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Abstract: In this paper, we consider the following Toda system: where ρ is a positive parameter which will go to zero, is a positive smooth function, and is a unit ball in . In [29], the authors construct a family of solutions of (0.1) for , such that the solutions blow up at the origin with mass , after some scaling with the limiting profile in , , where for . In that paper, the authors ask whether there exist blow-up solutions of mass , with the limiting profile such that is the sum of two bubbles and has one bubble. In this paper, we prove that such solution exists. [Copyright &y& Elsevier]

Abstract: In the present paper, an adaptive control of structural vibrations is presented. Based on earlier research, we claim that the periodical switching on of magneto-rheological controlled dampers results in the reduction of the amplitudes of vibrations more than does their permanent actuation. This statement, when applied to a moving load problem, was mathematically proved in earlier papers. In the present paper we determine the efficiency of such a control applied to a rotating shaft. The earlier mathematical analysis allows us to propose a control strategy. A finite element simulation together with the solution of the control problem shows that the dampers should act only during a short period of the highest displacements of the structure. The same conclusion is found in experimental tests. Although high frequency control with MR dampers is less efficient than in the theoretical investigations, we have found an amplitude reduction in the range of 10–20%. [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: Let M be an internally 4-connected binary matroid and N be an internally 4-connected proper minor of M. In our search for a splitter theorem for internally 4-connected binary matroids, we proved in the third paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, either M has a proper internally 4-connected minor with an N-minor such that , or, up to duality, M has a triangle T and an element e of T such that has an N-minor and has the property that one side of every 3-separation is a fan with at most four elements. The fourth paper in the series proved that, when we cannot find such a proper internally 4-connected minor of M, we can incorporate the triangle T into one of two substructures of M, a good bowtie or a good augmented 4-wheel. The goal of this paper is essentially to eliminate the need to consider good augmented 4-wheels by showing that, when M contains such a substructure, either it also contains a good bowtie, or, in an easily described way, we can obtain an internally 4-connected minor of M with an N-minor. [Copyright &y& Elsevier]

Abstract: In our quest to find a splitter theorem for internally 4-connected binary matroids, we proved in the preceding paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N either has a proper internally 4-connected minor with an N-minor such that or has, up to duality, a triangle T and an element e of T such that has an N-minor and has the property that one side of every 3-separation is a fan with at most four elements. This paper proves that, when we cannot find such a proper internally 4-connected minor of M, we can incorporate the triangle T into one of two substructures of M: a bowtie or an augmented 4-wheel. In the first of these, M has a triangle disjoint from T and a 4-cocircuit that contains e and meets . In the second, T is one of the triangles in a 4-wheel restriction of M with helpful additional structure. [Copyright &y& Elsevier]

Abstract: Text: This paper concerns the function , where is the argument of the Riemann zeta-function along the critical line. The main result is that which holds for all . Video: For a video summary of this paper, please click here or visit http://youtu.be/FldP0idE0aI. [Copyright &y& Elsevier]

Abstract: Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most by the Kahn–Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for , the number of independent sets in a graph with is at most that in . In this paper, we give a lower bound on the number of independent sets in a d-regular graph mirroring the upper bound in the Kahn–Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvinʼs conjecture, covering the case as well. We find it convenient to address this problem from the perspective of . From this perspective, we show that the number of complete subgraphs of a graph G on n vertices with , where with , is bounded above by the number of complete subgraphs in . [Copyright &y& Elsevier]

Abstract: In this paper some characterizations and properties of domains in with boundary accessible externally by some circular cones are given. It is shown that the boundary of such domains is also conically accessible from the interior. It is also characterized -accessible domains of , which are biholomorphically equivalent to the open Euclidean ball. The paper is also a continuation of a concept of V.V. Starkov (2011) [13] and refers to papers of D.A. Brannan and W.E. Kirwan (1969) [3], W. Ma and D. Minda (1991) [10], J. Stankiewicz (1966) [11], and T. Sugawa (2005) [15] connected with a planar case. [Copyright &y& Elsevier]

The purpose of this paper is to study the boundedness of operators of the form where is a function defined on a neighborhood of the origin in , satisfying , ψ is a cut-off function supported on a small neighborhood of , and K is a “multi-parameter singular kernel” supported on a small neighborhood of . We also study associated maximal operators. The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on ( ). The case when K is a Calderón–Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their work to the case when K is (for instance) given by a “product kernel”. Even when K is a Calderón–Zygmund kernel, our methods yield some new results. This is the second paper in a three part series. The first paper deals with the case , while the third paper deals with the special case when γ is real analytic. [ABSTRACT FROM AUTHOR]

Abstract: In an earlier paper (Turull, 2012 [12]) the author introduced the concept of endoisomorphisms and showed its natural connection with Clifford theory of finite groups. Associated with Clifford theory are module and character correspondences. In the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results. [Copyright &y& Elsevier]

Abstract: In a recent paper by Orazov et al. [On the dynamics of a novel ocean wave energy converter. Journal of Sound and Vibration329 (24) (2010) 5058–5069], a wave energy converter (WEC) was proposed. The converter features a mass modulation scheme and a simple model was used to examine its efficacy. The simple model did not adequately account for the momentum transfer which takes place during the mass modulation. The purpose of the present paper is to account for this transfer and to show that the WEC equipped with a novel and more general mass modulation scheme has the potential to improve its energy harvesting capabilities. [Copyright &y& Elsevier]

Abstract: This paper is mainly concerned with asymptotic studies of weighted bootstrap for - and -statistics. We derive the consistency of the weighted bootstrap - and -statistics, based on i.i.d. and non i.i.d. observations from some more general results which we first establish for sums of randomly weighted arrays of random variables. Some of the results in this paper significantly extend some well-known results on consistency of -statistics and also consistency of sums of arrays of random variables. We also employ a new approach to conditioning to derive a conditional central limit theorem (CLT) for weighted bootstrap - and -statistics, assuming the same conditions as the classical CLT for regular - and -statistics. [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]

In this paper we deal with noncoercive elliptic multipower systems of divergence type, which include p-Laplacian type operators as well as mean curvature operators and whose right hand sides depend on the product of both components of the solution and on a gradient factor. We prove that any nonnegative nontrivial entire weak solution (nonnecessarily radial) is constant. For nontrivial solutions we intend that both components are nontrivial. The paper improves former results due to Clément, Fleckinger, Mitidieri, de Thélin (2000) in [4], to Bidaut-Véron and Pohozaev (2001) in [3], where no gradient terms are considered. On the other hand the paper contains, as subcases, also some recent results of Filippucci (2011) [12] in which systems with gradient dependence for the forcing terms are treated but not of multipower type. The key ingredient of the proof technique is the method of test functions, employed by Mitidieri and Pohozaev for studying systems in Mitidieri and Pohozaev (1999) [17]. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions is required. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the Morse index and the symmetry-breaking for positive solutions of the following two-point boundary value problem are studied, where , , on , , for , and . The problem for the one-dimensional Hénon equation is a typical example, where and . This problem always has the unique positive even solution. It is well-known that if , then there is no positive non-even solution, and if is sufficiently large, then there exist positive non-even solutions. The result in this paper shows that if , then the Morse index of the positive least energy solution equals 1 and the Morse index of the positive even solution equals 2, and hence the positive least energy solution is non-even and symmetry-breaking phenomena occur. It is also shown that if and are sufficiently small, then there is no positive non-even solution and the Morse index of the even positive solution equals 1. [Copyright &y& Elsevier]

Abstract: Two definitions of the Clifford index for vector bundles on a smooth projective curve C of genus were introduced in a previous paper by the authors. In another paper the authors obtained results on one of these indices for bundles of rank 3. Here we extend these results in the case where C has classical Clifford index 3. In particular we prove Mercatʼs conjecture for bundles of rank 3 for and when C has classical Clifford index 3. We obtain complete results in the case of genus 7. [Copyright &y& Elsevier]

Abstract: Text: The aim of this paper is to establish some sharp bounds for the Glaisher–Kinkelin and Bendersky–Adamchik constants. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LmxI_0AW1vk. [Copyright &y& Elsevier]

Abstract: In this paper we deduce some new supercongruences modulo powers of a prime . Let . We show that and where denotes the Euler polynomial of degree , and stands for the Legendre symbol. The paper also contains some other results such as [Copyright &y& Elsevier]

Abstract: The study of the equivalence relations of particular multialgebras for which the factor multialgebras are universal algebras satisfying certain identities is a very important and intensively studied topic in multialgebra theory and not only. Our paper provides a general multialgebraic approach for the construction of all these relations and also some important and interesting properties concerning the construction of the corresponding factor universal algebra. One of the purposes of our approach is to improve some results concerning -relations of hyperrings and Krasner hyperrings and we will do this in the last sections of this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain existence of ground state solutions and ground state sign-changing solutions for the following Schrödinger equation where , , and are sign-changing functions satisfying very mild assumptions. The proofs rely on a new perturbed method which we design in this paper. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to establish certain inequalities between the numbers of convex polytopes in “containing” and “avoiding” zero provided that their vertex sets are subsets of a given finite set of points in . This paper is motivated by a question about these quantities raised by E. Boros and V. Gurvich in 2002. [Copyright &y& Elsevier]

Abstract: This is the second paper in a series devoted to studies of twisted tensor products of nonlocal vertex algebras. In this paper we introduce and study iterated twisted tensor products of nonlocal vertex algebras. Among the main results, we find conditions for constructing an iterated twisted tensor product of three factors, and prove that those conditions are enough for building an iterated twisted tensor product of any number of factors. And we also establish a universal property and give a characterization of an iterated twisted tensor product. Furthermore, we give an example of iterated twisted tensor product nonlocal vertex algebra. [Copyright &y& Elsevier]

Abstract: For a set of primes π, a group X is said to be π-decomposable if is the direct product of a π-subgroup and a -subgroup , where is the complementary of π in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: “Let π be a set of odd primes. If the finite group is a product of two π-decomposable subgroups and , then and this is a Hall π-subgroup of G.” We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. [Copyright &y& Elsevier]

Abstract: This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a closure operator to any representation which help us determine some criteria for the functions used in a representation. While linearly representable matroids can be viewed as having representations via matrices with only one row, we conclude this paper by an investigation of representations via matrices with only two rows. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the long-time behaviour of the solutions of the damped wave equation on . This equation has travelling front solutions of the form . Gallay and Joly have proved in Gallay and Joly (2009) [7] that when the nonlinearity is of bistable type, if the initial data are sufficiently close to the profile of a front for large , the solution of the damped wave equation converges uniformly on to a travelling front as . In this paper, we establish a global stability result under more general assumptions on the function , which include in particular nonlinearities of combustion type. We impose, however, more restrictive conditions on the initial data in the region . We also apply our method to the case of a monostable pushed front. [Copyright &y& Elsevier]

Abstract: The Grothendieck construction of a diagram of categories can be seen as a process to construct a single category by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category to the 2-category of small -categories for a fixed commutative ring . In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors are derived equivalent if there is a derived equivalence from to for all objects in satisfying some “-equivariance” conditions. In this paper we glue the derived equivalences between and together to obtain a derived equivalence between Grothendieck constructions and , which shows that if colax functors are derived equivalent, then so are their Grothendieck constructions. This generalizes and well formulates the fact that if two -categories with a -action for a group are “-equivariantly” derived equivalent, then their orbit categories are derived equivalent. As an easy application we see by a unified proof that if two -algebras and are derived equivalent, then so are the path categories and for any quiver ; so are the incidence categories and for any poset ; and so are the monoid algebras and for any monoid . Also we will give examples of gluing of many smaller derived equivalences together to have a larger derived equivalence. [Copyright &y& Elsevier]

Abstract: We gather together several bounds on the sizes of coefficients which can appear in factors of polynomials in ; we include a new bound which was latent in a paper by Mignotte, and a few improvements to some existing bounds. We compare these bounds, and for each bound give explicit examples where that bound is the best; thus showing that no one bound is universally better than the others. In the second part of the paper we give several concrete examples of factorizations where the factors have “unexpectedly” large coefficients. These examples help us understand why the bounds must be larger than you might expect, and greatly extend the collection published by Collins. [Copyright &y& Elsevier]

Abstract: In this paper we study transversals and matchings in 3-uniform hypergraphs. For , let be a -uniform hypergraph, and so every edge in has size . A transversal (or hitting set or vertex cover) in is a set of vertices in that has a nonempty intersection with every edge of , while the transversal number of is the minimum size of a transversal in . Let be the size of a maximum matching in . We observe that . We define to be -special if is a connected hypergraph with no isolated vertex satisfying and . We remark that -special hypergraphs include all projective planes, which are very well studied. Let denote the maximum order of a -special -uniform hypergraph. As a consequence of a result due to Gallai [T. Gallai, Neuer Beweis eines Tutteschen Satzes, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963) 135–139] we have . Lovász [L. Lovász, On minimax theorems of combinatorics (Doctoral thesis, in Hungarian), Mat. Lapok (NS) 26 (1975) 209–264] showed that for and , we have . Hanson and Toft [D. Hanson, B. Toft, On the maximum number of vertices in -uniform cliques, Ars Combin. 16A (1983) 205–216] showed that and . In this paper we show that for all , we have . [Copyright &y& Elsevier]

Abstract: The context of the paper is: a locally compact Hausdorff space ; the space , equipped with the uniform norm, of real continuous functions on which vanish at infinity; a linear subspace , of finite dimension , of the space ; and the set-valued metric projection of into the family of non-empty compact convex subsets of , defined by  the set of best uniform approximations to from . Those which are Chebyshev (that is the metric projection is single point valued) were characterised by Haar (1918). Those for which the metric projection is lower semi-continuous have been characterised by Wu Li (1989) and A.L. Brown (2005); the paper interprets and exploits the characterisation. A ‘Generalised Haar Condition’ which is necessary for to be lower semi-continuous is identified; in some circumstances it is also sufficient. The condition has a calculable determinantal form. The results of the paper include an essentially complete determination, in case is a compact space in which no net of components is convergent to a single point set, of those subspaces of for which is lower semi-continuous. Simple examples show that if the space is compact and totally disconnected, but not finite, the situation is essentially different. [Copyright &y& Elsevier]

Abstract: We investigate the problem of the maximum number of different cubic subwords (of the form ) in a given word. We also consider square subwords (of the form ). The problem of the maximum number of squares in a word is not well understood. Several new results related to this problem are presented in the paper. We consider two simple problems related to the maximum number of subwords which are squares or which are highly repetitive; then we provide a nontrivial estimation for the number of cubes. We show that the maximum number of squares such that is not a primitive word (nonprimitive squares) in a word of length is exactly , and the maximum number of subwords of the form , for , is exactly . In particular, the maximum number of cubes in a word is not greater than either. Using properties of occurrences of cubes, we improve this bound significantly. We show that the maximum number of cubes in a word of length is between and . 2 [2] In particular, we improve the lower bound from the conference version of the paper by Kubica et al. (2009) [18]. [Copyright &y& Elsevier]

Abstract: In this paper, the divergence theorem is proved by the Kurzweil–Henstock approach. The physical definition of the divergence of a vector field is used, instead of the usual definition used in mathematics papers. Sufficient conditions for the existence of the divergence of a vector field on -dimensional manifolds in are given. Concepts of strong differentiability are used in the sufficient conditions. [Copyright &y& Elsevier]

Abstract: This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations with Robin boundary conditions on an exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations. Our result in the general case and the special case is shorter than that of the initial–boundary value problem for one dimensional general quasilinear wave equations with Dirichlet boundary conditions. The results in this paper are not the trivial generalization of that in the case of Dirichlet boundary conditions. The lower bound estimates of life span of classical solutions to initial–boundary value problems are consistent with the actual physical meaning. The physical phenomenon also explains our results. [Copyright &y& Elsevier]

Abstract: In a recent paper, Duane, Garsia, and Zabrocki introduced a new statistic, “ndinv”, on a family of parking functions. The definition was guided by their study of a recursion on for a Macdonald eigenoperator, a modified Hall–Littlewood operator, and a composition of n. Using their newly introduced statistic, one can give a new interpretation for as a sum of parking functions counted by area and ndinv. This is a departure from the traditional sum, as stated by the shuffle conjecture, which q, t counts area and diagonal inversion number (dinv). Since their definition is necessarily recursive, they pose the problem of obtaining a non-recursive definition. In this paper, we solve this problem by giving an explicit formula for ndinv similar to the classical definition of dinv and prove it is equivalent to the ndinv of Duane, Garsia, and Zabrocki. [Copyright &y& Elsevier]

Abstract: The notation means the finite sum of all numbers as ranges over the integers that divide including and . The purpose of this paper is to define a theory of summability suggested by this type of sum and an associated theory of convergence. This paper studies five sequence spaces that are related to the arithmetical summability thus defined. [Copyright &y& Elsevier]

Abstract: In this paper, we study a suitable notion of solution for which a nonlinear elliptic problem governed by a general Leray–Lions operator is well posed for any diffuse measure data. In terms of the paper (Brezis et al., 2007, [10]), we study the notion of solution for which any diffuse measure is “good measure”. [Copyright &y& Elsevier]

Abstract: In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevichʼs topology, but irreducible with respect to Kambayashiʼs topology. Moreover, we give a counter-example of a supposed irreducibility criterion given in Shafarevich (1981) [Sha81] which is different from the counter-example given by Homma in Kambayashi (1996) [Kam96]. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich. [Copyright &y& Elsevier]

Abstract: Text: For any given two positive integers and , and any set A of nonnegative integers, let denote the number of solutions of the equation with . In this paper, we determine all pairs of positive integers for which there exists a set such that for all . We also pose several problems for further research. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY. [Copyright &y& Elsevier]

Abstract: In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sárközy to this kind of numeration systems. [Copyright &y& Elsevier]

Abstract: In a recent paper, the first author gave a connection between bilateral basic hypergeometric series and mock theta functions, which led to many new identities involving mock theta functions. This paper is a sequel, and our goal is to provide partition theoretic properties of new identities involving third or sixth order mock theta functions. [Copyright &y& Elsevier]

Abstract: We study sharp estimates of integral functionals for operators on the set of real trigonometric polynomials of degree in terms of the uniform norm of the polynomials and similar questions for algebraic polynomials on the unit circle of the complex plane. P. Erdös, A.P. Calderon, G. Klein, L.V. Taikov, and others investigated such inequalities. In this paper, we, in particular, show that the sharp inequality holds on the set for the Weyl fractional derivatives of order for . For (), this fact was proved by Lizorkin (1965) [12]. For and positive integer , the inequality was proved by Taikov (1965) [23]; however, in this case, the inequality follows from results of an earlier paper by Calderon and Klein (1951) [6]. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto–Ukaiʼs recent paper (see [Y. Morimoto, S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseudo-Differ. Oper. Appl. 1 (2010) 139–159]), but we extend the range of the index γ satisfying , and in this case we consider the kinetic factor in the form of instead of as Morimoto and Ukai did before. [Copyright &y& Elsevier]

Abstract: There are now a large number of papers in the spectroscopic literature which make use of multiple-valued (frequently double-valued) coordinate systems and the associated multiple-groups of the permutation-inversion group to deal with the symmetry properties of large-amplitude motions in molecules of high symmetry. The use of multiple-valued coordinate systems, and the resultant appearance of more minima on the potential surface than would be found on the surface for a single-valued coordinate system, can lead to conceptual confusion and questions of mathematical legitimacy. In the present paper we demonstrate that treatments using multiple-valued coordinate systems simply represent one scheme for applying the appropriate quantum mechanical boundary conditions to Schrödinger’s partial differential equation defined in a single-valued coordinate system. The demonstration is not general, but rather focuses on the specific example of the S1 electronic state of C2H2, which has local minima only for nonlinear configurations, and on the twofold and eightfold extended permutation–inversion groups recently introduced to simultaneously treat symmetry questions in trans-bent and cis-bent acetylene. Some discussion of the mathematical convenience lost by using a single-valued coordinate system is also presented. [Copyright &y& Elsevier]

Abstract: Let be the complete multipartite graph with parts, while each part contains vertices. The regular embeddings of complete graphs have been determined by Biggs (1971) , James and Jones (1985) and Wilson (1989) . During the past twenty years, several papers such as Du et al. (2007, 2010) , Jones et al. (2007, 2008) , Kwak and Kwon (2005, 2008) and Nedela et al. (2002) contributed to the regular embeddings of complete bipartite graphs and the final classification was given by Jones in 2010. Since then, the classification for general cases and has become an attractive topic in this area. In this paper, we deal with the orientable regular embeddings of for . We in fact give a reduction theorem for the general classification, namely, we show that if has an orientable regular embedding , then either and for some prime or and the normal subgroup of preserving each part setwise is a direct product of a 3-subgroup and an abelian -subgroup, where may be trivial. Moreover, we classify all the embeddings when and is abelian. We hope that our reduction theorem might be the first necessary approach leading to the general classification. [Copyright &y& Elsevier]

Abstract: There are several notions of the ‘dual’ of a word/tile substitution. We show that the most common ones are equivalent for substitutions in dimension one, where we restrict ourselves to the case of two letters/tiles. Furthermore, we obtain necessary and sufficient arithmetic conditions for substitutions being selfdual in this case. Since many connections between the different notions of word/tile substitution are discussed, this paper may also serve as a survey paper on this topic. [Copyright &y& Elsevier]

Abstract: In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation , defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrímsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Mélou and Steingrímsson asked if might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of , by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and . [Copyright &y& Elsevier]

Abstract: In this paper, the authors first show that the classical Hardy space can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups and , respectively, where is an elliptic operator with real constant coefficients of homogeneous order 2m (). Moreover, the authors also prove that can be characterized by the Riesz transforms if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space and give its characterizations by area integrals related to the semigroups and , respectively. Finally, authors give the boundedness of Riesz transforms, square functions and maximal functions associated with L. [Copyright &y& Elsevier]

Abstract: This paper considers stochastic population dynamics driven by Lévy noise. The contributions of this paper lie in that: (a) Using the Khasminskii–Mao theorem, we show that the stochastic differential equation associated with our model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such a model. [Copyright &y& Elsevier]

Abstract: Let q be a power of 2, n be a positive integer, and let be the finite field with elements. In this paper, we consider the existence of some specific elements in . The main results obtained in this paper are listed as follows: [(1)] There is an element ξ in such that both ξ and are primitive elements of if , and n is an odd number no less than 13 and . [(2)] For , and any odd n, there is an element ξ in such that ξ is a primitive normal element and is a primitive element of if either , and , or , and , . [Copyright &y& Elsevier]

Abstract: This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq–Navier–Stokes equations. The velocity in this system is related to the vorticity through the relations and , which reduces to the standard velocity–vorticity relation when . When either or , the velocity is more singular. The “quasi-velocity” determined by satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when and . In addition, the vorticity is shown to be globally bounded in several functional settings such as for in a suitable range. [Copyright &y& Elsevier]