Search

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]

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]

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]

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]

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]

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]

Perfect dyadic operators were first introduced in [1] , where a local T ( b ) theorem was proved for such operators. In [3] it was shown that for every singular integral operator T with locally bounded kernel on R n × R n there exists a perfect dyadic operator T such that T − T is bounded on L p ( d x ) for all 1 < p < ∞ . In this paper we show a decomposition of perfect dyadic operators on real line into four well known operators: two selfadjoint operators, paraproduct and its adjoint. Based on this decomposition we prove a sharp weighted version of the T ( 1 ) theorem for such operators, which implies A 2 conjecture for such operators with constant which only depends on ‖ T ( 1 ) ‖ BMO d , ‖ T ⁎ ( 1 ) ‖ BMO d and the constant in testing conditions for T . Moreover, the constant depends on these parameters at most linearly. In this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the Muckenhoupt class A ∞ d . [ABSTRACT FROM AUTHOR]

In a recent work, Girard proposed a new and innovative approach to computational complexity based on the proofs-as-programs correspondence. In a previous paper, the authors showed how Girard's proposal succeeds in obtaining a new characterization of co-NL languages as a set of operators acting on a Hilbert Space. In this paper, we extend this work by showing that it is also possible to define a set of operators characterizing the class L of logarithmic space languages. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel virtual interpolation point (VIP) method formulating discrete Stokes equations. We have formed virtual staggered structure for velocity and pressure from the actual computation node set. The VIP method by a point collocation scheme is well suited for meshfree scheme because the approximation comes from smooth kernels and kernels can be differentiated directly. This paper highlights our contribution to a stable flow computation without explicit structure of staggered grid. Our method eliminates the need to construct explicit staggered grid. Instead, virtual interpolation nodes play key roles in discretizing the conservative quantities of the Stokes equations. We have proved the inf–sup condition for VIP method with virtual structure of staggered grid and thus the existence and stability of discrete solutions follow. [ABSTRACT FROM AUTHOR]

This article closes a trilogy on the existence of singular functions with non-zero finite derivatives. In two previous papers, the authors had exhibited a continuous strictly increasing singular function from [ 0 , 1 ] into [ 0 , 1 ] with a derivative that takes non-zero finite values at two different zero-measure sets: first, at the points of an uncountable set; then at the points of a dense set in [ 0 , 1 ] . In the present paper, the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval ( a , b ) has Hausdorff dimension one. Another feature of this third article is the construction of the required function using the most paradigmatic of the singular functions: the Cantor–Lebesgue one. [ABSTRACT FROM AUTHOR]

König's paradox, that he presented for the first time in 1905, preserved the same structure in all his papers: there was a number that at the same time was and was not finitely definable. Still, he changed the way for forming it, and both its consequences and its solutions changed as well. In the present paper we are going to follow the story of König's paradox, that is an intriguing mix of labelling, solving, criticising an “object” from different viewpoints and for different aims. [ABSTRACT FROM AUTHOR]

In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier sequences interpolated by functions which are not entire and sums, averages and parametrized families of multiplier sequences. The main results include (i) the development of a ‘logarithmic’ multiplier sequence and (ii) several integral representations of a generalized Bessel-type function utilizing some ideas of G.H. Hardy and L.V. Ostrovskii. The explorations and analysis, augmented throughout the paper by a plethora of examples, led to a number of conjectures and intriguing open problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger equations with critical growth − Δ u + ( λ a ( x ) − δ ) u = | u | 2 ⁎ − 2 u , x ∈ R N , where N ≥ 4 , 2 ⁎ is the critical Sobolev exponent, a ( x ) ≥ 0 and its zero sets are not empty, λ > 0 is a parameter, δ > 0 is a constant such that the operator ( − Δ + λ a ( x ) − δ ) might be indefinite for λ large. We prove that if the zero sets of a ( x ) have several isolated connected components Ω 1 , ⋯ , Ω k such that the interior of Ω i ( i = 1 , 2 , … , k ) is not empty and ∂ Ω i ( i = 1 , 2 , … , k ) is smooth. Then for λ sufficiently large, the equation admits, for any i ∈ { 1 , 2 , ⋯ , k } , a solution which is trapped in a neighborhood of Ω i . The key ingredients of the paper are using a flow argument and a combination of global linking and local linking. [ABSTRACT FROM AUTHOR]

In this paper we consider finite energy ℓ -equivariant wave maps from R t , x 1 + 3 \ ( R × B ( 0 , 1 ) ) → S 3 with a Dirichlet boundary condition at r = 1 , and for all ℓ ∈ N . Each such ℓ -equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary ℓ -equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d = 2 ℓ + 3 , which are established in the companion paper [13] . [ABSTRACT FROM AUTHOR]

Exterior channel of energy estimates for the radial wave equation were first considered in three dimensions in [6] , and for the 5-dimensional case in [12] . In this paper we find the general form of the channel of energy estimate in all odd dimensions for the radial free wave equation. This will be used in the companion paper [11] to establish the soliton resolution for equivariant wave maps in R 3 exterior to the ball B ( 0 , 1 ) and in all equivariance classes. [ABSTRACT FROM AUTHOR]

This paper is devoted to exploring some analytic–geometric properties of the regularity and capacity associated with the so-called fractional dissipative operator ∂ t + ( − Δ ) α , naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation ( ∂ t + ( − Δ ) α ) u ( t , x ) = F ( t , x ) subject to u ( 0 , x ) = 0 . The methods used in this paper rely on effectively controlling the time-dependent non-local kernels and potentials with fractional order α ∈ ( 0 , 1 ) , dual representation of the capacity and Frostman type theorem from geometric measure theory. [ABSTRACT FROM AUTHOR]

Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer n has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number t is absent from the diagram then the partition is called a t -core. A partition is an ( s , t ) -core if it is both an s - and a t -core. Since the work of Anderson on ( s , t ) -cores, the topic has received growing attention. This paper expands the discussion to multiple-cores. More precisely, we explore ( s , s + 1 , … , s + k ) -core partitions much in the spirit of a recent paper by Stanley and Zanello. In fact, our results exploit connections between three combinatorial objects: multi-cores, posets and lattice paths (with a novel generalization of Dyck paths). Additional results and conjectures are scattered throughout the paper. For example, one of these statements implies a curious symmetry for twin-coprime ( s , s + 2 ) -core partitions. [ABSTRACT FROM AUTHOR]

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields. [ABSTRACT FROM AUTHOR]

A celebrated unresolved conjecture of Erdös 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 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 ) . For a fixed tournament H , define ξ ( H ) to be the supremum of all ϵ for which the following holds: for some n 0 and every n > n 0 every tournament with n ≥ n 0 vertices not containing H as a subtournament has a transitive subtournament of size at least n ϵ . We call ξ ( H ) the EH-supremum of H . The Erdös–Hajnal conjecture is true if and only if ξ ( H ) > 0 for every H . If the conjecture is false then the smallest counterexample has no nontrivial so-called homogeneous sets (to be defined below). Therefore of interest are EH-suprema of those tournaments. In [5] it was proven that there exists a constant η > 0 such that ξ ( H ) ≤ 4 h ( 1 + η log ⁡ ( h ) h ) for almost every h -vertex tournament H . However this result does not say anything about ξ ( H ) for an arbitrarily chosen tournament with no nontrivial homogeneous sets. We address that problem in this paper, proving that there exists C > 0 such that every h -vertex tournament H with no nontrivial homogeneous sets satisfies ξ ( H ) ≤ C log ⁡ ( h ) h . We will also give upper bounds on sizes of families of h -vertex tournaments with big EH-suprema. In [1] Alon, Pach and Solymosi proposed a procedure that produces bigger graphs satisfying the conjecture from smaller ones. All graphs obtained in such a way have nontrivial homogeneous sets. For a long time that was the only method to obtain infinite families of graphs satisfying the conjecture. Recently Berger, the author and Chudnovsky (see [2] ) constructed a new infinite family of tournaments (so-called galaxies , to be defined below) that satisfies the conjecture and with no nontrivial homogeneous sets. Therefore it cannot be obtained by the procedure described in [1] . In this paper we construct a new infinite family of tournaments satisfying the conjecture – the family of so-called constellations (to be defined below). These results extend the results of [2] since every galaxy is a constellation. [ABSTRACT FROM AUTHOR]

A method is presented, based upon the Stieltjes method (1884), for the determination of non-classical Gauss-type quadrature rules, and the associated sets of abscissae and weights. The method is then used to generate a number of quadrature sets, to arbitrary order, which are primarily aimed at deterministic transport calculations. The quadrature rules and sets detailed include arbitrary order reproductions of those presented by Abu-Shumays in [4,8] (known as the QR sets, but labelled QRA here), in addition to a number of new rules and associated sets; these are generated in a similar way, and we label them the QRS quadrature sets. The method presented here shifts the inherent difficulty (encountered by Abu-Shumays) associated with solving the non-linear moment equations, particular to the required quadrature rule, to one of the determination of non-classical weight functions and the subsequent calculation of various associated inner products. Once a quadrature rule has been written in a standard form, with an associated weight function having been identified, the calculation of the required inner products is achieved using specific variable transformations, in addition to the use of rapid, highly accurate quadrature suited to this purpose. The associated non-classical Gauss quadrature sets can then be determined, and this can be done to any order very rapidly. In this paper, instead of listing weights and abscissae for the different quadrature sets detailed (of which there are a number), the MATLAB code written to generate them is included as Appendix D . The accuracy and efficacy (in a transport setting) of the quadrature sets presented is not tested in this paper (although the accuracy of the QRA quadrature sets has been studied in [12,13] ), but comparisons to tabulated results listed in [8] are made. When comparisons are made with one of the azimuthal QRA sets detailed in [8] , the inherent difficulty in the method of generation, used there, becomes apparent, with the highest order tabulated sets showing unexpected anomalies. Although not in an actual transport setting, the accuracy of the sets presented here is assessed to some extent, by using them to approximate integrals (over an octant of the unit sphere) of various high order spherical harmonics. When this is done, errors in the tabulated QRA sets present themselves at the highest tabulated orders, whilst combinations of the new QRS quadrature sets offer some improvements in accuracy over the original QRA sets. Finally, in order to offer a quick, visual understanding of the various quadrature sets presented, when combined to give product sets for the purposes of integrating functions confined to the surface of a sphere, three-dimensional representations of points located on an octant of the unit sphere (as in [8,12] ) are shown. [ABSTRACT FROM AUTHOR]

Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. At the same time, direct extensions of this concept to infinite sets encounter some problems even in the case of countable sets, for instance, countable discrete Abelian groups. In an attempt to resolve these difficulties, we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. However, constructing a complete theory in the general case faces a set of obstacles related to the properties of the adjacency algebras and associated projection operators. To develop a theory of association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. Using the language of spectrally dual partitions, we prove that such groups support the construction of general Abelian (translation) schemes and establish properties of their spectral parameters (eigenvalues). Addressing the existence question of spectrally dual partitions, we show that they arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of p -adic numbers. This enables us to construct large classes of examples of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers (structural constants). We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group. Next we focus on the connection between schemes on zero-dimensional groups and harmonic analysis. We show that the eigenvalues have a natural interpretation in terms of Littlewood–Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal function systems on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelet bases on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and (non-Archimedean) harmonic analysis. We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose several classes of permutation polynomials with the form ( x p m − x + δ ) s + L ( x ) over the finite field F p 2 m , where p is an odd prime, and L ( x ) is a linearized polynomial with coefficients in F p . The main method used in this paper is to determine the number of solutions of some equations over finite fields of odd characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we improve upon some observations made in recent papers on the subject of increasingly flat interpolation. We shall establish that the corresponding Lagrange functions converge both for a finite set of functions (collocation matrix) and also for kernels (Fredholm matrix). In our analysis, we use a finite Maclaurin expansion of a multivariate function with remainder and some additional matrix theoretic facts. [ABSTRACT FROM AUTHOR]

We associate with every group G a set τ ( G ) of subgroups of G with 1 ∈ τ ( G ) . If H ∈ τ ( G ) , then we say that H is a τ-subgroup of G . If θ ( τ ( G ) ) ⊆ τ ( θ ( G ) ) for each epimorphism θ : G → G ⁎ , then we say that τ is a subgroup functor . We say also that a subgroup functor τ is: hereditary provided H ∈ τ ( E ) whenever H ≤ E ≤ G and H ∈ τ ( G ) ; regular provided for any group G , whenever H ∈ τ ( G ) is a p -group and N is a minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p ; Φ -regular (respectively Φ -quasiregular ) provided for any primitive group G , whenever H ∈ τ ( G ) is a p -group and N is a (respectively abelian) minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p . Let K ≤ H be subgroups of G and τ a subgroup functor. Then we say that: the pair ( K , H ) satisfies the F -supplement condition in G if G has a subgroup T such that H T = G and H ∩ T ⊆ K Z F ( T ) ; H is F τ -supplemented in G if for some τ -subgroup S ¯ of G ¯ contained in H ¯ the pair ( S ¯ , H ¯ ) satisfies the F -supplement condition in G ¯ , where G ¯ = G / H G and H ¯ = H / H G . In this paper we study the structure of a group G under the condition that some primary subgroups of G are F τ -supplemented in G . In particular, we prove the following result. Theorem A. Let F be a saturated formation containing the class U of all supersoluble groups, E a normal subgroup of G with G / E ∈ F , X = E or X = F ⁎ ( E ) , and τ a regular or hereditary Φ -regular subgroup functor. Suppose that every τ-subgroup of G contained in X is subnormally embedded in G. If every maximal subgroup of every non-cyclic Sylow subgroup of X is U τ -supplemented in G, then G ∈ F . Moreover, in the case when τ is regular, then every chief factor of G below E is cyclic. The results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results. [ABSTRACT FROM AUTHOR]

Martin-Löf defined algorithmic randomness, the use of martingales in this definition and several variants were explored by Schnorr, and Lutz introduced resource-bounded randomness. Ambos-Spies et al. have studied resource-bounded randomness under time-bounds and have shown that resource-bounded randomness implies resource-bounded genericity. While the genericity of Ambos-Spies is based on time-bound on finite-extension strategy, the genericity of Dowd, the main topic of this paper, is based on an analogy of the forcing theorem. For a positive integer r , an oracle D is r -generic in the sense of Dowd ( r -Dowd) if the following holds: If a certain formula F on an exponential-sized portion of D is a tautology then a polynomial-sized subfunction of D forces F to be a tautology. Here, r is the number of occurrences of query symbols in F . The relationship between resource-bounded randomness and Dowd-type genericity has been not clear so far. We show that there exists a primitive recursive function t ( n ) with the following property: Every t ( n ) -random set is r -Dowd for each positive integer r . A proof is done by means of constructing resource-bounded martingales. In our former paper, we left an open problem whether there exists a primitive recursive set which is r -Dowd for every positive integer r . In our recent work, we give an affirmative answer to the problem. The main theorem of the present paper gives an alternative proof of the affirmative answer. [ABSTRACT FROM AUTHOR]

In 2012, Sagan and Savage introduced the notion of st -Wilf equivalence for a statistic st and for sets of permutations that avoid particular permutation patterns. In this paper we consider inv-Wilf equivalence on sets of permutations that avoid two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns Π and Π ′ are inv-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of Π is equal to the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of Π ′ . In 2013, Cameron and Killpatrick gave the inversion generating function for Fibonacci tableaux which are in one-to-one correspondence with the set of permutations that simultaneously avoid the consecutive patterns 321 and 312. In this paper, we use the language of Fibonacci tableaux to study the inversion generating functions for permutations that avoid Π where Π is a set of three or more consecutive permutation patterns. In addition, we introduce the more general notion of strip tableaux which are a useful combinatorial object for studying consecutive pattern avoidance. We give the inversion generating functions for Π a subset of 4 or 5 consecutive permutation patterns and for all but one of the cases where Π is a subset of three consecutive permutation patterns. [ABSTRACT FROM AUTHOR]

In this paper we consider L ∞ -algebras equipped with complete descending filtrations. We prove that, under some mild conditions, an L ∞ quasi-isomorphism U : L → L ˜ induces a weak equivalence between the Deligne–Getzler–Hinich (DGH) ∞-groupoids corresponding to L and L ˜ , respectively. This paper may be considered as a modest addition to foundational paper [10] by Ezra Getzler. [ABSTRACT FROM AUTHOR]

Text Let N be the set of all nonnegative integers and k ≥ 2 be a fixed integer. For a set A ⊆ N , let r k ( A , n ) denote the number of solutions of a 1 + ⋯ + a k = n with a 1 , … , a k ∈ A . In this paper, we prove that for given positive integer u , there is a set A ⊆ N such that r k ( A , n ) ≥ 1 for all n ≥ 0 and the set of n with r k ( A , n ) = k ! u has density one. This generalizes recent results of Chen and Yang. Video For a video summary of this paper, please visit http://youtu.be/2fbKtDAOqQ0 . [ABSTRACT FROM AUTHOR]

This paper studies the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measures, which extends Feng and Huang's recent result on entropies [13] for pressures. More precisely, this paper defines the measure theoretic pressure P μ ( T , f ) for any Borel probability measure, and shows that P B ( T , f , K ) = sup ⁡ { P μ ( T , f ) : μ ∈ M ( X ) , μ ( K ) = 1 } , where M ( X ) is the space of all Borel probability measures, K ⊆ X is a non-empty compact subset and P B ( T , f , K ) is the Pesin–Pitskel topological pressure on K . Furthermore, if Z ⊆ X is an analytic subset, then P B ( T , f , Z ) = sup ⁡ { P B ( T , f , K ) : K ⊆ Z is compact } . This paper also shows that Pesin–Pitskel topological pressure can be determined by the measure theoretic pressure. [ABSTRACT FROM AUTHOR]

Let q ∈ ( 1 , 2 ) and x ∈ [ 0 , 1 q − 1 ] . We say that a sequence ( ϵ i ) i = 1 ∞ ∈ { 0 , 1 } N is an expansion of x in base q (or a q -expansion) if x = ∑ i = 1 ∞ ϵ i q − i . Let B ℵ 0 denote the set of q for which there exists x with exactly ℵ 0 expansions in base q . In [5] it was shown that min ⁡ B ℵ 0 = 1 + 5 2 . In this paper we show that the smallest element of B ℵ 0 strictly greater than 1 + 5 2 is q ℵ 0 ≈ 1.64541 , the appropriate root of x 6 = x 4 + x 3 + 2 x 2 + x + 1 . This leads to a full dichotomy for the number of possible q -expansions for q ∈ ( 1 + 5 2 , q ℵ 0 ) . We also prove some general results regarding B ℵ 0 ∩ [ 1 + 5 2 , q f ] , where q f ≈ 1.75488 is the appropriate root of x 3 = 2 x 2 − x + 1 . Moreover, the techniques developed in this paper imply that if x ∈ [ 0 , 1 q − 1 ] has uncountably many q -expansions then the set of q -expansions for x has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case. [ABSTRACT FROM AUTHOR]

Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m ≥ 2 , there exists a Cantor subset of the plane which is an attractor of some GIFS of order m , but is not an attractor of a GIFS of order m − 1 . In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS. [ABSTRACT FROM AUTHOR]