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Let Λ be a concealed canonical algebra and d the dimension vector of a Λ-module which is periodic respect to the action of the Auslander–Reiten translation τ . In the paper, we investigate the union of the closures of the orbits of the τ -periodic Λ-modules of dimension vector d . We show that this set is closed and regular in codimension one. [ABSTRACT FROM AUTHOR]

In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial differential equations. Our analysis leads to new integral equations of non-convolution type, which can be solved by classical methods. We derive solutions of these integral equations, which in turn lead to solutions of the associated Cauchy problems. [ABSTRACT FROM AUTHOR]

Given a linear equation L , a set A ⊆ [ n ] is L -free if A does not contain any ‘non-trivial’ solutions to L . In this paper we consider the following three general questions: (i) What is the size of the largest L -free subset of [ n ] ? (ii) How many L -free subsets of [ n ] are there? (iii) How many maximal L -free subsets of [ n ] are there? We completely resolve (i) in the case when L is the equation p x + q y = z for fixed p , q ∈ N where p ≥ 2 . Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L , thereby refining a special case of a result of Green (2005). We also give various bounds on the number of maximal L -free subsets of [ n ] for three-variable homogeneous linear equations L . For this, we make use of container and removal lemmas of Green (2005). [ABSTRACT FROM AUTHOR]

Let K be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea [25] , it has been proved that on the symbolic space X of K a natural augmented tree structure E exists; it is hyperbolic, and the hyperbolic boundary ∂ H X with the Gromov metric is Hölder equivalent to K . In this paper we consider certain reversible random walks with return ratio 0 < λ < 1 on ( X , E ) . We show that the Martin boundary M can be identified with ∂ H X and K . With this setup and a device of Silverstein [41] , we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate Θ ( ξ , η ) ≍ | ξ − η | − ( α + β ) , where α is the Hausdorff dimension of K and β depends on λ . For suitable β , the kernel defines a regular non-local Dirichlet form on K . This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5] ). [ABSTRACT FROM AUTHOR]

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala–Sturm (2014) [27] . In particular, it is shown that the qualitative non-degeneracy condition introduced by Cavalletti–Huesmann (2015) [6] implies that any essentially non-branching metric measure space has a unique transport maps whenever the initial measure is absolutely continuous. This generalizes a recently obtained result by Cavalletti–Mondino (2017) [8] on essentially non-branching spaces with the measure contraction condition MCP ( K , N ) . In the end we prove a measure rigidity result showing that any two essentially non-branching, qualitatively non-degenerate measures on a fixed metric spaces must be mutually absolutely continuous. This result was obtained under stronger conditions by Cavalletti–Mondino (2016) [7] . It applies, in particular, to metric measure spaces with generalized finite dimensional Ricci curvature bounded from below. [ABSTRACT FROM AUTHOR]

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form x + γ Tr q n / q ( x k ) is also presented, which generalizes two examples of [10] . [ABSTRACT FROM AUTHOR]

Recently, linear codes constructed from defining sets have been studied extensively. They may have excellent parameters if the defining set is chosen properly. Let m > 2 be a positive integer. For an odd prime p , let r = p m and Tr be the absolute trace function from F r onto F p . In this paper, we give a construction of linear codes by defining the code C D = { ( Tr ( a x ) ) x ∈ D : a ∈ F r } , where D = { x ∈ F r : Tr ( x ) = 1 , Tr ( x 2 ) = 0 } . Its complete weight enumerator and weight enumerator are determined explicitly by employing cyclotomic numbers and Gauss sums. However, we find that the code is optimal with respect to the Griesmer bound provided that m = 3 . In fact, it is MDS when m = 3 . Moreover, the codes presented have higher rate compared with other codes, which enables them to have essential applications in areas such as association schemes and secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this paper we study quasi-homogeneous operators, which include the homogeneous operators, in the Cowen–Douglas class. We give two separate theorems describing canonical models (with respect to equivalence under unitary and invertible operators, respectively) for these operators using techniques from complex geometry. This considerably extends the similarity and unitary classification of homogeneous operators in the Cowen–Douglas class obtained recently by the last author and A. Korányi. In a significant generalization of the properties of the homogeneous operators, we show that quasi-homogeneous operators are irreducible and determine which of them are strongly irreducible. Applications include the equality of the topological and algebraic K -group of a quasi-homogeneous operator and an affirmative answer to a well-known question of Halmos. [ABSTRACT FROM AUTHOR]

Large data sets are often modeled as being noisy samples from probability distributions μ in R D , with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well-approximated by k -dimensional planes, with k ≪ D , with k -dimensional manifolds isometrically embedded in R D being a special case. Samples from μ are furthermore corrupted by D -dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r , with a fixed center and varying r , and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M . As an application, we present an algorithm for estimating the intrinsic dimension of M . [ABSTRACT FROM AUTHOR]

In this paper, we present a new and simple proof of the classical Bernstein–Walsh inequality. Based on this proof, we give some improvements for this inequality in the case that the corresponding compact set is real. [ABSTRACT FROM AUTHOR]

The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al. and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of the proof we obtain a notion of cyclic descent set on these tableaux, which is rotated by the cyclic action on them. [ABSTRACT FROM AUTHOR]

Let β > 1 be a real number. For any x ∈ [ 0 , 1 ] , the run-length function r n ( x , β ) is defined as the length of the longest run of 0's amongst the first n digits in the β -expansion of x . Let { δ n } n ≥ 1 be a non-decreasing sequence of integers and define E ( { δ n } n ≥ 1 ) = { x ∈ [ 0 , 1 ] : lim sup n → ∞ r n ( x , β ) δ n = 1 } . In this paper, we show that dim H ⁡ E ( { δ n } n ≥ 1 ) = max ⁡ { 0 , 1 − lim inf n → ∞ δ n ⧸ n } . Using the same method, we also study a class of extremely refined subset of the exceptional set in Erdös–Rényi limit theorem. Precisely, we prove that if lim inf n → ∞ δ n n = 0 , then the set E max ( { δ n } n ≥ 1 ) = { x ∈ [ 0 , 1 ] : lim inf n → ∞ r n ( x , β ) δ n = 0 , lim sup n → ∞ r n ( x , β ) δ n = + ∞ } has full Hausdorff dimension. [ABSTRACT FROM AUTHOR]

The transcendental equation x e x = z and its solutions, described by the Lambert W function, often occur in physics and mathematics. In the last decades it turned out that the study of similar but more general equations is necessary in molecular physics, in the theory of general relativity and also in the description of Bose–Fermi mixtures as well as in some combinatorial problems. In this paper we offer a full description for the solution set of a one parameter generalization of the above mentioned equation. [ABSTRACT FROM AUTHOR]

In this paper, we extend the theory of special local Moufang sets. We construct a local Moufang set from every local Jordan pair, and we show that every local Moufang set satisfying certain (natural) conditions gives rise to a local Jordan pair. We also explore the connections between these two constructions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with subquadratic homogeneity elliptic problems with VMO-coefficients in divergence form. We obtain that the weak solution u is locally Hölder continuous besides a singular set by using A -harmonic approximation, where the Hölder exponent α ∈ ( 0 , 1 ) . The Hausdorff dimension of the singular set is less than n − p . [ABSTRACT FROM AUTHOR]

In this article, by introducing a new set and a new concept of φ − ( h , e ) -concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ − ( h , e ) -concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems. [ABSTRACT FROM AUTHOR]

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generalization of the concentricity inequality (which states that the sum of the inradius and circumradius is not greater than the diameter in general Minkowski spaces) for non-symmetric gauge bodies is derived and a strong connection between this new inequality, extremal sets of the generalized Bohnenblust inequality, and completeness of simplices is revealed. [ABSTRACT FROM AUTHOR]

This paper studies exact algorithms for the Maximum Induced Matching problem, in which an n -vertex graph is given and we are asked to find a set of maximum number of edges in the graph such that no pair of edges in the set have a common endpoint or are adjacent by another edge. This problem has applications in many different areas. We give several structural properties of the problem and show that the problem can be solved in O ⁎ ( 1.4231 n ) time and polynomial space or O ⁎ ( 1.3752 n ) time and exponential space. [ABSTRACT FROM AUTHOR]

In this paper we construct explicit sampling sets and present reconstruction algorithms for Fourier signals on finite vector spaces G , with | G | = p r for a suitable prime p . The two sampling sets have sizes of order O ( p t 2 r 2 ) and O ( p t 2 r 3 log ⁡ ( p ) ) respectively, where t is the number of large coefficients in the Fourier transform. The algorithms approximate the function up to a small constant of the best possible approximation with t non-zero Fourier coefficients. The fastest of the algorithms has complexity O ( p 2 t 2 r 3 log ⁡ ( p ) ) . [ABSTRACT FROM AUTHOR]

Many problems on graphs can be expressed in the following language: given a graph G = ( V , E ) and a terminal set T ⊆ V , find a minimum size set S ⊆ V which intersects all “structures” (such as cycles or paths) passing through the vertices in T . We refer to this class of problems as terminal set problems. In this paper, we introduce a general method to obtain faster exact exponential time algorithms for several terminal set problems. In the process, we break the O ⁎ ( 2 n ) barrier for the classic Node Multiway Cut , Directed Unrestricted Node Multiway Cut and Directed Subset Feedback Vertex Set problems. [ABSTRACT FROM AUTHOR]

In this paper universality limits are studied in connection with measures which exhibit power-type singular behavior somewhere in their support. We extend the results of Lubinsky for Jacobi measures supported on [ − 1 , 1 ] to generalized Jacobi measures supported on a compact subset of the real line, where the singularity can be located in the interior or at an endpoint of the support. The analysis is based upon the Riemann–Hilbert method, Christoffel functions, the polynomial inverse image method of Totik and the normal family approach of Lubinsky. [ABSTRACT FROM AUTHOR]

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree d , we construct a compact set E ⊂ R n of Hausdorff dimension n / d which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set E ⊂ R n , we study the angles determined by three-point subsets of E . The main result implies the existence of a compact set in R n of Hausdorff dimension n / 2 which does not contain the angle π / 2 . (This is known to be sharp if n is even.) We show that there is a compact set of Hausdorff dimension n / 8 which does not contain an angle in any given countable set. We also construct a compact set E ⊂ R n of Hausdorff dimension n / 6 for which the set of angles determined by E is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle ε close to any given angle. The main result can also be applied to distance sets. As a corollary we obtain a compact set E ⊂ R n ( n ≥ 2 ) of Hausdorff dimension n / 2 which does not contain rational distances nor collinear points, for which the distance set is Lebesgue null, moreover, every distance and direction is realised only at most once by E . [ABSTRACT FROM AUTHOR]

A right R -module is called R -projective if it is projective relative to the right R -module R R . A ring R is called right testing for projectivity if every right R -projective module is projective. Every right perfect ring is right testing and there are examples of local rings that are not right testing. In this paper an attempt is made to understand the class of right testing rings and several examples are provided to highlight the set theoretic obstacles in characterizing such rings. [ABSTRACT FROM AUTHOR]

This paper answers two open problems raised by Mahlon M. Day [4] and James C.S. Wong [17] : Does uniformly topological left amenability imply the existence of left translation continuous measures on a locally compact semitopological semigroup? Let T be a locally compact Borel subsemigroup of a locally compact semitopological semigroup S . Is the existence of a topological T -invariant mean M on S with M ( χ T ) > 0 enough to imply the topological left amenability of T ? We give a negative answer to the first question by providing a counterexample and a positive proof to the second. This example can also be used to show that the property ( α ) provided in Gerard L. Sleijpen [14] can not be removed in one of his results. [ABSTRACT FROM AUTHOR]

In this paper we obtain the pointwise gradient estimates via the nonlinear Wolff potentials for weak solutions of a class of non-homogeneous quasilinear elliptic equations with measure data. [ABSTRACT FROM AUTHOR]

In this paper, the stability of a certain class of time-varying systems evolving over nonuniformly spaced discrete domains is analyzed. Switched systems, used here in the context of dynamic equations over time scale domains, arise naturally in applications when a continuous time system is discretized via a sample-and-hold method with multiple sample rates. The stability of switched systems is typically deduced by appealing to certain interrelated properties of the subsystems (such as pairwise commutativity [24] , simultaneous diagonalization [7] , simultaneous triangularizability [8] , or other Lie algebraic conditions [1] ) which imply the existence of a common quadratic Lyapunov function. A novel approach is used here to determine the existence of quadratic Lyapunov functions which does not rely on how the subsystems interact with each other. This new method instead examines the role they each play in the aggregate system by way of singular value conditions. Results implying switched system stability and instability are developed under the two primary methods of examining such systems: how the system behaves during arbitrary switching and how it behaves under the influence of a particular switching signal. Several examples illuminating the results are provided throughout the text, as well as important insight discussing certain bounds on these results as they apply to stability theory in general. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the continuity property of the α -core with respect to data perturbations in different environments. We show that some collections of abstract economies (or normal games) with the nonempty α -core have the Hadamard well-posedness property. We also show that the α -core points of every game in a (dense) residual subset are essential. [ABSTRACT FROM AUTHOR]

This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz–Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the d -dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is derived. [ABSTRACT FROM AUTHOR]

This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive Banach spaces. A special case of the main result improves a classical application of the Mountain Pass Theorem in the fractal setting, given by Falconer and Hu in [14] . [ABSTRACT FROM AUTHOR]

Locally L 0 -convex modules were introduced in Filipovic et al. (2009) [10] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in Drapeau et al. (2016) [8] . In this paper we study locally L 0 -convex modules, and find exactly which subclass of locally L 0 -convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we establish a version of the so-called Jouini–Schachermayer–Touzi theorem for robust representation of conditional convex risk measures defined on a L ∞ -type module. [ABSTRACT FROM AUTHOR]

Let Φ be a family of functions analytic in some neighborhood of a complex domain Ω, and let T be a Hilbert space operator whose spectrum is contained in Ω ‾ . Our typical result shows that under some extra conditions, if the closed unit disc is complete K ′ -spectral for φ ( T ) for every φ ∈ Φ , then Ω ‾ is complete K -spectral for T for some constant K . In particular, we prove that under a geometric transversality condition, the intersection of finitely many K ′ -spectral sets for T is again K -spectral for some K ≥ K ′ . These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea–Beckermann–Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of T to a normal operator with spectrum in ∂Ω. As a key tool, we use the results from our previous paper [11] on traces of analytic uniform algebras. [ABSTRACT FROM AUTHOR]

Liouville copulas introduced in McNeil and Nešlehová (2010) are asymmetric generalizations of the ubiquitous Archimedean copula class. They are the dependence structures of scale mixtures of Dirichlet distributions, also called Liouville distributions. In this paper, the limiting extreme-value attractors of Liouville copulas and of their survival counterparts are derived. The limiting max-stable models, termed here the scaled extremal Dirichlet, are new and encompass several existing classes of multivariate max-stable distributions, including the logistic, negative logistic and extremal Dirichlet. As shown herein, the stable tail dependence function and angular density of the scaled extremal Dirichlet model have a tractable form, which in turn leads to a simple de Haan representation. The latter is used to design efficient algorithms for unconditional simulation based on the work of Dombry et al. (2016) and to derive tractable formulas for maximum-likelihood inference. The scaled extremal Dirichlet model is illustrated on river flow data of the river Isar in southern Germany. [ABSTRACT FROM AUTHOR]

This paper focuses on the following generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) , x ∈ R N , where N ≥ 3 , g ( s ) : R → R + is a nondecreasing function with respect to | s | . By using a change of variables and variational methods, we obtain the existence and multiplicity of nontrivial solutions for the above problem when the nonlinearity is superlinear but does not satisfy the Ambrosetti–Rabinowitz type condition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the Dirichlet boundary value problems of the form in a bounded smooth domain. Conditions – in the present paper are somewhat weaker than the famous Ambrosetti–Rabinowitz-type superquadratic condition. Here, we assume that the primitive of the nonlinearity g is either asymptotically quadratic or superquadratic as . [Copyright &y& Elsevier]

Abstract: This paper presents a set of distributed algorithms that support an Intrusion Detection System (IDS) model for Mobile Ad hoc NETworks (MANETs). The development of mobile networks has implicated the need of new IDS models in order to deal with new security issues in these communication environments. More conventional models have difficulties to deal with malicious components in MANETs. In this paper, we describe the proposed IDS model, focusing on distributed algorithms and their computational costs. The proposal employs fault tolerance techniques and cryptographic mechanisms to detect and deal with malicious or faulty nodes. The model is analyzed along with related works. Unlike studies in the references, the proposed IDS model admits intrusions and malice in their own algorithms. In this paper, we also present test results obtained with an implementation of the proposed model. [Copyright &y& Elsevier]

Abstract: In the paper Generalized roundness and negative type, Lennard, Tonge, and Weston show that the geometric notion of generalized roundness in a metric space is equivalent to that of negative type. Using this equivalent characterization, along with classical embedding theorems, the authors prove that for , fails to have generalized roundness q for any . It is noted, as a consequence, that the Schatten class , for , has maximal generalized roundness 0. In this paper, we prove that this result remains true for p in the interval . [Copyright &y& Elsevier]

A relational structure is homogeneous if every isomorphism between finite induced substructures has an extension to an automorphism of . A relational structure is indivisible if for every partition of the set of elements of , there is a copy of in or in . The structure is weakly indivisible if for every partition of the set of elements of for which some finite induced substructure of does not have a copy in , there exists a copy of in . The structure is age indivisible if for every partition of the elements of every finite induced substructure of has an embedding into or every finite induced substructure of has an embedding into . It follows that indivisibility implies weak indivisibility which in turn implies age indivisibility. There are many examples of countable infinite homogeneous structures which are weakly indivisible but not indivisible; see Sauer (2000) [17] and the end of Section 3.1 of this paper. In general, it is difficult to find structures which are age indivisible but not weakly indivisible. One such example (see Pouzet et al. (2011) [16]) is obtained via the inhomogeneous general linear group of vector spaces over finite fields. Finding such an example for homogeneous structures seems to be even more difficult. The only one that I obtained, together with L. Nguyen Van Thé, is a countable homogeneous metric subspace of the Hilbert sphere (see Nguyen Van Thé and Sauer (2010) [13]). Both of the examples above are relational structures with infinitely many relations. One of the still open questions then is: Are there countable infinite homogeneous structures with finite signature which are age but not weakly indivisible? We will show that the generic “free amalgamation” homogeneous structures are weakly indivisible; see Section 3.1. Countable “Urysohn metric spaces” (see the next section), with finite distance sets, are indivisible (see Sauer (2013) [19]). Countable Urysohn metric spaces are age indivisible, which follows from the Hales–Jewett theorem (Hales and Jewett (1963) [7]; see Delhommé et al. (2007) [2]). For non-Urysohn homogeneous countable metric spaces, except for the example mentioned above, the weak indivisibility question is completely open. We will show, in this paper, that a countable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible. [ABSTRACT FROM AUTHOR]

Abstract: We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new. [Copyright &y& Elsevier]

Abstract: In several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named -groups. Groups with all subgroups being -groups are called groups with the basis property. In the first part of this paper we correct and generalize some characterizations of groups with the basis property. In the second part we consider groups satisfying analogous conditions, but only for sets of generators of prime power orders. The class of groups introduced here is much larger then the class of groups with the basis property. For example, it contains all nilpotent groups. [Copyright &y& Elsevier]

Abstract: Given a subroup G of a stable (in the model-theoric sense) group Γ, in particular when Γ is a group of finite Morley rank, the traces on G of the definable subsets of Γ have a remarkable property: if the definable closure of G is connected, they are either supergeneric, or supergenerically complemented, in the sense of the definition given at the very beginning of this paper. An example of this situation is provided by the linear groups: for some n, G is a subgroup of , where K is a field that we may take algebraically closed; the definable sets in the sense of are its constructible subsets, i.e. the boolean combinations of a finite number of its Zariski closed subsets. For any group G, the supergeneric subsets of G form a filter of large sets, which, to my best knowledge, is defined here for the first time. This paper undertakes the study of supergenericity in a general context, with no hypotheses of a model-theoric nature, but with a special attention given to the very specific properties of genericity possessed by the definable subsets of a stable group. It can be read without any knowledge of Logic, provided that one is ready to skip the proofs of the theorems showing precisely that these definable sets have these properties. [Copyright &y& Elsevier]

Abstract: A ring R is left exact if, for every finitely generated left submodule , every left R-linear function from S to R extends to a left R-linear function from to R. The class of exact rings generalizes that of self-injective rings and has been introduced in a recent paper by Wilding, Johnson, and Kambites. In our paper we show that the group ring of a group G over a ring R is left exact if and only if R is left exact and G is locally finite. [Copyright &y& Elsevier]

Abstract: This paper studies the Cauchy problem for a parabolic–elliptic system in modeling chemotaxis as well as self-attracting particles. In the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying ω-limit sets. According to the results of this paper, any nonnegative radially symmetric bounded solution either stabilizes to a steady-state as , or oscillates between two steady-states. Moreover, a rather general class of nonnegative initial data, not necessarily radially symmetric, for which the associated solutions exhibit a complex oscillatory behavior is constructed; their ω-limit sets consist of a nontrivial topological continuum of steady-states. Besides the technical difficulties inherent to the lack of compactness of the resolvent operators, one has to add the challenge that the problem is utterly non-local. Consequently, thought the basic ideas on the foundations of this paper might be considered classical, most of the proofs throughout are extremely sophisticated and absolutely new in their full generality. [Copyright &y& Elsevier]

Abstract: We consider a generalized Gause prey–predator model with T-periodic continuous coefficients. In the case where the Poincaré map over time T is well defined, the result of the paper can be explained as follows: we locate a subset U of such that the topological degree equals to +1. The novelty of the paper is that the later is done under only continuity and (some) monotonicity assumptions for the coefficients of the model. A suitable integral operator is used in place of the Poincaré map to cope with possible nonuniqueness of solutions. The paper, therefore, provides a new framework for studying the generalized Gause model with functional differential perturbations and multi-valued ingredients. [Copyright &y& Elsevier]

Abstract: A graph is called locally-primitive if the vertex stabilizer is primitive on the neighbor set of for each vertex . In this paper, we classify locally-primitive Cayley graphs of dihedral groups, while 2-arc-transitive Cayley graphs of dihedral groups have been classified by a series of papers in the literature. [Copyright &y& Elsevier]

Abstract: Continuous analogs of the strong Szegő limit theorem may be formulated in terms of operators of the form for , where denotes the operator of multiplication by a suitably restricted mvf (matrix-valued function) acting on the space of vvfʼs (vector-valued functions) f that meet the constraint with and denotes the orthogonal projection onto the space of entire vvfʼs of exponential type⩽T that are subject to the same summability constraint. In this paper we study these operators for a more general class of Δ of the form , where , are mvfʼs in the Wiener plus algebra. This paper extends an earlier paper [6] by replacing the assumption that is an inner mvf for some by the less restrictive assumption that the Hankel operator with symbol is compact. We show that is a trace-class operator, that exists and is independent of Q and R when and . An example which shows that may depend on Q and R if these commutation conditions are not in force is furnished. [Copyright &y& Elsevier]

Abstract: This paper answers the conjecture of Adimurthi and Struwe [4], that the semilinear Trudinger–Moser functional (as well as functionals with more general critical nonlinearities) satisfies the Palais–Smale condition at all levels except , . In this paper we construct critical sequences at any level corresponding to a large family of distinct concentration profiles, indexed by closed subsets C of , that arise in the two-dimensional case instead of the “standard bubble” in higher dimensions. The paper uses the notion of concentration of [2,5] developed in the spirit of Solimini [14] and of [15]. [Copyright &y& Elsevier]

Abstract: The present paper contains a martingale representation theorem for set-valued martingales defined on a filtered probability space with a filtration generated by a Brownian motion. It is proved that such type martingales can be defined by some generalized set-valued stochastic integrals with respect to a given Brownian motion. The main result of the paper is preceded by short part devoted to the definition and some properties of generalized set-valued stochastic integrals. [Copyright &y& Elsevier]

Abstract: In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–Čech compactification . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–Čech compactification of . [Copyright &y& Elsevier]

Abstract: This paper deals with a nonlinear delay differential inclusion of evolution type involving m-dissipative operator and source term of multi-valued type in a Banach space. Under rather mild conditions, the -structure of -solution set is studied on compact intervals, which is then used to obtain the -property on non-compact intervals. Secondly, the result about the structure is furthermore employed to show the existence of -solutions for the inclusion (mentioned above) subject to nonlocal condition defined on right half-line. No nonexpansive condition on nonlocal function is needed. As samples of applications, we consider a partial differential inclusion with time delay and then with nonlocal condition at the end of the paper. [Copyright &y& Elsevier]

Abstract: Text: Let the prime factorization of n be with . A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is , where denotes the kth prime. In this paper I prove that all integers of the form ql are ordinary, where l is a square-free positive integer and q is a prime. This confirms a conjecture of Yong-Gao Chen. Video: For a video summary of this paper, please click here or visit http://youtu.be/WTY4wr8L_U0. [Copyright &y& Elsevier]