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In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

Stefan Kempisty was a Polish mathematician, working on the theory of real functions, set theory, integrals, interval functions and the theory of surface area. In 1919 he defended his Ph.D. thesis, On semi-continuous functions , at the Jagiellonian University in Cracow under the supervision of Kazimierz Żorawski. In December 1924 he did his habilitation at the University of Warsaw and continued his work at the Stefan Batory University in Vilnius. Kempisty published over forty scientific papers, three textbooks and one monograph. Kempisty's name in mathematics appears in connection with the definition of quasi-continuous functions, different kinds of continuity of functions of several variables, the classification of Baire, Young and Sierpiński functions, interval functions, and Denjoy or Burkill integrals. This paper is prepared for a wide range of readers. It is an abridged version of the article written in Polish by the same authors (cf. Jóźwik et al., 2017), where can be found more detailed information. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in S n (123) and S n (132). We first study the distribution of consecutive pattern γ -matches in S n (123) and S n (132) for each length 3 consecutive pattern γ. Then we extend our methods to study the joint distributions of multiple consecutive patterns. Some more general cases are discussed in this paper as well. [ABSTRACT FROM AUTHOR]

Though the central role of Gösta Mittag-Leffler in the promotion of specialized, research-oriented mathematics at Stockholms Högskola is widely acknowledged, the specific social and technical means by which he sought to cultivate a fledgling research community there during the early- to mid-1880s have received little attention. In particular, a detailed study of the relationship of his own research activity to that of his first Swedish students is absent from the existing literature. Through the juxtaposition of their research activities and unpublished correspondence, this paper explores Mittag-Leffler's active and deliberate efforts to engage his students Ivar Bendixson and Edvard Phragmén in open problems within his own research agenda, support them through his institutional connections, and instill within them norms concerning research ideologies, practices of communication and criticism, and frameworks for shared knowledge. It also illuminates the extent to which his teachings took root in at least one student to emerge from his program, who would perpetuate the mathematical practices set in place by his teacher and set forth on the international stage to promote his newly-acquired system of values. [ABSTRACT FROM AUTHOR]

Abstract Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when | P | = 2. However, despite various attempts, characterising this number in terms of agreement forests for | P | > 2 remains elusive. In this paper, we characterise the minimum hybridisation number for when P is of arbitrary size and consists of not necessarily binary trees. Building on our previous work on cherry-picking sequences, we first establish a new characterisation to compute the minimum hybridisation number in the space of tree-child networks. Subsequently, we show how this characterisation extends to the space of all rooted phylogenetic networks. Moreover, we establish a particular hardness result that gives new insight into some of the limitations of agreement forests. [ABSTRACT FROM AUTHOR]

Abstract: The present paper deals with counting the number and orders of regular semisimple elements, together with the number of primary classes of . The approach used in this paper depends essentially on partitions of positive integers ⩽n. We give the numbers of regular semisimple elements and primary classes of for and see that the number of regular semisimple elements is an integral polynomial in q, while the number of primary classes is a rational polynomial in q. [Copyright &y& Elsevier]

Abstract: [E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the -Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon''s theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result. [Copyright &y& Elsevier]

Abstract: Difference systems of sets (DSS) were introduced in 1971 by Levenstein for the construction of codes for synchronization, and are closely related to cyclic difference families. In this paper, algebraic constructions of difference systems of sets using functions with optimum nonlinearity are presented. All the difference systems of sets constructed in this paper are perfect and optimal. One conjecture on difference systems of sets is also presented. [Copyright &y& Elsevier]

Abstract: In this paper the concept of a contraction for multi-valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved. Presented results generalize and improve the recent results of Y. Feng, S. Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112], D. Klim, D. Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139] and several others. The method used in the proofs of our results is new and is simpler than methods used in the corresponding papers. Two examples are given to show that our results are genuine generalization of the results of Feng and Liu and Klim and Wardowski. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence of a solution to the scalar equilibrium problem in generalized convex space. As applications, we derive results on the best approximations and simultaneous approximations. The results of this paper generalize some known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we determine all connected 5-arc transitive cubic Cayley graphs on the alternating group ; there are only two such graphs (up to isomorphism). By earlier work of the authors, these are the only two non-normal connected cubic arc-transitive Cayley graphs for finite nonabelian simple groups, and so this paper completes the classification of such non-normal Cayley graphs. [Copyright &y& Elsevier]

Abstract: The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655–666]. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the conjecture saying that, for any connected locally finite graph and any vertex-transitive group of automorphisms of , at least one of the following assertions holds: (1) There exists an imprimitivity system of on with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph in the induced group of automorphisms is finite. (2) The graph is hyperbolic (i.e., for some positive integer , the graph defined by and contains the regular tree of valency 3). Our approach to the conjecture consists in fixing a finite permutation group and considering the conjecture under the assumption that the stabilizer of a vertex of in induces on the neighborhood of the vertex a group permutation isomorphic to . In the paper we elaborate a method (the modified track method) which allows us to prove the conjecture for many groups . The paper consists of two parts. The present first part of the paper involves results on which the modified track method arguments are based, and a few first applications of the method. The second part is devoted to applications of the modified track method. [Copyright &y& Elsevier]

Abstract: We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild–Kostant–Rosenberg theorem. The main contributions of the present paper are:[] we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds; [] we give a proof of the fact that the natural Chern character map becomes, after the HKR isomorphism, the usual one ; and [] we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture. [Copyright &y& Elsevier]

Abstract: In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed examples of a distance graph on the real line R and difference graphs on the real plane R whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space R , whose chromatic number is a positive integer in the Zermelo–Fraenkel-choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory. [Copyright &y& Elsevier]

The paper presents a formula for the γ-interior of a set under special conditions for γ :expX→expX, more general than those in the previous paper [Acta Math. Hungar. 80 (1998) 89–93]. There are also some applications. [Copyright &y& Elsevier]

We develop upper and lower bound arguments for counting acceptance modes of communication protocols. A number of separation results for counting communication complexity classes is established. This extends the investigation of the complexity of communication between two processors in terms of complexity classes initiated by Babai et al. (Proceedings of the 27th IEEE FOCS, 1986, pp. 337–347) and continued in several papers (e.g., J. Comput. System Sci. 41 (1990) 402; 49 (1994) 247; Proceedings of the 36th IEEE FOCS, 1995, pp. 6–15).In particular, it will be shown that for all pairs of distinct primes p and q the communication complexity classes MODpPcc and MODqPcc are incomparable with regard to inclusion. The same is true for PPcc and MODmPcc, for any number m⩾2. Moreover, non-determinism and modularity are incomparable to a large extend. On the other hand, if m=p1ℓ1·…·prℓr is the prime decomposition of m⩾2, then the complexity classes MODmPcc and MODρ(m)Pcc coincide, where ρ(m)=p1·…·pr.The results are obtained by characterizing the modular and probabilistic communication complexity in terms of the minimum rank of matrices ranging over certain equivalence classes. Methods from algebra and analytic geometry are used.This paper is the completely revised and strongly extended version of the conference paper Damm et al. (Proc. 9th Ann. STACS, pp. 281–291) where a subset of the results was presented. [Copyright &y& Elsevier]

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider 𝑛 pairwise disjoint s element sets and form all (s2)n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least 𝑛ds, where ds=1/c⌈s/2⌉ is defined in the paper and tends to e−1 as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi–Tóth bound on an old Erdo&x030B;s problem: we prove that n distinct points in the plane determine Ω(n4e5e−1−ϵ) distinct distances, where ϵ>0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables. [Copyright &y& Elsevier]

Let N be the set of positive integers and A a subset of N. For n∈N, let p(A,n) denote the number of partitions of n with parts in A. In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any N∈N and B⊂{1,2,…,N}, there is a unique set A=A0(B,N), such that p(A,n) is even for n>N. Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) considered σ(A,n)=∑d∣n,d∈Ad, and proved that for all k≥0, the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F2[z] with f(0)=1, by constructing a set A such that the generating function fA of A is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in F2[z] with P(0)=Q(0)=1, then for all k≥0 the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. [Copyright &y& Elsevier]

This article provides a detailed discussion of Otto Ho¨lder's ideas on the foundations of mathematics. It focuses on a paper he published in 1892 written in reaction to a book published in 1891 by Robert Graßmann, which Ho¨lder saw as an attempt to axiomatize arithmetic. Ho¨lder's paper is important for at least three reasons: First, it represents what might be called Ho¨lder's research manifesto on the foundations of mathematics, containing a wealth of ideas which Ho¨lder gradually developed in a variety of publications until the end of his life. Second, Ho¨lder's analysis of R. Graßmann's foundational ideas provides an important assessment of the contribution of Hermann and Robert Graßmann to the axiomatization of arithmetic, a contribution which, though often mentioned, is itself still not widely acknowledged and not fully understood. Third, the effort of exposing the weak spots in R. Graßmann's ideas led Ho¨lder to formulate the main problems confronting formal axiomatics: independence of the axioms, consistency, completeness, and the issue of the relationship between pure mathematics and its applications. The first part of this paper presents R. Graßmann's ideas on the foundations of mathematics as outlined in two closely related works published in 1872 and 1891. The second part focuses on Ho¨lder's analysis of these ideas. [Copyright &y& Elsevier]

In this paper, we prove some results on the sum-product problem over arbitrary fields which improve and generalize results given by Hegyvári and Hennecart [5] . More precisely, we prove that, for related pairs of two-variable functions f ( x , y ) and g ( x , y ) , if A and B are two sets in an arbitrary field F with | A | = | B | , then max ⁡ { | f ( A , B ) | , | g ( A , B ) | } ≫ | A | 1 + c , for some c > 0 . [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 H be an atomic monoid. The set of distances Δ ( H ) of H is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d but u 1 ⋅ … ⋅ u k cannot be written as a product of ℓ irreducible elements for any ℓ ∈ N with k < ℓ < k + d . It is well-known (and easy to show) that, if Δ ( H ) is nonempty, then min ⁡ Δ ( H ) = gcd ⁡ Δ ( H ) . In this paper we show conversely that for every finite nonempty set Δ ⊂ N with min ⁡ Δ = gcd ⁡ Δ there is a finitely generated Krull monoid H such that Δ ( H ) = Δ . [ABSTRACT FROM AUTHOR]

In this paper, we give an algebraic construction of a new infinite family of Cameron–Liebler line classes with parameter x = q 2 − 1 2 for q ≡ 5 or 9 ( mod 12 ) , which generalizes the examples found by Rodgers in [26] through a computer search. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets in AG ( 2 , q ) , which is closely related to our Cameron–Liebler line classes. [ABSTRACT FROM AUTHOR]

Given a k × l ( 0 , 1 ) -matrix F , we denote by fs ( m , F ) the largest number for which there is an m × fs ( m , F ) ( 0 , 1 ) -matrix with no repeated columns and no induced submatrix equal to F . A conjecture of Anstee, Frankl, Füredi and Pach states that, regarding F as fixed, fs ( m , F ) = O ( m k ) . The main results of this paper are that for k = 2 , fs ( m , F ) = O ( m 2 + o m ( 1 ) ) and for k ≥ 3 that fs ( m , F ) = O ( m ( 5 / 3 ) k − 1 + o m ( 1 ) ) . [ABSTRACT FROM AUTHOR]

Abstract: We say the sets of nonnegative integers and are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set by using analytic tools. [Copyright &y& Elsevier]

Abstract: In this paper, we determine all sets A of integers such that, for any integral-valued polynomial which has no fixed divisor, for all integers and n, there are infinitely many integers and a choice of such that . The earlier result shows that is such a set. [Copyright &y& Elsevier]

Abstract: A system of distinct representatives (SDR) of a family is a sequence of distinct elements with for . Let denote the number of SDRs of a family ; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer , a family is called a -family if the union of any sets in the family contains at least elements. The famous Hall’s theorem says that if and only if is a -family. Denote by the minimum number of SDRs in a -family. The problem of determining and those families containing exactly SDRs was first raised by Chang [G.J. Chang, On the number of SDR of a -family, European J. Combin. 10 (1989) 231–234]. He solved the cases when and gave a conjecture for . In this paper, we solve the conjecture. [Copyright &y& Elsevier]

Abstract: In this paper we obtain a bilinear analogue of Fefferman–Steinʼs vector valued inequality for classical Hardy–Littlewood maximal function. Also, we prove the boundedness of bilinear Hardy–Littlewood maximal operator from , where , by applying the method of rotations. [Copyright &y& Elsevier]

Abstract: By now the -spectra of self-conformal measures satisfying the so-called open set condition are well understood. However, if the open set condition is not satisfied, then almost nothing is known. In this paper we compute the mixed -spectra for self-conformal measures, which have bound distortion property and satisfy the weak separation condition. [Copyright &y& Elsevier]

Abstract: In this paper we generalize a recent result of Wittmann on densities of the 4-rank of class groups of imaginary quadratic function fields to the ℓ-rank of class groups of imaginary ℓ-cyclic extensions of where ℓ is any odd prime dividing . [Copyright &y& Elsevier]

Abstract: In this paper, we study rank two semifields of order that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in Marino et al. (in press) for . Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with ; for any , such that and for any with . Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields. [Copyright &y& Elsevier]

Abstract: This paper is a continuation of previous work by Győri, Sárközy, and the author, concerning the maximal number of integers that can be selected from so that none of them divides the product of others. [Copyright &y& Elsevier]

Abstract: In this paper, the invertibility of nonnegative Hamiltonian operator with unbounded entries is studied, and the sufficient conditions for the everywhere defined bounded invertibility of nonnegative Hamiltonian operator are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Let be a simple graph of order . The domination polynomial of is the polynomial , where is the number of dominating sets of of size . A root of is called a domination root of . We denote the set of distinct domination roots by . Two graphs and are said to be -equivalent, written as , if . The -equivalence class of is . A graph is said to be -unique if . In this paper, we show that if a graph has two distinct domination roots, then . Also, if is a graph with no pendant vertex and has three distinct domination roots, then . Also, we study the -equivalence classes of some certain graphs. It is shown that if , then is -unique, and if , then consists of exactly two graphs. [ABSTRACT FROM AUTHOR]

Abstract: We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erdős and Nathanson showed that for every there exists a minimal asymptotic basis A of order h with , where denotes the density of A. Erdős and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order such that . Moreover, they asked if there exists a minimal asymptotic basis with . In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition [Copyright &y& Elsevier]

Abstract: In this paper, we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption). [Copyright &y& Elsevier]

Abstract: Quantum finite automata have been studied intensively since their introduction in late 1990s as a natural model of a quantum computer working with finite-dimensional quantum memory space. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover communicates with a quantum-automaton verifier through a common communication cell. Our quantum interactive proof systems are juxtaposed to Dwork–Stockmeyer''s classical interactive proof systems whose verifiers are two-way probabilistic finite automata. We demonstrate strengths and weaknesses of our systems by studying how various restrictions on the behaviors of quantum-automaton verifiers affect the power of quantum interactive proof systems. [Copyright &y& Elsevier]

Abstract: SDPS-sets are very nice sets of points in dual polar spaces which themselves carry the structure of dual polar spaces. They were introduced in [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211] because they gave rise to new valuations and hyperplanes of dual polar spaces. In the present paper, we show that the symplectic dual polar space , , has up to isomorphisms a unique SDPS-set. [Copyright &y& Elsevier]

Abstract: In this paper we study a distance-regular graph of diameter such that for any given pair of vertices at distance there exists a strongly closed subgraph of diameter containing them. We prove several inequalities for intersection numbers of . We show that if the equalities hold in some of these inequalities, then is either the Odd graph, the doubled Odd graph, the doubled Grassmann graph, the Hamming graph or the dual polar graph. [Copyright &y& Elsevier]

Abstract: The semifields of order which are two-dimensional over their left nucleus and six-dimensional over their center have been geometrically partitioned into six classes by using the associated linear sets in . One of these classes has been partitioned further into three subclasses. In this paper the generic multiplication is determined for each of these three subclasses, and several examples of new semifields are constructed that belong to these subclasses. For two of the subclasses, no examples were previously known. [Copyright &y& Elsevier]

Abstract: Let be a union-closed family of subsets of an m-element set A. Let and for let denote the number of sets in that contain a. Frankl''s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element with . Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280–288] we verify the conjecture for the particular case when and . Moreover, for these “large” families we prove an even stronger version via averaging. Namely, the sum of the , for all , is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices. [Copyright &y& Elsevier]

Abstract: Let G be a block-transitive automorphism group of a 2- design . It has been shown that the pairs fall into three classes: those where G is unsolvable and is flag-transitive, those where G is a subgroup of , and those where G is solvable and is of small order. Not much is known about the latter two classes. In this paper, we investigate the existence of 2- designs admitting a block-transitive automorphism group . Using Weil''s theorem on character sums, the following theorem is proved: if a prime power q is large enough and then there is a 2- design which has a block-transitive, but nonflag-transitive automorphism group G. Moreover, using computers, some concrete examples are given when q is small. [Copyright &y& Elsevier]

Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic . It is proved in this paper that if the p-envelope of adL in DerL contains a torus of maximal dimension whose centralizer in adL acts nontriangulably on L, then and L is isomorphic to one of the Melikian algebras . In conjunction with [A. Premet, H. Strade, Simple Lie algebras of small characteristic V. The non-Melikian case, J. Algebra 314 (2007) 664–692, Theorem 1.2], this implies that, up to isomorphism, any finite-dimensional simple Lie algebra over an algebraically closed field of characteristic is either classical or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5. This result finally settles the classification problem for finite-dimensional simple Lie algebras over algebraically closed fields of characteristic . [Copyright &y& Elsevier]

Abstract: In this paper we study reachability relations on vertices of digraphs, informally defined as follows. First, the weight of a walk is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed in the direction opposite to their orientation. Then, a vertex is -related to a vertex if there exists a 0-weighted walk from to whose every subwalk starting at has weight in the interval . Similarly, a vertex is -related to a vertex if there exists a 0-weighted walk from to whose every subwalk starting at has weight in the interval . For all positive integers , the relations and are equivalence relations on the vertex set of a given digraph. We prove that, for transitive digraphs, properties of these relations are closely related to other properties such as having property , the number of ends, growth conditions, and vertex degree. [Copyright &y& Elsevier]

Abstract: Let R be a commutative ring and denote its zero-divisor graph. In this paper we investigate the crosscap number of the non-orientable compact surface which can be embedded and illustrate all finite commutative rings R (up to isomorphism) such that is projective. [Copyright &y& Elsevier]

Abstract: The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms. [Copyright &y& Elsevier]

Abstract: In many problems in analysis, dynamics, and in their applications, it is important to subdivide objects under consideration into simple pieces, keeping control of high-order derivatives. It is known that semi-algebraic sets and mappings allow for such a controlled subdivision: this is the “ reparametrization theorem” which is a high-order quantitative version of the well-known results on the existence of a triangulation of semi-algebraic sets. In a -version we just require in addition that each simplex be represented as an image, under the “reparametrization mapping”, of the standard simplex, with all the derivatives of up to order k uniformly bounded. The main result of this paper is, that if we reparametrize all the set A but its small part of a size , we can do much more: not only to “kill” the derivatives, but also to bound uniformly the analytic complexity of the pieces, while their number remains of order . [Copyright &y& Elsevier]

Abstract: This paper consists of three interconnected parts. Parts I, III study the relationship between the cohomology of a reductive group G and that of a Levi subgroup H. For example, we provide a sufficient condition, arising from Kazhdan–Lusztig theory, for a natural map to be surjective, given irreducible G-modules and corresponding irreducible H-modules . In cohomological degree , the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer [D. Hemmer, A row removal theorem for the quiver of symmetric groups and Schur algebras, Proc. Amer. Math. Soc. 133 (2) (2005) 403–414 (electronic)] for , and both extend and improve upon the latter when our condition is met. Part II obtains results on Lusztig character formulas (LCFs) for reductive groups, obtaining new necessary and sufficient conditions for such formulas to hold. In the special case of , these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group. This work on improves upon [B. Parshall, L. Scott, Quantum Weyl reciprocity for cohomology, Proc. London Math. Soc. 90 (3) (2005) 655–688], which established only sufficient, rather than necessary and sufficient, conditions for the validity of the LCF in terms of the symmetric group. [Copyright &y& Elsevier]

Abstract: This paper is concerned with chaos induced by regular snap-back repellers. One new criterion of chaos induced by strictly coupled-expanding maps in compact sets of metric spaces is established. By employing this criterion, the nondegenerateness assumption in the Marotto theorem established in 1978 is weakened. In addition, it is proved that a regular snap-back repeller and a regular homoclinic orbit to a regular expanding fixed point in finite-dimensional spaces imply chaos in the sense of Li–Yorke. An illustrative example is provided with computer simulations. [Copyright &y& Elsevier]