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Let \(X\) be a complete, connected and finitely connected Alexandrov space without boundary of dimension two whose curvature is bounded below by a constant and admitting total curvature. Denote by \(F_p\) the union of all rays in \(X\) emanating from the point \(p\in X\) and by \(A(p)\) the set of the directions of rays at \(p\). One of the two results of this paper is the following: Every \(p\in X\) has the property that \(\mu(A(p))= 2\pi\chi (X)-C(X-F_p)\), where \(\mu(A(p))\) is the Lebesgue measure of \(A(p)\), \(\chi(X)\) is the Euler characteristic of \(X\) and \(C(U)\) denotes the total excess of the subset \(U\subset X\). The Riemannian case of this was proved by \textit{K. Shiga} [Tsukuba J. Math. 6, 41-50 (1982; Zbl 0531.53035)]. A well-known theorem of Cohn-Vossen says that for a non-compact complete surface M the following inequality holds \(\int K\leq 2\pi e(M)\), provided the total curvature \(\int K\) exists. Here K is the Gaussian curvature and e(M) is the Euler characteristic of M. In this paper the author proves that 2\(\pi\) e(M)-\(\int K\) is an upper bound for the measure of all initial unit vectors of all the geodesic rays (length minimising) emanating from a given point. He also proves that if M is homeomorphic to \({\mathbb{R}}^ 2\), then 2\(\pi\)-\(\int K_+\) is a lower bound for this measure, where \(K_+=\max(K,0)\). The method of proof is application of the Gauss-Bonnet formula to a well chosen sequence of compact domains exhausting M. The proof relies on classical ideas of Cohn-Vossen and Busemann.

Let \(X\) be a complete, connected and finitely connected Alexandrov space without boundary of dimension two whose curvature is bounded below by a constant and admitting total curvature. Denote by \(F_p\) the union of all rays in \(X\) emanating from the point \(p\in X\) and by \(A(p)\) the set of the directions of rays at \(p\). One of the two results of this paper is the following: Every \(p\in X\) has the property that \(\mu(A(p))= 2\pi\chi (X)-C(X-F_p)\), where \(\mu(A(p))\) is the Lebesgue measure of \(A(p)\), \(\chi(X)\) is the Euler characteristic of \(X\) and \(C(U)\) denotes the total excess of the subset \(U\subset X\). The Riemannian case of this was proved by \textit{K. Shiga} [Tsukuba J. Math. 6, 41-50 (1982; Zbl 0531.53035)]. Verschiedene Entwicklungen von Funktionen nach Ableitungen nichtganzer Ordnung.

The authors determine all integers \(x\) such that \(x^2-1\) has only prime factors smaller than 100. They write \(x^2-dy^2=1\), and solve the Pell equation by first computing the regulator of the ring of integers of the corresponding quadratic field and then from the regulator obtaining a compact representation of the fundamental solution. For errata to this paper see ibid. 83, No. 285, 337 (2014; Zbl 1285.11057). Errata to [Math. Comput. 80, No. 273, 429--435 (2011; Zbl 1221.11080)]. In Theorem 1e the number 16167 should be 16223. Odd and even \(x-s\) missing from the previous tables are supplied.

The authors determine all integers \(x\) such that \(x^2-1\) has only prime factors smaller than 100. They write \(x^2-dy^2=1\), and solve the Pell equation by first computing the regulator of the ring of integers of the corresponding quadratic field and then from the regulator obtaining a compact representation of the fundamental solution. For errata to this paper see ibid. 83, No. 285, 337 (2014; Zbl 1285.11057). The authors give a complete characterization of the asymptotic behaviour as \(t \to \infty\) of non-negative solutions \(u = u(x,t)\) of the equation \(u_t - \Delta u + f(u) = 0\), where the spatial variable runs through the whole space \(\mathbb{R}^N\), \[ f(u) = u + \sum_{j=1}^m b_ju^{r_j} - \sum_{i=1}^n a_iu^{p_i}, \quad 1 < r_j < p_i \leq N/(N - 2), \;a_i,b_j > 0. \] Using a recent result of \textit{C.-C. Chen} and \textit{C.-S. Lin} [Commun. Partial Differ. Equations 16, No. 8/9, 1549-1572 (1991; Zbl 0753.35034)] on the uniqueness (up to a spatial shift) of the least energy solution (the so-called ground state) \(w_g\) to the stationary equation \(- \Delta w + f(w) = 0, \;w\not \equiv 0, x \in \mathbb{R}^N\), they prove: for any non-negative function \(\bar u\) with a compact support in \(\mathbb{R}^N\), \(\bar u \not \equiv 0\) there exists a constant \(\alpha_c = \alpha_c(\bar u) > 0\) such that the solution \(u\) with \(u(0) = \alpha_c\bar u\) converges to a spatial shift of the ground state \(w_g\). If \(u(0) = \alpha \bar u\) and \(0 \leq \alpha < \alpha_c\) then \(u\) tends to zero or, in case \(\alpha > \alpha_c\), \(u\) blows up at a finite time. The proof is based on a combination of the concentrated compactness and comparison theorems for parabolic equations.

Kripke semantics for predicate modal logic provides a translation of the formulas of this logic into classical predicate logic. Attempts to use this translation directly for automated deduction were hopeless: too many irrelevant clauses were generated. The author proposed earlier [Lect. Notes Comput. Sci. 310, 500-516 (1988; Zbl 0647.03010)] a more sophisticated version of the translation and has shown how to include term rewriting to make proof search more efficient for most popular logics. The paper under review presents a translation in a more general situation and gives some advice on strategies of resolution and paramodulation, but no new results on the use of term rewriting. [For the entire collection see Zbl 0638.00038.] A resolution calculus for the quantified versions of the modal logics T, S4, S5, B, D, D4 and DB is presented. It presupposes a syntax transformation, similar to the skolemization in predicate logic, that eliminates the modal operators from modal logic formulae and shifts the modal context information to the term level. The formulae in the transformed syntax can be translated into conjunctive normal form such that a clause based modal resolution calculus is definable without any additional inference rule, but with special modal unification algorithms. The method can be applied to first-order modal logics with the two operators \(\square\) and \(\diamond\) and with standard constant-domain possible worlds semantics with flexible constant and function symbols, where the accessibility relation is serial and may have any combination of the following properties; reflexivity, symmetry, transitivity. Extensions to other systems, especially the nonserial ones K, K4 and KB are possible.

Kripke semantics for predicate modal logic provides a translation of the formulas of this logic into classical predicate logic. Attempts to use this translation directly for automated deduction were hopeless: too many irrelevant clauses were generated. The author proposed earlier [Lect. Notes Comput. Sci. 310, 500-516 (1988; Zbl 0647.03010)] a more sophisticated version of the translation and has shown how to include term rewriting to make proof search more efficient for most popular logics. The paper under review presents a translation in a more general situation and gives some advice on strategies of resolution and paramodulation, but no new results on the use of term rewriting. We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate. As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.

Let \({\mathcal C}_n\) be a local quasi-analytic subring of the ring of germs of \(C^\infty\) functions on \(\mathbb R^n\) and let \({\mathcal C}=\{{\mathcal C}_n,\;n\in\mathbb N\}.\) Let us denote by \(\hat{\cdot}:{\mathcal C}_n \rightarrow\mathbb R[[x_1,\dots,x_n]]\) the map which associates to each \(f\in{\mathcal C}_n\) its Taylor expansion at the origin. Assume that, for each \(n\in\mathbb N,\) \({\mathcal C}_n\) is closed upon taking derivatives, \({\mathcal C}_n\) is closed under division by coordinates and \({\mathcal C}_n\) contains the ring of polynomials \(\mathbb R[x_1,\dots,x_n]\). Moreover suppose that the system \({\mathcal C}\) is closed under composition and the implicit function theorem holds. Let \(\varphi=(\varphi_1,\dots,\varphi_k)\) with \(\varphi_i\in {\mathcal C}_n,\) \(i=1,\dots,k,\) and \(\varphi_i(0)=0\). The author considers the morphism \(\phi:{\mathcal C}_k\rightarrow{\mathcal C}_n\) defined by composition of \(\varphi\) and its extension to the completion \(\hat{\phi}:\mathbb R[[y_1,\dots,y_k]]\rightarrow \mathbb R[[x_1,\dots,x_n]].\) The main result of the paper is the following: if for each \(\varphi\) as above with generic rank rk\((\varphi)=k\) and for each \(\psi\in\mathbb R[[y_1,\dots,y_k]]\) so that \(\hat{\phi}(\psi)=\hat{g},\) with \(g\in {\mathcal C}_n,\) there exists \(\beta\in {\mathcal C}_k\) such that \(\hat{\beta}=\psi,\) then \({\mathcal C}_n\subset \mathbb R\{x_1,\dots,x_n\},\) i.e. the system is analytic. The proof uses the results of \textit{A. Elkhadiri} and \textit{H. Sfouli} [Stud. Math. 185, No.~1, 83--86 (2008; Zbl 1144.26032)] in an essential way. The paper is devoted to an extension of the result by \textit{L. C. Childress} [Can. J. Math. 28, 938--953 (1976; Zbl 0355.32009)] to an arbitrary quasianalytic ring of germs of smooth functions. Namely, it is considered a system \(({\mathcal C}_n)_{n\in {\mathbb N}}\) of quasianalytic subrings of the ring of of germs, at the origin of \({\mathbb R}^n\), of \({\mathcal C}^{\infty}\)-functions, such that each \({\mathcal C}_n\) contains the local ring of germs of Nash functions, and is closed under taking partial derivatives, as well as the whole system \(({\mathcal C}_n)_{n\in {\mathbb N}}\) is closed under composition. It is proved that if Weierstrass division theorem holds for \({\mathcal C}_3\), then for each \(n\in {\mathbb N}\), \({\mathcal C}_n\) is contained in the ring of germs, at the origin of \({\mathbb R}^n\), of real analytic functions.

Let \({\mathcal C}_n\) be a local quasi-analytic subring of the ring of germs of \(C^\infty\) functions on \(\mathbb R^n\) and let \({\mathcal C}=\{{\mathcal C}_n,\;n\in\mathbb N\}.\) Let us denote by \(\hat{\cdot}:{\mathcal C}_n \rightarrow\mathbb R[[x_1,\dots,x_n]]\) the map which associates to each \(f\in{\mathcal C}_n\) its Taylor expansion at the origin. Assume that, for each \(n\in\mathbb N,\) \({\mathcal C}_n\) is closed upon taking derivatives, \({\mathcal C}_n\) is closed under division by coordinates and \({\mathcal C}_n\) contains the ring of polynomials \(\mathbb R[x_1,\dots,x_n]\). Moreover suppose that the system \({\mathcal C}\) is closed under composition and the implicit function theorem holds. Let \(\varphi=(\varphi_1,\dots,\varphi_k)\) with \(\varphi_i\in {\mathcal C}_n,\) \(i=1,\dots,k,\) and \(\varphi_i(0)=0\). The author considers the morphism \(\phi:{\mathcal C}_k\rightarrow{\mathcal C}_n\) defined by composition of \(\varphi\) and its extension to the completion \(\hat{\phi}:\mathbb R[[y_1,\dots,y_k]]\rightarrow \mathbb R[[x_1,\dots,x_n]].\) The main result of the paper is the following: if for each \(\varphi\) as above with generic rank rk\((\varphi)=k\) and for each \(\psi\in\mathbb R[[y_1,\dots,y_k]]\) so that \(\hat{\phi}(\psi)=\hat{g},\) with \(g\in {\mathcal C}_n,\) there exists \(\beta\in {\mathcal C}_k\) such that \(\hat{\beta}=\psi,\) then \({\mathcal C}_n\subset \mathbb R\{x_1,\dots,x_n\},\) i.e. the system is analytic. The proof uses the results of \textit{A. Elkhadiri} and \textit{H. Sfouli} [Stud. Math. 185, No.~1, 83--86 (2008; Zbl 1144.26032)] in an essential way. The dynamical equation of the slider-crank mechanism is established by using Lagrange equation and Newton's second law. The slider-crank mechanism with an open crack rod is investigated and then establishes the equivalent mechanics model by a massless torsional spring to simulate the influence of the crack in the rod, and the mechanism of a cracked rod is divided into two subsystems. The dynamical equation of the slider-crank mechanism with a crack rod is established. Comparing the dynamic analysis results between with and without crack in the rod, the results show that the existence of the crack leads to a great change in the motion characteristics of the slider. The calculated maximum Lyapunov exponent is positive, which shows that the movement of the slider in the crank slider mechanism with a cracked rod is chaotic.

In the paper under review the author discusses the thesis of \textit{B. Vitrac} (1993) mentioned in the title. Recently \textit{B. Vitrac} published a French translation of Euclid's Elements [Euklid. Les Eléments. Volume I. Livres I--IV: Géométrie plane. Traduits du texte grec de Heiberg et avec commentaires par Bernard Vitrac. Paris (1990; Zbl 0755.51003); Volume II. Livres V--VI: Proportions et similitude. Livres VII--IX: Arithmétique. Paris (1994)]. Vols. III and IV in preparation. The book is the first part of the prepared four books devoted to Euclid's Elements. The reviewed book is written by Bernard Vitrac, who translated the Greek text of I. L. Heiberg and gave extensive commentaries and numerous foot-notes to it. The book is devoted to the first four Euclid's books i.e. to the plaine geometry without the theory of proportion. For every book the author gives definitions, propositions enriched by commentaries and notices. The book contains a comprehensive and very interesting introduction (136 pp.) written by Maurice Caveing. In the introduction are given numerous informations on the Euclidean tradition in ancient Greece, the influence of the Elements of Greek mathematics, the history of the Books as well as of its translations. The analysis of the form of the Elements is added, too. The appendix contains tables and deductive schemes of the Elements elaborated by I. Mueller. The reviewed book is excellent elaborated and will be valuable for all those who are interested in the history of mathematics.

In the paper under review the author discusses the thesis of \textit{B. Vitrac} (1993) mentioned in the title. Recently \textit{B. Vitrac} published a French translation of Euclid's Elements [Euklid. Les Eléments. Volume I. Livres I--IV: Géométrie plane. Traduits du texte grec de Heiberg et avec commentaires par Bernard Vitrac. Paris (1990; Zbl 0755.51003); Volume II. Livres V--VI: Proportions et similitude. Livres VII--IX: Arithmétique. Paris (1994)]. Vols. III and IV in preparation. Fast creating of the present territory, a few of the wonders are broadly seen to flourishing the significant fulfillment to the behavior of the information to the punter. The prerequisite from client satisfaction is made as a design to achieve sociopsycho-economic knowledge. The substance structure approach gathers the informational collection to accomplish the objective of consumer necessity. The system contains the client fulfillment to online market conduct to watch the customer request on every single tick of client, based on every single arrangement of snap and hunt information, thereby constructing the model. The representation supports the sociopsycho behavior to watch the request limits of the customers with few ways to achieve the state of a client. Earlier frameworks do not thoroughly investigate the human sociopsycho based Web-based promoting. The designed structure provides investigating the human sociopsycho completely with the assistance of soft set theoretic hypothesis technique. The strategy assesses every single substance of the client request and investigates it.

This is the continuation of the two earlier works of the first author [ibid. 4, No. 2, 153-177 (1992; Zbl 0756.45004)]. Let \(\Omega \subset \mathbb{R}^n\) be an unbounded open set and let \(X\) be the space of bounded and continuous functions on \(\overline \Omega\). For \(p \geq 0\), let \(W_p (s) = (1 + |s |)^p\) and let \(X_p\) denote the weighted space \(X_p : = \{x \in X : |x |^p_\infty : = |w_p x |_\infty < \infty\}\). In the space \(X_p\), the integral equation \[ x(s) - \int_\Omega k (s,t) x(t)dt = y(s),\;s \in \overline \Omega \] is considered. An application to acoustics is given. In the space of bounded continuous functions the integral equation \(x(s)+\int^ \infty_{-\infty}k(s,t)x(t)dt=y(s)\), \(s\in(- \infty,+\infty)\), is studied. The class of equations includes an equation of Wiener-Hopf type. An application to a boundary integral equation arising in harmonic sound propagation is given.

This is the continuation of the two earlier works of the first author [ibid. 4, No. 2, 153-177 (1992; Zbl 0756.45004)]. Let \(\Omega \subset \mathbb{R}^n\) be an unbounded open set and let \(X\) be the space of bounded and continuous functions on \(\overline \Omega\). For \(p \geq 0\), let \(W_p (s) = (1 + |s |)^p\) and let \(X_p\) denote the weighted space \(X_p : = \{x \in X : |x |^p_\infty : = |w_p x |_\infty < \infty\}\). In the space \(X_p\), the integral equation \[ x(s) - \int_\Omega k (s,t) x(t)dt = y(s),\;s \in \overline \Omega \] is considered. An application to acoustics is given. Dasselbe Problem, das die höhere Mathematik durch den Begriff des Unendlichkleinen metodisch formulirt und löst, drängt sich schon in die Elementarmathematik, bei der Definition geometrischer Grundgebilde, bei Messung gleichartiger Grössen durch einander, bei der Rectification und Quadratur des Kreises etc. ein. Eine Hineinziehung höherer Reihen, der Differential- und Integralrechnung, der analytischen Geometrie etc. in den Elementarunterricht ist unthunlich. Aber das Verhältnis des Stetigen und Unstetigen zu einander muss auch hier schon bestimmt werden. Der Selbsterhaltungstrieb des Geistes verlangt die Herstellung einer Beziehung zwischen den vollkommen verschiedenen Begriffen des Stetigen und Unstetigen. durch den axiomatischen Grenzbegriff wird diese Forderung erfüllt. Der Grenzbegriff ist das discursive Correlat des in der Anschauung vorhandenen Stetigen, ja die Definition des Stetigen selbst.

The Gauss-Bonnet theorem expresses the Euler characteristic of a surface with geodesic boundary as \(2\pi\) times the integral of the Gaussian curvature. Its extension to higher dimension due to Chern involves the Pfaffian polynomial in the curvature tensor. The Euler characteristic may be interpreted as the index of the Dirac operator \(d+d^*\) acting on even-dimensional forms. Generalizations of the Gauss-Bonnet-Chern formula hold for Dirac-type operators, the index being given by the integral of a locally-defined form minus the eta-invariant of the boundary. The local integrand can be obtained as the small-time limit of the pointwise super-trace of the associated heat operator. The present paper extends the above theory to complete Riemannian manifolds \(X\) with edge metrics. Edge manifolds are interiors of a manifold with boundary \(\bar{X}\). An edge metric roughly looks like \(x^{-2}(dx^2+dy^2) +dz^2\) where \(x\) is a boundary-defining function for the boundary of \(\bar{X}\), and \(y,z\) are multi-variables on the boundary, \(y\) corresponding to the base and \(z\) to the fibers of a fibration of \(\partial \bar{X}\). For \(b\)-metrics, which are an extremal case of edge metrics in that there are no \(y\) variables, this program has been completed by \textit{R.~B.~Melrose} [The Atiyah-Patodi-Singer index theorem. Wellesley, MA: A. K. Peters, Ltd. (1993; Zbl 0796.58050)]. For \(b\)-metrics, the local integrand in the index formula is in general not integrable, while the supertrace of the heat kernel is no longer constant in time. The eta invariant arises as the variation of the heat supertrace between \(0\) and \(\infty\). In this context, the index (viewed as the supertrace of the projector onto the kernel of the Dirac operator) equals a regularized integral of the local Atiyah-Singer integrand, minus the eta invariant of the boundary. For the present case of edge metrics, the null-space of the Dirac operator may be infinite-dimensional. Albin defines the index as the regularized integral of the local supertrace of the projector onto the null-space. Then, by following Melrose's approach, he proves a regularized index theorem in which all three terms are regularized. This general theorem is then refined for the Gauss-Bonnet operator. In this case, the eta contribution vanishes, while the local integrand is the Pfaffian of the curvature. By using a soft Gauss-Bonnet theorem, the regularized integral of the Pfaffian is also identified with the Euler characteristic minus the finite part of a transgression form. The index is thus linked to the Euler characteristic. The method of proof is by constructing a heat-edge calculus in the spirit of Melrose, i.e., a space of conormal distributions on a blown-up space, which is rather involved due to the sheer number of boundary faces, for instance to prove composition one needs a triple heat-edge space with fourteen hyperfaces. The regularization procedure used here depends usually on the choice of the boundary-defining function. Further results of this paper address the independence on the boundary-defining function of the renormalized index for conformally compact manifolds, which are another particular case of edge metrics for which there are no \(z\) variables. This case is particularly interesting since it contains hyperbolic space and some hyperbolic manifolds of infinite volume. Das Atiyah-Patodi-Singer Theorem berechnet den Index eines Dirac- Operators einer kompakten Riemannschen Mannigfaltigkeit \(X\) mit Rand. Dieser Index ist durch das \(\widehat A\)-Geschlecht von \(X\) und der spektralen Asymmetrie des Dirac-Operators auf dem Rand \(\partial X\) gegeben. Rein analytische Beweise sind mit Methoden der Wärmeleitungskerne seit den 70-iger Jahren bekannt, und im Fall \(\partial X = \emptyset\) bewiesen Patodi, Gilkey und Getzler (1986) diese Formeln auch lokal. Seither gibt es viele Arbeiten, in denen die Index- und Spektraltheorie elliptischer Operatoren einerseits auf gewisse Klassen offener Mannigfaltigkeiten und andererseits auf kompakte singuläre ``Mannigfaltigkeiten'' übertragen wird. Das Ziel des vorliegenden Buches besteht darin, die hierfür benötigte Analysis im Fall einer kompakten Mannigfaltigkeit mit Ecken und Kanten bereitzustellen und die eingangs skizzierte Theorie in dieser Kategorie zu entwickeln. Dabei betrachtet der Autor kompakte Mannigfaltigkeiten \(X\) mit Rand, auf deren Inneren \(\text{Int} (X)\) eine vollständige Riemannsche Metrik \(g\) mit \(g=(dx^ 2/x)+h\) nahe des Randes \(\partial X\) vorliegt \((b\)-Metriken). Für diese Klasse von Räumen werden nun die Differentialgeometrie, die Theorie der charakteristischen Klassen, die Hodge-Theorie und die Dirac-Operatoren behandelt. Weil die Kerne \(K(x,y)\) der zu betrachtenden Operatoren auf \(X \times X\) definiert sind, wird im Kapitel 4 ein spezielles Produkt \(X^ 2_ b\to X^ 2\) mittels einer Aufblasung konstruiert. Dies beruht im wesentlichen auf der Beobachtung, daß der Grenzwert \((x-x')/(x+x')\) entlang jeder positiven Geraden \(l \subset R^ 2\) existiert. Nach diesen geometrischen Vorbereitungen entwickelt der Autor in den Kapiteln 5, 6 und 7 des Buches den vollen Kalkül der Pseudo- Differentialoperatoren, der relativen Indextheoreme sowie der Wärmeleitungskerne in der Kategorie der \(b\)-Metriken. Die abschließenden beiden Kapitel beinhalten einerseits den Beweis des lokalen Indexsatzes nach der Getzler-Methode sowie Anwendungen des APS- Theorems (Euler Charakteristik, Signatur-Theorem, Analytische Torsion). Das Buch ist sehr sorgfältig geschrieben, in sich selbst verständlich und jedem Mathematiker, der sich für die Analysis auf Räumen mit Singularitäten interessiert, sehr zu empfehlen.

The Gauss-Bonnet theorem expresses the Euler characteristic of a surface with geodesic boundary as \(2\pi\) times the integral of the Gaussian curvature. Its extension to higher dimension due to Chern involves the Pfaffian polynomial in the curvature tensor. The Euler characteristic may be interpreted as the index of the Dirac operator \(d+d^*\) acting on even-dimensional forms. Generalizations of the Gauss-Bonnet-Chern formula hold for Dirac-type operators, the index being given by the integral of a locally-defined form minus the eta-invariant of the boundary. The local integrand can be obtained as the small-time limit of the pointwise super-trace of the associated heat operator. The present paper extends the above theory to complete Riemannian manifolds \(X\) with edge metrics. Edge manifolds are interiors of a manifold with boundary \(\bar{X}\). An edge metric roughly looks like \(x^{-2}(dx^2+dy^2) +dz^2\) where \(x\) is a boundary-defining function for the boundary of \(\bar{X}\), and \(y,z\) are multi-variables on the boundary, \(y\) corresponding to the base and \(z\) to the fibers of a fibration of \(\partial \bar{X}\). For \(b\)-metrics, which are an extremal case of edge metrics in that there are no \(y\) variables, this program has been completed by \textit{R.~B.~Melrose} [The Atiyah-Patodi-Singer index theorem. Wellesley, MA: A. K. Peters, Ltd. (1993; Zbl 0796.58050)]. For \(b\)-metrics, the local integrand in the index formula is in general not integrable, while the supertrace of the heat kernel is no longer constant in time. The eta invariant arises as the variation of the heat supertrace between \(0\) and \(\infty\). In this context, the index (viewed as the supertrace of the projector onto the kernel of the Dirac operator) equals a regularized integral of the local Atiyah-Singer integrand, minus the eta invariant of the boundary. For the present case of edge metrics, the null-space of the Dirac operator may be infinite-dimensional. Albin defines the index as the regularized integral of the local supertrace of the projector onto the null-space. Then, by following Melrose's approach, he proves a regularized index theorem in which all three terms are regularized. This general theorem is then refined for the Gauss-Bonnet operator. In this case, the eta contribution vanishes, while the local integrand is the Pfaffian of the curvature. By using a soft Gauss-Bonnet theorem, the regularized integral of the Pfaffian is also identified with the Euler characteristic minus the finite part of a transgression form. The index is thus linked to the Euler characteristic. The method of proof is by constructing a heat-edge calculus in the spirit of Melrose, i.e., a space of conormal distributions on a blown-up space, which is rather involved due to the sheer number of boundary faces, for instance to prove composition one needs a triple heat-edge space with fourteen hyperfaces. The regularization procedure used here depends usually on the choice of the boundary-defining function. Further results of this paper address the independence on the boundary-defining function of the renormalized index for conformally compact manifolds, which are another particular case of edge metrics for which there are no \(z\) variables. This case is particularly interesting since it contains hyperbolic space and some hyperbolic manifolds of infinite volume. In the past, many refinements have been proposed to select equilibria in cheap talk games. Usually, these refinements were motivated by a discussion of how rational agents would reason in some particular cheap talk games. In this paper, we propose a new refinement and stability measure that is intended to predict actual behavior in a wide range of cheap talk games. According to our Average Credible Deviation Criterion (ACDC), the stability of an equilibrium is determined by the frequency and size of credible deviations. ACDC organizes the results from several cheap talk experiments in which behavior converges to equilibrium, even in cases where other criteria do not make a prediction.

The minimization method for a quadratic functional with linear constraints, which is proposed in this paper, generalizes the technique previously published by the first author (see [\textit{A. A. Tret'yakov}, Russ. J. Numer. Anal. Math. Model. 25, No. 3, 279--288 (2010; Zbl 1193.65102)]). The aim of the authors is to obtain the solution in a finite number of steps, not using a priori information. The algorithm constructs a sequence of iterative approximations of the gradient method with step doubling. Each iteration generates the set of active constraints and forms the equivalent Kuhn-Tucker system that gives the minimum condition. The iterative approximations are initial points of a sequence of projections. The final point of this sequence gives the required point of minimum after a finite number of iterations. The number of iterations at each step polynomially depends on the dimension of the problem. The author proposes a gradient projection method for solving systems of linear equations on a nonnegative orthant of the form \[ A\cdot z- b= 0_m,\qquad z\geq 0_n. \] Using the least squares method, the problems are reduced to unconstrained minimization of constrained residuals for the original or for the alternative system. The methods obtain the solution in a finite number of steps starting calculations from an arbitrary point.

The minimization method for a quadratic functional with linear constraints, which is proposed in this paper, generalizes the technique previously published by the first author (see [\textit{A. A. Tret'yakov}, Russ. J. Numer. Anal. Math. Model. 25, No. 3, 279--288 (2010; Zbl 1193.65102)]). The aim of the authors is to obtain the solution in a finite number of steps, not using a priori information. The algorithm constructs a sequence of iterative approximations of the gradient method with step doubling. Each iteration generates the set of active constraints and forms the equivalent Kuhn-Tucker system that gives the minimum condition. The iterative approximations are initial points of a sequence of projections. The final point of this sequence gives the required point of minimum after a finite number of iterations. The number of iterations at each step polynomially depends on the dimension of the problem. This paper presents a systematic discussion on the variational principles for the vibration of a piezoelectric plate. It is shown that there exist four types of variation formulations depending on the internal energy, electric enthalpy, mechanical enthalpy, and enthalpy, respectively. The formulation depending on the internal energy is in a positive definite form which leads to a few properties of the resonance frequency.

The first author has revived the theory of corings [in Algebr. Represent. Theory 5, No. 4, 389-410 (2002; Zbl 1025.16017)]. One of the main aims was the development of a Galois theory for corings, providing a new and elegant approach to Hopf-Galois theory. In the final section of his paper, the first author mentioned that a dual theory can be developed: the category of bicomodules over a coalgebra \(C\) over a field is a monoidal category. A \(C\)-ring \(\mathcal A\) is a monoid in this category. \(C\)-rings can be constructed from so-called matrix ring contexts. Matrix ring contexts are closely related to Morita-Takeuchi contexts: if one of the maps in the contexts is bijective, then one can construct a context of the other type by replacing this map by its inverse. If two coalgebras \(C\) and \(D\) are connected by a matrix ring context, then there is a \(C\)-ring structure on the cotensor product of \(C\) and \(D\). Modules over \(C\)-rings are introduced in the natural way. Then it is possible to introduce the notion of \(\mathcal A\)-coendomorphism coalgebra of a right module, quasi-finite and injective as a \(C\)-comodule. Now take a right \(\mathcal A\)-module \(M\), and assume that \(M\) is a quasi-finite injector; then there is a matrix ring context connecting \(C\) and the \(\mathcal A\)-coendomorphism coalgebra of \(M\), and there is a natural \(C\)-ring morphism \(\beta\) from \(\mathcal A\) to the associated \(C\)-ring. If \(\beta\) is an isomorphism, then \(M\) is called a Galois module for \(\mathcal A\); if, moreover, \(M\) is an injective module of the \(\mathcal A\)-coendomorphism coalgebra of \(M\), then \(M\) is called a principal Galois module. Characterizations of (principal) Galois modules are given. \(C\)-rings can be constructed starting from (weak) entwining structures, and it is investigated when \(C\) is a Galois module for this \(C\)-ring. Finally, a generalized dual version of the Kreimer-Takeuchi Theorem is given. Let \(A\) be a ring and \(\mathcal C\) an \(A\)-coring. First it is studied when the forgetful functor \(F\) from the category of right \(\mathcal C\)-comodules (\(\mathbb{M}^{\mathcal C}\)) to the category of right \(A\)-modules (\(\mathbb{M}_A\)) and the induction functor \(-\otimes_A{\mathcal C}\colon\mathbb{M}_A\to\mathbb{M}^{\mathcal C}\) are separable. Secondly, necessary and sufficient conditions for the forgetful functor \(F\) to be Frobenius are given. When \(A\) itself is a right \(\mathcal C\)-comodule (i.e. when \(\mathcal C\) has a grouplike element) it is defined the subring of coinvariants \(B\) of \(A\) and then is studied the induction functor \(-\otimes_BA\) from the category of right \(B\)-modules to the category of right \(\mathcal C\)-comodules. In this case it is also defined the notion of Galois coring and it is proved that if \(\mathcal C\) is a flat left \(A\)-module, then \(-\otimes_BA\) is an equivalence if and only if \(A\) is a faithfully flat left \(B\)-module and \(\mathcal C\) is Galois. Finally, the notion of \(C\)-ring is introduced as a dualisation of the notion of \(A\)-coring and some dual results of the previous sections are presented.

The first author has revived the theory of corings [in Algebr. Represent. Theory 5, No. 4, 389-410 (2002; Zbl 1025.16017)]. One of the main aims was the development of a Galois theory for corings, providing a new and elegant approach to Hopf-Galois theory. In the final section of his paper, the first author mentioned that a dual theory can be developed: the category of bicomodules over a coalgebra \(C\) over a field is a monoidal category. A \(C\)-ring \(\mathcal A\) is a monoid in this category. \(C\)-rings can be constructed from so-called matrix ring contexts. Matrix ring contexts are closely related to Morita-Takeuchi contexts: if one of the maps in the contexts is bijective, then one can construct a context of the other type by replacing this map by its inverse. If two coalgebras \(C\) and \(D\) are connected by a matrix ring context, then there is a \(C\)-ring structure on the cotensor product of \(C\) and \(D\). Modules over \(C\)-rings are introduced in the natural way. Then it is possible to introduce the notion of \(\mathcal A\)-coendomorphism coalgebra of a right module, quasi-finite and injective as a \(C\)-comodule. Now take a right \(\mathcal A\)-module \(M\), and assume that \(M\) is a quasi-finite injector; then there is a matrix ring context connecting \(C\) and the \(\mathcal A\)-coendomorphism coalgebra of \(M\), and there is a natural \(C\)-ring morphism \(\beta\) from \(\mathcal A\) to the associated \(C\)-ring. If \(\beta\) is an isomorphism, then \(M\) is called a Galois module for \(\mathcal A\); if, moreover, \(M\) is an injective module of the \(\mathcal A\)-coendomorphism coalgebra of \(M\), then \(M\) is called a principal Galois module. Characterizations of (principal) Galois modules are given. \(C\)-rings can be constructed starting from (weak) entwining structures, and it is investigated when \(C\) is a Galois module for this \(C\)-ring. Finally, a generalized dual version of the Kreimer-Takeuchi Theorem is given. For a positive integer \(q\), let \(L(q)\) be the set of \(k\) integers, smallest in absolute value, and symmetric about 0. A connected, simple \((p,q)\)-graph \(G = (V, E)\) is said to be super edge-graceful if there is a bijection \(f : E\to L(q)\) inducing a bijection \(f^* : V\to L(p)\) via \(f^*(u) = \sum_{\{u,v\}\in E}f(u,v)\). Let \(T(n;(a_1,a_2,\dots,a_n))\) be the tree obtained by amalgamating the path \(P_n\) at each vertex \(u_t\), \(t = 1, 2,\dots,n\), with a path of length \(a_t\). We validate a conjecture of Lee and Wei that the trees \(T(2m+3; (02^{2m+1}0))\) and \(T(2m+2; (02^{2m}1))\) are super edge-graceful by giving in each case a large lower bound, exponential in \(m\), of the number of super edge-graceful labelings of the tree.

The indicatory property was defined in [\textit{S. Watanabe} and \textit{F. Ikeda}, JP J. Geom. Topol. 4, No. 2, 141--146 (2004; Zbl 1065.53058)]. The paper studies basic properties of indicatory Riemannian spaces defined by tensor fields of type \((0,k)\), \(k\in\{2,3,4\}\). In Finsler spaces, the indicatory tensors (the angular metric tensor \(h_{ij}\), the Cartan tensor \(C_{ijk}\), the \((v)hv\)-torsion tensor \(P_{ijk}\) of the Cartan connection and the \(T\)-tensor \(T_{hijk}\) etc.) play the important role. Most of the familiar indicatory tensors of Finsler spaces vanish in the Riemannian case, but by considering the indicatorized tensor of a Riemannian tensor, we obtain indicatory tensors in Riemannian spaces. The purpose of the present paper is to study indicatory properties of special Riemannian spaces.

The indicatory property was defined in [\textit{S. Watanabe} and \textit{F. Ikeda}, JP J. Geom. Topol. 4, No. 2, 141--146 (2004; Zbl 1065.53058)]. The paper studies basic properties of indicatory Riemannian spaces defined by tensor fields of type \((0,k)\), \(k\in\{2,3,4\}\). Official trials were conducted of a number of e-voting systems in the UK in 2002/3 and 2007 during local government elections, yet none of these test systems were subsequently used in any further elections, and all trials were suspended in 2008. We describe these trials, concentrating on the second more extensive 2007 trial, and how their results were received. Based on these events, we consider the key challenges involved in introducing current e-voting systems into the present system of UK national and local elections, and what general implications this may have for achieving practical take-up of e-voting within the UK.

In [\textit{L. Pirio} and \textit{J.-M. Trépreau}, Int. Math. Res. Not. 2015, No. 13, 4449--4504 (2015; Zbl 1406.32014)], it has been proved that for \(r>1\), \(n\geq 2\) and \(d\geq (r+1)(n-1)+2\), a \(d\)-web of type \((r,n)\) with maximal rank is algebraizable in the classical sense, except maybe when \(n\geq 3\) and \(d=(r+2)(n-1)+1\). In the present paper, one considers this particular case. Under these hypotheses on \(n\) and \(d\), one constructs some examples of `\textit{exceptional algebraic webs}': these are generalized algebraic webs of maximal rank that are not algebraizable in the classical sense. Let \(n\geq 2\), \(r\geq 2\), and \(d\geq (r+1)(n-1)+2\) be integers. We show that a \(d\)-web of codimension \(r\) on a germ of variety of dimension \(rn\) is algalizable in the classical sense if it is of maximum rank, except perhaps when \(n\geq 3\) and \(d=(r+2)(n-1)+1\). In the latter case, we show that it is algebraic, but in a generalized sense.

In [\textit{L. Pirio} and \textit{J.-M. Trépreau}, Int. Math. Res. Not. 2015, No. 13, 4449--4504 (2015; Zbl 1406.32014)], it has been proved that for \(r>1\), \(n\geq 2\) and \(d\geq (r+1)(n-1)+2\), a \(d\)-web of type \((r,n)\) with maximal rank is algebraizable in the classical sense, except maybe when \(n\geq 3\) and \(d=(r+2)(n-1)+1\). In the present paper, one considers this particular case. Under these hypotheses on \(n\) and \(d\), one constructs some examples of `\textit{exceptional algebraic webs}': these are generalized algebraic webs of maximal rank that are not algebraizable in the classical sense. Unabhängig von \textit{E. M. Wright} (vgl. das nachstehende Referat), beweist Verf. -- in den Bezeichnungen des vorstehenden Referates - \[ N(k)\leqq \dfrac{k^2+k}{2}+ 1. \]

The main results of the paper are concerned around \textit{J. Herzog}'s question [see ``Homological properties of the modules of differentials'', Colecao Atas Soc. Brasileira Mat. 14 (1981)]: Let P be a regular local ring containing the rationals, let I be an ideal, and set \(R=P/I\). Then does the vanishing of the cotangent modules \(T_ i(R/P,R)\) for all \(i\geq 2\) imply that I is a complete intersection? Let I be in the linkage class of a complete intersection. Let \(\omega_ R\) denote the canonical module of R. Then it is shown: \((1)\quad I\quad is\) a Gorenstein ideal, provided \(T_ 4(R/P,R)=0\) or \(T_ 2(R/P,\omega_ R)=0\). \((2)\quad I\quad is\) a complete intersection, if in addition \(T_ 3(R/P,R)=0.\) There are similar results for upper cotangent modules, e.g., the equivalence of the following conditions is shown: \((a)\quad T^ 6(R/P,R)=T^{10}(R/P,R)=0.\quad (b)\quad R\quad is\) Gorenstein and \(T^{10}(R/P,R)=0\). \((c)\quad I\quad is\) a complete intersection. In order to prove most of his results the author uses a structure theorem for ideals in the linkage class of a complete intersection [see \textit{C. Huneke} and the author, Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003)]. Two unmixed ideals I, J in a Cohen-Macaulay local ring R are said to be linked (written \(I\sim J)\) if there is a regular sequence \(x=x_ 1,...,x_ g\) in \(I\cap J\) such that \((x):I=J\) and \((x):J=I\). Two ideals are in the same linkage class if there is a finite chain of links from one to the other; ideals in the linkage class of a complete intersection are called licci. While many properties are invariants of a linkage class (or at least of the even linkage class of an ideal), better results seem to require more control over the intermediate steps of the linking chain. The authors prove a remarkable theorem asserting that given I, there is a universal chain of links \(IS\sim L^ 1(I)\sim...\sim L^ n(I)\) in a suitable extension \(S=R[X]_{{\mathfrak m}_ R[X]}\). Universality means that if \(I\sim I_ 1\sim...\sim I_ n\) is any chain of links in R, then \((S,L^ i(I))\) is essentially a deformation of \((R,I_ i)\) for every i, that is, obtained by deformation, localization, and automorphism. In particular, since I is linked to itself in two steps each \(L^{2k}(I)\) is essentially a deformation of I. The result is an explicit construction based on the idea of generic linkage. If \(I=(f_ 1,...,f_ n)=(f)\) then \(\alpha =fX\) is a regular sequence in \(R'=R[X]\) where X is an \(n\times g\) matrix of indeterminates. Then \(L_ 1(f)=(\alpha):IR'\) is called the first generic link, and \(L^ 1(f)=(\alpha):IS\) is the first universal link. By definition \(L^ i(I)=L^ 1(L^{i-1}(I))\) for \(i>0.\) The utility of the authors' result has two main aspects. First, since it is an explicit construction, they are able to actually compute the non- Gorenstein and non-complete intersection loci of the second universal link (this is the critical step) and to prove that the defining ideals of these loci have prime radicals. This enables them to establish sharp smoothability results. Secondly, there is the philosophy, amply demonstrated here, that if (S,J) is essentially a deformation of (R,I) then S/J is ``at least as good'' as R/I. The authors prove that if R/I is Cohen-Macaulay, Gorenstein, complete intersection, \((R_ k)\), respectively, then so is S/J. The multiplicity, Cohen-Macaulay type, embedding codimension of S/J, and deviation of J are all less than or equal to the corresponding values for R/I and I. Applications are to ideals in a regular local ring, which for definiteness we shall take to be power series over a field. The authors demonstrate that with respect to smoothability all licci ideals, independent of codimension, behave either like a perfect ideal of codimension two or a Gorenstein ideal of codimension three: If R/I is not Gorenstein then (R,I) is smoothable in codimension three but not four; and if it is Gorenstein but not a complete intersection, then (R,I) is smoothable in codimension six but not seven. Further applications are to homogeneous ideals in \(R=k[X_ 1,..,X_ n]\), expressing linkage properties in terms of the twists that appear in a minimal homogeneous resolution \(0\to F_ g\to...\to F_ 1\to F_ 0\) of R/I. If \(F_ i=\otimes^{\beta_ i}_{j=1}R(-n_{ij}),\quad then\) (after localizing at the irrelevant maximal ideal) I is not licci if \(\max_ j\{n_{gj}\}\leq (g-1)d\), where \(d=\min_ j\{n_{1j}\}\). With some additional assumptions the authors also prove that the multiplicity e(R/I) is minimal over all ideals in the even linkage class of I. If, on the other hand, I is licci, \(n_{1j}=d\) for all j, and \(n_{gj}\) are also independent of j, then R/I has small class group, i.e. \(Cl(R/I)={\mathbb{Z}}[K_{R/I}]\), if the completion \((R/I)^{\wedge}\) is rigid.

The main results of the paper are concerned around \textit{J. Herzog}'s question [see ``Homological properties of the modules of differentials'', Colecao Atas Soc. Brasileira Mat. 14 (1981)]: Let P be a regular local ring containing the rationals, let I be an ideal, and set \(R=P/I\). Then does the vanishing of the cotangent modules \(T_ i(R/P,R)\) for all \(i\geq 2\) imply that I is a complete intersection? Let I be in the linkage class of a complete intersection. Let \(\omega_ R\) denote the canonical module of R. Then it is shown: \((1)\quad I\quad is\) a Gorenstein ideal, provided \(T_ 4(R/P,R)=0\) or \(T_ 2(R/P,\omega_ R)=0\). \((2)\quad I\quad is\) a complete intersection, if in addition \(T_ 3(R/P,R)=0.\) There are similar results for upper cotangent modules, e.g., the equivalence of the following conditions is shown: \((a)\quad T^ 6(R/P,R)=T^{10}(R/P,R)=0.\quad (b)\quad R\quad is\) Gorenstein and \(T^{10}(R/P,R)=0\). \((c)\quad I\quad is\) a complete intersection. In order to prove most of his results the author uses a structure theorem for ideals in the linkage class of a complete intersection [see \textit{C. Huneke} and the author, Ann. Math., II. Ser. 126, 277-334 (1987; Zbl 0638.13003)]. [For the entire collection see Zbl 0627.00012.] Let S be a metrizable Choquet simplex and E the set of all extreme points of S. Given \(x\in S\), let \(\mu_ x\) be the unique maximal measure representing the point x and let face(x) be the smallest face of S containing x. By the solution of the Dirichlet problem for a boundary function f on E we understand the affine function \(u_ f\) defined on \(D_ f=\{x\in S:f\in L\) \(1(\mu_ x)\}\) (which is always a face of S) by \(u_ f(x)=\mu_ x(f).\) The aim of this paper is to describe the points \(x\in S\) for which an analogue of the Harnack inequality is satisfied on face(x), i.e., for any compact set \(K\subset face(x)\) there is a number \(\alpha_ K\) such that for every continuous affine function f:face(x)\(\to [0,\infty)\) we have \(\sup_{y\in K}f(y)\leq \alpha_ Kf(x)\). The author proves that this is the case iff the restriction to face(x) of the solution of the Dirichlet problem is continuous for every boundary function from L \(1(\mu_ x)\).

This paper considers representations of the fundamental group \(\Gamma_g\) of the closed surface of genus \(g\) in \(\text{PSL}(2,\mathbb{R})\). Each such representation has an associated Euler class. The authors show that the Euler class of a discrete representation (i.e., one whose image is a discrete subgroup of \(\text{PSL}(2,\mathbb{R})\)) is essentially determined by its image, and that there exist discrete representations for every genus and possible Euler class. It was shown by \textit{W. M. Goldman} [Invent. Math. 93, No. 3, 557-607 (1988; Zbl 0655.57019)] that the components of Euler class \(2-2g\) and \(2g-2\) comprise the injective discrete representations; here it is shown that the non-faithful representations form a dense subset in all the other components. Let \(S\) be a closed oriented surface of genus \(g>1\) and let \(\pi\) denote its fundamental group. Let \(G\) be a semisimple Lie group. The purpose of this paper is to investigate the global properties of the space \(\Hom(\pi,G)\) of all representations \(\pi\to G\), when \(G\) is locally isomorphic to either PSL(2,\({\mathbb{C}})\) or PSL(2,\({\mathbb{R}})\). The main results of this paper may be summarized as follows: Theorem A. (i) Let \(G\) be the \(n\)-fold covering group of PSL(2,\({\mathbb{R}})\). Then the number of connected components of \(Hom(\pi,G)\) is given by the following formula: \[ 2n^{2g}+(4g-4)/n-1\quad\text{if} \quad n| 2g-2;\quad 2[(2g-2)/n]+1\quad\text{if}\quad n\nmid 2g-2. \] For example \(\Hom(\pi,\text{SL}(2,\mathbb{R}))\) has \(2^{2g+1}+2g-3\) components. (ii) Let \(G=SO(3)\) or PSL(2,\({\mathbb{C}})\). Then \(\Hom(\pi,G)\) has exactly two connected components. If \(G=SU(2)\) or SL(2,\({\mathbb{C}})\) then \(\Hom(\pi,G)\) is connected.

This paper considers representations of the fundamental group \(\Gamma_g\) of the closed surface of genus \(g\) in \(\text{PSL}(2,\mathbb{R})\). Each such representation has an associated Euler class. The authors show that the Euler class of a discrete representation (i.e., one whose image is a discrete subgroup of \(\text{PSL}(2,\mathbb{R})\)) is essentially determined by its image, and that there exist discrete representations for every genus and possible Euler class. It was shown by \textit{W. M. Goldman} [Invent. Math. 93, No. 3, 557-607 (1988; Zbl 0655.57019)] that the components of Euler class \(2-2g\) and \(2g-2\) comprise the injective discrete representations; here it is shown that the non-faithful representations form a dense subset in all the other components. Die Statik der starren Systeme in theoretischer und insbesondere analytischer Richtung ist in zwei Artikeln der Enzyklopädie von \textit{Timerding} und \textit{Jung} behandelt; daher werden in dem gegenwärtigen Referate nur die graphischen Methoden besprochen. Außerdem wird die Beschränkung auf die starren Systeme festgehalten; die auf elastische Körper bezüglichen graphischen Methoden werden in den Referaten über Elastizität und Festigkeitslehre folgen. Inhaltsübersicht. I. Grundzüge der graphischen Statik. A. Das ebene Kräftesystem. B. Anwendungen. C. Das räumliche Kräftesystem. II. Die bestimmten Fachwerke. Allgemeine Theorie. A. Ebene Fachwerke. B. Räumliche Fachwerke. III. Spezielle Fachwerksträger.

The authors study the stabilization of control systems modelled by an abstract retarded functional differential equation with infinite delay. The results reported in the paper extend those given by the first author in [Syst. Control Lett. 44, No. 1, 35--43 (2001; Zbl 0986.93057)] for the finite delay case. It is shown that if the poles of the controlled system located in the right half complex plane are finite and the translation semigroup is uniformly stable, then the stabilizing problem can be reduced to the stabilizing one for a finite-dimensional system. The equations under consideration model problems such as heat conduction in materials with memory or population dynamics for spatially distributed populations. An ample bibliography contains 51 items. We study the asymptotic stabilization of semilinear distributed parameter control systems with delay. Assuming that the semigroup of operators associated with the uncontrolled and undelayed equation is compact, we prove that a well-known rank condition, which implies the feedback stabilization for lumped systems, can be extended to this type of systems.

The authors study the stabilization of control systems modelled by an abstract retarded functional differential equation with infinite delay. The results reported in the paper extend those given by the first author in [Syst. Control Lett. 44, No. 1, 35--43 (2001; Zbl 0986.93057)] for the finite delay case. It is shown that if the poles of the controlled system located in the right half complex plane are finite and the translation semigroup is uniformly stable, then the stabilizing problem can be reduced to the stabilizing one for a finite-dimensional system. The equations under consideration model problems such as heat conduction in materials with memory or population dynamics for spatially distributed populations. An ample bibliography contains 51 items. We consider Gauss sums of the form \[ G_n(a) = \sum_{x \in \mathbb{F} _{p^m}} \chi(x^n) \] with a nontrivial additive character \(\chi \neq \chi_0\) of a finite field \(\mathbb{F} _{p^m}\) of \( p^m\) elements of characteristic \(p\). The classical bound \(| G_n(a)| \leq(n-1) p^{m/2}\) becomes trivial for \(n \geq p^{m/2} + 1\). We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on \(| G_n(a)|\) which is nontrivial for the values of \(n\) of order up to \(p^{m/2 + 1/6}\). We also show that for almost all primes one can obtain a bound which is nontrivial for the values of \(n\) of order up to \(p^{m/2 + 1/2}\).

Let \(X\) be a compact Kähler manifold of dimension \(n\). By the Hodge decomposition theorem the \(n\)-th complex de Rham cohomology group \(H^n(X,\mathbb{Z})\otimes\mathbb{C}\) of \(X\) can be written as the direct sum \(H^n(X,\mathbb{Z})\otimes\mathbb{C} =\sum_{p+q=n}H^{p,q}(X)\), where \(H^{p,q}(X)\) denotes the cohomology group of \((p,q)\)-forms of \(X\). A \((1,1)\)-form \(\omega\) on \(X\) is called a polarization of \(X\) if \([\omega]\) is the first Chern class of an ample line bundle over \(X\). The pair \((X,\omega)\) is called a polarized algebraic variety. Using \(\omega\) we can define the primitive cohomology \(P^n(X,\mathbb{C})\) and then put \(H_\mathbb{Z}=P^n(X,\mathbb{C})\cap H^n(X,\mathbb{Z})\) and \(H^{p,q}=P^n(X,\mathbb{C})\cap H^{p,q}(X)\) for \(0\leq p,q \leq n\). Set \(H =H_\mathbb{Z}\otimes\mathbb{C}\) and we call \(\{H^{p,q}\}\) the Hodge decomposition of \(H\). The authors define a filtration \(\{F^p\}\) of \(H\) by a sequence of subspaces \(0\subset F^n\subset\dots \subset F^1=H\) such that \(H^{p,q}=F^p\cap\overline F^q\), \(F^p\oplus \overline{F^{n-p+1}}=H\). Next, using the filtrations of \(H\) the authors define polarized Hodge structure and the classifying space \(D\) for the polarized Hodge structure and then Hodge bundles over the classifying space \(D\). The authors show that \(D\) can be written as a homogeneous space \(D=G/V\), where \(V\) is a compact subgroup of a semi-simple Lie group \(G\). In the case of polarized Calabi-Yau manifolds, following the result of \textit{G. Tian} [Adv. Ser. Math. Phys. 1, 629--646 (1987; Zbl 0696.53040)], the Weil-Petersson metric can be defined by the curvature of the first Hodge bundle. This implies that the Weil-Petersson metric can be defined without the detailed knowledge of the Calabi-Yau manifolds. The authors obtain, among other things, the following result: Let \(M\) be a moduli space of polarized Calabi-Yau manifolds. Let \(\omega_{WP}\) be the Kähler form to the Weil-Petersson metric. Then the metric \(\omega_\mu = \mu\cdot\omega_{WP}+\text{Ric}(\omega_{WP})\) is Kählerian for \(\mu > \dim M + 1\) and the curvature tensor of \(\omega_\mu\) is given explicitly. The authors also prove that the Ricci and the holomorphic sectional curvature are all negatively bounded by the constant \(-1/(m+4)\), where \(m\) is the complex dimension of the moduli space. [For the entire collection see Zbl 0651.00012.] It is proved that a Kähler manifold X with trivial canonical bundle has a smooth Kuranishi space, or, in other words, the deformation of X is unobstructed. According to the author, this result was first proved in Bogomolov's preprint, but with a much more complicated proof. The present proof is by constructing a vector (0,1)-form \(\phi\) (t) with the integrability condition for the deformation based on an isomorphism \(T_ x \overset \sim \rightarrow \Omega^{n-1}.\) As an application the author starts studying the period mapping for such X. It is proved that the Peterson-Weil metric on the universal polarized deformation space is the pull-back of an invariant metric on the period space D.

Let \(X\) be a compact Kähler manifold of dimension \(n\). By the Hodge decomposition theorem the \(n\)-th complex de Rham cohomology group \(H^n(X,\mathbb{Z})\otimes\mathbb{C}\) of \(X\) can be written as the direct sum \(H^n(X,\mathbb{Z})\otimes\mathbb{C} =\sum_{p+q=n}H^{p,q}(X)\), where \(H^{p,q}(X)\) denotes the cohomology group of \((p,q)\)-forms of \(X\). A \((1,1)\)-form \(\omega\) on \(X\) is called a polarization of \(X\) if \([\omega]\) is the first Chern class of an ample line bundle over \(X\). The pair \((X,\omega)\) is called a polarized algebraic variety. Using \(\omega\) we can define the primitive cohomology \(P^n(X,\mathbb{C})\) and then put \(H_\mathbb{Z}=P^n(X,\mathbb{C})\cap H^n(X,\mathbb{Z})\) and \(H^{p,q}=P^n(X,\mathbb{C})\cap H^{p,q}(X)\) for \(0\leq p,q \leq n\). Set \(H =H_\mathbb{Z}\otimes\mathbb{C}\) and we call \(\{H^{p,q}\}\) the Hodge decomposition of \(H\). The authors define a filtration \(\{F^p\}\) of \(H\) by a sequence of subspaces \(0\subset F^n\subset\dots \subset F^1=H\) such that \(H^{p,q}=F^p\cap\overline F^q\), \(F^p\oplus \overline{F^{n-p+1}}=H\). Next, using the filtrations of \(H\) the authors define polarized Hodge structure and the classifying space \(D\) for the polarized Hodge structure and then Hodge bundles over the classifying space \(D\). The authors show that \(D\) can be written as a homogeneous space \(D=G/V\), where \(V\) is a compact subgroup of a semi-simple Lie group \(G\). In the case of polarized Calabi-Yau manifolds, following the result of \textit{G. Tian} [Adv. Ser. Math. Phys. 1, 629--646 (1987; Zbl 0696.53040)], the Weil-Petersson metric can be defined by the curvature of the first Hodge bundle. This implies that the Weil-Petersson metric can be defined without the detailed knowledge of the Calabi-Yau manifolds. The authors obtain, among other things, the following result: Let \(M\) be a moduli space of polarized Calabi-Yau manifolds. Let \(\omega_{WP}\) be the Kähler form to the Weil-Petersson metric. Then the metric \(\omega_\mu = \mu\cdot\omega_{WP}+\text{Ric}(\omega_{WP})\) is Kählerian for \(\mu > \dim M + 1\) and the curvature tensor of \(\omega_\mu\) is given explicitly. The authors also prove that the Ricci and the holomorphic sectional curvature are all negatively bounded by the constant \(-1/(m+4)\), where \(m\) is the complex dimension of the moduli space. Die Substitution \(X =f(x,y,y')\), \(Y=\varphi(x,y,y')\) ordnet jeder Funktion \(y=\psi(x)\) eine Funktion \(Y\) von \(X\), die ``Konsequente'', zu; umgekehrt entspricht einer Kurve \(Y=\psi (X)\), die durch einen Punkt \(P\) geht, eine ``Antezedente'', die durch denselben Punkt geht, nämlich die durch \(P\) gehende Lösung der Differentialgleichung \[ \psi[f(x,y,y')]=\varphi(x,y,y'). \] Über die Reihe der sukzessiven Antezedenten und Konsequenten und deren Grenze stellt Verf. zwei Theoreme auf.

Let \(f:X\rightarrow X\) be a map of a continuum (i.e., a compact, connected metric space) to itself. By \(\Omega(f)\), the nonwandering points of \(f\), we mean those \(x\in X\) such that for any open neighborhood \(U\) of \(x\), there exists \(y\in U\) and \(n\in\mathbb N\) such that \(f^n(y)\in U\). We call \(x\) an eventually periodic point if for some \(m\in\mathbb N\), \(f^m(x)\) is a periodic point of \(f\). The set of such \(x\) is denoted \(EP(f)\). One says that \(X\) has the nonwandering-eventually-periodic property, abbreviated \(\Omega\)EP-property, if for every map \(f:X\rightarrow X\), \(\Omega(f)\subset\)cl\((EP(f))\). In Section 2, the authors point out that arcs and trees have the \(\Omega\)EP-property but that neither the null-comb [\textit{J. Charatonik} and \textit{A. Illanes}, Topology Appl. 144, 109--132 (2004; Zbl 1057.54025)] nor the topologist's sine curve has the \(\Omega\)EP-property. In Sections 4 and 5, the authors construct respectively the ``basic function'' \(\varphi:[0,1]\rightarrow[0,1]\) and the ``advanced function'' \(\psi:[0,1]\rightarrow[0,1]\) by modifying \(\varphi\). The latter is used to prove that neither the harmonic fan (cone over the convergent sequence) nor the topologist's sine curve has the \(\Omega\)EP-property. In Section 5 they prove: Theorem 3: Let \(X\) be a nonlocally connected continuum with a point \(p\in X\) such that \(X\) is not connected \textit{im kleinen} at \(p\). If there is an arc \(A\) in \(X\) containing \(p\), then the continuum \(X\) does not have the \(\Omega\)EP-property. Section 9 ends the paper with several questions. Let \(X\) be a metric space and let \(f: X\to X\). A point \(x\in X\) is said to be: (1) a periodic point of \(f\) if there is \(n> 0\) such that \(f^n(x)= x\); if moreover, \(f^k(x)\neq x\) for \(1\leq k< n\), then \(x\) is called a periodic point of \(f\) of period \(n\); (2) an eventually periodic point of \(f\) of period \(n> 0\) if there is \(m\geq 0\) such that \(f^m(x)\) is a periodic point of \(f\) of period \(n\); (3) an eventually periodic point of \(f\) if there is \(n> 0\) such that \(x\) is an eventually periodic point of \(f\) of period \(n\); (4) a non-wandering point of \(f\) if for every open set \(U\) containing \(x\) there is \(y\in U\) and \(n> 0\) such that \(f^n(y)\in U\). A space \(X\) is said to have the \(\Omega\)EP-property if for each continuous function \(f: X\to X\) the set of all non-wandering points of \(f\) is contained in the closure of the set of all eventually periodic points of \(f\). A tree is a graph which contains no simple closed curve. A dendrite is a locally connected and uniquely arcwise connected metric continuum. The authors extend to dendrites several results obtained earlier for trees [\textit{H. Hosaka} and \textit{H. Kato}, Topology Appl. 81, No. 1, 35--46 (1997; Zbl 0908.54023)]. For example, they prove the following Theorem. A dendrite \(X\) has the \(\Omega\)EP-property if and only if \(X\) does not contain a topological copy of the planar dendrite \(W= [0, 1]\times\{0\}\cup \bigcup\{\{1/n\}\times [0, 1/n]: n\geq 1\}\).

Let \(f:X\rightarrow X\) be a map of a continuum (i.e., a compact, connected metric space) to itself. By \(\Omega(f)\), the nonwandering points of \(f\), we mean those \(x\in X\) such that for any open neighborhood \(U\) of \(x\), there exists \(y\in U\) and \(n\in\mathbb N\) such that \(f^n(y)\in U\). We call \(x\) an eventually periodic point if for some \(m\in\mathbb N\), \(f^m(x)\) is a periodic point of \(f\). The set of such \(x\) is denoted \(EP(f)\). One says that \(X\) has the nonwandering-eventually-periodic property, abbreviated \(\Omega\)EP-property, if for every map \(f:X\rightarrow X\), \(\Omega(f)\subset\)cl\((EP(f))\). In Section 2, the authors point out that arcs and trees have the \(\Omega\)EP-property but that neither the null-comb [\textit{J. Charatonik} and \textit{A. Illanes}, Topology Appl. 144, 109--132 (2004; Zbl 1057.54025)] nor the topologist's sine curve has the \(\Omega\)EP-property. In Sections 4 and 5, the authors construct respectively the ``basic function'' \(\varphi:[0,1]\rightarrow[0,1]\) and the ``advanced function'' \(\psi:[0,1]\rightarrow[0,1]\) by modifying \(\varphi\). The latter is used to prove that neither the harmonic fan (cone over the convergent sequence) nor the topologist's sine curve has the \(\Omega\)EP-property. In Section 5 they prove: Theorem 3: Let \(X\) be a nonlocally connected continuum with a point \(p\in X\) such that \(X\) is not connected \textit{im kleinen} at \(p\). If there is an arc \(A\) in \(X\) containing \(p\), then the continuum \(X\) does not have the \(\Omega\)EP-property. Section 9 ends the paper with several questions. In this paper, we establish the \(M^2\)-type sharp maximal function estimates for the Toeplitz type operators associated to some multiplier operators. As an application, we obtain the boundedness of the operators on Lebesgue and Morrey spaces.

Actually considered is an IBV-problem for the linearized system of Navier-Stokes equations for a viscous compressible fluid in two state dimensions. The linearization is done around a steady state solution, the domain is a rectangle and its boundary is controlled. With a performance functional inspired from the article of \textit{M. D. Gunzburger} and \textit{S. Manservisi} [SIAM J. Control Optimization 39, No. 2, 594--634 (2000; Zbl 0991.49002)], one formulates an optimal control problem. The adjoint system to the linearized model is studied first and under some special assumptions on the steady state solution, the existence and uniqueness of the strong solution is proved. Further, this result is used to show the existence of a unique solution for the linearized system. Under the same assumptions on steady state solutions, the existence of an unique optimal control is proved and necessary and sufficient conditions of optimality in form of an optimality system are obtained. The authors consider optimal flow control problems where velocity is controlled by suction-blowing on part of the boundary. There are no state or control constraints and the aim is to minimize a quadratic functional measuring deviation or a target velocity and incorporating a penalization term to limit the size of the control. Results include existence, first order optimality conditions and numerics (by discretization). Among recent works on the subject, see \textit{J. Málek} and \textit{T. Roubíček} [Sequeira, Adélia (ed.) et al., Applied Nonlinear Analysis. In honor of the 70th birthday of Professor Jindřich Necǎs, New York, NY: Kluwer Academic/Plenum Publishers (ISBN 0-306-46303-2/hbk). 355-372 (1999; Zbl 0962.49017)], where, although the control is distributed, the cost functionals are more general and the necessary conditions are shown to be sufficient in certain cases. For the reduction of boundary control to distributed control even for time dependent flows see the reviewer and \textit{S. S. Sritharan} [Proc. R. Soc. Lond., Ser. A 439, No. 1, 81-102 (1992; Zbl 0786.76063)].

Actually considered is an IBV-problem for the linearized system of Navier-Stokes equations for a viscous compressible fluid in two state dimensions. The linearization is done around a steady state solution, the domain is a rectangle and its boundary is controlled. With a performance functional inspired from the article of \textit{M. D. Gunzburger} and \textit{S. Manservisi} [SIAM J. Control Optimization 39, No. 2, 594--634 (2000; Zbl 0991.49002)], one formulates an optimal control problem. The adjoint system to the linearized model is studied first and under some special assumptions on the steady state solution, the existence and uniqueness of the strong solution is proved. Further, this result is used to show the existence of a unique solution for the linearized system. Under the same assumptions on steady state solutions, the existence of an unique optimal control is proved and necessary and sufficient conditions of optimality in form of an optimality system are obtained. In this paper, we get a classification of possible values of the Ricci curvature signature of left invariant Riemannian metrics on three-dimensional Lie groups, that is a specification of some results of \textit{J. W. Milnor} [Adv. Math. 21, 293--329 (1976; Zbl 0341.53030)]. As an auxiliary problem, we classify three-dimensional nonunimodular metric Lie algebras.

This is an attractive survey paper dealing with the problem how to discretize a manifold by selecting a large finite subset of its points. Natural candidates for such good point sets are point sets that minimize a certain energy functional. Suppose that the compact manifold \(A\) is embedded in a Euclidean space. A minimal \(s\)-energy configuration is a subset \(\omega_N = \{x_1, \ldots, x_N\}\) of \(A\) that minimizes the Riesz \(s\)-energy \[ E_s(\omega_N) := \sum_{i \neq j} \frac{1}{|x_i -x_j|^s} \] among all \(N\)-point subsets of \(A\). The paper focuses on two questions. How are minimal \(s\)-energy configurations distributed for large \(N\)? How does the asymptotic behavior of the \(N\)-point minimal \(s\)-energy \[ \mathcal E_s(A,N) = \inf_{\omega_N \subset A} E_s(\omega_N) \] depends on \(A\) and \(s\)? Suppose \(A\) has Hausdorff dimension \(d\). Then for \(0 < s < d\), classical tools of potential theory can be used from which it follows that \[ \lim_{N \to \infty} \frac{\mathcal E_s(A,N)}{N^2} = I_s[\lambda_{s}] \] where \(\lambda_{s}\) minimizes the \(s\)-energy integral \[ I_s[\mu] = \iint \frac{1}{|x-y|^s} d\mu(x) d\mu(y) \] over all probability measures \(\mu\) supported on \(A\). In addition, any sequence of minimal \(s\)-energy configurations is distributed according to the measure \(\lambda_s\) as \(N \to \infty\). Recent deep results of the authors [Adv. Math. 193, No. 1, 174--204 (2005; Zbl 1192.49048)] show that the situation is quite different for \(s > d\). If \(\mathcal H_d(A)\) is the \(d\)-dimensional Hausdorff measure of \(A\) and \(\mathcal H_d(A) > 0\), then \[ \lim_{N\to \infty} \frac{\mathcal E_s(A,N)}{N^{1+s/d}} = \frac{C_{s,d}}{\mathcal H_d(A)^{s/d}} \] where \(C_{s,d}\) is a constant that is independent of \(A\). Moreover, any sequence of minimal \(s\)-energy configurations is uniformly distributed (as \(N \to \infty\)) with respect to \(d\)-dimensional Hausdorff measure on \(A\). It is an open problem to describe the constants \(C_{s,d}\). The paper contains beautiful illustrations, created by R. Womersley, of near optimal energy configurations on the sphere and the torus. The numerical experiments for the sphere indicate that the majority of points are surrounded by nearly regular hexagonal Voronoi cells. In the vast sea of regular hexagons a small number of defects occur consisting of pentagonal and heptagonal Voronoi cells. The defects appear along 12 scars centered at the vertices of an inscribed icosahedron. At present there is no rigorous explanation of this intriguing fact. For a compact set \(A\) in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) \(N\)-point configurations that minimize the Riesz \(s\)-energy (corresponding to the potential \(1/t^s\)) over all \(N\)-point subsets of \(A\), where \(s>0\). For a large class of manifolds \(A\) having finite, positive \(d\)-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as \(N\) tends to infinity with \(s\) fixed) equal to \(d\)-dimensional Hausdorff measure whenever \(s>d\) or \(s=d\). In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the \(d\)-dimensional sphere. An outline of the proof along with illustrative examples for the torus has also appeared in Notices Am. Math. Soc. 51, No. 10, 1186--1194 (2004; Zbl 1095.49031).

This is an attractive survey paper dealing with the problem how to discretize a manifold by selecting a large finite subset of its points. Natural candidates for such good point sets are point sets that minimize a certain energy functional. Suppose that the compact manifold \(A\) is embedded in a Euclidean space. A minimal \(s\)-energy configuration is a subset \(\omega_N = \{x_1, \ldots, x_N\}\) of \(A\) that minimizes the Riesz \(s\)-energy \[ E_s(\omega_N) := \sum_{i \neq j} \frac{1}{|x_i -x_j|^s} \] among all \(N\)-point subsets of \(A\). The paper focuses on two questions. How are minimal \(s\)-energy configurations distributed for large \(N\)? How does the asymptotic behavior of the \(N\)-point minimal \(s\)-energy \[ \mathcal E_s(A,N) = \inf_{\omega_N \subset A} E_s(\omega_N) \] depends on \(A\) and \(s\)? Suppose \(A\) has Hausdorff dimension \(d\). Then for \(0 < s < d\), classical tools of potential theory can be used from which it follows that \[ \lim_{N \to \infty} \frac{\mathcal E_s(A,N)}{N^2} = I_s[\lambda_{s}] \] where \(\lambda_{s}\) minimizes the \(s\)-energy integral \[ I_s[\mu] = \iint \frac{1}{|x-y|^s} d\mu(x) d\mu(y) \] over all probability measures \(\mu\) supported on \(A\). In addition, any sequence of minimal \(s\)-energy configurations is distributed according to the measure \(\lambda_s\) as \(N \to \infty\). Recent deep results of the authors [Adv. Math. 193, No. 1, 174--204 (2005; Zbl 1192.49048)] show that the situation is quite different for \(s > d\). If \(\mathcal H_d(A)\) is the \(d\)-dimensional Hausdorff measure of \(A\) and \(\mathcal H_d(A) > 0\), then \[ \lim_{N\to \infty} \frac{\mathcal E_s(A,N)}{N^{1+s/d}} = \frac{C_{s,d}}{\mathcal H_d(A)^{s/d}} \] where \(C_{s,d}\) is a constant that is independent of \(A\). Moreover, any sequence of minimal \(s\)-energy configurations is uniformly distributed (as \(N \to \infty\)) with respect to \(d\)-dimensional Hausdorff measure on \(A\). It is an open problem to describe the constants \(C_{s,d}\). The paper contains beautiful illustrations, created by R. Womersley, of near optimal energy configurations on the sphere and the torus. The numerical experiments for the sphere indicate that the majority of points are surrounded by nearly regular hexagonal Voronoi cells. In the vast sea of regular hexagons a small number of defects occur consisting of pentagonal and heptagonal Voronoi cells. The defects appear along 12 scars centered at the vertices of an inscribed icosahedron. At present there is no rigorous explanation of this intriguing fact. This study aims to develop techniques for the meaning acquisition of the NL phrases. Our approach is based on modeling meaning extraction by analyzing lexical component, syntactic component and semantic component of the phrases written or spoken in NL. To assure necessary semantic precision we adopt a double-level interpretation of the NL phrases using formal language with well defined semantics. This fact is essential for our approach.

A Catalan triangulation of the Möbius band is an abstract simplicial complex that uses no interior vertices. The authors present a generating function for the number of triangulations with \(n\) vertices analogous to that for Catalan triangulations of the disk [see, for example, \textit{D. D. Sleator}, \textit{R. E. Tarjan} and \textit{W. P. Thurston}, Rotation distance, triangulations, and hyperbolic geometry, J. Am. Math. Soc. 1, No. 3, 647-681 (1988; Zbl 0653.51017)]. They also show that any two such triangulations are connected by a sequence of diagonal flips. Let T be a binary tree on n points. Collapse an internal edge to a point (thereby creating a node with three children) and then re-expand this node in the alternative possible way to create two nodes with two children each, giving a new tree T'. Such an operation is called a rotation; it is a common operation used in ``balancing'' trees. The rotation distance between two binary trees on n nodes is the minimum number of rotations required to convert one into the other. The authors show that the maximum rotation distance (taken over all pairs of trees on n nodes) is at most 2n-6 for \(n\geq 11\); moreover, this bound is sharp for sufficiently large n. The proof proceeds by translating the problem into a problem on triangulations of polyhedra by tetrahedra. Particular examples are constructed using hyperbolic geometry.

A Catalan triangulation of the Möbius band is an abstract simplicial complex that uses no interior vertices. The authors present a generating function for the number of triangulations with \(n\) vertices analogous to that for Catalan triangulations of the disk [see, for example, \textit{D. D. Sleator}, \textit{R. E. Tarjan} and \textit{W. P. Thurston}, Rotation distance, triangulations, and hyperbolic geometry, J. Am. Math. Soc. 1, No. 3, 647-681 (1988; Zbl 0653.51017)]. They also show that any two such triangulations are connected by a sequence of diagonal flips. The electric current flowing across two parallel plates that touch each other over a circular contact is calculated. The change in the physical properties due to temperature rise caused by Joule heating is taken into account so that the current is not linearly proportional to the voltage difference. The resistance is calculated and an accurate approximation for the resistance, based on minimizing the power, is presented.

The type-II hidden symmetries are extra symmetries appearing when the number of variables in PDEq is reduced by a variable transformation found from a Lie symmetry of this PDE. The main of the goal of the article is to show that the provenance of the type II Lie point hidden symmetries found for differential equations can be explained by considering some weak symmetries or conditional symmetries of the original PDEq. The author focus his attention in weak symmetries of the PDEq with special differential constraint in order to determine the source of these type-II hidden symmetries. The main new result is that it can be identified the PDEq from which the type-II hidden symmetries are inherited by using as differential constraint the ''side condition'' from which the reduction has been derived. The author explain why some PDEq derived by \textit{B. Abraham-Shrauner} and \textit{K. S. Govinder} [J. Nonlinear Math. Phys. 13, No. 1--4, 612--622 (2006; Zbl 1110.35321)] by guessing do not gain the whole set of Lie symmetries. The provenance of Type II hidden point symmetries of differential equations reduced from nonlinear partial differential equations is analyzed. The hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. These Type II hidden symmetries do not arise from contact symmetries or nonlocal symmetries as in the case of ordinary differential equations. The Lie point symmetries of a model equation and the two-dimensional Burgers equation and their descendants are used to identify the hidden symmetries. The significant new result is the provenance of the Type II Lie point hidden symmetries found for differential equations reduced from partial differential equations. Two methods for determining the source of the hidden symmetries are developed.

The type-II hidden symmetries are extra symmetries appearing when the number of variables in PDEq is reduced by a variable transformation found from a Lie symmetry of this PDE. The main of the goal of the article is to show that the provenance of the type II Lie point hidden symmetries found for differential equations can be explained by considering some weak symmetries or conditional symmetries of the original PDEq. The author focus his attention in weak symmetries of the PDEq with special differential constraint in order to determine the source of these type-II hidden symmetries. The main new result is that it can be identified the PDEq from which the type-II hidden symmetries are inherited by using as differential constraint the ''side condition'' from which the reduction has been derived. The author explain why some PDEq derived by \textit{B. Abraham-Shrauner} and \textit{K. S. Govinder} [J. Nonlinear Math. Phys. 13, No. 1--4, 612--622 (2006; Zbl 1110.35321)] by guessing do not gain the whole set of Lie symmetries. This paper builds on the outcomes of the Cornerstone Maths pilot study, a USA/UK collaboration, which is now in a phase of scaling to over 100 schools. We describe the vision for the project and its iterative design, both informed by a twenty-year history of research on dynamic digital technologies. The resulting intervention builds on our understanding of some of the constraints to the widespread use of dynamic digital technologies by pupils in mathematics, which relate to accessibility, teacher development, curriculum alignment and the need to support the instrumentation process for teachers. The accompanying research agenda is concerned with evaluating models for scaling that are mindful of the `grain size' of analysis and the necessary re-alignment of the design principles of the innovation to take account of implementation imperatives.

Authors' abstract: The Harnack inequality established in \textit{M. Röckner} and \textit{F.-Y. Wang} [J.~Funct.~Anal.~203, No. 1, 237--261 (2003; Zbl 1059.47051)] for generalized Mehler semigroup is improved and generalized. As applications, log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U processes with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbation with linear growth are also investigated. Let \(E\) be a real separable Hilbert space with a Borel \(\sigma\)-field \(L(E)\). Let \(T_t\), \(t\geq 0\), be a \(C_0\)-semigroup of bounded linear operators on \(E\). Let further \(\mu_t\), \(t\geq 0\), be a family of probability measures on \(E\) satisfying \[ \mu_{t+s}= (\mu_t\circ T^{-1}_s)* \mu_s,\quad t,s\geq 0. \] The generalized Mehler semigroup (also called generalized Ornstein--Uhlenbeck semigroup) associated with \(T_t\), \(\mu_t\) is defined by \[ P_tf(x)= \int_E f(T_tx+ y)\mu_t(dy), \text{ forall bounded }f\in L(E),\text{ and all }x\in E, \] (introduced by \textit{V. I. Bogachev, M. Röckner} and \textit{B. Schmuland} [Probab. Theory Relat. Fields 105, 193--225 (1996; Zbl 0849.60066)]). Under certain assumptions, the authors prove here Harnack type inequalities for such semigroups, which in particular imply upper bound estimates for the transition density. Poincaré and log-Sobolev inequalities are also proved in terms of estimates for the square field operators. The results are supported by concrete examples.

Authors' abstract: The Harnack inequality established in \textit{M. Röckner} and \textit{F.-Y. Wang} [J.~Funct.~Anal.~203, No. 1, 237--261 (2003; Zbl 1059.47051)] for generalized Mehler semigroup is improved and generalized. As applications, log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U processes with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbation with linear growth are also investigated. The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points \(\Theta\) and a countable family of Lipschitz curves \(\mathcal{T}\) such that outside \(\mathcal{T}\cup \Theta\) the solution is continuous, and for all points in \(\mathcal{T}\setminus\Theta\) the solution has left and right limit. This extends the corresponding structural result in [\textit{A. Bressan} and \textit{P. G. LeFloch}, Indiana Univ. Math. J. 48, No. 1, 43--84 (1999; Zbl 0932.35139)] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. \(L^1_{\operatorname{loc}}\)-convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems.

Let \(F\) be an infinite field of characteristic not equal to \(2\), \(G\) an algebraic group over \(F\), \(V\) an algebraic \(F\)-representation (i.e. an algebraic group morphism \(G\to\text{GL}_F(V)\)), \(F(V)\) the field of rational functions, and \(F(V)^G\) the invariant field of \(F(V)\) with respect to the natural action of \(G\) on \(F(V)\). Also, let \(K\) and \(K'\) be extensions of \(F\). We say that \(K\) and \(K'\) are stably isomorphic over \(F\), if the rational function fields \(K(X_1,\dots,X_a)\) and \(K'(Y_1,\dots,Y_b)\) are isomorphic over \(F\), for some \(a,b\in\mathbb{N}\). The extension \(K/F\) is called retract rational, if \(K\) is the field of fractions of an \(F\)-algebra domain \(S\), and there is a localized polynomial ring \(F[X](1/s)=F[x_1,\dots,x_n](1/s)\) with \(F\)-algebra maps \(f\colon S\to F[X](1/s)\) and \(g\colon F[X](1/s)\to S\), such that the composition \(g\circ f\) is the identity of \(S\). It has been proved by \textit{D. J. Saltman} [see Isr. J. Math. 47, 165-215 (1984; Zbl 0546.14013)] that if \(K\) and \(K'\) are stably isomorphic and \(K/F\) is retract rational, then \(K'\) is retract rational. In particular, stably isomorphic to a rational extension implies retract rational (but not conversely). The paper under review considers the problem of whether every extension of \(F\) stably isomorphic to \(F(V)^G\) is retract rational. The significance of the problem is due to the fact that if \(G=\text{PGL}_n\), \(\text{PO}_n\) or \(\text{PSp}_n\) (the projective, respectively, linear, orthogonal and symplectic, group of order \(n\), with \(n\) even in the symplectic case), and \(V\) is a direct sum of \(r\) isomorphic copies of the full matrix \(F\)-vector space \(M_n(F)\), on which the action of \(G\) is induced by diagonal conjugation, then \(F(V)^G\) is the centre of the generic division algebra \(UD(F,n,r)\), \(U_t(F,n,r)\) (with orthogonal involution), and \(UD_s(F,n,r)\) (with symplectic involution), respectively. The paper shows that the answer is affirmative, if \(G=\text{PO}_{8m}\) or \(\text{PSp}_{8m}\), in case \(m\) is an odd positive integer. This is achieved by proving the equivalent lifting property for the class of central simple \(F\)-algebras of dimension \(64m^2\) with involution. Let \(F\) be a field. A field extension \(K/F\) is retract rational if and only if \(K\) is the quotient field of an \(F\)-algebra domain \(S\) such that there exists a localized polynomial ring \(P = F[x_1,\ldots,x_n](1/w)\) and \(F\)-algebra maps \(Y\colon S\to P\), \(\psi\colon P\to S\) such that \(\psi \circ Y\) is the identity on \(S\). For \(K/F\), rational implies stably rational implies retract rational implies unirational. This extraordinarily rich paper relates retract rationality of field extensions arising from certain generic constructions, to lifting of classes of algebras from residue fields to local rings, both generally and for two special classes of algebras, Galois extensions and Azumaya algebras. Let \((T,M,K)\) denote a local ring \(T\) with maximal ideal \(M\) and residue field \(T/M=K\), containing \(F\). The author considers lifting of \(G\)-Galois extensions of \(K\), \(G\) a finite group, and obtains the following: For \(V\) a \(G\)-module which is a finite dimensional vector space over \(F\) such that the map from \(G\) to \(End_F(V)\) is \(1-1\), let \(F_t(V)\) be the symmetric algebra on \(V\), \(F_t(V)\) be the quotient field. Then \(F_t(V)^G\) is retract rational over \(F\) if and only if for all \((T,M,K)\) any \(G\)-Galois extension \(L/K\) lifts to a \(G\)-Galois extension \(S/T\) with \(S\otimes_TK=L\). As examples, the author proves: (*) If \(G\) is finite abelian of exponent \(2^rm\), \(m\) odd, then \(F_t(V)^G/F\) is retract rational if and only if \(F(\rho)/F\) is cyclic, where \(\rho\) is a primitive \(2^r\)-th root of unity, or \(F\) has characteristic 2. Moreover, if \(\mathrm{Gal}(F(\rho)/F)=H\) is not cyclic and \(A\) is any finite abelian group containing a cyclic group of order \(2^r\), then there exists \((T,M,K)\) and an \(A\)-Galois extension \(L/K\) which does not lift to \(T\). The author also considers lifting Azumaya \(K\)-algebras of certain types. To do this he considers function fields of algebraic tori, \(Q(L/K,N)\), defined as follows: \(G\) is a finite group, \(N\) is a \(G\)-module which is a finitely generated free \(\mathbb Z\)-module, \(L/F\) a Galois extension with group \(G\), \(L[N]\) the group algebra, then \(Q(L/F,N)=L[N]^G\). These fields have been studied recently by \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [Ann. Sci. Éc. Norm. Supér. IV. Sér. 10, 175--229 (1977; Zbl 0356.14007)]. The author defines a certain \(N=M_2(G)\) generated by images of ``general'' 2-cocycle. Then for all local algebras \((T,M,K)\) and all \(G\)-Galois extensions \(S/T\), the map on relative Brauer groups \(\mathrm{Br}(S/T)\to \mathrm{Br}((S/MS)/K)\) is surjective if and only if \(Q(L/F, M_2(G))\) is retract rational, if and only if all the Sylow subgroups of \(G\) are cyclic. This is based on a general criterion for retract rationality of \(Q(L/F,N)\). The author is also interested in the center \(Z(F,n,r)\) of the generic division algebra \(\mathrm{UD}(F,n,r)\). He shows \(Z(F,n,r)\) is retract rational if and only if for all \((T,M,K)\) and every central simple \(K\)-algebra \(A\) of degree \(n\), there exists an Azumaya \(T\)-algebra \(B\) such that \(B\otimes_T K=A\). Using this, he shows that \(Z(F,p,r)\) is retract rational for \(p\) prime. This is obtained by using the fact that \(F_t(V)^{S_p}/F\) is rational for \(S_p\) the symmetric group, hence given \((T,M,K)\), \(S_p\)-Galois extensions \(L/K\) lift, hence crossed products lift. This last argument illustrates a recurring theme of the paper, namely, that lifting problems for Azumaya algebras and for Galois extensions are closely related. This is explicitly shown in section 4 of this paper, where the author studies liftings of Galois extensions of order \(q\), where \(q=2^r\). Let \(F'=F(\rho)\) where \(\rho\) is a primitive \(q\)-th root of unity, let \(H=\mathrm{Gal}(F'/F)\), \(C_q= \) cyclic group of order \(q\). Given \((T,M,K)\), let \(T'=T\otimes_FF', K'=K\otimes_FF'\). Then (**) a given \(C_q\)-Galois extension \(L/K\) lifts to \(S/T\) if and only if a certain corresponding crossed product \(A=D(K'/K, H,c)\) lifts to a crossed product over \(T\). A related approximation result is the following: let \(K\) be a field with valuations \(v_1,\ldots,v_n\), \(L_i/K C_q\)-Galois extensions, \(A_i\) the corresponding crossed products. Then there is a \(C_q\)-Galois extension \(L/K\) with \(L_i=L\otimes_KK_i\) for all \(i\) if and only if there is a crossed product \(A=D(K'/K,H,c)\) with \(A\otimes_KK_i=A_i\). This connection between Azumaya algebras and Galois extensions reduces the Grunwald-Wang theorem to a straightforward computation of Hasse invariants of abelian crossed products. The results (*) are also obtained by relating lifting for \(G\)-Galois extensions to lifting for crossed products, using (**), and then applying the theory developed for \(Q(F(\rho)/F, M_ 2(H))\), \(H=\mathrm{Gal}(F(\rho)F)\) to solve the problem of lifting crossed products. The author's results on relating retract rationality and lifting are formulated in quite general setting, involving modules with linear structure, called \(F\)-classes. An \(F\)-class \(\mathcal C\) consists of objects \(\mathcal M/R\), consisting of a commutative \(F\)-algebra \(R\), an \(R\)-module \(M\), and multilinear maps of specified types on \(M\). An \(F\)-class has the lifting property if for all \((T,M,K)\), any \(\mathcal M/K\) in the class lifts to \(\mathcal M'/T\). The author defines representing, densely representing and local projective objects in an \(F\)-class, and proves the general result that if \(\mathcal M/R\) is densely representing for \(\mathcal C\) and is locally projective, then the class has the lifting property if and only if the quotient field of \(F\) is retract rational. This theorem underlines the results in this paper and also the result of the author's sequel [Invent. Math. 77, 71--84 (1984; Zbl 0546.14014)].

Let \(F\) be an infinite field of characteristic not equal to \(2\), \(G\) an algebraic group over \(F\), \(V\) an algebraic \(F\)-representation (i.e. an algebraic group morphism \(G\to\text{GL}_F(V)\)), \(F(V)\) the field of rational functions, and \(F(V)^G\) the invariant field of \(F(V)\) with respect to the natural action of \(G\) on \(F(V)\). Also, let \(K\) and \(K'\) be extensions of \(F\). We say that \(K\) and \(K'\) are stably isomorphic over \(F\), if the rational function fields \(K(X_1,\dots,X_a)\) and \(K'(Y_1,\dots,Y_b)\) are isomorphic over \(F\), for some \(a,b\in\mathbb{N}\). The extension \(K/F\) is called retract rational, if \(K\) is the field of fractions of an \(F\)-algebra domain \(S\), and there is a localized polynomial ring \(F[X](1/s)=F[x_1,\dots,x_n](1/s)\) with \(F\)-algebra maps \(f\colon S\to F[X](1/s)\) and \(g\colon F[X](1/s)\to S\), such that the composition \(g\circ f\) is the identity of \(S\). It has been proved by \textit{D. J. Saltman} [see Isr. J. Math. 47, 165-215 (1984; Zbl 0546.14013)] that if \(K\) and \(K'\) are stably isomorphic and \(K/F\) is retract rational, then \(K'\) is retract rational. In particular, stably isomorphic to a rational extension implies retract rational (but not conversely). The paper under review considers the problem of whether every extension of \(F\) stably isomorphic to \(F(V)^G\) is retract rational. The significance of the problem is due to the fact that if \(G=\text{PGL}_n\), \(\text{PO}_n\) or \(\text{PSp}_n\) (the projective, respectively, linear, orthogonal and symplectic, group of order \(n\), with \(n\) even in the symplectic case), and \(V\) is a direct sum of \(r\) isomorphic copies of the full matrix \(F\)-vector space \(M_n(F)\), on which the action of \(G\) is induced by diagonal conjugation, then \(F(V)^G\) is the centre of the generic division algebra \(UD(F,n,r)\), \(U_t(F,n,r)\) (with orthogonal involution), and \(UD_s(F,n,r)\) (with symplectic involution), respectively. The paper shows that the answer is affirmative, if \(G=\text{PO}_{8m}\) or \(\text{PSp}_{8m}\), in case \(m\) is an odd positive integer. This is achieved by proving the equivalent lifting property for the class of central simple \(F\)-algebras of dimension \(64m^2\) with involution. This paper concerns the inviscid, heat-conductive and resistive compressible MHD system in a horizontally periodic flat strip domain. The global well-posedness of the problem around an equilibrium with the positive constant density and temperature and a uniform non-horizontal magnetic field is established, and the solution decays to the equilibrium almost exponentially. Our result reveals the strong stabilizing effect of the transversal magnetic field and resistivity as the global well-posedness of compressible inviscid heat-conductive flows in multi-D is unknown.

Let \(G\) be a group, \(A\) a \(G\)-graded ring. The author obtains necessary and sufficient conditions for the smash product \(A \# G^*\) to be prime and simple, generalizing the results for finite \(G\) obtained in [\textit{S. Montgomery} and \textit{D. S. Passman}, J. Algebra 115, 92-124 (1988; Zbl 0639.16002)]. Let \(G\) be a finite group and let \(A\) be a \(G\)-graded ring with 1. From the theory of Hopf algebras the smash product \(A{\#}G^*\), an extension ring of \(A\) is introduced. The authors introduce the Connes and the strong Connes subgroup of \(G\) and relate them to the ideal structure of \(A{\#}G^*\). From this criteria for \(A{\#}G^*\) to be prime or simple are derived. Now let \(A\) be a \(K\)-algebra and \(G\) an Abelian group acting as \(K\)-algebra automorphisms on \(A\). If \(K\) contains certain roots of unity then the skew group ring \(AG\) has a smash product structure \(A{\#}\widehat G^*\) where \(\widehat G=\Hom(G,K^.)\) is the dual group of \(G\). If \(A\) is a prime ring, the authors determine the Connes subgroup of \(\widehat G\) in terms of the action of \(G\) on \(A\).

Let \(G\) be a group, \(A\) a \(G\)-graded ring. The author obtains necessary and sufficient conditions for the smash product \(A \# G^*\) to be prime and simple, generalizing the results for finite \(G\) obtained in [\textit{S. Montgomery} and \textit{D. S. Passman}, J. Algebra 115, 92-124 (1988; Zbl 0639.16002)]. A real \(n \times n\) matrix \textit{M} is said to be a \textbf{P}-matrix if all its principal minors are positive. In a recent paper \textit{C. B. Chua} and \textit{P. Yi} [SIAM J. Optim. 20, No. 5, 2560--2583 (2010; Zbl 1229.90229)] described this property in terms of norm: There exists a \(\gamma > 0\) such that for all nonnegative diagonal matrices \textit{D} and vectors \textit{x}, \(\| M x + D x \| \geq \gamma \| x \|\). In this paper, we introduce a block version of this property for a linear transformation defined on a product of normed or inner product spaces. In addition to relating this to (real) positive stability and positive principal minor properties, we study the invariance of this property by principal subtransformations and Schur complements. We also specialize this property to \textbf{Z}-transformations and to Euclidean Jordan algebras.

The author continues his work on the almost sure central limit theorem (ASCLT) [Studia Sci. Math. Hungar. 31, No. 1-3, 197-202 (1996)]. Firstly, he extends an ASCLT for a sequence of \(\varphi\)-mixing random variables to the \(\rho\)-mixing case, where \(\varphi^{1/2} (*)\) is replaced by \(\rho (*)\). Secondly, he gives sufficient conditions for \[ (1/ \log n) \sum^n_{k = 1} I(a_k \leq S_k \leq b_k)/k P(a_k \leq S_k \leq b_k) \Rightarrow 1\;\text{a.s.} \] This is a local ASCLT for \(\rho\)-mixing sequences similarly to results by \textit{E. Csáki}, \textit{A. Földes} and \textit{P. Révész} [Probab. Theory Relat. Fields 97, No. 3, 321-337 (1993; Zbl 0792.60028)]. Let \(S_ 1,S_ 2,\dots\) be a sequence of sums of i.i.d. random variables. The properties of the logarithmic average \({1 \over \log n} \sum^ n_{k=1} {I\{a_ k \leq S_ k<b_ k\} \over kP(a_ k \leq S_ k<b_ k)}\) will be studied under some conditions.

The author continues his work on the almost sure central limit theorem (ASCLT) [Studia Sci. Math. Hungar. 31, No. 1-3, 197-202 (1996)]. Firstly, he extends an ASCLT for a sequence of \(\varphi\)-mixing random variables to the \(\rho\)-mixing case, where \(\varphi^{1/2} (*)\) is replaced by \(\rho (*)\). Secondly, he gives sufficient conditions for \[ (1/ \log n) \sum^n_{k = 1} I(a_k \leq S_k \leq b_k)/k P(a_k \leq S_k \leq b_k) \Rightarrow 1\;\text{a.s.} \] This is a local ASCLT for \(\rho\)-mixing sequences similarly to results by \textit{E. Csáki}, \textit{A. Földes} and \textit{P. Révész} [Probab. Theory Relat. Fields 97, No. 3, 321-337 (1993; Zbl 0792.60028)]. The next generation network (NGN) is the mainstream framework of the existing telecomm networks. NGN is a service-driven network. To simulate its service-supporting environment, an service-oriented simulating model is presented based on CPN Tools, the modeling, analyzing, simulating platform of colored Petri net (CPN). The model employs the hierarchical feature of CPN and the skeleton of object-oriented CPN. To reflect the transitional characteristic of the existing network, it contains the fields of intelligent networks and NGN, which are connected with NGW. The main kinds of functional entities of the two fields are modeled as pages representing objects, while the protocols as the messages between the objects. The model can be used for studying the service creation, running and interaction in the existing telecom networks.

The object of study is the parabolic equation \(\partial_t u-\nabla\cdot a(x,t,u,\nabla u)=b(x,t,u,\nabla u)\), where \(a\) is assumed to be degenerately elliptic, with what is called ``\(p \)-growth'' with respect to the gradient variable at \(0\) and \(\infty\). It is proven that on balls, where a certain Wolff potential is sufficiently small, there exists a local weak solution for which the space Hessian exists almost everywhere, and the space gradient is in \(L^q\) locally for \(q\) arbitarily large. The required a priori estimates are obtained by a finite number of Moser type iteration steps. In a second theorem, a similar property is proven for all weak solutions. Finally, the authors give sufficient conditions for the boundedness of the gradients of solutions, again in terms of Wolff potentials. Related results on elliptic equations and systems have been obtained by \textit{F. Duzaar} and \textit{G. Mingione} [Ann.\ Inst.\ Henri Poincaré, Anal.\ Non Linéaire 27, No.\ 6, 1361--1396 (2010; Zbl 1216.35063)]. In this paper the authors study the existence and regularity of solutions to nonhomogeneous \(p\)-Laplacean type systems and equations, on a bounded Lipschitz domain \(\Omega \) in \(\mathbb R^n\). They establish an \(L^\infty\) estimate of the gradient via suitable nonlinear potentials of the right-hand side. Also they show that when the right-hand side function belongs to the limiting Lorentz space \(L(n,1)\), the gradient of the solution is locally bounded independently of \(p\). The a priori estimates derived imply the existence of locally Lipschitz regular solutions to certain degenerate systems with critical growth of the type arising when considering geometric analysis problems.

The object of study is the parabolic equation \(\partial_t u-\nabla\cdot a(x,t,u,\nabla u)=b(x,t,u,\nabla u)\), where \(a\) is assumed to be degenerately elliptic, with what is called ``\(p \)-growth'' with respect to the gradient variable at \(0\) and \(\infty\). It is proven that on balls, where a certain Wolff potential is sufficiently small, there exists a local weak solution for which the space Hessian exists almost everywhere, and the space gradient is in \(L^q\) locally for \(q\) arbitarily large. The required a priori estimates are obtained by a finite number of Moser type iteration steps. In a second theorem, a similar property is proven for all weak solutions. Finally, the authors give sufficient conditions for the boundedness of the gradients of solutions, again in terms of Wolff potentials. Related results on elliptic equations and systems have been obtained by \textit{F. Duzaar} and \textit{G. Mingione} [Ann.\ Inst.\ Henri Poincaré, Anal.\ Non Linéaire 27, No.\ 6, 1361--1396 (2010; Zbl 1216.35063)]. New formulas for the calculation of the volume and surface moments of arbitrary dimensional polytopes are presented. The formulas are especially convenient for polygonal representations of 3D objects.

A (2-dimensional) surface \(M\) in the Lorentzian space \(\mathbb{R}^n_1\) is called time-like if the induced scalar product on \(TM\) is indefinite. The authors identify \(\mathbb{R}^2\) with the ring \(\mathbb{D}\) of double (\(\equiv\) split complex) numbers \(a+jb\), \(j^2=1\), consider the standard embedding of \(\mathbb{R}^n_1\) into \(\mathbb{D}^n\), and ``complexify'' the tangent and normal spaces of \(M\). If \(x=x(u,v)\) is a local parametrization of \(M\) over a domain in \(\mathbb{R}^2\cong\mathbb{D}\), they introduce the function \(\Phi(t)=x_u+jx_v\), \(t=u+jv\), with values in \(\mathbb{D}^n\) and show that the time-like surfaces are locally determined by functions \(\Phi(t)\) of a variable \(t\in\mathbb{D}\) with values in \(\mathbb{D}^n\) satisfying certain conditions. Many properties of the time-like surfaces are characterized by means of the function \(\Phi\). For example, the authors prove that a time-like surface is minimal if and only if \(\Phi\) is a holomorphic function. The Gauss curvature of a minimal time-like surface is expressed by means of the function \(\Phi\). The authors obtain also a local description of the minimal time-like surfaces in terms of the primitive \(\Psi=x+jy\) of \(\Phi\), where \(y\) is the harmonic conjugate function of \(x\). Isothermal coordinates for which the scalar square of the normal component \(\Phi'^{\perp}\) of the the function \(\Phi'\) equals \(1\) are called canonical by the authors. This notion extends a similar notion introduced by \textit{G. Ganchev} and \textit{V. Milousheva} [Isr. J. Math. 196, 413--433 (2013; Zbl 1281.53022)] in the case \(n=4\). The authors of the paper under review prove existence and uniqueness of canonical coordinates for the so-called minimal time-like surfaces of general type, a notion defined by mean of the function \(\Phi\). A geometric interpretation of the canonical coordinates is given by means of the hyperbola of normal curvature of a minimal time-like surface. The local theory of time-like surfaces with zero mean curvature in the 4-dimensional Minkowski space \(\mathbb R_1^4\) is studied. Special isothermal (canonical) parameters are introduced on any time-like surface with zero mean curvature free of flat points. Using the canonical parameters and canonical directions of such a surface one introduces a moving frame field at each point of the surface. It is shown that the Frenet-type derivative formulas with respect to this geometric frame field determine two invariant functions \(\mu\) and \(\nu\). The fundamental result of the paper is that any time-like surface with zero mean curvature free of flat points is determined up to a motion in \(\mathbb R_1^4\) by two natural partial differential equations satisfied by the invariants \(\mu\) and \(\nu\) and conversely. Equivalently, these equations can be expressed in functions of the Gauss curvature \(K\) and the normal curvature \(\chi\) as unknown quantities. The results are applied to the class of time-like rotational surfaces of Moore type. The meridian curves of all such surfaces are obtained. It is shown that for this class of surfaces the curvatures \(K\) and \(\chi\) are functions of only one variable. The respective surfaces determine a one-parameter family of solutions to the system of natural partial differential equations describing time-like surfaces with zero mean curvature. Finally, a survey of the background systems of partial differential equations is given for the cases of the surfaces with zero mean curvature in Euclidean space \(\mathbb R^4\) or Minkowski space \(\mathbb R_1^4\) in terms of the Gauss curvature \(K\) and the normal curvature \(\chi\).

A (2-dimensional) surface \(M\) in the Lorentzian space \(\mathbb{R}^n_1\) is called time-like if the induced scalar product on \(TM\) is indefinite. The authors identify \(\mathbb{R}^2\) with the ring \(\mathbb{D}\) of double (\(\equiv\) split complex) numbers \(a+jb\), \(j^2=1\), consider the standard embedding of \(\mathbb{R}^n_1\) into \(\mathbb{D}^n\), and ``complexify'' the tangent and normal spaces of \(M\). If \(x=x(u,v)\) is a local parametrization of \(M\) over a domain in \(\mathbb{R}^2\cong\mathbb{D}\), they introduce the function \(\Phi(t)=x_u+jx_v\), \(t=u+jv\), with values in \(\mathbb{D}^n\) and show that the time-like surfaces are locally determined by functions \(\Phi(t)\) of a variable \(t\in\mathbb{D}\) with values in \(\mathbb{D}^n\) satisfying certain conditions. Many properties of the time-like surfaces are characterized by means of the function \(\Phi\). For example, the authors prove that a time-like surface is minimal if and only if \(\Phi\) is a holomorphic function. The Gauss curvature of a minimal time-like surface is expressed by means of the function \(\Phi\). The authors obtain also a local description of the minimal time-like surfaces in terms of the primitive \(\Psi=x+jy\) of \(\Phi\), where \(y\) is the harmonic conjugate function of \(x\). Isothermal coordinates for which the scalar square of the normal component \(\Phi'^{\perp}\) of the the function \(\Phi'\) equals \(1\) are called canonical by the authors. This notion extends a similar notion introduced by \textit{G. Ganchev} and \textit{V. Milousheva} [Isr. J. Math. 196, 413--433 (2013; Zbl 1281.53022)] in the case \(n=4\). The authors of the paper under review prove existence and uniqueness of canonical coordinates for the so-called minimal time-like surfaces of general type, a notion defined by mean of the function \(\Phi\). A geometric interpretation of the canonical coordinates is given by means of the hyperbola of normal curvature of a minimal time-like surface. Let P be the probability distribution on \({\mathbb{R}}^{m+1}\) of a random vector \((X,Y)\in {\mathbb{R}}^{m+1}\) with \(X\in {\mathbb{R}}\), \(Y\in {\mathbb{R}}^ m\) and let \(R=\sum^{\infty}_{n=0}P^{n*}\) be the renewal measure associated with P. Let \(x\in {\mathbb{R}}\), \(y\in {\mathbb{R}}^ m\). When x is large, \(R(A+(x,y))\) is approximated by \(\hat R(A+(x,y))\), where the measure \(\hat R\) is the product of \(\nu^{-1}\) times Lebesgue measure on the x-axis and an m-dimensional Gaussian measure for the y-coordinates. Here \(\nu =E X>0\). Let \(F_ a\) be the set of positive Borel functions \(f: {\mathbb{R}}^{m+1}\to [0,1]\) with \(f(x,y)=0\) for \(x\not\in [a,a+1].\) The paper proves two estimates for \(\int fdR-\int fd\hat R\) as \(a\to \infty\), involving integrals of the oscillation of f over exponentially decreasing distance with coefficients bounded by order terms, uniformly with respect to \(f\in F_ a\). It is allowed that \(Y=(Y_ 1,Y_ 2)\) where \(Y_ 2\) is arithmetic and \(E \exp(is X+ip\cdot Y_ 1+iq\cdot Y_ 2)\) satisfies a strengthened version of Cramér's condition for \(| s| +\| p\| \to \infty\). There are some moment conditions and \(Y-\nu^{-1}X E Y\) is required to have a finite nonsingular covariance matrix. A corollary gives a global limit theorem for \(R([a,a+1]\times B)\). Proofs apply elaborate Fourier techniques. The paper extends previous results of the author for \(m=1\) in Probab. Theory Relat. Fields 80, No.1, 1-20 (1988; Zbl 0638.60053).

The paper investigates a generalizations of Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions by introducing a notion of \(c\)-differential uniformity: Given a \(p\)-ary \((n, m)\)-function \(f : \mathbb{F}_p^n\rightarrow \mathbb{F}_p^m\), and \(c\in\mathbb{F}_p^m\), the \(c\)-derivative of \(f\) with respect to \(a\) is the function \(_cD_af(x) = f(x + a)- cf(x)\). For an \((n, n)\)-function \(f\), and \(a, b \in F_p^n\), let \(_c\Delta_f (a, b) := |\{x \in F_p^n : f(x + a)- cf(x) = b\}|\), and \(_c\Delta_f :=\max\{_c\Delta_f (a, b) : a, b\in F_p^n, (a, c) \neq (0, 1)\}\). We say that \(f\) is differentially \((c, \delta)\)-uniform if \(_c\Delta_f\le\delta\in\mathbb{N}\). Authors focus on the special cases \(\delta = 1\) and \(\delta = 2\), i.e. PcN and APcN functions. For quadratic PcN and APcN functions authors provide their characterizations and a correspondence between Dembowski-Ostrom polynomials and APcN maps. They show that it is possible to construct several classes of APcN and PcN functions using the AGW criterion [\textit{A. Akbary} et al., Finite Fields Appl. 17, No. 1, 51--67 (2011; Zbl 1281.11102)]. Paper concludes with nonexistence results for some exceptional monomial APcN and PcN functions using connections with algebraic curves and Galois theory tools. The main topic is finding nontrivial classes of permutation polynomials, which induce permutations of the set of elements of a finite group, with finite fields being of particular interest. The authors quote Lemma 2.1 from [\textit{M. E. Zieve}, Int. J. Number Theory 4, No. 5, 851--857 (2008; Zbl 1204.11180)] as Theorem 1.1, which reduces the question as to whether a certain type of polynomial induces a permutation of \(\mathbb{F}_{q}\) to whether another, related polynomial permutes a smaller set. In this theorem, endomorphisms of \(\mathbb{F}_{q}\) of the form \(x\mapsto x^e\) for some integer \(e\) play a role. Analogous results with arbitrary endomorphisms are treated in [\textit{M. E. Zieve}, Additive number theory. Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson. New York, NY: Springer, 355--361 (2010; Zbl 1261.11081)]. Taking this approach further, the authors prove a fundamental lemma which contains theorem 1.1 and other constructions of permutation polynomials as special cases. This lemma allows constructions of permutation polynomials of a set \(A\) out of a bijection between two subsets of \(A\). The case of \(A\) being a finite field is of particular interest, but this lemma can also be applied in the more general context of a finite group. Here it is shown that under certain conditions, a permutation of a finite group can be constructed from two endomorphisms of the group. This is applied to the multiplicative group of a finite field, with endomorphisms of the form \(x\mapsto x^s\), to elliptic curves over finite fields, with endomorphisms of the form multiplication by \(m\) or the Frobenius map, and to the additive group of a finite field, with endomorphisms given by additive polynomials. This last one is considered the most interesting case.

The paper investigates a generalizations of Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions by introducing a notion of \(c\)-differential uniformity: Given a \(p\)-ary \((n, m)\)-function \(f : \mathbb{F}_p^n\rightarrow \mathbb{F}_p^m\), and \(c\in\mathbb{F}_p^m\), the \(c\)-derivative of \(f\) with respect to \(a\) is the function \(_cD_af(x) = f(x + a)- cf(x)\). For an \((n, n)\)-function \(f\), and \(a, b \in F_p^n\), let \(_c\Delta_f (a, b) := |\{x \in F_p^n : f(x + a)- cf(x) = b\}|\), and \(_c\Delta_f :=\max\{_c\Delta_f (a, b) : a, b\in F_p^n, (a, c) \neq (0, 1)\}\). We say that \(f\) is differentially \((c, \delta)\)-uniform if \(_c\Delta_f\le\delta\in\mathbb{N}\). Authors focus on the special cases \(\delta = 1\) and \(\delta = 2\), i.e. PcN and APcN functions. For quadratic PcN and APcN functions authors provide their characterizations and a correspondence between Dembowski-Ostrom polynomials and APcN maps. They show that it is possible to construct several classes of APcN and PcN functions using the AGW criterion [\textit{A. Akbary} et al., Finite Fields Appl. 17, No. 1, 51--67 (2011; Zbl 1281.11102)]. Paper concludes with nonexistence results for some exceptional monomial APcN and PcN functions using connections with algebraic curves and Galois theory tools. Recent developments in pattern recognition and image processing are bringing about a new generation of real-time trackers. The RTV system described here is an example.

A first result proved in this paper shows that for an arbitrary monic univariate complex polynomial whose coefficients are in \(C^k(I)\) for some compact real interval \(I\) there exist an integer \(k\geq 1\) and a rational number \(p > 1\), both depending only on the degree, such that any continuous choice of roots is absolutely continuous with distributional derivatives in \(L^q\) for all \(1\leq q < p\). Moreover, this regularity of roots is uniform. Based on this uniformity, a multiparameter version of this result is next obtained. The proofs depend on three essential ingredients. The authors first establish formulas for the roots of a universal monic polynomial in terms of its coefficients. These formulas, in which the derivation of Hironaka's resolution of singularities is employed, allows one to reduce the problem to radicals of functions. The proof is completed using a recent result of \textit{M. Ghisi} and \textit{M. Gobbino} [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12, 1001--1021 (2013; Zbl 1317.26010)] giving the optimal regularity for radicals of functions. The paper ends with an explicit treatment of cubic polynomials. In this case, the authors give a detailed description of the roots and compute the bounds \(k\), \(p\). The Glaeser inequality \(|v'(x)|^2 \leq 2 |v(x)| \sup |v''(x)|\) is generalized for derivatives of higher order, namely the inequality of the form \(|v'(x)|^{k+\alpha} \leq C(k) |v(x)|^{k+\alpha -1}H_{\alpha}(v^{(k)},R)\) is shown, where \(C(k)\) is a constant and \(H_{\alpha}(v^{(k)},R)\) is the \(\alpha\)-Hölder constant of \(v^{(k)}\) in \(R\). The main result concerns the regularity of roots, where a similar inequality is shown between the derivative of \(f\) and a Hölder constant of higher order root of \(f\). Using this result some properties of the root of the function \(f\) are shown.

A first result proved in this paper shows that for an arbitrary monic univariate complex polynomial whose coefficients are in \(C^k(I)\) for some compact real interval \(I\) there exist an integer \(k\geq 1\) and a rational number \(p > 1\), both depending only on the degree, such that any continuous choice of roots is absolutely continuous with distributional derivatives in \(L^q\) for all \(1\leq q < p\). Moreover, this regularity of roots is uniform. Based on this uniformity, a multiparameter version of this result is next obtained. The proofs depend on three essential ingredients. The authors first establish formulas for the roots of a universal monic polynomial in terms of its coefficients. These formulas, in which the derivation of Hironaka's resolution of singularities is employed, allows one to reduce the problem to radicals of functions. The proof is completed using a recent result of \textit{M. Ghisi} and \textit{M. Gobbino} [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12, 1001--1021 (2013; Zbl 1317.26010)] giving the optimal regularity for radicals of functions. The paper ends with an explicit treatment of cubic polynomials. In this case, the authors give a detailed description of the roots and compute the bounds \(k\), \(p\). We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on \(M_{2} (\mathbb{C})\) and \(M_{3}(\mathbb{C})\). Our main tool stems from the classical Grothendieck theorem on the tensor product of Banach spaces and is an older and more general version of the Choi-Jamiołkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting, we show that in the \(M_{2}(\mathbb{C})\) case, a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, \(2P_{x}\) with \(x\) a maximally entangled vector and \(p\otimes \mathbb1\) with \(p\) rank 1 projector. We indicate problems with passing this result to the \(M_{3}(\mathbb{C})\) case. Among similarities, in both \(M_{2}(\mathbb{C})\) and \(M_{3}(\mathbb{C})\) cases, any unital positive map represented by self-adjoint unitary is unitarily equivalent to the transposition map. Consequently, we obtain a large family of exposed maps. Furthermore, for \(M_{3}(\mathbb{C})\), there appears a new non-trivial class of maps represented by Choi matrices with square equal to a projector. We examine this case. We also investigate a convex structure of the Choi map, the first example of non-decomposable map. As a result, the nature of the Choi map will be explained.{ \copyright 2013 American Institute of Physics} Editorial remark. Part I has appeared [ibid. 53, No. 2, 023515 (2012; Zbl 1275.47086)].

Toth's ``Glimpses'' offer selected material that connect algebra and geometry (but also number-theoretic and topological topics), at a level that is meant to ``fill the gap between undergraduate and graduate mathematics studies''. The concept was to present this in a rather informal style in a discussion that goes far enough to reach interesting results and effects, but to stop before things get ``really technical''. The first edition was reviewed in Zbl 0892.00002. This second edition is a revised and substantially expanded version, so for example it includes a detailed treatment of the solution of the cubic and quartic, as well as a long new chapter on Klein's famous work on the quintic and the icosahedron. For the third edition, the reviewer would suggest to add more references and suggestions for further reading, and to name the authors of the many web pages listed, to revise some of the topological material (starting with a usable and consistent definition of ``polyhedron'' and ``surface''), and perhaps to modernize the treatment of regular polytopes (via reflection groups), as well as some of the visualization efforts. The book emerged of a ``bridge course'' that facilitates the transition between undergraduate and graduate studies. This is one of the few texts to explore the subtle and sometimes puzzling connections between number theory, classical geometry and modern algebra. Among other material it contains sections about the five Platonic solids, Euler characteristic and the famous Four Colour Theorem. Material from this volume can be taught the traditional way using slides or over 160 computer -- generated images, accessible via the World Wide Web, respectively. The book is written in a very informal, lively style, with 170 illustrations, including 18 in full colour and hints for selected problems.

Toth's ``Glimpses'' offer selected material that connect algebra and geometry (but also number-theoretic and topological topics), at a level that is meant to ``fill the gap between undergraduate and graduate mathematics studies''. The concept was to present this in a rather informal style in a discussion that goes far enough to reach interesting results and effects, but to stop before things get ``really technical''. The first edition was reviewed in Zbl 0892.00002. This second edition is a revised and substantially expanded version, so for example it includes a detailed treatment of the solution of the cubic and quartic, as well as a long new chapter on Klein's famous work on the quintic and the icosahedron. For the third edition, the reviewer would suggest to add more references and suggestions for further reading, and to name the authors of the many web pages listed, to revise some of the topological material (starting with a usable and consistent definition of ``polyhedron'' and ``surface''), and perhaps to modernize the treatment of regular polytopes (via reflection groups), as well as some of the visualization efforts. Component-trees model the structure of grey-level images by considering their binary level-sets obtained from successive thresholdings. They also enable to define anti-extensive filtering procedures for such images. In order to extend this image processing approach to any (grey-level or multivalued) images, both the notion of component-tree, and its associated filtering framework, have to be generalised. In this article we deal with the generalisation of the component-tree structure. We define a~new data structure, the \textit{component-graph}, which extends the notion of component-tree to images taking their values in any (partially or totally) ordered set. The component-graphs are declined in three variants, of increasing richness and size, whose structural properties are studied.

The author gives a down-to-earth introduction to the theory of \(p\)-adic families of modular forms, and presents an elementary proof of \textit{D. Wan}'s result [Invent. Math. 133, No.~2, 449--463 (1998; Zbl 0907.11016)] that the Newton polygon of the \(U_p\)-operator acting on \(S_k(\Gamma_1(N_p))\) is bounded below by an explicit quadratic lower bound which is independent of \(k\) (Theorem 3). Let \(p\geq 5\) be a fixed prime and let \(N\) be a fixed positive integer not divisible by \(p\). Let \(K\) be a finite extension of \(\mathbb{Q}_p\). For a nonnegative integer \(k\), denote by \(M_k(Np,K)\) the space of classical modular forms of weight \(k\) on \(\Gamma_1 (N) \cap \Gamma_0(p)\) whose Fourier coefficients belong to \(K\). We have the direct sum decomposition \[ M_k(Np,K) =\oplus_{\alpha\in \mathbb{Q}_{>0}} M_k(Np, K)^{(\alpha)}, \] under the action of the Atkin \(U\)-operator, where each root of the characteristic polynomial of \(U\) on \(M_k (Np,K)^{(\alpha)}\) has \(p\)-adic valuation \(\alpha\). Let \(d(k,\alpha): =\dim M_k (Np,K)^{ (\alpha)}\). Let \(m(\alpha)\) be the smallest non-negative integer (or \(+\infty\) if it does not exist) such that whenever \(k_1\) and \(k_2\) are two positive integers \(\geq 2\alpha+2\) satisfying \(k_1 \equiv k_2 \pmod {p^{m (\alpha)} (p-1)}\), then we have the equality \(d(k_1, \alpha)= d(k_2, \alpha)\). Conjecture (Gouvêa-Mazur): \(m(\alpha) \leq[\alpha]\). The author gives a quadratic upper bound for \(m(\alpha)\) (Theorem 1.1). The proof is based on Katz's theory of overconvergent \(p\)-adic modular forms, and recent work of Coleman and Gouvêa-Mazur.

The author gives a down-to-earth introduction to the theory of \(p\)-adic families of modular forms, and presents an elementary proof of \textit{D. Wan}'s result [Invent. Math. 133, No.~2, 449--463 (1998; Zbl 0907.11016)] that the Newton polygon of the \(U_p\)-operator acting on \(S_k(\Gamma_1(N_p))\) is bounded below by an explicit quadratic lower bound which is independent of \(k\) (Theorem 3). We develop a dynamic control strategy for minimizing energy expenditure in a time-varying wireless network with adaptive transmission rates. The algorithm operates without knowledge of traffic rates or channel statistics, and yields average power that is arbitrarily close to the minimum possible value achieved by an algorithm optimized with complete knowledge of future events. Proximity to this optimal solution is shown to be inversely proportional to network delay. We then present a similar algorithm that solves the related problem of maximizing network throughput subject to peak and average power constraints. The techniques used in this paper are novel and establish a foundation for stochastic network optimization.

The paper is a collection of results on real-valued Baire one functions, defined over an arbitrary topological space $X$. The main results, as it seems, are Theorems 5.4 and 5.6 which characterise $B_1$-embeddings ($B_1^*$-embeddings) in $X$, i.e., subspaces $Y\subset X$ that allow any (resp.\ bounded) Baire one function on $Y$ to be extended to a (resp.\ bounded) Baire one function on the entire $X$. These and other results arrive as a consequence of various considerations concerning properties of Baire one functions themselves, their zero sets, and pairs of subsets of $X$ they can separate (in Urysohn's sense). \par Most of, if not all, the content are rediscoveries of well-known (some even classical) theorems. E.g., the main result, Theorem 5.4, can be found in [\textit{O. F. K. Kalenda} and \textit{J. Spurný}, Topology Appl. 149, No. 1--3, 195--216 (2005; Zbl 1075.54011)], Proposition 7 there. The paper consists of six sections: Introduction, Preliminaries, Extension of Baire-one functions, Counterexamples and questions, Extension of mappings of the first Borel class, Extension of Baire-one functions on compact convex sets. In the introduction the authors describe the history of the problem, the main result and give the summary of two last sections. Section 2 contains necessary definitions and propositions. In Section 3 the authors prove their main result on extending Baire-one functions (Theorem 13): Let \(Y\) be a Lindelöf hereditarily Baire subset of a completely regular space \(X\) and \(f\) be a Baire-one function on \(Y\). Then there exists a Baire-one function \(g\) on \(X\) such that \(f=g\) on \(Y\), \(\inf (f(Y)_=\inf (g(X))\), \(\sup (f(Y))=\sup(g(X))\). In Section 4 the authors collect several examples showing that the assumptions of their main theorem cannot be weakened in a natural way. They also collect here some open questions. In Section 5 the authors show that once it is possible to extend Baire-one functions, the extension theorem can be obtained even for \(F_\sigma\)-measurable mappings with values in Polish spaces. The last section of the paper is devoted to an application in convex analysis (which motivated the authors research).

The paper is a collection of results on real-valued Baire one functions, defined over an arbitrary topological space $X$. The main results, as it seems, are Theorems 5.4 and 5.6 which characterise $B_1$-embeddings ($B_1^*$-embeddings) in $X$, i.e., subspaces $Y\subset X$ that allow any (resp.\ bounded) Baire one function on $Y$ to be extended to a (resp.\ bounded) Baire one function on the entire $X$. These and other results arrive as a consequence of various considerations concerning properties of Baire one functions themselves, their zero sets, and pairs of subsets of $X$ they can separate (in Urysohn's sense). \par Most of, if not all, the content are rediscoveries of well-known (some even classical) theorems. E.g., the main result, Theorem 5.4, can be found in [\textit{O. F. K. Kalenda} and \textit{J. Spurný}, Topology Appl. 149, No. 1--3, 195--216 (2005; Zbl 1075.54011)], Proposition 7 there. JFM 37.0946.02

The author presents a complete solution of the Jacobian problem for vector fields on the plane by proving the following theorem: Suppose that \(v\) is a vector field in \(\mathbb{R}^2\) such that \(v(0)=0\), and suppose that \(\text{div } v< 0\) and \(J(v)> 0\) on the whole plane, where \(J(v)\) is the Jacobian (that is, the determinant of the differential) of \(v\). Then \(0\) is a globally attracting point for the phase curves of \(v\). For the particular case of a polynomial vector field, this result had been obtained by \textit{G. H. Meisters} and \textit{C. Olech} [Analyse mathématique et applications, Contrib. Honneur J.-L. Lions, 373-381 (1988; Zbl 0668.34048)]. [For the entire collection see Zbl 0651.00008.] \textit{L. Markus} and \textit{H. Yamabe} [Osaka Math. J. (2)12, 305-317 (1960; Zbl 0096.288)] stated the conjecture: If the eigenvalues \(\alpha_ 1(x),...,\alpha_ n(x)\) of the Jacobian matrix \(J(x)\equiv f'(x)\) of a class \(C^ 1\) vector field f: \(R^ n\to R^ n\) all have negative real parts at every x in \(R^ n\) and if \(f(0)=0\), then every solution of the differential equation \(x'=F(x)\) tends to the origin as \(t\to +\infty\). In this paper this conjecture is settled in the affirmative for two- dimensional polynomial vector fields f: \(R^ 2\to R^ 2\). The proof makes use of the following geometric lemma (reminiscent of, but distinct from, Harnack's Theorem in complex algebraic geometry): Lemma 1. If P(x,y) is a real polynomial and if grad \(P\equiv (P_ x,P_ y)\neq (0,0)\) for each (x,y) in \(R^ 2\), then, for each a in R, each connected component of the curve \(P(x,y)=a\) tends to infinity in both directions, and the number of connected components of \(p(x,y)=a\) does not exceed 4 times the degree of P. Use is also made of an older result of \textit{C. Olech} [Contrib. Diff. Equations 1, 389-400 (1963; Zbl 0136.086); Theorem 1] which states that the origin is a globally asymptotically stable rest point under the additional condition \(\| f^*\| \geq \sigma >0\) for \(\| x\| \geq r>0\) where \(f^*(x)=f(x+x_ 1)-a\) for appropriate vector \(x_ 1\) and real a.

The author presents a complete solution of the Jacobian problem for vector fields on the plane by proving the following theorem: Suppose that \(v\) is a vector field in \(\mathbb{R}^2\) such that \(v(0)=0\), and suppose that \(\text{div } v< 0\) and \(J(v)> 0\) on the whole plane, where \(J(v)\) is the Jacobian (that is, the determinant of the differential) of \(v\). Then \(0\) is a globally attracting point for the phase curves of \(v\). For the particular case of a polynomial vector field, this result had been obtained by \textit{G. H. Meisters} and \textit{C. Olech} [Analyse mathématique et applications, Contrib. Honneur J.-L. Lions, 373-381 (1988; Zbl 0668.34048)]. [For the entire collection see Zbl 0651.00008.] The study of propagation of singularities of nonlinear hyperbolic equations has been given recently an impetus by several excellent researchers. The developed technique especially for the equation \(\square u+R(u,u',u'')=0,\) where \(\square\) is the wave operator in \({\mathbb{R}}^{n+1}\) and R is a quadratic form in \((u,u',u'')\), leads to estimates of the \(L^ 2\) norm of \(\hat Z^ Iu\) with respect to x, hence to \(L^ 1\) estimates for R. Here \(\hat Z^ I\) is the composition of an arbitrary number of operators of first order of the type \(\hat Z=\lambda_ jx_ j\partial_ k-\lambda_ kx_ k\partial_ j,\) \(0\leq j<k\leq n\), \(x\in {\mathbb{R}}^{n+1}\), \(x=(x_ 0,x_ 1,...,x_ n)\), \(x_ 0=t\), \(\lambda =(\lambda_ 0,\lambda_ 1,...,\lambda_ n)\) is the vector (1,-1,...,-1). In addition one can use for \(\hat Z\) the radial operator \(\sum x_ j\partial_ j\). In this paper, first, some basic \(L^ 1\), \(L^{\infty}\) estimates are obtained and a translation invariant estimates for the solution \(u=E*f\) of the Cauchy problem: \(\square u=f\) when \(x_ 0\geq 0\), \(u=u_ 0\) and \(\partial_ 0u=u_ 1\) when \(x_ 0=0\), are given, too. Second, the problem of the ``control'' of \(\hat Z^ If\) for all products of \(| I| \leq s\) operators of the type mentioned above is successfully analysed and a notion of continuous weight (\(\hat Z\) continuous) with respect to the metric \(dq^ 2/\hat q^ 2+dt^ 2/t^ 2+| d\omega |^ 2\) is associated. This permits to make precise the global behaviour (for big t and \(| x|)\) of the ``small'' solutions of the nonlinear wave equation in different cases under some initial conditions exactly formulated in the last paragraph of the paper. The obtained estimate for \(| \hat Z^ Iu(x)|\) is the same as the one obtained by \textit{S. Klainerman} [Commun. Pure Appl. Math. 36, 321-332 (1985; Zbl 0635.35059)] for \(n=4\), but it is much better for \(n>4\) away from the light cone, where no improvement is possible.

The author comments on some of the results obtained by \textit{A. Barron}, \textit{L. Birgé} and \textit{P. Massard} [Probab. Theory Relat. Fields 113, No. 3, 301-413 (1999; see the preceding entry, Zbl 0946.62036)] via the results on sharp minimax estimation, where the squared integrated risk converges with optimal rate and the best constant. The authors develop performance bounds for criteria of model selection, using recent theory for sieves. The model selection criteria are based on an empirical loss or contrast function with an added penalty term roughly proportional to the number of parameters needed to describe the model divided by the number of observations. Most of the presented examples involve density or regression estimation settings, and the authors focus on the problem of estimating the unknown density or regression function. It is shown that the quadratic risk of the minimum penalized empirical contrast estimator is bounded by an index of the accuracy of the sieve. The connection between model selection via penalization and adaptation in the minimax sense is pointed out. Such illustrations of the introduced method as penalized maximum likelihood, projection or least squares estimation are provided. The models involve commonly used finite dimensional expansions such as piecewise polynomials with fixed or variable knots, trigonometric polynomials, wavelets, neural nets, and related nonlinear expansions defined by superposition of ridge functions.

The author comments on some of the results obtained by \textit{A. Barron}, \textit{L. Birgé} and \textit{P. Massard} [Probab. Theory Relat. Fields 113, No. 3, 301-413 (1999; see the preceding entry, Zbl 0946.62036)] via the results on sharp minimax estimation, where the squared integrated risk converges with optimal rate and the best constant. The authors define classes of second-order (semi-strictly) \((F, \alpha,\rho,d)\)-V-type I and second-order (strictly) pseudoquasi \((F,\tilde \alpha,\tilde \rho,d)\)-V-type I functions. They study the multiobjective program \[ \min (f_i(x)+ (x^TB_ix)^{\frac{1}{2}}, i=1, \ldots, k) \] subject to \(x \in X =\{x: g(x) \leqq 0\}\), where \(f_i(x)\) are twice differentiable on \(X\) and \(B_i\) are \(n\times n\) positive semidefinite matrices. For both Mangasarian type and Mond-Weir type duals of this problem they prove weak, strong, and strict converse duality results.

Let \(R\) denote a Noetherian ring, let \(\mathfrak a \subset R\) be an ideal of \(R\) and let \(M\) be a finitely generated \(R\)-module. For an integer \(i \geq 0\) let \(H^i_{\mathfrak a}(M)\) denote the \(i\)-th local cohomology module of \(M\) with respect to \(\mathfrak a.\) The aim of this note is to show that the set of associated primes \(\text{ Ass}_R H^i_{\mathfrak a}(M)\) is finite, whenever all of the modules \(H^j_{\mathfrak a}(M)\) for \(j < i\) are finitely generated. This generalizes the corresponding result for the special case of \(i \leq 2\) shown by \textit{M. Brodmann, C. Rotthaus} and \textit{R. Y. Sharp} [``On annihilators and associated primes of local cohomology modules'', J. Pure Appl. Algebra 153, No. 3, 197-227 (2000)]. The full result was shown independently with a different method by \textit{K. Khashyarmanesh} and \textit{Sh. Salarian} [Commun. Algebra 27, No. 12, 6191-6198 (1999; Zbl 0940.13013)]. But in fact the authors of the present paper show a slightly more general result: Under the same assumption as before let \(N \subseteq H^i_{\mathfrak a}(M)\) denote a finitely generated submodule. Then the set \(\text{ Ass}_R (H^i_{\mathfrak a}(M)/N)\) is finite. This turns out by a clever induction argument. The main theorem of this paper is: Let \(I\) be an ideal of a Noetherian ring \(A\), and let \(M\) be a finitely generated \(A\)-module. For a positive integer \(n\), the following conditions are equivalent: (i) \(H^i_I (M)\) is finitely generated for \(i=1, \dots,n-1;\) (ii) there exists an unconditioned strong \(d\)-sequence of length \(n\) on \(M\) which is an \(I\)-filter regular sequence on \(M\). \((x_1, \dots, x_n\) is an unconditioned strong \(d\)-sequence on \(M\) if \(x_1^{i_1}, \dots, x_n^{i_n}\) is a \(d\)-sequence in any order for all positive integers \(i_j\).) An application to associated primes of \(H^i_I(M)\) is given.

Let \(R\) denote a Noetherian ring, let \(\mathfrak a \subset R\) be an ideal of \(R\) and let \(M\) be a finitely generated \(R\)-module. For an integer \(i \geq 0\) let \(H^i_{\mathfrak a}(M)\) denote the \(i\)-th local cohomology module of \(M\) with respect to \(\mathfrak a.\) The aim of this note is to show that the set of associated primes \(\text{ Ass}_R H^i_{\mathfrak a}(M)\) is finite, whenever all of the modules \(H^j_{\mathfrak a}(M)\) for \(j < i\) are finitely generated. This generalizes the corresponding result for the special case of \(i \leq 2\) shown by \textit{M. Brodmann, C. Rotthaus} and \textit{R. Y. Sharp} [``On annihilators and associated primes of local cohomology modules'', J. Pure Appl. Algebra 153, No. 3, 197-227 (2000)]. The full result was shown independently with a different method by \textit{K. Khashyarmanesh} and \textit{Sh. Salarian} [Commun. Algebra 27, No. 12, 6191-6198 (1999; Zbl 0940.13013)]. But in fact the authors of the present paper show a slightly more general result: Under the same assumption as before let \(N \subseteq H^i_{\mathfrak a}(M)\) denote a finitely generated submodule. Then the set \(\text{ Ass}_R (H^i_{\mathfrak a}(M)/N)\) is finite. This turns out by a clever induction argument. Two theorems related to the existence of positive solutions of the boundary value problem \[ - \Delta \bigl[ P(t - 1) \Delta y(t - 1) \bigr] = f \bigl( t,y(t) \bigr) \] are proved. In particular when \(f(t,y(t)) = 0\) it is shown that the above problem has only the trivial solution and the corresponding Green's function exists.

A long-standing problem in the study of fuzzy logics is to understand and justify the motivation behind them. Two traditional approaches have tried to tackle this problem. A semantical approach aims to develop a framework for dealing with statements which are not necessarily fully true or fully false and modifies classical propositional logic accordingly. The idea is to associate with propositions many truth values, rather than restricting the values to 0 and 1: all the elements from the real unit interval are permitted. Several connectives can be used to combine many-valued propositions generalizing the classical case. However, the choice of which connectives to use seems rather arbitrary and to be made ad hoc. A syntactical approach sees fuzzy logics as members of the family of substructural logics, since they too lack certain structural rules. Still, even in this case, it seems hard to comprehend what fuzzy logics exactly reason about. The author proposes a new approach to the problem introducing an alternative semantics. The focus is on two prominent fuzzy logics: Łukasiewicz logic and Hájek's Basic Logic BL, i.e., the logic of continuous t-norms and their residual implications. The author relies on Boolean algebras and Heyting algebras and defines certain equivalence relations (called a-equivalence relations) on them. Such relations define equivalence classes that are interpreted as expressing ambiguity of crisp properties. The definition of certain operations over the set of equivalence classes allows him to define the structure of an MV-algebra (in the case of Boolean algebras) and the structure of a BL-algebra (in the case of a Heyting algebra). All totally ordered MV-algebras and all totally ordered Heyting algebras can be shown to be constructed in this way. Łukasiewicz logic and BL formulas are evaluated in the constructed MV-algebras and BL-algebras (respectively), making the interpretation of the new semantics explicit. Several examples of such constructions are given, while the specific proofs of the characterization can be found in [\textit{T. Vetterlein}, ``Residuated lattices arising from equivalence relations on Boolean and Brouwerian algebras'', Math. Log. Q. 54, No. 4, 350--367 (2008; Zbl 1182.03110)]. Residuated lattices are frequently used in the syntax-free representation of vague propositions: the monoidal operation plays the role of a conjunction; the order reflects the amount of information contained in a proposition. The author is concerned with the problem of representing (classes of) residuated lattices by means of more familiar structures. Building on his previous work, in this paper he uses uses Boolean algebras \(B\) and dual Heyting algebras \(H\), equipped with a special kind of equivalence relation \(\sim\), called s-equivalence. Taking the quotient of \(B\) by \(\sim\) one gets an MV-algebra; by similarly taking the quotient of \(H\) by \(\sim\) one gets a residuated lattice. Further conditions ensure that \(H{/\!\sim}\) is a BL-algebra.

A long-standing problem in the study of fuzzy logics is to understand and justify the motivation behind them. Two traditional approaches have tried to tackle this problem. A semantical approach aims to develop a framework for dealing with statements which are not necessarily fully true or fully false and modifies classical propositional logic accordingly. The idea is to associate with propositions many truth values, rather than restricting the values to 0 and 1: all the elements from the real unit interval are permitted. Several connectives can be used to combine many-valued propositions generalizing the classical case. However, the choice of which connectives to use seems rather arbitrary and to be made ad hoc. A syntactical approach sees fuzzy logics as members of the family of substructural logics, since they too lack certain structural rules. Still, even in this case, it seems hard to comprehend what fuzzy logics exactly reason about. The author proposes a new approach to the problem introducing an alternative semantics. The focus is on two prominent fuzzy logics: Łukasiewicz logic and Hájek's Basic Logic BL, i.e., the logic of continuous t-norms and their residual implications. The author relies on Boolean algebras and Heyting algebras and defines certain equivalence relations (called a-equivalence relations) on them. Such relations define equivalence classes that are interpreted as expressing ambiguity of crisp properties. The definition of certain operations over the set of equivalence classes allows him to define the structure of an MV-algebra (in the case of Boolean algebras) and the structure of a BL-algebra (in the case of a Heyting algebra). All totally ordered MV-algebras and all totally ordered Heyting algebras can be shown to be constructed in this way. Łukasiewicz logic and BL formulas are evaluated in the constructed MV-algebras and BL-algebras (respectively), making the interpretation of the new semantics explicit. Several examples of such constructions are given, while the specific proofs of the characterization can be found in [\textit{T. Vetterlein}, ``Residuated lattices arising from equivalence relations on Boolean and Brouwerian algebras'', Math. Log. Q. 54, No. 4, 350--367 (2008; Zbl 1182.03110)]. In this study we propose a unified semiparametric approach to estimate various indices of treatment effect under the density ratio model, which connects two density functions by an exponential tilt. For each index, we construct two estimating functions based on the model and apply the generalized method of moments to improve the estimates. The estimating functions are allowed to be non smooth with respect to parameters and hence make the proposed method more flexible. We establish the asymptotic properties of the proposed estimators and illustrate the application with several simulations and two real data sets.

The paper deals with two-level optimization problems. A function F(x,y) must be minimized with respect to x and subject to some constraints on x. The vector y, being a function of x, is the result of another minimization; minimize f(x,y), subject to some (combined) constraints on x and y. The problem can be viewed as a Stackelberg problem in game theory: the leader must choose x. After x has become known, the follower chooses y. In the current paper more than one follower is allowed; they do, however, not interact with each other. Each follower minimizes a function \(f(x,y_ p)\), subject to some constraints on x and \(y_ p\) only; \(p=1,...,P\), where P is the number of followers. Two propositions and three corollaries give some properties of the solution (-spaces) under some convexity conditions. It is well-known that, even for quadratic F and f, and affine constraints, the solution space may admit many local minima. See for instance \textit{A. Bagchi}, ``Stackelberg differential games in economic models'' (1984; Zbl 0543.90095). An algorithm is presented which will find the global optimum for strictly convex F, quadratic f and affine constraints. Numerical experience is reported with respect to a range of sample problems (not given explicitly). The algorithm presented, compared to two other methods known in the literature, yields favorable results. This is one of very few monographs written on the subject of Stackelberg differential game theory and its application to economic problems. The focus of the monograph is on two economic models. The first is concerned with a regional investment allocation problem, and the second involves a world industrialization model. In addition, some background material on Stackelberg games and some other economic applications is also presented. The monograph is 195 pages long and contains the following eight chapters: 1. Introduction and Overview. 2. Game Theory and Economics. 3. Mathematical Formulation of Differential Games. 4. Solutions of Hierarchical Differential Games. 5. Application to Regional Investment Allocation Problem. 6. Stackelberg Games in Linear Programming Context. 7. Application to a World Industrialization Model. 8. Econometric Models: Adaptive Games. In addition, there are 72 references on the subject. The monograph is very well written, and is essential reading to system theorists and economists interested in the application of differential games to economics.

The paper deals with two-level optimization problems. A function F(x,y) must be minimized with respect to x and subject to some constraints on x. The vector y, being a function of x, is the result of another minimization; minimize f(x,y), subject to some (combined) constraints on x and y. The problem can be viewed as a Stackelberg problem in game theory: the leader must choose x. After x has become known, the follower chooses y. In the current paper more than one follower is allowed; they do, however, not interact with each other. Each follower minimizes a function \(f(x,y_ p)\), subject to some constraints on x and \(y_ p\) only; \(p=1,...,P\), where P is the number of followers. Two propositions and three corollaries give some properties of the solution (-spaces) under some convexity conditions. It is well-known that, even for quadratic F and f, and affine constraints, the solution space may admit many local minima. See for instance \textit{A. Bagchi}, ``Stackelberg differential games in economic models'' (1984; Zbl 0543.90095). An algorithm is presented which will find the global optimum for strictly convex F, quadratic f and affine constraints. Numerical experience is reported with respect to a range of sample problems (not given explicitly). The algorithm presented, compared to two other methods known in the literature, yields favorable results. We present a discontinuous finite element method for the Mindlin-Reissner plate model based on continuous piecewise second-degree polynomials for transverse displacements and discontinuous piecewise linear approximations for the rotations. We prove convergence, uniformly in the thickness of the plate, and thus show that locking is avoided. Lastly, we present numerical results.

There is a well-developed theory that examines the spectrum of the Laplace operator on manifolds. Less is known about the spectrum of the Laplace operator on forms. This paper uses Seiberg-Witten theory to derive an upper bound on the first eigenvalue of the Hodge Laplacian on co-exact \(1\)-forms on a wide class of \(3\)-manifolds. This result improves prior results that began with the Seiberg-Witten proof of the adjunction inequality. After establishing this upper bound, it is applied to give a new proof of an inequality first established for hyperbolic manifolds by \textit{J. F. Brock} and \textit{N. M. Dunfield} [Invent. Math. 210, No. 2, 531--558 (2017; Zbl 1379.57023)]. This is an example of a result that follows easily once one applies one key idea. In this case after establishing the standard inequality that implies compactness results for the Seiberg-Witten moduli via a Weitzenböck formula, Lin applies the Bochner formula to the form represented by the quadratic term in the first Seiberg-Witten formula. The result is a very clean and clear proof of this interesting result. By Mostow rigidity, every closed hyperbolic 3-manifold supports a unique hyperbolic structure (up to isometry). So all invariants of the manifold defined by the hyperbolic metric are topological invariants. In particular, norms of harmonic representatives of (de Rham) cohomology classes are topological invariants. In this paper, the authors compare two norms on the first cohomology of closed hyperbolic 3-manifolds: the Thurston norm defined by topology and the \(L^2\) harmonic norm defined by geometry. In particular, the authors prove that, for any closed hyperbolic 3-manifold \(M\) and any \(\phi\in H^1(M;\mathbb{R})\), the following inequality holds: \[ \frac{\pi}{\sqrt{\text{vol}(M)}}\|\phi\|_{Th}\leq \|\phi\|_{L^2}\leq \frac{10\pi}{\sqrt{\text{inj}(M)}}\|\phi\|_{Th}. \] Here \(\text{vol}(M)\) is the hyperbolic volume of \(M\) and \(\text{inj}(M)\) is the injectivity radius of \(M\). The proof of \(\|\phi\|_{L^2}\geq \frac{\pi}{\sqrt{\text{vol}(M)}}\|\phi\|_{Th}\) uses two auxiliary norms: the least area norm and the harmonic \(L^1\)-norm. The least area norm is closely related with the Thurston norm, while the harmonic \(L^1\)-norm is closely related with the harmonic \(L^2\)-norm. The authors prove the desired inequality by proving these two auxiliary norms are equal to each other. For the other inequality \(\|\phi\|_{L^2}\leq \frac{10\pi}{\sqrt{\text{inj}(M)}}\|\phi\|_{Th}\), the authors need a control of \(L^{\infty}\)-norms of harmonic forms by their \(L^2\)-norms. They first get such a control on balls in \(\mathbb{H}^3\): for any harmonic 1-form \(\alpha\) in a ball \(B=B(r)\) with radius \(r\) and center \(p\), \[ |\alpha_p|\leq \frac{1}{\sqrt{\nu(r)}}\|\alpha\|_{L^2(B)} \] holds, where \(\nu(r)\) is an explicit function of \(r\). By using a deep result of \textit{M. Culler} and \textit{P. B. Shalen} [Isr. J. Math. 190, 445--475 (2012; Zbl 1257.57022)] that 0.29 is a Margulis number for all hyperbolic \(3\)-manifolds with positive \(b_1\), and applying the above estimation on balls, the authors derive that \[ \|\alpha\|_{L^{\infty}}\leq \frac{5}{\sqrt{\text{inj}(M)}}\|\alpha\|_{L^2} \] for any harmonic 1-form on a closed hyperbolic 3-manifold \(M\). Then this inequality implies the desired inequality by a simple calculation. The authors also give examples to show that some quantitative aspects of the above inequalities are sharp. For the inequality \(\|\phi\|_{L^2}\geq \frac{\pi}{\sqrt{\text{vol}(M)}}\|\phi\|_{Th}\), only the constant \(\pi\) can be improved; for the inequality \(\|\phi\|_{L^2}\leq \frac{10\pi}{\sqrt{\text{inj}(M)}}\|\phi\|_{Th}\), there are examples that \(\frac{\|\phi_n\|_{L^2}}{\|\phi_n\|_{Th}}\to \infty\) like \(\sqrt{-\log{(\text{inj}(M_n))}}\) as \(\text{inj}(M_n)\to 0\). Another interesting family of closed hyperbolic 3-manifolds is also constructed in this paper. The authors construct a family of hyperbolic 3-manifolds \(M_n\) with universal lower bound on the injectivity radii, with \(b_1(M_n)=1\), and the Thurston norm of the generators of \(H^1(M_n;\mathbb{Z})\) has exponential growth with respect to the volume of manifold. Since the routine estimate implies that there is always an integer cohomology class whose Thurston norm is bounded by an exponential function of the volume, this family of examples shows that this estimate can not be substantially improved. This family of examples is also related with the famous conjecture that the asymptotic growth of homological torsion of congruence covers of a fixed arithmetic hyperbolic 3-manifold equals its hyperbolic volume divided by \(6\pi\).

There is a well-developed theory that examines the spectrum of the Laplace operator on manifolds. Less is known about the spectrum of the Laplace operator on forms. This paper uses Seiberg-Witten theory to derive an upper bound on the first eigenvalue of the Hodge Laplacian on co-exact \(1\)-forms on a wide class of \(3\)-manifolds. This result improves prior results that began with the Seiberg-Witten proof of the adjunction inequality. After establishing this upper bound, it is applied to give a new proof of an inequality first established for hyperbolic manifolds by \textit{J. F. Brock} and \textit{N. M. Dunfield} [Invent. Math. 210, No. 2, 531--558 (2017; Zbl 1379.57023)]. This is an example of a result that follows easily once one applies one key idea. In this case after establishing the standard inequality that implies compactness results for the Seiberg-Witten moduli via a Weitzenböck formula, Lin applies the Bochner formula to the form represented by the quadratic term in the first Seiberg-Witten formula. The result is a very clean and clear proof of this interesting result. In one of his notebooks, Ramanujan stated but did not prove an identity for a ratio of gamma functions involving two separate variables. Such an identity was referred to as a reciprocity theorem due to the symmetry relations between the variables. The two variable result was later proved and generalized to several variables by \textit{B. C. Berndt} and \textit{D. Koukoulopoulos} [Proc. Am. Math. Soc. 137, No. 7, 2369--2373 (2009; Zbl 1287.33001)]. In this paper, we state and prove similar reciprocity theorems for ratios of \(q\)-gamma functions involving two and more variables.

The author considers the definable set \(S\) [see \textit{L. van den Dries}, Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series. 248. Cambridge: Cambridge Univ. Press (1998; Zbl 0953.03045)] and studies the scalar curvature measure on \(S\) -- a generalization of the integral scalar curvature measure of a Riemannian space. The main result of the paper states that if the definable set \(S\) is an Aleksandrov space of curvature \(\geq\kappa\) relative to the induced intrinsic metric of \(S\), then \(\text{scal}( S,\cdot) \geq km(m-1)\text{Vol}(\cdot) \). This book gives a beautiful introduction to aspects of o-minimality, in the spirit of Grothendieck's `tame topology'. Although, as the author comments, the subject was developed in close contact with model theory, no model-theoretic background is needed, and many of the methods come from real algebraic geometry. Much of the material is not previously published. The book begins with a definition of o-minimality, a proof that o-minimal ordered groups and ordered fields are divisible abelian and real closed respectively, and a proof that the real field \(({\mathbb{R}},<,+,\cdot)\) is o-minimal (via the Tarski -Seidenberg Theorem which is proved via a cell decomposition). The o-minimal Monotonicity and Cell Decomposition Theorems (developed in papers of Pillay and Steinhorn, and one also with Knight) are then proved in Ch. 3. Dimension and Euler characteristic and their basic properties are introduced in the next chapter. In Ch. 5 the author shows that in an o-minimal structure any definable family of definable sets is a Vapnik-Cervonenkis class (a notion from probability theory, relevant also to neural networks). This is equivalent to the fact that o-minimal structures do not have the independence property. In the next two chapters some basic point set topology is developed, followed (for o-minimal expansions of fields) by some theory of differentiation: a Mean Value Theorem, an Implicit Function Theorem, and a Cell Decomposition with \(C^1\)-cells and maps. A Triangulation Theorem is proved in Ch. 8, via a Good Directions Lemma. This leads to a proof that in an o-minimal expansion of an ordered field, two definable sets have the same dimension and Euler characteristic if and only if there is a definable bijection between them. Under the same assumptions, a Trivialisation Theorem is proved in Section 9. It follows that given a definable family of definable sets, the sets fall into finitely many embedded definable homeomorphism types. This and Wilkie's proof of the o-minimality of the reals with exponentiation are applied to prove a conjecture of Benedetti and Risler: roughly speaking, if we consider semialgebraic subsets of \({\mathbb{R}}^n\) defined by a bounded number of polynomial equalities and inequalities, and the polynomials are built from monomials by a bounded number of additions, then the semialgebraic sets fall into finitely many embedded homeomorphism types. Finally, in Ch. 10 the author moves from definable sets to definable spaces, given by an atlas of charts, and constructs definable quotients. The book is an elegant and lucid account, well-suited to a beginning graduate student, with a number of exercises. No attempt is made to cover recent material on o-minimality, for example on o-minimal expansions of the reals, or on the Trichotomy Theorem of Peterzil and Starchenko and its applications to definable groups.

The author considers the definable set \(S\) [see \textit{L. van den Dries}, Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series. 248. Cambridge: Cambridge Univ. Press (1998; Zbl 0953.03045)] and studies the scalar curvature measure on \(S\) -- a generalization of the integral scalar curvature measure of a Riemannian space. The main result of the paper states that if the definable set \(S\) is an Aleksandrov space of curvature \(\geq\kappa\) relative to the induced intrinsic metric of \(S\), then \(\text{scal}( S,\cdot) \geq km(m-1)\text{Vol}(\cdot) \). Implicit surfaces are often used in computer graphics. They can be easily modeled and rendered, and many objects are composed of them in our daily life. In this paper, based on the concept of virtual objects, a novel method of real-time rendering is presented for reflection and refraction on implicit surface. The method is used to construct virtual objects from real objects quickly, and then render the virtual objects as if they were real objects except for one more step of merging their images with the real objects' images. Characteristics of implicit surfaces are used to compute virtual objects effectively and quickly. GPUs (Graphics Processing Units) are used to compute virtual vertices quickly and further accelerate the computing and rendering processes. As a result, realistic effects of reflections and refractions on implicit surfaces are rendered in real time.

The author studies the Keller-Segel-Navier-Stokes system \[\begin{cases} n_t + u \cdot \nabla n = \Delta n - \chi \nabla \cdot (n \nabla c) + \rho n - \mu n^2, & \quad (x,t) \in \Omega \times (0,\infty),\\ c_t + u \cdot \nabla c = \Delta c -c +n, & \quad (x,t) \in \Omega \times (0,\infty), \\ u_t + (u \cdot \nabla) u = \Delta u + \nabla P + n \nabla \phi + f(x,t), & \quad (x,t) \in \Omega \times (0,\infty), \\ \nabla \cdot u =0, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases}\] endowed with homogeneous Neumann boundary conditions for \(n\) and \(c\) as well as homogeneous Dirichlet boundary conditions for \(u\) and initial data \((n_0, c_0, u_0) \in C^0(\bar{\Omega}) \times W^{1,\infty} (\Omega) \times (W^{2,2} (\Omega) \cap W^{1,2}_0 (\Omega) \cap L^2_\sigma (\Omega))\) such that \(n_0\) and \(c_0\) are nonnegative with \(n_0 \not\equiv 0\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^3\) is a bounded convex domain with smooth boundary, \(\rho \in \mathbb{R}\), \(\chi, \mu >0\), \(\phi \in W^{2,\infty} (\Omega)\), and \(f \in C^1 (\bar{\Omega} \times [0,\infty); \mathbb{R}^3)\) is such that \[\sup\limits_{t>0} \int_t^{t+1} \| f(\cdot, s)\|_{L^p (\Omega)}^q \, ds < \infty\] for some \(p > \frac{3}{2}\) and \(q > \frac{2p}{2p-3}\). Here, \(n\) denotes the density of a population, \(c\) the concentration of a chemoattractant, \(u\) the fluid velocity field and \(P\) the correponding pressure. Assuming the above conditions, the author proves the following result: For any \(\omega >0\) there are \(\eta, \kappa >0\) depending on \(\omega\) such that if in addition \[\rho < \eta \cdot \min \left\{\mu, \mu^{\frac{3}{2}+\omega} \right\} \quad\mbox{and}\quad \limsup\limits_{t \to \infty} \int_t^{t+1} \| f(\cdot, s)\|_{L^{\frac{3}{2} + \omega} (\Omega)}^2 \, ds < \kappa,\] then the global generalized solution to the above problem from [\textit{M. Winkler}, J. Funct. Anal. 276, No. 5, 1339--1401 (2019; Zbl 1408.35132)] is a classical solution to this problem in \(\overline{\Omega} \times [t_0,\infty)\) for some \(t_0 >0\) large enough. Moreover, any such solution is eventually uniformly bounded in the sense that \[\limsup_{t \to \infty} \left\{ \| n(\cdot,t)\|_{L^\infty(\Omega)} + \| c(\cdot,t)\|_{L^\infty(\Omega)} + \| u(\cdot,t)\|_{L^\infty(\Omega)} \right\} \le C\] for some constant \(C = C(\omega,f)\). The proofs rely on the analysis of global classical solutions \((n_\varepsilon, c_\varepsilon, u_\varepsilon, P_\varepsilon)\) to appropriate approximate problems. For these solutions and some of their spatial derivatives appropriate uniform estimates in certain \(L^p\) spaces are derived which in a first step lead to an appropriate eventual smallness property of \(\nabla c_\varepsilon\). In a second step, an appropriately constructed entropy-type functional leads to further bounds on \(n_\varepsilon\) and \(\nabla c_\varepsilon\). In conjunction with the regularity theory for parabolic and Stokes problems they imply certain Hölder estimates for the solution components leading to the claimed results. The Keller-Segel-Navier-Stokes system \[ \left\{ \begin{aligned} n_t + u \cdot {\nabla} n & = {\Delta} n - \chi {\nabla} \cdot(n {\nabla} c) + \rho n - \mu n^2, \\ c_t + u \cdot {\nabla} c & = {\Delta} c - c + n, \\ u_t +(u \cdot {\nabla}) u & = {\Delta} u + {\nabla} P + n {\nabla} \phi + f(x, t), \\ {\nabla} \cdot u & = 0, \end{aligned}\right. \tag{\(\star\)} \] is considered in a bounded convex domain \({\Omega} \subset \mathbb{R}^3\) with smooth boundary, where \(\phi \in W^{1, \infty}({\Omega})\) and \(f \in C^1(\bar{{\Omega}} \times [0, \infty))\), and where \(\chi > 0, \rho \in \mathbb{R}\) and \(\mu > 0\) are given parameters. It is proved that under the assumption that \(\sup_{t > 0} \int_t^{t + 1} \| f(\cdot, s) \|_{L^{\frac{6}{5}}({\Omega})} d s\) be finite, for any sufficiently regular initial data \((n_0, c_0, u_0)\) satisfying \(n_0 \geq 0\) and \(c_0 \geq 0\), the initial-value problem for \((\star)\) under no-flux boundary conditions for \(n\) and \(c\) and homogeneous Dirichlet boundary conditions for \(u\) possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in \(L^1({\Omega}) \times L^6({\Omega}) \times L^2({\Omega}; \mathbb{R}^3)\). Moreover, under the explicit hypothesis that \(\mu > \frac{\chi \sqrt{\rho_+}}{4}\), these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying \[ (n(\cdot, t), c(\cdot, t)) \rightarrow(\frac{\rho_+}{\mu}, \frac{\rho_+}{\mu}) \text{in } L^1({\Omega}) \times L^p({\Omega})\quad \text{for all } p \in [1, 6) \text{as } t \rightarrow \infty . \] Finally, under an additional condition on temporal decay of \(f\) it is shown that also the third solution component equilibrates in that \(u(\cdot, t) \rightarrow 0\) in \(L^2({\Omega}; \mathbb{R}^3)\) as \(t \rightarrow \infty\).

The author studies the Keller-Segel-Navier-Stokes system \[\begin{cases} n_t + u \cdot \nabla n = \Delta n - \chi \nabla \cdot (n \nabla c) + \rho n - \mu n^2, & \quad (x,t) \in \Omega \times (0,\infty),\\ c_t + u \cdot \nabla c = \Delta c -c +n, & \quad (x,t) \in \Omega \times (0,\infty), \\ u_t + (u \cdot \nabla) u = \Delta u + \nabla P + n \nabla \phi + f(x,t), & \quad (x,t) \in \Omega \times (0,\infty), \\ \nabla \cdot u =0, & \quad (x,t) \in \Omega \times (0,\infty), \end{cases}\] endowed with homogeneous Neumann boundary conditions for \(n\) and \(c\) as well as homogeneous Dirichlet boundary conditions for \(u\) and initial data \((n_0, c_0, u_0) \in C^0(\bar{\Omega}) \times W^{1,\infty} (\Omega) \times (W^{2,2} (\Omega) \cap W^{1,2}_0 (\Omega) \cap L^2_\sigma (\Omega))\) such that \(n_0\) and \(c_0\) are nonnegative with \(n_0 \not\equiv 0\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^3\) is a bounded convex domain with smooth boundary, \(\rho \in \mathbb{R}\), \(\chi, \mu >0\), \(\phi \in W^{2,\infty} (\Omega)\), and \(f \in C^1 (\bar{\Omega} \times [0,\infty); \mathbb{R}^3)\) is such that \[\sup\limits_{t>0} \int_t^{t+1} \| f(\cdot, s)\|_{L^p (\Omega)}^q \, ds < \infty\] for some \(p > \frac{3}{2}\) and \(q > \frac{2p}{2p-3}\). Here, \(n\) denotes the density of a population, \(c\) the concentration of a chemoattractant, \(u\) the fluid velocity field and \(P\) the correponding pressure. Assuming the above conditions, the author proves the following result: For any \(\omega >0\) there are \(\eta, \kappa >0\) depending on \(\omega\) such that if in addition \[\rho < \eta \cdot \min \left\{\mu, \mu^{\frac{3}{2}+\omega} \right\} \quad\mbox{and}\quad \limsup\limits_{t \to \infty} \int_t^{t+1} \| f(\cdot, s)\|_{L^{\frac{3}{2} + \omega} (\Omega)}^2 \, ds < \kappa,\] then the global generalized solution to the above problem from [\textit{M. Winkler}, J. Funct. Anal. 276, No. 5, 1339--1401 (2019; Zbl 1408.35132)] is a classical solution to this problem in \(\overline{\Omega} \times [t_0,\infty)\) for some \(t_0 >0\) large enough. Moreover, any such solution is eventually uniformly bounded in the sense that \[\limsup_{t \to \infty} \left\{ \| n(\cdot,t)\|_{L^\infty(\Omega)} + \| c(\cdot,t)\|_{L^\infty(\Omega)} + \| u(\cdot,t)\|_{L^\infty(\Omega)} \right\} \le C\] for some constant \(C = C(\omega,f)\). The proofs rely on the analysis of global classical solutions \((n_\varepsilon, c_\varepsilon, u_\varepsilon, P_\varepsilon)\) to appropriate approximate problems. For these solutions and some of their spatial derivatives appropriate uniform estimates in certain \(L^p\) spaces are derived which in a first step lead to an appropriate eventual smallness property of \(\nabla c_\varepsilon\). In a second step, an appropriately constructed entropy-type functional leads to further bounds on \(n_\varepsilon\) and \(\nabla c_\varepsilon\). In conjunction with the regularity theory for parabolic and Stokes problems they imply certain Hölder estimates for the solution components leading to the claimed results. This is a prospective on the research in the intersection of mathematics and artificial intelligence that I see as having been the most important over the past 10 years and that I think should be pursued vigorously during this decade. Part of this is drawn from my personal research agenda, part is from vast readings, and part is from my editorial position with the Annals of Mathematics and Artificial Intelligence.

This is a continuation of previous work by the authors on the theory of isomanifolds [Tsagas-Sourlas, Isomanifolds..., ibid. 12, No. 1, 1-65 (1995; see the paper above)]. In the present paper they study isomappings between isomanifolds and also introduce the notion of an isotensor field and isodistribution. The notion of isodifferential calculus was introduced by \textit{R. M. Santilli} [ibid. 8, No. 3, 169-266 (1991; Zbl 0755.70006) and ibid. 8, No. 4, 275-399 (1991; Zbl 0772.70012)]. The purpose of the present paper is to develop the theory of differentiable isomanifolds and associated isotensor fields and isoexterior algebras. \{Reviewer's remark. The list of thirty-one references is for the most part obsolete. The authors never use them and they get cited only in the introduction, but without any explanation of their relevance. On the other hand, the authors do refer to some papers, e.g. by Santilli, which however, are not listed in the references\}.

This is a continuation of previous work by the authors on the theory of isomanifolds [Tsagas-Sourlas, Isomanifolds..., ibid. 12, No. 1, 1-65 (1995; see the paper above)]. In the present paper they study isomappings between isomanifolds and also introduce the notion of an isotensor field and isodistribution. Verf. beweist den bekannten Induktionssatz für natürliche Zahlen. Die Ausgangspunkte für seine Überlegungen will er in einer späteren Arbeit angeben.

The semigroup \((P^t)_t\) of any one-dimensional diffusion in the general sense of Mandl satisfies the variation diminishing property: The sign changes of \(\mu P^t\) decreases in \(t\). The proof given by the reviewer [in: Markov processes and control theory. Math. Res. 54, 164-177 (1989; Zbl 0694.60072)], relies on an approximation of diffusions by birth and death processes, for which the statement is easier but still involved. The authors establish the result by an interesting new approach: Start with \(n\) signed particles and let them independently run. If two particles of opposite sign meet, then annihilate both. With this interpretation the sign changes, counted as sign changes of the particles ordered on the real line, are obviously pointwise decreasing in time. If \(n\) tends to infinity, then the law of large numbers implies in the limit the variation diminishing property. The given proof uses essentially the path continuity (in the right metric) and exhibits clearly the variation diminishing property as a consequence of the nearest neighbor property of birth and death processes or diffusions. The given interpretation simplifies and points out the underlying structure of many applications, like in partial differential equation, Sturm-Liouville theory, curve shrinking problems and control theory. [For the entire collection see Zbl 0682.00024.] The transition kernel of a one-dimensional diffusion process applied to a function smoothes the function. There are different attempts to express and exploit this smoothening, one is introduced here with help of the variation diminishing property. The article explains the basic ideas, the connection to total positivity, monotone likelihood quotient and Polya frequency functions. One consequence of the variation diminishing property is the unimodality of the density distribution functions of passage times [the author, Ann. Probab. 8, 853-859 (1980; Zbl 0446.60025)]. Another is the strong result, that the transition probabilities have a derivative with respect to the speed measure (and this derivative is unimodal).

The semigroup \((P^t)_t\) of any one-dimensional diffusion in the general sense of Mandl satisfies the variation diminishing property: The sign changes of \(\mu P^t\) decreases in \(t\). The proof given by the reviewer [in: Markov processes and control theory. Math. Res. 54, 164-177 (1989; Zbl 0694.60072)], relies on an approximation of diffusions by birth and death processes, for which the statement is easier but still involved. The authors establish the result by an interesting new approach: Start with \(n\) signed particles and let them independently run. If two particles of opposite sign meet, then annihilate both. With this interpretation the sign changes, counted as sign changes of the particles ordered on the real line, are obviously pointwise decreasing in time. If \(n\) tends to infinity, then the law of large numbers implies in the limit the variation diminishing property. The given proof uses essentially the path continuity (in the right metric) and exhibits clearly the variation diminishing property as a consequence of the nearest neighbor property of birth and death processes or diffusions. The given interpretation simplifies and points out the underlying structure of many applications, like in partial differential equation, Sturm-Liouville theory, curve shrinking problems and control theory. We study an effective fermion model on a square lattice to investigate the cooperation and competition of superconductivity and anti-ferromagnetism. In addition to particle tunneling and on-site interaction, a bosonic excitation mediated attractive interaction is also included in the model. We assume that the attractive interaction is mediated by spin fluctuations and excitations of Bose-Einstein condensation (BEC) in electronic systems and Bose-Fermi mixtures on optical lattices, respectively. Using an effective mean-field theory to treat both superconductivity and anti-ferromagnetism at equal footing, we study a single effective model relevant for both systems within the Landau energy functional approach and a linearized theory. Within our approaches, we find possible co-existence of superconductivity and anti-ferromagnetism for both electronic and cold-atomic models. Our linearized theory shows while spin fluctuations favor d-wave superconductivity and BEC excitations favor s-wave superconductivity.

Let \(\text{PG}(5,q)\) be the projective space of dimension \(5\) over the finite field with \(q\) elements. Let \(\mathcal K\) be the Klein quadric of order \(q\), i.e.\ the hyperbolic quadric contained in \(\text{PG}(5,q)\). A \(k\)-cap of \(\mathcal K\) is a set of \(k\) points of \(\mathcal K\) no three of which are collinear. If \(q\) is odd, then the maximal size of a cap contained in \(\mathcal K\) is \(q^3+q^2+q+1\), where \((q^3+q^2+q+1)\)-caps correspond via the Klein correspondence to linear complexes of \(\text{PG}(3,q)\). For even \(q\), \textit{G. L. Ebert, K. Metsch} and \textit{T. Szőnyi} [Geom. Dedicata 70, No. 2, 181-196 (1998; Zbl 0911.51013)] gave a construction of caps contained in \(\mathcal K\) having bigger size. Under more, they construct maximal caps of size \(q^3+q^2\pm \sqrt{2q}(q+1)-q+1\), where \(q\) is an odd power of 2. In the present paper, the author gives a new construction of such caps as a union of \(q+1\) Suzuki-Tits ovoids and one elliptic quadric (the only intersection points lying in the elliptic quadric). This construction has a more group-theoretical nature than the original construction. A \(k\)-cap in a finite projective space \(\pi\) is a set of \(k\) points, no three of which are collinear. For \(\pi=\)PG\((5,q)\), \(q\) a power of \(2\), the authors construct a cap \(C\) of size \(q^3+2q^2+1\) which is maximally embedded in a Klein quadric \(K\) of \(\pi\); this means that \(C\) is contained in \(K\) and \(C\) cannot be extended to a larger cap by adding a further point of \(K\). The deficiency to the theoretically upper bound \((q^2+1)(q+2)\) of the cardinality of such a cap is equal to \(q+1\), and, in fact, any cap which is maximally embedded in \(K\) having more than \(q^3+2q^2+1\) points necessarily is a \((q^2+1)(q+2)\)-cap. It is not known, however, whether \((q^2+1)(q+2)\)-caps can be embedded in a Klein quadric. The situation \(\pi=PG(5,q)\), \(q\) odd, has been previously investigated in \textit{D. G. Glynn}, Geom. Dedicata 26, No. 3, 273-280 (1988; Zbl 0645.51012). Concerning higher dimensional spaces in arbitrary characteristic, in the present paper, the authors also provide a result on caps embedded in line Grassmannians.

Let \(\text{PG}(5,q)\) be the projective space of dimension \(5\) over the finite field with \(q\) elements. Let \(\mathcal K\) be the Klein quadric of order \(q\), i.e.\ the hyperbolic quadric contained in \(\text{PG}(5,q)\). A \(k\)-cap of \(\mathcal K\) is a set of \(k\) points of \(\mathcal K\) no three of which are collinear. If \(q\) is odd, then the maximal size of a cap contained in \(\mathcal K\) is \(q^3+q^2+q+1\), where \((q^3+q^2+q+1)\)-caps correspond via the Klein correspondence to linear complexes of \(\text{PG}(3,q)\). For even \(q\), \textit{G. L. Ebert, K. Metsch} and \textit{T. Szőnyi} [Geom. Dedicata 70, No. 2, 181-196 (1998; Zbl 0911.51013)] gave a construction of caps contained in \(\mathcal K\) having bigger size. Under more, they construct maximal caps of size \(q^3+q^2\pm \sqrt{2q}(q+1)-q+1\), where \(q\) is an odd power of 2. In the present paper, the author gives a new construction of such caps as a union of \(q+1\) Suzuki-Tits ovoids and one elliptic quadric (the only intersection points lying in the elliptic quadric). This construction has a more group-theoretical nature than the original construction. We present a transformation that helps price American options on assets that are modeled by a diffusion as well as a jump component. The presence of a jump component circumvents some shortcomings of the Black-Scholes model. The proposed transformation essentially transforms the arising free-boundary partial integro-differential equation (PIDE) into a sequence of fixed-boundary PIDEs which are much easier to solve. Finally, we provide numerical results illustrating convergence of the scheme and comparisons to other methods.

A generalized indefinite string is a triple \((L, \omega , v)\) where \(L \in(0, \infty ]\), \(\omega \) is a real distribution in \(H_{\mathrm{loc}}^{-1} [0, L)\) and \(v\) is a non-negative Borel measure on the interval \([0, L)\). Associated with this string is the differential equation \(-f''= z \omega f + z ^2 vf \), where \(z\) is the complex spectral parameter. This differential equation gives rise to a self-adjoint linear relation \(T\) on the Hilbert space \(\mathcal{H} = \dot H_0^1 [0, L) \times L^2 ([0, L); v)\), where a pair \((f,g)\in \mathcal{H}\times \mathcal{H}\) belongs to \(T\) if and only if \(-f_1 '' = \omega g_1 + vg_2\) and \(vf_2 = vg_1\). The main result is as follows: Let \(\alpha \) and \(\eta \) be real numbers such that \(\alpha \ge0\) and \(\eta >0\). Let \(\rho \) be the square root of the Radon-Nikodým derivative of \(v\) and let \(v_s\) be the singular part of \(v\). Put \(\beta =\mathrm{sgn}\,\alpha \). If \[ \int_0^\infty |\mathsf{w}(x) - c|^2 x^\beta \, dx+\int_0^\infty \left| \rho (x) - \frac \eta { 1+2\alpha x}\right| x^\beta \, dx + \int_{ [0,\infty )} x^\beta \, dv_s (x) < \infty \] for some real number \(c\), where \(\mathsf{w}\) is the normalized anti-derivative of the distribution \(\omega \), then the essential spectrum of the linear relation \(T\) associated with the string \((\infty,\omega ,v)\) is the set \((-\infty , -\alpha /\eta ] \cup [\alpha /\eta , \infty )\), and the absolutely continuous spectrum is essentially supported on this set. The Weyl-Titchmarsh function is employed for the proofs, and the general case is considered as a perturbation of the case \(\omega =0\) and \(v=(1+2\alpha x)^{-2}\), for which a fundamental system can be found explicitely. As an example, the isospectral conservative Camassa-Holm flow is considered. This flow is described by the differential equation \( -g'' + \frac 14 g = z \omega g + z^2 v g\), where \(\omega = u - u''\) in the distributional sense for some real valued function \(u\in H^1_{\mathrm{loc}}[0,\infty )\). This problem can be transformed into a string problem, and under the assumptions that \(u\in H^1[0,\infty )\), \(v_s\) is finite and \(\rho -1\in L^2[0,\infty)\) the assumptions of the general theorem are satified, and the essential spectrum is \((-\infty , -1 /2 ] \cup [1 /2 , \infty )\). The paper mainly aims at investigating the absolutely continuous part of the spectrum of generalized indefinite strings associated with the ordinary differential equation of the form \[-f''=z\omega f +z^2 v f\text{ on }[0, L),\] where \(z\) is a complex spectral parameter. Spectral problems of this type are of interest for at least two reasons. Firstly, they constitute a canonical model for operators with simple spectrum; secondly, they are of relevance in connection with certain completely integrable nonlinear wave equations (most prominently, the Camassa-Holm equation), for which these kinds of spectral problems arise as isospectral problems. The proposed approach follows the ideas of Deift and Killip and are based on two main ingredients. The first ingredient is a continuity property for the correspondence between generalized indefinite strings and their associated Weyl-Titchmarsh functions. The relevance of the Weyl-Titchmarsh function here stems from the fact that a measure \(\mu\) in a certain integral representation for this function is a spectral measure for the ordinary differential equation of the generalized indefinite strings. The second ingredient is a so-called trace formula, which provides a relation between the spectral/scattering data and the coefficients in the differential equation, and hence allows to control the spectral measure by means of the coefficients. In regard to the conservative Camassa-Holm flow, the presented results concern the absolutely continuous spectrum of the corresponding isospectral problem \[-g''+\frac{1}{4}g = z \omega g + z^2 v g, \quad \omega = u-u''\] on the real line. It is shown that under the assumption that \(u\) is a real-valued function such that \(u-1\) belongs to \(H^1(\mathbb R)\) and \(v\) is a non-negative finite Borel measure on \(\mathbb R\), the essential spectrum and the absolutely continuous spectrum of a self-adjoint realization of the isospectral problem above coincide with the interval \( [1/4, \infty)\).

A generalized indefinite string is a triple \((L, \omega , v)\) where \(L \in(0, \infty ]\), \(\omega \) is a real distribution in \(H_{\mathrm{loc}}^{-1} [0, L)\) and \(v\) is a non-negative Borel measure on the interval \([0, L)\). Associated with this string is the differential equation \(-f''= z \omega f + z ^2 vf \), where \(z\) is the complex spectral parameter. This differential equation gives rise to a self-adjoint linear relation \(T\) on the Hilbert space \(\mathcal{H} = \dot H_0^1 [0, L) \times L^2 ([0, L); v)\), where a pair \((f,g)\in \mathcal{H}\times \mathcal{H}\) belongs to \(T\) if and only if \(-f_1 '' = \omega g_1 + vg_2\) and \(vf_2 = vg_1\). The main result is as follows: Let \(\alpha \) and \(\eta \) be real numbers such that \(\alpha \ge0\) and \(\eta >0\). Let \(\rho \) be the square root of the Radon-Nikodým derivative of \(v\) and let \(v_s\) be the singular part of \(v\). Put \(\beta =\mathrm{sgn}\,\alpha \). If \[ \int_0^\infty |\mathsf{w}(x) - c|^2 x^\beta \, dx+\int_0^\infty \left| \rho (x) - \frac \eta { 1+2\alpha x}\right| x^\beta \, dx + \int_{ [0,\infty )} x^\beta \, dv_s (x) < \infty \] for some real number \(c\), where \(\mathsf{w}\) is the normalized anti-derivative of the distribution \(\omega \), then the essential spectrum of the linear relation \(T\) associated with the string \((\infty,\omega ,v)\) is the set \((-\infty , -\alpha /\eta ] \cup [\alpha /\eta , \infty )\), and the absolutely continuous spectrum is essentially supported on this set. The Weyl-Titchmarsh function is employed for the proofs, and the general case is considered as a perturbation of the case \(\omega =0\) and \(v=(1+2\alpha x)^{-2}\), for which a fundamental system can be found explicitely. As an example, the isospectral conservative Camassa-Holm flow is considered. This flow is described by the differential equation \( -g'' + \frac 14 g = z \omega g + z^2 v g\), where \(\omega = u - u''\) in the distributional sense for some real valued function \(u\in H^1_{\mathrm{loc}}[0,\infty )\). This problem can be transformed into a string problem, and under the assumptions that \(u\in H^1[0,\infty )\), \(v_s\) is finite and \(\rho -1\in L^2[0,\infty)\) the assumptions of the general theorem are satified, and the essential spectrum is \((-\infty , -1 /2 ] \cup [1 /2 , \infty )\). Bericht über einige sich auf die Entstehung und das Widerstandsgesetz der turbulenten Bewegungszustände beziehende Untersuchungen sowie über einen vom Verf. herrührenden neuen, sich im Rahmen der Methoden der kinetischen Gastheorie bewegenden Ansatz, der an Hand eines ausführlich durchgerechneten Problems illustriert wird.

In the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), let \(K\) be a compact convex subset, and let \(K_1\) be the Steiner symmetral of \(K\) with respect to the hyperplane \(e_1^{\perp}\), orthogonal to a unit vector \(e_1\in\mathbb{R}^n\). Namely, \(K_1\) is obtained by translating all the chords of \(K\) in direction \(e_1\), so that their centers belong to \(e_1^{\perp}\). From a result by \textit{C. O. Kiselman} [J. Lond. Math. Soc., II. Ser. 33, 101--109 (1986; Zbl 0655.52004)], it follows that understanding the smoothness of \(K_1\) is a non trivial problem, since the Steiner symmetral of a convex body of class \(\mathcal{C}^{\infty}\) need not even be of class \(\mathcal{C}^2\). In this paper the main theorem shows that if \(K\) has \(\mathcal{C}^2\) boundary, and positive Gauss curvature, then the same holds for its Steiner symmetral. As a corollary, it follows that the orthogonal projection of \(K\) onto \(e_1^{\perp}\) also has, in the hyperplane \(e_1^{\perp}\), \(\mathcal{C}^2\) boundary and positive Gauss curvature. The question whether the theorem can be easily obtained from the corollary is left as an open problem, as well as the investigation of the same property for convex bodies having higher regularity class. Let A be a convex body in \({\mathbb{R}}^ 3\) and let \(\pi\) be a linear surjection of \({\mathbb{R}}^ 3\) onto \({\mathbb{R}}^ 2\). The author studies the smoothness of the boundary of \(\pi\) (A) given specified smoothness assumptions on the boundary of A. He proves that if A has boundary of class \(C^ 2\) with Lipschitz continuous second derivatives, then the boundary of \(\pi\) (A) is twice differentiable. If A has real-analytic boundary, then the boundary of \(\pi\) (A) is of class \(C^{2+\epsilon}\) for some \(\epsilon >0\). Moreover, he shows that these results cannot be much improved. The set A may have a \(C^{\infty}\) boundary with the boundary of \(\pi\) (A) not of class \(C^ 2\). In the real-analytic case, the boundary of \(\pi\) (A) may be exactly of class \(C^{2+2/q}\) for any odd integer \(q\geq 3\).

In the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), let \(K\) be a compact convex subset, and let \(K_1\) be the Steiner symmetral of \(K\) with respect to the hyperplane \(e_1^{\perp}\), orthogonal to a unit vector \(e_1\in\mathbb{R}^n\). Namely, \(K_1\) is obtained by translating all the chords of \(K\) in direction \(e_1\), so that their centers belong to \(e_1^{\perp}\). From a result by \textit{C. O. Kiselman} [J. Lond. Math. Soc., II. Ser. 33, 101--109 (1986; Zbl 0655.52004)], it follows that understanding the smoothness of \(K_1\) is a non trivial problem, since the Steiner symmetral of a convex body of class \(\mathcal{C}^{\infty}\) need not even be of class \(\mathcal{C}^2\). In this paper the main theorem shows that if \(K\) has \(\mathcal{C}^2\) boundary, and positive Gauss curvature, then the same holds for its Steiner symmetral. As a corollary, it follows that the orthogonal projection of \(K\) onto \(e_1^{\perp}\) also has, in the hyperplane \(e_1^{\perp}\), \(\mathcal{C}^2\) boundary and positive Gauss curvature. The question whether the theorem can be easily obtained from the corollary is left as an open problem, as well as the investigation of the same property for convex bodies having higher regularity class. The authors consider semilinear elliptic boundary value problems of the types \[ \begin{aligned} -\Delta u&=f(x,u,\nabla u) + \mu \quad \text{in}\quad \Omega, \\ u&=0 \quad \text{on} \quad \Gamma_0,\\ \partial_\nu u&=g(x,u) + \sigma_1 \quad \text{on}\quad \Gamma_1, \end{aligned} \] and \[ \begin{aligned} -\Delta u&=h(x,u) + \mu \quad \text{in}\quad \Omega, \\ u&=\sigma_0 \quad \text{on}\quad \Gamma_0, \\ \partial_\nu u&=\sigma_1 \quad \text{on}\quad \Gamma_1, \end{aligned} \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\) \((n\geq 2)\), \(\Gamma:=\Gamma_0\cup \Gamma_1\) denotes its boundary, \(\Gamma_0\) and \(\Gamma_1\) are open in \(\Gamma\) and disjoint. Moreover, \(\mu\), \(\sigma_0\), \(\sigma_1\) are bounded Radon measures, and \(f\), \(g\), \(h\) are continuous functions satisfying some growth restrictions. The authors show that those problems can be put in a functional-analytical framework, and using this fact they derive existence, regularity and multiplicity results by methods from nonlinear functional analysis.

A conjecture of Deligne states that the special values of motivic \(L\)-functions may be expressed as algebraic numbers times certain periods. This expository article describes the situation for algebraic Hecke characters, that is cohomological automorphic representations of \(\mathrm{GL}_1\). The first 5 sections systematically develop the theory of algebraic Hecke characters; they require only basic algebraic number theory, knowledge of the adeles, and a modicum of sheaf cohomology theory. The final section, which is conceptually and technically more demanding, concerns the ratio between successive critical values of the Hecke \(L\)-function attached to an algebraic Hecke character. \textit{G. Harder} [Invent. Math. 89, 37--118 (1987; Zbl 0629.10023)] established arithmeticity results for these ratios through the study of Eisenstein cohomology for arithmetic subgroups of \(\mathrm{GL}_2\) over a number field \(F\): such a ratio, suitably normalized, is algebraic and satisfies a reciprocity law that describes explicitly the action of the absolute Galois group. As this paper explains in detail, the precise reciprocity law requires taking into account a subtle matter regarding a signature. The author remarks that ``the results of this article show that whereas the analytic theory of \(L\)-functions of automorphic forms on \(\mathrm{GL}(1)\) is not sensitive to the inner structure of the base field, the arithmetic theory\dots crucially depends on the arithmetic structure of the base field.'' This article will be of high value for those seeking a clear treatment of these important topics. The author has recently established general results on the critical values of ratios of Rankin-Selberg \(L\)-functions for \(\mathrm{GL}_n\times \mathrm{GL}_{n'}\) over an imaginary quadratic base field [\textit{A. Raghuram}, ``Eisenstein cohomology for \(\mathrm{GL}(N)\) and the special values of Rankin-Selberg \(L\)-functions over a totally imaginary field'', Preprint, \url{arXiv:2207.03393}]. The final section of this paper is an exposition of these results when \(n=n'=1\). Let F be a number field and \(G=GL(2,F)\). This paper lays a careful foundation for studying the Eisenstein cohomology of arithmetic groups by giving a precise and general treatment for arithmetic subgroups \(\Gamma\) of G. It includes and extends results from several earlier papers of the same author. The cohomology of \(\Gamma\) with coefficients in a rational representation of G is studied, with particular care given to \({\mathbb{Q}}\)- structures on the various cohomology groups. Adelic methods are used throughout by taking limits as \(\Gamma\) shrinks. The cohomology of the boundary of the Borel-Serre compactification of the locally symmetric spaces attached to the \(\Gamma\) 's is described in terms of algebraic Hecke characters on the maximal split torus. The image I of the whole cohomology in the boundary is computed in terms of these algebraic Hecke characters. The answer depends on special values and poles of the Hecke L-functions attached to the characters. The main method here is to represent cohomology classes ``induced from the boundary'' by differential forms which are Eisenstein series as defined by Selberg and Langlands. This provides a map back from I to the cohomology which is almost completely defined over \({\mathbb{Q}}\). As a corollary, one obtains results about the algebraicity of ratios of special values of L-functions and the transformation law of those ratios under the Galois groups of \({\mathbb{Q}}.\) In the last section, the Eisenstein cohomology classes are integrated over cycles attached to quadratic extensions of F, again with care given to the \({\mathbb{Q}}\)-structures. A similar algebraicity corollary is deduced.

A conjecture of Deligne states that the special values of motivic \(L\)-functions may be expressed as algebraic numbers times certain periods. This expository article describes the situation for algebraic Hecke characters, that is cohomological automorphic representations of \(\mathrm{GL}_1\). The first 5 sections systematically develop the theory of algebraic Hecke characters; they require only basic algebraic number theory, knowledge of the adeles, and a modicum of sheaf cohomology theory. The final section, which is conceptually and technically more demanding, concerns the ratio between successive critical values of the Hecke \(L\)-function attached to an algebraic Hecke character. \textit{G. Harder} [Invent. Math. 89, 37--118 (1987; Zbl 0629.10023)] established arithmeticity results for these ratios through the study of Eisenstein cohomology for arithmetic subgroups of \(\mathrm{GL}_2\) over a number field \(F\): such a ratio, suitably normalized, is algebraic and satisfies a reciprocity law that describes explicitly the action of the absolute Galois group. As this paper explains in detail, the precise reciprocity law requires taking into account a subtle matter regarding a signature. The author remarks that ``the results of this article show that whereas the analytic theory of \(L\)-functions of automorphic forms on \(\mathrm{GL}(1)\) is not sensitive to the inner structure of the base field, the arithmetic theory\dots crucially depends on the arithmetic structure of the base field.'' This article will be of high value for those seeking a clear treatment of these important topics. The author has recently established general results on the critical values of ratios of Rankin-Selberg \(L\)-functions for \(\mathrm{GL}_n\times \mathrm{GL}_{n'}\) over an imaginary quadratic base field [\textit{A. Raghuram}, ``Eisenstein cohomology for \(\mathrm{GL}(N)\) and the special values of Rankin-Selberg \(L\)-functions over a totally imaginary field'', Preprint, \url{arXiv:2207.03393}]. The final section of this paper is an exposition of these results when \(n=n'=1\). We are considering a semilinear singular perturbation reaction-diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is \(\epsilon\)-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

For \(G\) a compact Lie group and \(X\) a compact \(G\)-space, the Atiyah-Segal theorem (Theorem 1.1) identifies the topological \(K\)-theory of the Borel space \(X_G=X\times_GE_G\) with the completion of the equivariant \(K\)-theory of \(X\) with respect to the augmentation ideal, assuming a finiteness condition on the equivariant \(K\)-groups. In this paper, the author gives an elegant answer to the same question in algebraic geometry, using Morel-Voevodsky's \(\mathbb{A}^1\)-homotopy theory [\textit{F. Morel} and \textit{V. Voevodsky}, Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to produce a model for the Borel space (see Section 3.1). The main theorem (Theorem 8.7) gives a positive answer to the completion problem for a connected split reductive algebraic group acting over a scheme under a condition of being filtrable (see Section 5.2), condition always satisfied for smooth projective schemes (Theorem 6.4). For non-projective schemes, the author shows that the result still holds at the level of \(K_0\) (Theorem 9.10), and more generally for singular schemes by working with equivariant \(G\)-theory (i.e. \(K\)-theory of coherent sheaves); for higher \(K\)-groups the completion theorem fails and the author gives a counterexample on complex numbers (Theorem 10.1). In this paper the authors develop \(\mathbb{A}^1\)-homotopy theory of schemes -- a homotopy theory of algebraic varieties where the affine line plays the role of the unit interval. In the three chapters and nine paragraphs the authors present: A homotopy category of a site with interval; the \(\mathbb{A}^1\)-homotopy category of schemes over a base, classifying spaces of algebraic groups. First, they give a number of general results about simplicial sheaves on sites which are latter applied to the study of the homotopy category of schemes. Then, the authors study the basic properties of the \(\mathbb{A}^1\)-homotopy category \({\mathcal H}(S)\) of smooth schemes over a base scheme \(S\) with interval \(((Sm/S)_{Nis},\mathbb{A}^1)\) where \(Sm/S\) is the category of smooth schemes (of finite type) over \(S\) and Nis refers to the Nisnevich topology. They discuss the properties of the homotopy category of simplicial sheaves on \((Sm/S)_{Nis}\), then they prove three theorems with a major role in further applications of their constructions. Finally the authors consider some examples of topological realization functors. The last chapter is dedicated to applications of the general technique developed above. The main results are: A geometrical construction of a space which represents in \({\mathcal H}(S)\) the functor \(H^1_{et}(-,G)\) for étale group schemes \(G\) of order prime to \(\text{char} (S)\), the second result shows that algebraic \(K\)-theory of a regular scheme \(S\) can be described in terms of morphisms in \({\mathcal H}(S)\) with values in the infinite Grassmannian and the third result shows how one can use \(\mathbb{A}^1\)-homotopy theory together with basic functoriality for simplicial sheaves on smooth sites to give a definition of Quillen-Thomason \(K\)-theory for all Noetherian schemes.

For \(G\) a compact Lie group and \(X\) a compact \(G\)-space, the Atiyah-Segal theorem (Theorem 1.1) identifies the topological \(K\)-theory of the Borel space \(X_G=X\times_GE_G\) with the completion of the equivariant \(K\)-theory of \(X\) with respect to the augmentation ideal, assuming a finiteness condition on the equivariant \(K\)-groups. In this paper, the author gives an elegant answer to the same question in algebraic geometry, using Morel-Voevodsky's \(\mathbb{A}^1\)-homotopy theory [\textit{F. Morel} and \textit{V. Voevodsky}, Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to produce a model for the Borel space (see Section 3.1). The main theorem (Theorem 8.7) gives a positive answer to the completion problem for a connected split reductive algebraic group acting over a scheme under a condition of being filtrable (see Section 5.2), condition always satisfied for smooth projective schemes (Theorem 6.4). For non-projective schemes, the author shows that the result still holds at the level of \(K_0\) (Theorem 9.10), and more generally for singular schemes by working with equivariant \(G\)-theory (i.e. \(K\)-theory of coherent sheaves); for higher \(K\)-groups the completion theorem fails and the author gives a counterexample on complex numbers (Theorem 10.1). We introduce here new modified cosine and sine sums as \[ \frac{a_0}{2} + \sum^n_{k=1} \sum^n_{j=k} \Delta (a_j \cos jx) \] and \[ \sum^n_{k=1} \sum^n_{j=k}\Delta (a_j \sin jx) \] and study their integrability and \(L^1\) -convergence. The \(L^1\)-convergence of cosine and sine series have been obtained as corollary. In this paper, we have been able to remove the necessary and sufficient condition \(a_k\log k = o(1)\) as \(k \rightarrow \infty \) for the \(L^1\)-convergence of cosine and sine series.

The paper under review regards singular holomorphic foliations \(\mathcal F\) on the complex projective plane \(\mathbb P^2\). Roughly speaking, the Baum-Bott map associates to a foliation the Baum-Bott indexes of its singularities. See also [\textit{A. Lins Neto} et al., Compos. Math. 142, No. 6, 1549--1586 (2006; Zbl 1109.37041)]. The author studies the fibers of the Baum-Bott map in the space of foliations of degree two on \(\mathbb P^2\). In detail, the set of singular holomorphic foliations \(\{\mathcal F\}\) of degree \(d\) and having only non-degenerate singularities is denoted by \(\mathbb F\mathrm{ol}_{nd}(d,2)\). It is an open Zariski subset of the complex projective space, comprised by all the singular holomorphic foliations of degree \(d\). A foliation \(\mathcal F\) in \(\mathbb F\mathrm{ol}_{nd}(d,2)\) has \(d^2+d+1:=N(d)\) isolated singularities. Hence, the \textit{Baum-Bott} map is \[ \mathcal{BB}_d:\mathbb F\mathrm{ol}_{nd}(d,2) \longrightarrow\mathbb C^{N(d)}/S_{N(d)} \] \[ \mathcal F\longmapsto\mathcal{BB}_d(\mathcal F)= [BB(\mathcal F,p_1),\dots, BB(\mathcal F,p_{N(d)})], \] where the singular points of \(\mathcal F\) are \(\{p_1,\dots,p_{N(d)}\}\subset\mathbb P^2\) and \([\dots]\) means the class under \(S_{N(d)}\), the group of permutations of order \(N(d)\). Given \(\mathcal F\) the author denotes its fiber \(\mathcal{BB}_d^{-1}(\mathcal{BB}_d(\mathcal F))\) by \(F_d(\mathcal F)\). Furthermore, the group of holomorphic automorphisms of \(\mathbb P^2\) determines a natural action \(\Psi\), given by \[ (T,\mathcal F) \in\mathrm{Aut}(\mathbb P^2)\times \mathbb F\mathrm{ol}_{nd}(d,2) \mapsto{\Psi} T^\ast(\mathcal F)\in \mathbb F\mathrm{ol}_{nd}(d,2), \] leaving invariant the Baum-Bott indexes. In particular, when \(d=2\), the dimension of the generic fiber of \(\mathcal{BB}_2\) is \(8\), coinciding with \(\dim_{\mathbb C}(\mathrm{Aut}(\mathbb P^2))\). Whence the generic fiber is a finite union of \(\Psi\)-orbits. The main result is as follows: The generic fiber \(F_d (\mathcal F)\) of \(\mathcal{BB}_2\) contains exactly 240 \(\Psi\)-orbits. The paper explores very interesting examples and provides related problems. Authors' abstract: Our main result says that the generic rank of the Baum-Bott map for foliations of degree \(d\), \(d\geq 2\), of the projective plane is \(d^2+d\). This answers a question of Gómez-Mont and Luengo and shows that there are no other universal relations among the Baum-Bott indexes of a foliation of \(\mathbb{P}^2\) besides the Baum-Bott formula. We also define the Camacho-Sad field for foliations on surfaces and prove its invariance under the pull-back by meromorphic maps. As an application, we prove that a generic foliation of degree \(d\geq 2\) is not the pull-back of a foliation of smaller degree. In Appendix A, we show that the monodromy of the singular set of the universal foliation with very ample cotangent bundle is the full symmetric group.

The paper under review regards singular holomorphic foliations \(\mathcal F\) on the complex projective plane \(\mathbb P^2\). Roughly speaking, the Baum-Bott map associates to a foliation the Baum-Bott indexes of its singularities. See also [\textit{A. Lins Neto} et al., Compos. Math. 142, No. 6, 1549--1586 (2006; Zbl 1109.37041)]. The author studies the fibers of the Baum-Bott map in the space of foliations of degree two on \(\mathbb P^2\). In detail, the set of singular holomorphic foliations \(\{\mathcal F\}\) of degree \(d\) and having only non-degenerate singularities is denoted by \(\mathbb F\mathrm{ol}_{nd}(d,2)\). It is an open Zariski subset of the complex projective space, comprised by all the singular holomorphic foliations of degree \(d\). A foliation \(\mathcal F\) in \(\mathbb F\mathrm{ol}_{nd}(d,2)\) has \(d^2+d+1:=N(d)\) isolated singularities. Hence, the \textit{Baum-Bott} map is \[ \mathcal{BB}_d:\mathbb F\mathrm{ol}_{nd}(d,2) \longrightarrow\mathbb C^{N(d)}/S_{N(d)} \] \[ \mathcal F\longmapsto\mathcal{BB}_d(\mathcal F)= [BB(\mathcal F,p_1),\dots, BB(\mathcal F,p_{N(d)})], \] where the singular points of \(\mathcal F\) are \(\{p_1,\dots,p_{N(d)}\}\subset\mathbb P^2\) and \([\dots]\) means the class under \(S_{N(d)}\), the group of permutations of order \(N(d)\). Given \(\mathcal F\) the author denotes its fiber \(\mathcal{BB}_d^{-1}(\mathcal{BB}_d(\mathcal F))\) by \(F_d(\mathcal F)\). Furthermore, the group of holomorphic automorphisms of \(\mathbb P^2\) determines a natural action \(\Psi\), given by \[ (T,\mathcal F) \in\mathrm{Aut}(\mathbb P^2)\times \mathbb F\mathrm{ol}_{nd}(d,2) \mapsto{\Psi} T^\ast(\mathcal F)\in \mathbb F\mathrm{ol}_{nd}(d,2), \] leaving invariant the Baum-Bott indexes. In particular, when \(d=2\), the dimension of the generic fiber of \(\mathcal{BB}_2\) is \(8\), coinciding with \(\dim_{\mathbb C}(\mathrm{Aut}(\mathbb P^2))\). Whence the generic fiber is a finite union of \(\Psi\)-orbits. The main result is as follows: The generic fiber \(F_d (\mathcal F)\) of \(\mathcal{BB}_2\) contains exactly 240 \(\Psi\)-orbits. The paper explores very interesting examples and provides related problems. Disasters are usually managed through a four-phase cycle including mitigation, preparedness, response and recovery. The first two phases happen before a disaster and the last two after it. This survey focuses on casualty management (CM), which is one of the actions taken in the response phase of a disaster. Right after a severe disaster strikes, we may be confronted with a large number of casualties in a very short period of time. These casualties are in need of urgent treatment and their survival depends on a rapid response. Therefore, managing resources in the first few hours after a disaster is critical and efficient CM can significantly increase the survival rate of casualties. Uncertainty in the location of a disaster, disruption to transportation networks, scarcity of resources and possible deaths of rescue and medical teams due to the disaster in such situations make it hard to manage casualties. In this survey, we focus on CM for disasters where the following five steps are taken, respectively: (i) Resource dispatching/search and rescue, (ii) on-site triage, (iii) on-site medical assistance, (iv) transportation to hospitals and (v) triage and comprehensive treatment. With a special focus on Operations Research (OR) techniques, we categorize the existing research papers and case studies in each of these steps. Then, by critically observing and investigating gaps, trends and the practicality of the extant research studies, we suggest future directions for academics and practitioners.

The authors make use of the Banach contraction mapping principle to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition introduced in [\textit{G. Mastroeni}, in: Equilibrium problems and variational models, Nonconvex Optim. Appl. 68, 289--298 (2003; Zbl 1069.49009)]. Then, they apply the algorithm to strongly monotone Lipschitzian variational inequalities. As a consequence, they obtain a new linearly convergent derivative-free algorithm for strongly monotone complementarity problems. The linear convergence rate allows the algorithm to be coupled with inexact proximal point methods for solving monotone (not necessarily strongly monotone) problems satisfying the Lipschitz-type condition mentioned above. Finally, the authors propose a line-search free algorithm for the strong monotone problem which does not require the Lipschitz-type condition. Applications to a Nash-Cournot market equilibrium model are discussed in section 6 and some preliminary computational results are reported. The auxiliary problem principle introduced by \textit{G. Cohen} [J. Optimization Theory Appl. 32, 277--305 (1980; Zbl 0417.49046); ibid. 59, No. 2, 325--334 (1988; Zbl 0628.90066)] is extended to a general equilibrium problem. In particular, applications to variational inequalities and to convex optimization problems are analysed.

The authors make use of the Banach contraction mapping principle to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition introduced in [\textit{G. Mastroeni}, in: Equilibrium problems and variational models, Nonconvex Optim. Appl. 68, 289--298 (2003; Zbl 1069.49009)]. Then, they apply the algorithm to strongly monotone Lipschitzian variational inequalities. As a consequence, they obtain a new linearly convergent derivative-free algorithm for strongly monotone complementarity problems. The linear convergence rate allows the algorithm to be coupled with inexact proximal point methods for solving monotone (not necessarily strongly monotone) problems satisfying the Lipschitz-type condition mentioned above. Finally, the authors propose a line-search free algorithm for the strong monotone problem which does not require the Lipschitz-type condition. Applications to a Nash-Cournot market equilibrium model are discussed in section 6 and some preliminary computational results are reported. After a brief review of stability conditions and some wall-crossing phenomena, the author proves that certain ``dual'' moduli spaces of Bridgeland stable objects are isomorphic. Following this, the author describes a sequence of flips of the secant varieties of each Veronese surface \(\nu_d\left(\mathbb{P}^2\right) \subset | \mathcal{O}_{\mathbb{P}^2}(d) |\) by embedding \(\text{bl}_{\nu_d\left(\mathbb{P}^2\right)}| \mathcal{O}_{\mathbb{P}^2}(d) |\) into a suitable moduli space of Bridgeland stable objects on \(\mathbb{P}^2\).

The main result of the book consist in (1) the identification and analysis of those categories of parallelism which are far beyond the ones typically associated with PROLOG (namely AND- and OR-parallelism), and (2) the derivation of MMLOP logic-based language suitable both for the specification of parallel problems and the parallel evaluation of logic programs. Organization and contents of the book: Chapters 1 and 2 overview some necessary foundations from the perspective of the two basic standpoints in the book: logic (\textit{W. Bibel's} Connection method [J. Autom. Reasoning 6, No. 3, 287-297 (1990; Zbl 0702.68091)], its specialized variant called model elimination, and the logic programming language LOP), and parallelism (the parallel specification language UNITY). Chapter 3 discusses the state of the art in the field of parallel logic systems: the languages used for programming, the computational models applied to language evaluation, the abstract machines to execute and code during the evaluation, and the hardware for implementing the systems. Chapter 4 investigates various parallelism categories, with their grains, and their formal specification in the UNITY language. Chapter 5 exposes the parallel logic language MMLOP, having two major sources of parallelism: the problem-induced and the execution-oriented one. MMLOP expresses parallelism constructs of two main classes: modules and meta- evaluation. These are the underlying notions for the MMLOP computational model, described in Chapter 6. Implications of the particular computational model called spanning setters (coming from spanning sets met in connection method) are investigated on three classes of hardware architectures in Chapter 7. Conclusions are summarized in Chapter 8, the book ending with a rich bibliography of 483 basic titles. Quadratic proofs of pigeonhole formulas are presented using connection method proof techniques. The interest of this result derives from the fact that for this class of formulas exponential lower bounds are known for the length of resolution refutations.

The main result of the book consist in (1) the identification and analysis of those categories of parallelism which are far beyond the ones typically associated with PROLOG (namely AND- and OR-parallelism), and (2) the derivation of MMLOP logic-based language suitable both for the specification of parallel problems and the parallel evaluation of logic programs. Organization and contents of the book: Chapters 1 and 2 overview some necessary foundations from the perspective of the two basic standpoints in the book: logic (\textit{W. Bibel's} Connection method [J. Autom. Reasoning 6, No. 3, 287-297 (1990; Zbl 0702.68091)], its specialized variant called model elimination, and the logic programming language LOP), and parallelism (the parallel specification language UNITY). Chapter 3 discusses the state of the art in the field of parallel logic systems: the languages used for programming, the computational models applied to language evaluation, the abstract machines to execute and code during the evaluation, and the hardware for implementing the systems. Chapter 4 investigates various parallelism categories, with their grains, and their formal specification in the UNITY language. Chapter 5 exposes the parallel logic language MMLOP, having two major sources of parallelism: the problem-induced and the execution-oriented one. MMLOP expresses parallelism constructs of two main classes: modules and meta- evaluation. These are the underlying notions for the MMLOP computational model, described in Chapter 6. Implications of the particular computational model called spanning setters (coming from spanning sets met in connection method) are investigated on three classes of hardware architectures in Chapter 7. Conclusions are summarized in Chapter 8, the book ending with a rich bibliography of 483 basic titles. The algorithm for computing the spatial relationship is the key of a spatial database and the Geographical Information System (GIS). This paper focuses on the qualitative representation and computing of the directional relationship in the spatial relationship and presents an efficient algorithm to compute directional relationship between regions composed of sets of polygons in linear time.

A lower bound for the Hausdorff dimension of the non-self-similar fractal \(A_{F,T}\) generated by a certain family of constructions \(F=(f_ 1,\dots,f_ p)\) (hyperbolic iterated function system) and a square matrix \(T\) is found. The answer is given in terms of the lower bounds \(r_ i\), \(1\leq i\leq p\), of the Lipschitz constants of the functions belonging to \(F\). Namely, \[ \dim_ H A_{F,T} \geq d(R,T) \] where \(R=\text{diag}\{r_ 1,\dots,r_ p\}\) and \(d(R,T)=d\) is determined uniquely in such a way that the spectral norm of \(TR^ d\) is equal to 1. By the same token, the author proves a conjecture of \textit{D. B. Ellis} and \textit{M. G. Branton} [Lect. Notes Math. 1342, 158-171 (1988; Zbl 0683.58033)]. [For the entire collection see Zbl 0653.00011.] This paper is divided into three sections. In the first section we define the notion of an iterated function system and discuss the relationship between attractors of these systems and symbol spaces. In the second section we recall the definition of Hausdorff-Besikovitch dimension and give an elementary proof of an estimate for the dimension of the attractor for a hyperbolic iterated function system. The Markov attractor of a hyperbolic iterated function system is discussed in the third section. In particular, we show that the symbol space \(\Sigma^+_ M\), where M is a Markov transition matrix, has Hausdorff-Besikovitch dimension \(\ln \| M\| /\ln 2\), where \(\| M\|\) is the spectral radius of M. We obtain estimates for the Markov attractor of more general hyperbolic iterated function systems.

A lower bound for the Hausdorff dimension of the non-self-similar fractal \(A_{F,T}\) generated by a certain family of constructions \(F=(f_ 1,\dots,f_ p)\) (hyperbolic iterated function system) and a square matrix \(T\) is found. The answer is given in terms of the lower bounds \(r_ i\), \(1\leq i\leq p\), of the Lipschitz constants of the functions belonging to \(F\). Namely, \[ \dim_ H A_{F,T} \geq d(R,T) \] where \(R=\text{diag}\{r_ 1,\dots,r_ p\}\) and \(d(R,T)=d\) is determined uniquely in such a way that the spectral norm of \(TR^ d\) is equal to 1. By the same token, the author proves a conjecture of \textit{D. B. Ellis} and \textit{M. G. Branton} [Lect. Notes Math. 1342, 158-171 (1988; Zbl 0683.58033)]. According to standard theory founded on \textit{J. C. Harsanyi} [``Cardinal utility in welfare economics and in the theory of risk-taking'', J. Polit. Econ. 61, 434--435 (1953; \url{doi:10.1086/257416}); ``Cardinal welfare, individualistic ethics and interpersonal comparisons of utility'', ibid. 63, 309--321 (1955; \url{doi:10.1086/257678})] a social welfare function can be appropriately based on the individual's approach to choice under uncertainty. We investigate how people really do rank distributions in terms of welfare. According to Harsanyi, the evaluation can be done from the standpoint of an uninvolved external judge, a public official, for example, or by a person who knows that she holds one of the positions in society, with an equal chance for any of the available positions. Are these two structures to be viewed differently? We use a questionnaire experiment to focus on the two different interpretations of the Harsanyi approach. There are important, systematic differences that transcend the cultural background of respondents.