quantum mechanics illustrates the value of the principle of mathematical beauty, Dirac said. In each case, the mathematics involved in the theory is more beautiful than the mathematics of the theory it superseded. He even speculated that mathematics and physics will eventually become one, `every branch of pure mathematics having its physical application, its importance in physics being proportional to [its] interest in mathematics'. So he urged theoreticians to take beauty as their principal guide, even though this way of coming up with new theories `has not yet been applied successfully'.'' We can certainly agree with Farmelo that, as a physicist, Dirac had been well served by mathematics, and his own words illuminate the work of many present-day researchers: ``If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until this happened. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.'' The discreet strangest man still makes his voice heard.

In his youth, Paul Adrien Maurice Dirac (1902--1985), who won the Nobel Price in physics in 1933 (together with Erwin Schrödinger), happened to be somewhat of a solitary figure, with one exception: his classmate Herbert Charles Wiltshire (1902--1968) at his young years, at school and later at Bristol University. As to this all, view [\textit{G. Farmelo}, The strangest man. The hidden life of Paul Dirac, mystic of the atom. New York, NY: Basic Books; London: Faber and Faber (2009; Zbl 1230.01028)]. The aim of the paper under review is to describe the life and whereabouts of Wiltshire, more or less contrary those things of Dirac. Wiltshire made a career all over the world in electrical engineering (for instance, in Argentina, in England, in the U.S.A., and in particular after World War 2 at Cranfield, the department of aircraft economics and production). Later again, Wiltshire made visits to several universities in the U.S.A., such as Columbia University, Harvard, Cornell, M.I.T., etc. The author has given a fine overview of Wiltshire's life and work, with nice details. The (later in time) restoration of the contacts between Wiltshire and Dirac is also mentioned in the paper. The reader should take knowledge of the bibliography. Let me finish with quoting paraphrased parts of the author's conclusion: ``Paul Dirac left a legacy in the foundations of Quantum Mechanics, but his classmate took a different route. Wiltshire left behind a flourishing Work Study school with an international reputation at Cranfield. It has been unique in British higher education as an institution concerned only with postgraduate work, but one that had no recognized university status. It received a Royal Charter in 1969 (one year after the death of Wiltshire). Its name was changed to `The Cranfield Institute of Technology' and nowadays it is the Cranfield University.'' An approximate boundary condition, which renders consideration of the field surrounding an optical fibre unnecessary, is derived. A comparison with the exact theory for a circular fibre shows that the approximation yields good results when the frequency is not too low.

\textit{B. Daniel} [Trans. Am. Math. Soc. 361, No.~12, 6255--6282 (2009; Zbl 1213.53075)], gave a necessary and sufficient condition for an \(n\)-dimensional Riemannian manifold to be isometrically immersed in \(S^n \times {\mathbb R}\) or \(H^n \times {\mathbb R}\) in terms of its first and second fundamental forms and some other conditions. In the present paper the authors generalise this to higher codimensions. The necessary and sufficient conditions are the three equations of Gauss, Codazzi and Ricci, together with two conditions on the vector field on \(M\) which is obtained by projection of the vector field \(\partial/\partial t\) associated to the \({\mathbb R}\)-factor in the target space. The author gives a necessary and sufficient condition for an \(n\)-dimensional, simply-connected Riemannian manifold with prescribed shape operator \(S\) to admit an isometric immersion into \(S^n\times\mathbb R\) or \(H^n\times\mathbb R\), where \(S^n\) and \(H^n\) are the \(n\)-dimensional sphere and hyperbolic space, respectively, in terms of the (prescribed) horizontal and normal projections \(T\) and \(\nu\) of \({\partial\over\partial t}\), where \(t\) is the ``height'' of the hypersurface immersion (Thm 3.3). He then uses this result to obtain an analogue of the usual associated family of minimal surfaces in \(\mathbb R^3\), obtained by rotating the shape operator, for minimal surfaces in \(S^2\times\mathbb R\) and \(H^2\times\mathbb R\) (Thm 4.2).

\textit{B. Daniel} [Trans. Am. Math. Soc. 361, No.~12, 6255--6282 (2009; Zbl 1213.53075)], gave a necessary and sufficient condition for an \(n\)-dimensional Riemannian manifold to be isometrically immersed in \(S^n \times {\mathbb R}\) or \(H^n \times {\mathbb R}\) in terms of its first and second fundamental forms and some other conditions. In the present paper the authors generalise this to higher codimensions. The necessary and sufficient conditions are the three equations of Gauss, Codazzi and Ricci, together with two conditions on the vector field on \(M\) which is obtained by projection of the vector field \(\partial/\partial t\) associated to the \({\mathbb R}\)-factor in the target space. The groups \(QF\), \(QT\), \(\bar{Q}T\), \(\bar{Q}V\) and \(QV\) are groups of quasiautomorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups \(F\), \(T\) and \(V\). We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type \(F_{\infty}\). Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.

This paper deals with \(Q\)-states Potts model with Kac potential on \({\mathbb Z}^d\) \((Q \geq 3\), \(d \geq 2)\). In this model, the Hamiltonian in a finite region \( \Lambda \) with boundary conditions \(\sigma_{\Lambda^c}\) is given by \[ H_{\gamma,\Lambda}(\sigma_{\Lambda}| \sigma_{\Lambda^c}) = -\frac{1}{2}\sum_{i,j\in\Lambda}\gamma^dJ(\gamma(i-j)){\mathbf 1}_{\{\sigma(i)=\sigma(j)\}} - \sum_{i\in\Lambda, j\in \Lambda^c}\gamma^dJ(\gamma(i-j)){\mathbf 1}_{\{\sigma(i)=\sigma(j)\}}, \] where \( J(x) \) is supposed to be a spherically symmetric nonnegative function supported by the unit ball with bounded derivative. The result is: There exists \( \widetilde\gamma > 0\) such that for all \(\gamma \in (0, \widetilde\gamma)\), there exists \(\beta_c(\gamma)\) at which there are \(Q+1\) mutually independent Gibbs states. It indicates the first order phase transition of the model for sufficient large but finite range of interaction. This is contrast to the case of \( Q=3\), \(d=2 \) nearest neighbor interaction, for which available results show the second order phase transition. The method of the paper is based on the idea of perturbing around the mean field model (corresponding the limit \(\gamma \to 0\)) [\textit{J. L. Lebowitz} and \textit{O. Penrose}, J. Math. Phys. 7, 98--113 (1966; Zbl 0938.82520)] and the Pirogov-Sinai approach. The authors study the statistical mechanics of a classical system of particles in \(\nu\) dimensions when the interaction is given by a pair potential of the form \(q({r})+\gamma^\nu\varphi(\gamma{r})\). The problem of the limit \(\gamma\rightarrow 0\) is given particular attention. This problem had been investigated in detail for a particular one-dimensional system by \textit{M. Kac, G. E. Uhlenbeck} and \textit{P. Hemmer} [same J. 4, 216-228 (1963; Zbl 0938.82507)]. Let \(\rho\) be the density, \(a^0(\rho)\) the free energy density for pair potential \(q({r})\), \(a(p,\gamma)\) the free energy density for pair potential \(q({r})+\gamma^\nu\varphi(\gamma{r})\) and \(\alpha=\int\varphi({r}) d{r}\). It is asserted that, for suitable \(\varphi\) (e.g., \(\varphi\leq 0\) or \(\varphi\) of positive type), \(\lim_{\gamma\rightarrow 0}a(p,\gamma)\) exists and is the maximal convex function of \(\rho\) not exceeding \(a^0(p)+\frac 1{2}\alpha p^2\). A behaviour analogous to that of a first-order phase transition may thus arise. In the course of the proof, upper and lower bounds to \(a(p,\gamma)\), valid for \(\gamma\neq 0\), are obtained.

This paper deals with \(Q\)-states Potts model with Kac potential on \({\mathbb Z}^d\) \((Q \geq 3\), \(d \geq 2)\). In this model, the Hamiltonian in a finite region \( \Lambda \) with boundary conditions \(\sigma_{\Lambda^c}\) is given by \[ H_{\gamma,\Lambda}(\sigma_{\Lambda}| \sigma_{\Lambda^c}) = -\frac{1}{2}\sum_{i,j\in\Lambda}\gamma^dJ(\gamma(i-j)){\mathbf 1}_{\{\sigma(i)=\sigma(j)\}} - \sum_{i\in\Lambda, j\in \Lambda^c}\gamma^dJ(\gamma(i-j)){\mathbf 1}_{\{\sigma(i)=\sigma(j)\}}, \] where \( J(x) \) is supposed to be a spherically symmetric nonnegative function supported by the unit ball with bounded derivative. The result is: There exists \( \widetilde\gamma > 0\) such that for all \(\gamma \in (0, \widetilde\gamma)\), there exists \(\beta_c(\gamma)\) at which there are \(Q+1\) mutually independent Gibbs states. It indicates the first order phase transition of the model for sufficient large but finite range of interaction. This is contrast to the case of \( Q=3\), \(d=2 \) nearest neighbor interaction, for which available results show the second order phase transition. The method of the paper is based on the idea of perturbing around the mean field model (corresponding the limit \(\gamma \to 0\)) [\textit{J. L. Lebowitz} and \textit{O. Penrose}, J. Math. Phys. 7, 98--113 (1966; Zbl 0938.82520)] and the Pirogov-Sinai approach. Es wird durch Aufstellung der Gleichung der vier Verzweigungspunkte der Weyr'sche Satz (cf. die Note des Herrn Weyr zu Herrn Le Paige's Abhandl. über die singulären Elemente cubischer Involutionen; hier im Jahrbuche besprochen auf p. 484, JFM 12.0484.03) bewiesen, dass es immer zwei cubische Involutionen mit gemeinsamen Doppelpunkten giebt. Sollen die cubischen Involutionen auf einer cubischen Raumcurve liegen, und sind vier Doppelpunkte als vier Punkte der Raumcurve gegebene, so erhält man die beiden cubischen Involutionen dadurch, dass man in den gegebenen vier Punkte die vier Tangenten zieht, die beiden Strahlen \(g\) und \(h\) zieht, deren jeder diese vier Tangenten schneidet, und die beiden einstufigen Systeme von Punkttripeln aufsucht, welche auf der Raumcurve durch die beiden Ebenenbüschel ausgeschnitten werden, deren Träger \(g\) und \(h\) sind. Die zweimal vier Verzweigungspunkte sind dann einfache Schnittpunkte der durch \(g\) und \(h\) an die Raumcurve gelegten Tangentialebenen. Dem Beweise des Weyr'schen Satzes folgen noch andere algebraische Untersuchungen über die cubischen Involutionen, welche den Verfasser in einfaster Art zu genometrischen Darstellung vieler Invarianten von algebraischen Formen führen.

The concept of sign-central matrices was introduced by \textit{T. Ando} and \textit{R. A. Brualdi} [Linear Algebra Appl. 208-209, 283-295 (1994; Zbl 0818.15017)]. A \((0,1,-1)\)-matrix is sign-central if all matrices with the same sign pattern is central (i.e. the origin belongs to the convex hull of its columns). In the present paper the notion of nearly sign-central matrix is defined. A matrix \(A\) is said to be nearly sign-central if there is a nonzero sign pattern vector \(\alpha\) such that the matrix \([A\alpha]\) is sign-central. Some properties of these matrices are discussed and the main result (Theorem 2.7) gives a characterization of nearly sign-central matrices. Let \(A = (a_{ij})\) be a real \(m \times n\) matrix. The sign pattern of \(A\) is the \(m \times n\) \((0,1, -1)\) matrix \(\text{sign} A = (\text{sign} a_{ij})\), where \(\text{sign} a = 0\) (if \(a = 0)\), \(+ 1\) (if \(a > 0)\), and \(- 1\) (if \(a < 0)\). The qualitative class of \(A\) is the set \(Q(A)\) of all real matrices with the same sign pattern as \(A\). Let \(C(B)\) be the convex polytope determined by the columns of \(B\). The matrix \(B\) is said to be central if the convex polytope \(C(B)\) contains the origin of \(\mathbb{R}^ m\). The matrix \(A\) is said to be sign-central, if each matrix \(B\) in \(Q(A)\) is central. An \(m \times (m + 1)\) matrix \(A\) is an \(S\)-matrix if for each matrix \(B\) in \(Q(A)\), \(C(B)\) is a simplex containing the origin in its interior relative to the real space \(\mathbb{R}^ m\). In this paper the authors give a more transparent and natural proof of the combinatorial characterization given by \textit{G. V. Davydov} and \textit{I. M. Davydova} [Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 9(340), 85-88 (1990; Zbl 0725.90072)]. Under some minimality assumption, they investigate minimal sign-central matrices and given some methods for their construction. In particular, they show that the number of columns of a minimal sign-central matrix with \(m\) nonzero rows is between \(m + 1\) and \(2^ m\), and that those with \(m + 1\) columns are precisely the \(m \times (m + 1)\) \(S\)-matrices.

The concept of sign-central matrices was introduced by \textit{T. Ando} and \textit{R. A. Brualdi} [Linear Algebra Appl. 208-209, 283-295 (1994; Zbl 0818.15017)]. A \((0,1,-1)\)-matrix is sign-central if all matrices with the same sign pattern is central (i.e. the origin belongs to the convex hull of its columns). In the present paper the notion of nearly sign-central matrix is defined. A matrix \(A\) is said to be nearly sign-central if there is a nonzero sign pattern vector \(\alpha\) such that the matrix \([A\alpha]\) is sign-central. Some properties of these matrices are discussed and the main result (Theorem 2.7) gives a characterization of nearly sign-central matrices. This paper considers off-line synthesis of stabilizing static feedback control laws for discrete-time piecewise affine (PWA) systems. Two of the problems of interest within this framework are: (i) incorporation of the \(\mathcal{S}\)-procedure in synthesis of a stabilizing state feedback control law and (ii) synthesis of a stabilizing output feedback control law. Tackling these problems via (piecewise) quadratic Lyapunov function candidates yields a bilinear matrix inequality at best. A new solution to these problems is proposed in this work, which uses infinity norms as Lyapunov function candidates and, under certain conditions, requires solving a single linear program. This solution also facilitates the computation of piecewise polyhedral positively invariant (or contractive) sets for discrete-time PWA systems.

The authors study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, where the latttice is not necessarily regular (two-dimensional graphs are admitted). The main goal of the paper is to propose a general setting accommodating many models of interest and prove a general result concerning the decay of correlations. In section 6 a proof is given of the McBryan-Spencer-Koma-Tasaki theorem concerning algebraic decay of correlations, generalizing earlier works on this subject, [\textit{A. Naddaf}, Commun. Math. Phys. 184, No. 2, 387--395 (1997; Zbl 0873.60086); \textit{T. Koma} and \textit{H. Tasaki}, ``Decay of superconducting and magnetic correlations in one- and two-dimensional Hubbard models'', Phys. Rev. Lett. 68, No. 21, 3248--3251 (1992; \url{doi:10.1103/PhysRevLett.68.3248})]. This main theorem (Theorem 6.1), formulated in the context of general models with \(\mathrm{U}(1)\) symmetry, makes various other theorems of Section 2--4 of the paper, its straightforward applications. Exemplary models are the Heisenberg, Hubbard, and t-J models, and certain models of random loops. In the introduction, a brief exposition is given of earlier results on the absence of coontinuous symmetry breaking, that of the absence of long range order and incongruencies between the logarithmic decay of two-point correlations and the expected -- by various reasons -- power-law decay. An open problem remains to find a proof for bosonic systems, like, e.g., the Bode-Hubbard model, where the present method does not generalize in a straightforward way. . We extend the method of complex translations which was originally employed by \textit{O. A. McBryan} and \textit{T. Spencer} [ibid. 53, 299-302 (1977)] to obtain a decay rate for the two point function in two-dimensional \(SO(n)\)-symmetric models with non-analytic Hamiltonians for \(n\geq 2\).

The authors study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, where the latttice is not necessarily regular (two-dimensional graphs are admitted). The main goal of the paper is to propose a general setting accommodating many models of interest and prove a general result concerning the decay of correlations. In section 6 a proof is given of the McBryan-Spencer-Koma-Tasaki theorem concerning algebraic decay of correlations, generalizing earlier works on this subject, [\textit{A. Naddaf}, Commun. Math. Phys. 184, No. 2, 387--395 (1997; Zbl 0873.60086); \textit{T. Koma} and \textit{H. Tasaki}, ``Decay of superconducting and magnetic correlations in one- and two-dimensional Hubbard models'', Phys. Rev. Lett. 68, No. 21, 3248--3251 (1992; \url{doi:10.1103/PhysRevLett.68.3248})]. This main theorem (Theorem 6.1), formulated in the context of general models with \(\mathrm{U}(1)\) symmetry, makes various other theorems of Section 2--4 of the paper, its straightforward applications. Exemplary models are the Heisenberg, Hubbard, and t-J models, and certain models of random loops. In the introduction, a brief exposition is given of earlier results on the absence of coontinuous symmetry breaking, that of the absence of long range order and incongruencies between the logarithmic decay of two-point correlations and the expected -- by various reasons -- power-law decay. An open problem remains to find a proof for bosonic systems, like, e.g., the Bode-Hubbard model, where the present method does not generalize in a straightforward way. . An overview is given of a number of recent developments in SAT and SAT Modulo Theories (SMT). In particular, based on our framework of Abstract DPLL and Abstract DPLL Modulo Theories, we explain our DPLL(T) approach to SMT. Experimental results and future projects are discussed within BarcelogicTools, a set of logic-based tools developed by our research group in Barcelona. At the 2005 SMT competition, BarcelogicTools won all four categories it participated in (out of the seven existing categories).

Let \(a\) be a positive integer. Let \(R\) be a ring such that any rational prime \(p\leq a\) is invertible in \(R\). For example \(R\) is a ring of characteristic greater than \(a\). The result is a construction of an explicit pairing \(\langle\cdot,\cdot\rangle\) from \(\text{Sym}^a R^2\times \text{Sym}^a R^2\) to \(R\), where \(\text{Sym}^a R^2\) is the \(a\)th symmetric algebra of the ring \(R\), with the property that \(\langle x\alpha,y\alpha\rangle=\det(\alpha)^a\langle x,y\rangle\) for all two-by-two matrices \(\alpha\) with entries in \(R\), with the induced right action on the symmetric power module. The proof uses elementary linear algebra. The result is used by \textit{R. Taylor} [Invent. Math. 98, No. 2, 265--280 (1989; Zbl 0705.11031)] in the context of associating Galois representations to modular eigenforms of weight \(a + 2\), where \(R\) is a field of characteristic greater than \(a\). The author studies the conjecture concerning 2-dimensional Galois representations over totally real number fields \(F\) attached to Hilbert cusp forms \(f\) over \(F\), which are Hecke eigenforms. This conjecture was already established for \(F=\mathbb Q\) (Eichler, Shimura, Deligne, Deligne--Serre respectively for weight 2, \(\geq 2\), 1) and for fields of odd degree \((F:\mathbb Q)\) by Ohta and, using Shimura curves, by Rogawski-Tunnell. In case of even degree \((F:\mathbb Q)\) however, additional assumptions on the automorphic representation \(\pi_ f\) attached to the form \(f\) were needed. The content of this paper is a proof of the conjecture for arbitrary totally real fields \(F\) of even degree, in case the weights of the forms are \(\geq 2\). The author proceeds along the lines of Wiles' approach, who had proved the conjecture for even fields \(F\), in case the form is ordinary at the relevant prime. The main part of the paper is the proof of congruences between the given form \(f\) and a suitable newform, for which the conjecture is already established. Then using Wiles' method of `pseudo-representations' the desired representation is constructed.

Let \(a\) be a positive integer. Let \(R\) be a ring such that any rational prime \(p\leq a\) is invertible in \(R\). For example \(R\) is a ring of characteristic greater than \(a\). The result is a construction of an explicit pairing \(\langle\cdot,\cdot\rangle\) from \(\text{Sym}^a R^2\times \text{Sym}^a R^2\) to \(R\), where \(\text{Sym}^a R^2\) is the \(a\)th symmetric algebra of the ring \(R\), with the property that \(\langle x\alpha,y\alpha\rangle=\det(\alpha)^a\langle x,y\rangle\) for all two-by-two matrices \(\alpha\) with entries in \(R\), with the induced right action on the symmetric power module. The proof uses elementary linear algebra. The result is used by \textit{R. Taylor} [Invent. Math. 98, No. 2, 265--280 (1989; Zbl 0705.11031)] in the context of associating Galois representations to modular eigenforms of weight \(a + 2\), where \(R\) is a field of characteristic greater than \(a\). I incorporate an exchange rate target zone with intramarginal interventions in a small open economy model. Using the method of undetermined coefficients, I solve for the price level and the nominal exchange rate to determine how price shocks from the large economy affect the small open economy. The results show that the behaviour of inflation transmission within the band differs from the behavior of inflation transmission at the edge of the band of the target zone. Foreign shocks can affect local prices in both cases but the central bank can respond through market interventions within the band while it cannot do so at the edge. Near the edge of the band, a central bank has to intervene to stop the exchange rate from breaching the band. My model predicts that if the interventions are robust, then the exchange rate is mean reverting and an exchange rate target zone can insulate an economy from foreign price shocks. Based on the model, central bank interventions contribute to long-run price stability in a target zone regime. Finally, I empirically test the model using unit root and cointegration tests, and present some policy implications.