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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]

Abstract: Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant and for each there are cones with normal constant . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results. [Copyright &y& Elsevier]

Abstract: We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system , where is the set of abstract origamis and is a binary relation on , that models fold. An abstract origami is a structure , where is a set of faces constituting an origami, and and are binary relations on , each representing adjacency and superposition relations between the faces. We then address representation and transformation of abstract origamis and further reasoning about the construction for computational purposes. We present a labeled hypergraph of origami and define fold as algebraic graph transformation. The algebraic graph-theoretic formalism enables us to reason about origami in two separate domains of discourse, i.e. pure combinatorial domain where symbolic computation plays the main role and geometrical domain . We detail the program language for the algebraic graph rewriting and graph rewriting algorithms for the fold, and show how fold is expressed by a set of graph rewrite rules. [Copyright &y& Elsevier]

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]

Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou [28] in even characteristic. In 2016, L. Qu [23] proposed a new approach to constructing quadratic planar functions over F 2 n . Very recently, D. Bartoli and M. Timpanella [4] characterized the condition on coefficients a , b such that the function f a , b (x) = a x 2 2 m + 1 + b x 2 m + 1 ∈ F 2 3 m [ x ] is a planar function over F 2 3 m by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in [23] , we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over F q k , where q = 2 m with m sufficiently large (see Theorem 1.1). The first and last classes of them are over F q 2 and F q 4 respectively, while the other two classes are over F q 3 . One class over F q 3 is an extension of f a , b (x) investigated in [4] , while our proofs seem to be much simpler. In addition, although the planar binomial over F q 2 of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in [23]. [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]

Abstract In this paper, we study the following classical question of extremal set theory: what is the maximum size of a family of subsets of [ n ] such that no s sets from the family are pairwise disjoint? This problem was first posed by Erdős and resolved for n ≡ 0 , − 1 (mod s) by Kleitman in the 60s. Very little progress was made on the problem until recently. The only result was a very lengthy resolution of the case s = 3 , n ≡ 1 (mod 3) by Quinn, which was written in his PhD thesis and never published in a refereed journal. In this paper, we give another, much shorter proof of Quinn's result, as well as resolve the case s = 4 , n ≡ 2 (mod 4). This complements the results in our recent paper, where, in particular, we answered the question in the case n ≡ − 2 (mod s) for s ≥ 5. [ABSTRACT FROM AUTHOR]

Abstract Let π be a permutation of { 1 , 2 , ... , n }. If we identify a permutation with its graph, namely the set of n dots at positions (i , π (i)) , it is natural to consider the minimum L 1 (Manhattan) distance, d (π) , between any pair of dots. The paper computes the expected value (and higher moments) of d (π) when n → ∞ and π is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when d is fixed and n → ∞ , the probability that d (π) ≥ d + 2 tends to e − d 2 − d. The minimum jump mj (π) of π , defined by mj (π) = min 1 ≤ i ≤ n − 1 ⁡ | π (i + 1) − π (i) | , is another natural measure in this context. The paper computes the asymptotic moments of mj (π) , and the asymptotic probability that mj (π) ≥ d + 1 for any constant d. [ABSTRACT FROM AUTHOR]

In 1973, Erdős et al. introduced the anti-Ramsey number for a graph G in K n , which is defined to be the maximum number of colors in an edge-coloring of K n which does not contain any rainbow G . This is always regarded as one of rainbow generalizations of the classic Ramsey theory. Since then the anti-Ramsey numbers for several special graph classes in complete graphs have been determined. Also, the researchers generalized the host graph for the anti-Ramsey number from the complete graph to general graphs, including bipartite graphs, complete split graphs, planar graphs, and so on. In this paper, we study the anti-Ramsey number of matchings in the complete split graph. Since the complete split graph contains the complete graph as a subclass, the results in this paper cover the previous results about the anti-Ramsey number of matchings in the complete graph. [ABSTRACT FROM AUTHOR]

A classical problem, originated by Cohn's 1966 paper [1] , is to characterize the integral domains R satisfying the property: ( GE n ) “every invertible n × n matrix with entries in R is a product of elementary matrices”. Cohn called these rings generalized Euclidean, since the classical Euclidean rings do satisfy ( GE n ) for every n > 0 . Important results on algebraic number fields motivated a natural conjecture: a non-Euclidean principal ideal domain R does not satisfy ( GE n ) for some n > 0 . We verify this conjecture for two important classes of non-Euclidean principal ideal domains: (1) the coordinate rings of special algebraic curves, among them the elliptic curves having only one rational point; (2) the non-Euclidean PID's constructed by a fixed procedure, described in Anderson's 1988 paper [2] . [ABSTRACT FROM AUTHOR]

Allen's Interval Algebra is among the leading formalisms in the area of qualitative temporal reasoning. However, its applications are restricted to linear flows of time. While there is some recent work studying relations between intervals on branching structures, there is no rigorous study of the first-order theory of branching time. In this paper, we approach this problem under a general definition of time structures, namely, tree-like lattices. Allen's work proved that meets is expressively complete in the linear case. We also prove that, surprisingly, it remains complete for all unbounded tree-like lattices. This does not generalize to the case of all tree-like lattices, for which we prove that the smallest complete set of relations has cardinality three. We provide in this paper a sound and complete axiomatic system for both the unbounded and general case, in Allen's style, and we classify minimally complete and maximally incomplete sets of relations. [ABSTRACT FROM AUTHOR]

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m -ovoids and i -tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of non-autonomous degenerate p -Laplacian equations ∂ t u − div ( a ( x ) | ∇ u | p − 2 ∇ u ) + λ u + f ( u ) = g ( x , t ) in Ω, where a ( x ) is allowed to vanish on a nonempty closed subset with Lebesgue measure zero, g ( x , t ) ∈ L l o c p ′ ( R ; D − 1 , p ′ ( Ω , a ) ) and Ω an unbounded domain in R N . We first establish the well-posedness of these equations by constructing a compact embedding. Then we show the existence of the minimal pullback D μ -attractor, and prove that it indeed attracts the D μ class in L 2 + δ -norm for any δ ∈ [ 0 , ∞ ) . Our results extend some known ones in previously published papers. [ABSTRACT FROM AUTHOR]

Text In a series of papers C. Dartyge and A. Sárközy (partly with other coauthors) studied pseudorandom measures of subsets. In this paper we extend the theory of C. Dartyge and A. Sárközy to several dimensions. We introduce measure for multi-dimensional pseudorandom subsets, and study the connection between measures of different orders. Large families of multi-dimensional pseudorandom subsets are given by using the squares in F q . Video For a video summary of this paper, please visit https://youtu.be/7iH9x7nyyfQ . [ABSTRACT FROM AUTHOR]

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A I stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A I is free if the root system is of classical type or G 2 and conjectured that this is also the case for all types. This was established only very recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement A I is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro–Steinberg–Kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated Weyl group. Our first aim in this paper is to investigate a stronger freeness property of the A I . We show that all A I are inductively free, with the possible exception of some cases in type E 8 . In the same paper from 2006, Sommers and Tymoczko define a Poincaré polynomial I ( t ) associated with each ideal I which generalizes the Poincaré polynomial W ( t ) for the underlying Weyl group W . Solomon showed that W ( t ) satisfies a product decomposition depending on the exponents of W for any Coxeter group W . Sommers and Tymoczko showed in a case by case analysis in types A n , B n and C n , and some small rank exceptional types that a similar factorization property holds for the Poincaré polynomials I ( t ) generalizing the formula of Solomon for W ( t ) . They conjectured that their multiplicative formula for I ( t ) holds in all types. In our second aim to investigate this conjecture further, the same inductive tools we develop to obtain inductive freeness of the A I are also employed. Here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions. [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]

• Definition of the Hybrid Fuzzy Stochastic approach for considering soil uncertainties on seismic site-response analysis. • Numerical analyses on all four types of soil classified in accordance to European seismic code. • Determination of the Gradual Elastic Response Spectrum for engineering design of structures. The analysis of the seismic site response is conventionally carried out by the study of the one-dimensional amplification of vertically propagating shear waves through a horizontal soil profile with equivalent-linear elastic properties. Site response analysis requires the specification of the input ground motion and the dynamic characterization of the soil deposit. Whilst the stochastic approach is commonly used to model seismic excitations, the use of probability density functions for describing the soil properties is consistent only when precise information based on a large amount of data from soil surveys are available. Conversely, a non-probabilistic approach based on fuzzy set theory would be more appropriate for dealing with uncertainties that are just expressed by vague, imprecise, qualitative, or incomplete information supplied by engineering judgment. In this paper, we address a hybrid fuzzy-stochastic 1D site response analysis approach: we consider probability models for the seismic input and fuzzy intervals for dealing with soil uncertainties; the problem boundary values are defined as convex normal fuzzy sets and described by means of membership functions. Zadeh's extension principle, in combination with an efficient implementation of the Differential Evolution Algorithm for global minimization and maximization, is used to perform fuzzy computations. Results are presented as fuzzy median value of the largest peaks of the peak ground acceleration at the surface by considering four types of soil classified in accordance with the European seismic building code. Finally, elastic response spectra defined in terms of gradual functions are proposed in order to evaluate the influence of the soil uncertainties on the seismic response of structures. [ABSTRACT FROM AUTHOR]

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles. Applications are given for simultaneous Diophantine approximation, Khintchine-Groshev theorem for rectangles and higher dimensional shrinking target problems. [ABSTRACT FROM AUTHOR]

In this paper we consider the question of well quasi-order for classes defined by a single obstruction within the classes of all graphs, digraphs and tournaments, under the homomorphic image ordering (in both its standard and strong forms). The homomorphic image ordering was introduced by the authors in a previous paper and corresponds to the existence of a surjective homomorphism between two structures. We obtain complete characterisations in all cases except for graphs under the strong ordering, where some open questions remain. [ABSTRACT FROM AUTHOR]

Text Let H n be the n -th harmonic number and let u n be its numerator. For any prime p , let J p be the set of positive integers n with p | u n . In 1991, Eswarathasan and Levine conjectured that J p is finite for any prime p . It is clear that the p -adic valuation of H n is not less than − ⌊ log p ⁡ n ⌋ . Let T p be the set of positive integers n such that the p -adic valuation of H n is equal to − ⌊ log p ⁡ n ⌋ . Recently, Carlo Sanna proved that | J p ∩ [ 1 , x ] | < 129 p 2 / 3 x 0.765 and that there exists S p ⊆ T p with δ ( S p ) > 0.273 , where δ ( X ) denotes the logarithmic density of the set X of positive integers. He also commented that with his methods δ ( S p ) > 1 / 3 − ε cannot be achieved. In this paper, we improve these results. For example, two of our results are: (a) | J p ∩ [ 1 , x ] | ≤ 3 x 2 / 3 + 1 / ( 25 log ⁡ p ) ; (b) δ ( T p ) exists and 1 − ( 2 log ⁡ p ) − 1 ≤ δ ( T p ) ≤ 1 − ( p log ⁡ p ) − 1 for all primes p ≥ 13 . In particular, δ ( T p ) > 0.63 for all primes p . Video For a video summary of this paper, please visit https://youtu.be/3ujCuVwH8k8 . [ABSTRACT FROM AUTHOR]

The main object of study in the paper is the distance from a point to a line in the Riemannian manifold associated with the Heston model. We reduce the problem of computing such a distance to certain minimization problems for functions of one variable over finite intervals. One of the main ideas in this paper is to use a new system of coordinates in the Heston manifold and the level sets associated with this system. In the case of a vertical line, the formulas for the distance to the line are rather simple. For slanted lines, the formulas are more complicated, and a more subtle analysis of the level sets intersecting the given line is needed. We also find simple formulas for the Heston distance from a point to a level set. As a natural application, we use the formulas obtained in the present paper in the study of the small maturity limit of the implied volatility in the Heston model. [ABSTRACT FROM AUTHOR]

Text Let A = { a 1 , … , a k } be a finite multiset of positive real numbers. Consider the sequence of all positive integers multiples of all a i 's, and note the multiplicity of each term in this sequence. This sequence of multiplicities is called the resonance sequence generated by { a 1 , … , a k } . Two multisets are called combinatorially equivalent if they generate the same resonance sequence. The paper gives a complete criterion of classification of multisets with 3 elements up to combinatorial equivalence, by showing this is equivalent to a system of equations depending directly of the numbers in the multisets. Video For a video summary of this paper, please visit https://youtu.be/rf12nhySOJQ . [ABSTRACT FROM AUTHOR]

In the paper [1] , the first author used the analytic methods and the properties of Gauss sums to study the computational problem of the fourth power mean of the generalized Kloosterman sums for any primitive character χ mod q , and give an exact computational formula for it. In this paper, we considered the same problem for non-primitive character χ mod q , and solved it completely. [ABSTRACT FROM AUTHOR]

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations with respect to semimartingale integrators. We present new connections between their solutions. In particular, we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set-valued stochastic equations. We also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. The results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases. [ABSTRACT FROM AUTHOR]

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by e i t : up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials. [ABSTRACT FROM AUTHOR]

Abstract This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor. [ABSTRACT FROM AUTHOR]

Abstract Let R be the homogeneous coordinate ring of the Grassmannian G = Gr (2 , n) defined over an algebraically closed field of characteristic p > 0. In this paper we give a completely characteristic free description of the decomposition of R , considered as a graded R p -module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of G and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that R provides a "noncommutative resolution" for R p when p ≥ n − 2 , generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirković on Frobenius decompositions for partial flag varieties. We are hopeful that it is an instance of a more general " p -uniformity" principle. [ABSTRACT FROM AUTHOR]

Abstract Based on classical results of Rees and on multivariate Hilbert polynomials, we define new mixed multiplicities of two arbitrary ideals in a local ring (A , m) and use them to express the local degrees of all varieties appearing in the Gaffney–Gassler construction of Segre cycles. We prove that the classical mixed multiplicities of m and an arbitrary ideal I , which are a special case of the new ones, are equal to the generalized Samuel multiplicities of an ideal in the Rees algebra R I (A). This equality is used to improve a result of Jeffries, Montaño and Varbaro on the degree of the fiber cone of an ideal. We conclude the paper with formulas (and their inverses) which express the degrees of Segre classes of subschemes of arbitrary projective varieties by generalized Samuel multiplicities or by classical mixed multiplicities. Using the mixed multiplicities of balanced rational normal scrolls, which have been computed by Hoang and Lam, we find the mixed multiplicities of all rational normal scrolls as well as their Segre classes and their generalized Samuel multiplicities. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we first determine the minimum possible size of an F q -linear set of rank k in PG (1 , q n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an F q -linear set of rank k in PG (2 , q n). In the case k = n , this confirms a conjecture by Sziklai in [9]. [ABSTRACT FROM AUTHOR]

Abstract Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szántó. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds. [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 A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in R d and those in S d is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis. Among other results, we derive a combinatorial characterization of graphs that can be realized as infinitesimally rigid frameworks in the plane with a given set of points collinear. This extends a result by Jackson and Jordán, which deals with the case when three points are collinear. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the problem of whether the best approximation to any x in a real Hilbert space X from the closed convex set K ≔ C ∩ D can be characterized by the best approximation to a perturbation x − l of x from the set C for some l in a certain cone in X. The set C is a closed convex subset of X and D ≔ { x ∈ X : g j (x) ≤ 0 , ∀ j = 1 , 2 , ... , m } , where the functions g j : X ⟶ R (j = 1 , 2 , ... , m) are continuously Fréchet differentiable that are not necessarily convex. We show under suitable conditions that this "perturbation property" is characterized by the strong conical hull intersection property of C and D at the point x 0 ∈ K. We prove this by first establishing a dual cone characterization of a nearly convex set. Our result shows that the convex geometry of K is critical for the characterization rather than the representation of D by convex inequalities, which is commonly assumed for the problems of best approximation from a convex set. In the special case where the set D is convex, we show that the Lagrange multiplier characterization of best approximation holds under the standard Slater's constraint qualification together with a non-degeneracy condition. The lack of representation of D by convex inequalities is supplemented by the non-degeneracy condition, but the characterization, even in this special case, allows applications to problems with quasi-convex functions g j , j = 1 , 2 , ... , m , as they guarantee the convexity of D. Simple numerical examples illustrate the nature of our assumptions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove the existence of an initial trace T u for any positive solution u to the semilinear fractional diffusion equation (H) ∂ t u + (− Δ) s u + f (t , x , u) = 0 in (0 , + ∞) × R N , where N ≥ 1 , the operator (− Δ) s with s ∈ (0 , 1) is the fractional Laplacian, f : R + × R N × R + → R is a Caratheodory function satisfying f (t , x , u) u ≥ 0 for all (t , x , u) ∈ R + × R N × R + and R + = [ 0 , + ∞). We define the regular set of the trace T u as an open subset of R u ⊂ R N carrying a nonnegative Radon measure ν u such that lim t → 0 ⁡ ∫ R u u (t , x) ζ (x) d x = ∫ R u ζ d ν u , ∀ ζ ∈ C 0 2 (R u) , and the singular set S u = R N ∖ R u as the set points a such that lim sup t → 0 ∫ B ρ (a) u (t , x) d x = + ∞ for any ρ > 0. We also study the reverse problem of constructing a positive solution to (H) with a given initial trace (S , ν) , where S ⊂ R N is a closed set and ν is a positive Radon measure on R = R N ∖ S and develop the case f (t , x , u) = t β u p with β > − 1 and p > 1. [ABSTRACT FROM AUTHOR]

Abstract In this paper we answer several questions raised by Sidorov on the set B 2 of bases in which there exist numbers with exactly two expansions. In particular, we prove that the set B 2 is closed, and it contains both infinitely many isolated and accumulation points in (1 , q K L) , where q K L ≈ 1.78723 is the Komornik–Loreti constant. Consequently we show that the second smallest element of B 2 is the smallest accumulation point of B 2. We also investigate the higher order derived sets of B 2. Finally, we prove that there exists a δ > 0 such that dim H ⁡ (B 2 ∩ (q K L , q K L + δ)) < 1 , where dim H denotes the Hausdorff dimension. [ABSTRACT FROM AUTHOR]

Abstract A celebrated unresolved conjecture of Erdős and Hajnal states that for every undirected graph H there exists ϵ (H) > 0 such that every undirected graph on n vertices that does not contain H as an induced subgraph contains a clique or stable set of size at least n ϵ (H) . The conjecture has a directed equivalent version stating that for every tournament H there exists ϵ (H) > 0 such that every H -free n -vertex tournament T contains a transitive subtournament of order at least n ϵ (H) . We say that a tournament is prime if it does not have nontrivial homogeneous sets. So far the conjecture was proved only for some specific families of prime tournaments (Berger et al., 2014; Choromanski, 2015 [3]) and tournaments constructed according to the so-called substitution procedure (Alon et al., 2001). In particular, recently the conjecture was proved for all five-vertex tournaments (Berger et al. 2014), but the question about the correctness of the conjecture for all six-vertex tournaments remained open. In this paper we prove that all but at most one six-vertex tournament satisfy the Erdős–Hajnal conjecture. That reduces the six-vertex case to a single tournament. [ABSTRACT FROM AUTHOR]

Generally speaking, the probability of every qualified set is the same and fixed in conventional secret image sharing (SIS) for general access structure (GAS). In this paper, first we introduce generalized GAS (GGAS), which allows the user to assign probability to every qualified set. Then we design a SIS scheme for GGAS by Chinese remainder theorem (CRT). On one hand, for any qualified set, we can decode the secret image at pre-assigned probability. When we collect all the shares, we can losslessly decode the secret image. On the other hand, for any forbidden set, we will decode nothing of the secret image. We only employ modular operation to decode the secret image, which may be suitable for real-time and green computing scenarios. Experimental results and analyses are realized to examine the effectiveness of our scheme. [ABSTRACT FROM AUTHOR]

Abstract The following result gives the flavor of this paper: Let t , k and q be integers such that q ≥ 0 , 0 ≤ t < k and t ≡ k (mod 2) , and let s ∈ [ 0 , t + 1 ] be the unique integer satisfying s ≡ q + k − t − 2 2 (mod (t + 2)). Then for any integer n such that n ≥ max ⁡ { k , 1 2 (t + 2) k 2 + q − s t + 2 k − t 2 + s } and any function f : [ n ] → { − 1 , 1 } with | ∑ i = 1 n f (i) | ≤ q , there is a set B ⊆ [ n ] of k consecutive integers with | ∑ y ∈ B f (y) | ≤ t. Moreover, this bound is sharp for all the parameters involved and a characterization of the extremal sequences is given. This and other similar results involving different subsequences are presented, including decompositions of sequences into subsequences of bounded weight. [ABSTRACT FROM AUTHOR]

Abstract Let G ⩽ Sym (Ω) be a finite almost simple primitive permutation group with socle G 0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G , denoted b (G) , is the minimal size of a base. We say that G is standard if G 0 = A n and Ω is an orbit of subsets or partitions of { 1 , ... , n } , or if G 0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G 0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b (G) ⩽ 7 for every non-standard group G , with equality if and only if G is the Mathieu group M 24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b (G) = 6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce BBI spaces (“big balls of itself”), which based on the notion of BPI spaces (“big pieces of itself”) used by David and Semmes to study self-similarity. We prove that the “self-similar” construction described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior. We apply this result to self-conformal sets satisfying the WSC and prove that positive Lebesgue measure implies nonempty interior for such sets. This generalizes Zerner's corresponding result for self-similar sets. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes P . A chief factor H / K of G is said to be σ-central (in G ) if the semidirect product ( H / K ) ⋊ ( G / C G ( H / K ) ) is a σ i -group for some i ∈ I . The group G is said to be σ-nilpotent if either G = 1 or every chief factor of G is σ -central. Let G N σ be the σ-nilpotent residual of G , that is, the intersection of all normal subgroups N of G with σ -nilpotent quotient G / N . Then we say that G is σ-supersoluble if each chief factor of G below G N σ is cyclic. In this paper we study properties of σ -supersoluble groups and also consider some applications of such groups in the theory of generalized CLT -groups. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of Lelong numbers of m -subharmonic currents of bidimension ( p , p ) on an open subset of C n , when m + p ≥ n . In the special case of m -subharmonic function φ , we give a relationship between the Lelong numbers of d d c φ and the mean values of φ on spheres and balls. As an application we study the integrability exponent of φ . We express the integrability exponent of φ in terms of volume of sub-level sets of φ and we give a link between this exponent and its Lelong number. [ABSTRACT FROM AUTHOR]

A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved in this paper that a cycle base of a permutation group of degree n can be constructed in polynomial time in n . [ABSTRACT FROM AUTHOR]

Let p be a prime and let q be a power of p . In this paper, by using generalized Reed–Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance-separable (MDS) codes with parameters [ [ t q , t q − 2 d + 2 , d ] ] q for any 1 ≤ t ≤ q , 2 ≤ d ≤ ⌊ t q + q − 1 q + 1 ⌋ + 1 , and [ [ t ( q + 1 ) + 2 , t ( q + 1 ) − 2 d + 4 , d ] ] q for any 1 ≤ t ≤ q − 1 , 2 ≤ d ≤ t + 2 with ( p , t , d ) ≠ ( 2 , q − 1 , q ) . Our quantum MDS codes have flexible parameters, and have minimum distances larger than q 2 + 1 when t > q 2 . Furthermore, it turns out that our constructions generalize and improve some previous results. [ABSTRACT FROM AUTHOR]

Code obfuscation is a set of transformations that make code programs harder to understand. The goal of code obfuscation is to make reverse engineering of programs infeasible, while maintaining the logic on the program. Originally, it has been used to protect intellectual property. However, recently code obfuscation has been also used by malware writers in order to make cyber threats easily able to evade antimalware scanners. As a matter of fact, metamorphic and polymorphic viruses exhibit the ability to obfuscate their code as they propagate. In this paper we propose a model checking-based approach which is able to identify the most widespread obfuscating techniques, without making any assumptions about the nature of the obfuscations used. We evaluate the proposed method on a real-world dataset obtaining an accuracy equal to 0.9 in the identification of obfuscation techniques. [ABSTRACT FROM AUTHOR]

In this paper we define the concept of a near-infinity concentrated norm on a Banach space X with a boundedly complete Schauder basis. When ‖ ⋅ ‖ is such a norm, we prove that ( X , ‖ ⋅ ‖ ) has the fixed point property (FPP); that is, every nonexpansive self-mapping defined on a closed, bounded, convex subset has a fixed point. In particular, P.K. Lin's norm in ℓ 1 [14] and the norm ν p ( ⋅ ) (with p = ( p n ) and lim n ⁡ p n = 1 ) introduced in [3] are examples of near-infinity concentrated norms. When ν p ( ⋅ ) is equivalent to the ℓ 1 -norm, it was an open problem as to whether ( ℓ 1 , ν p ( ⋅ ) ) had the FPP. We prove that the norm ν p ( ⋅ ) always generates a nonreflexive Banach space X = R ⊕ p 1 ( R ⊕ p 2 ( R ⊕ p 3 … ) ) satisfying the FPP, regardless of whether ν p ( ⋅ ) is equivalent to the ℓ 1 -norm. We also obtain some stability results. [ABSTRACT FROM AUTHOR]

A new approach to problems of the Uncertainty Principle in Harmonic Analysis, based on the use of Toeplitz operators, has brought progress to some of the classical problems in the area. The goal of this paper is to develop and systematize the function theoretic component of the Toeplitz approach by introducing a partial order on the set of inner functions induced by the action of Toeplitz operators. We study connections of the new order with some of the classical problems and known results. We discuss remaining problems and possible directions for further research. [ABSTRACT FROM AUTHOR]

Let n ∈ N , R be a binary relation on [ n ] , and C 1 ( i , j ) , … , C n ( i , j ) ∈ Z , for i , j ∈ [ n ] . We define the exponential system of equations E ( R , ( C k ( i , j ) i , j , k ) to be the system X i Y 1 C 1 ( i , j ) ⋯ Y n C n ( i , j ) = X j , for ( i , j ) ∈ R , in variables X 1 , … , X n , Y 1 , … , Y n . The aim of this paper is to classify precisely which of these systems admit a monochromatic solution ( X i , Y i ≠ 1 ) in an arbitrary finite colouring of the natural numbers. This result could be viewed as an analogue of Rado's theorem for exponential patterns. [ABSTRACT FROM AUTHOR]

This paper presents an approximate direct method for solving a class of fractional optimal control problems by combining the epsilon and Ritz methods and utilizing multiscaling basis functions. By applying the epsilon method, we transform the given optimal control problem into a variational problem. Then, we solve the variational problem approximately by the Ritz direct method with multiscaling polynomial basis functions. The choice of multiscaling basis functions provides a high rate of convergence for the Ritz method. The convergence of the derived approximate minimal to the exact one is extensively discussed. At the end some illustrative test examples are included to demonstrate the efficiency of the technique. [ABSTRACT FROM AUTHOR]

Let Γ be a discrete group. A C*-algebra A is an exotic C*-algebra (associated to Γ) if there exist proper surjective C*-quotients C ⁎ ( Γ ) → A → C r ⁎ ( Γ ) which compose to the canonical quotient C ⁎ ( Γ ) → C r ⁎ ( Γ ) . In this paper, we show that a large class of exotic C*-algebras has poor local properties. More precisely, we demonstrate the failure of local reflexivity, exactness, and local lifting property. Additionally, A does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when A is from the class of exotic C*-algebras defined by Brown and Guentner (see [8] ). In order to achieve the main results of this paper, we prove a result which implies the factorization property for the class of discrete groups which are algebraic subgroups of locally compact amenable groups. [ABSTRACT FROM AUTHOR]