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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]

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]

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 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]

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]

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]

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]

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]

In this paper we show a proof by explicit bijections of the famous Kirkman–Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by k subsets of { 1 , 2 , … , n } with cardinality ≥ 2 and the set of partitions of { 1 , 2 , … , n + k − 1 } into k blocks with cardinality ≥ 2 . A bijection between these two sets can be obtained from Péter L. Erdős and L.A. Székely result in Erdős and Székely (1989); to make this paper self contained we describe another explicit bijection that is a variant of their bijection. [ABSTRACT FROM AUTHOR]

The main object of this paper is to determine the maximum number of { 0 , ± 1 } -vectors subject to the following condition. All vectors have length n , exactly k of the coordinates are +1 and one is −1, n ≥ 2 k . Moreover, there are no two vectors whose scalar product equals the possible minimum, −2. Thus, this problem may be seen as an extension of the classical Erdős–Ko–Rado theorem. Rather surprisingly there is a phase transition in the behaviour of the maximum at n = k 2 . Nevertheless, our solution is complete. The main tools are from extremal set theory and some of them might be of independent interest. [ABSTRACT FROM AUTHOR]

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

In this paper, we present a complete classification of all imaginary n -quadratic fields of class number 1. This classification is done using theoretical methods to narrow down the list of potential number fields of class number 1. The final result is produced through computations in the Sage mathematics software [19] . [ABSTRACT FROM AUTHOR]

In this paper, we derive an explicit combinatorial formula for the number of k -subset sums of quadratic residues over finite fields. [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this paper we show that p can be expressed as p = a 2 ± a b − b 2 with integers a and b if and only if p is congruent to 0, 1 or −1 ( mod 5 ) and p can be expressed as p = a 2 ± a b + b 2 with integers a and b if and only if p is congruent to 0, 1 ( mod 3 ) . [ABSTRACT FROM AUTHOR]

Abstract: In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition some left and right columns are fixed. The main result is a simple linear relation between the number of alternating sign matrices where the top row as well as the left and the right column is fixed and the number of alternating sign matrices where the two top rows and the bottom row are fixed. This may be seen as a first indication for the fact that the refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows as well as left and right columns can possibly be reduced to the refined enumerations where only some top and bottom rows are fixed. For the latter numbers we provide a system of linear equations that conjecturally determines them uniquely. [Copyright &y& Elsevier]

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]

Let A t , k ( n ) denote the number of partition k -tuples of n where each partition is t -core. In this paper, we establish formulas of A t , k ( n ) for some values of t and k by employing the method of modular forms, which extends Wang's result for t = 3 and k = 2 , 3 . [ABSTRACT FROM AUTHOR]

A signed graph is a pair ( G , Σ ) where G is a graph and Σ is a set of edges of G . A cycle of G is balanced if it contains an even number of edges of Σ , and unbalanced otherwise. A blocking pair of ( G , Σ ) is a pair of vertices { s , t } such that every unbalanced cycle intersects at least one of s or t . In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no − K 5 minor. [ABSTRACT FROM AUTHOR]

A finite set X in a complex sphere is called a complex spherical 2-code if the number of inner products between two distinct vectors in X is equal to 2. In this paper, we characterize the tight complex spherical 2-codes by doubly regular tournaments or skew Hadamard matrices. We also give certain maximal 2-codes relating to skew-symmetric D -optimal designs. To prove them, we show the smallest embedding dimension of a tournament into a complex sphere by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel matrix. [ABSTRACT FROM AUTHOR]

The canonical structure of DNA has four bases – Thymine (T), Adenine (A), Cytosine (C), and Guanine (G) – and DNA codes are regarded as words over the alphabet set Σ = { A , C , G , T } , satisfying certain combinatorial conditions. Good DNA codes are desirable for DNA computation, DNA microarray technologies and molecular barcodes, etc. One of the main tasks in DNA code designing is to build more codewords and better GC -content for given fixed word length n . Existing heuristic methods work well for small n . In this paper, we present a systematic method for constructing good DNA codes for large n by using irreducible cyclic codes. Being different from traditional DNA constructions, our method is based on algebraic number theory rather than classical heuristic algorithms and the conventional coding theory. Furthermore, comparing with the traditional DNA codes, our codes have larger number of codewords and better GC -content. As far as we know, it is the very first time to utilize irreducible cyclic codes for constructing a type of DNA codes. [ABSTRACT FROM AUTHOR]

Set agreement and renaming are two tasks that allow processes to coordinate, even when agreement is impossible. In k -set agreement , n processes must decide on at most k of their input values. While n -set agreement is trivially wait-free solvable by each process deciding on its input, ( n − 1 ) -set agreement is not wait-free solvable. In M -renaming , processes must decide on distinct names in a range of size M . For any number n of processes, ( 2 n − 1 ) -renaming is wait-free solvable, but surprisingly, ( 2 n − 2 ) -renaming is wait-free solvable if and only if n is not a prime power; the only previous lower bound on the number of names necessary for renaming, when n is not a prime power, is n + 1 . In adaptive renaming, M decreases when the number p of participants in the execution decreases. It is known that ( 2 p − 1 ) -adaptive renaming is wait-free solvable, while ( 2 p − ⌈ p n − 1 ⌉ ) -adaptive renaming is not. This paper presents counting-based proofs for the above mentioned impossibility results: n processes can wait-free solve neither ( n − 1 ) -set agreement nor ( 2 p − ⌈ p n − 1 ⌉ ) -adaptive renaming; if n is a prime power, n processes cannot wait-free solve ( 2 n − 2 ) -renaming. For an arbitrary number of processes, we give a lower bound for renaming, by reduction from renaming for a different number of processes, and relying on the distribution of prime numbers. Our proofs combine simple operational properties of a restricted set of executions with elementary counting arguments to show the existence of an execution violating the task’s conditions. This makes the proofs easier to understand, verify, and, we hope, extend. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complexity of deciding determinism of unary languages. First, we give a method to derive a set of arithmetic progressions from a regular expression E over a unary alphabet, and establish relations between numbers represented by these arithmetic progressions and words in L ( E ) . Next, we define a problem relating to arithmetic progressions and investigate the complexity of this problem. Then by a reduction from this problem we show that deciding determinism of unary languages is coNP -complete. Finally, we extend our derivation method to expressions with counting, and prove that deciding whether an expression over a unary alphabet with counting defines a deterministic language is in Π 2 p . We also establish a tight upper bound for the size of the minimal DFA for expressions with counting. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over Q with prime conductor p . For each positive integer n we put K n : = Q ( E [ p n ] ) . The aim of this paper is to estimate the order of the p -Sylow group of the ideal class group of K n . We give a lower bound in terms of the Mordell–Weil rank of E ( Q ) . As an application of our result, we give an example such that p 2 n divides the class number of the field K n in the case of p = 5077 for each positive integer n . [ABSTRACT FROM AUTHOR]

A natural question for matroids follows: Which matroids N guarantee that, when present as minors in a larger matroid M , their duals are present as minors? The main results of this paper focus on this question with the additional constraint that N and M are 3-connected. We also solve the problem with other connectivity and representability constraints. [ABSTRACT FROM AUTHOR]

We discuss a new companion to Capparelli's identities. Capparelli's identities for m = 1 , 2 state that the number of partitions of n into distinct parts not congruent to m , − m modulo 6 is equal to the number of partitions of n into distinct parts not equal to m , where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not congruent to 3 − m modulo 3, and 3 l + 1 and 3 l + 2 do not appear together as consecutive parts for any integer l has the same number of elements as the above mentioned Capparelli's partitions of n . In this study we also extend the work of Alladi, Andrews and Gordon by providing a complete set of generating functions for the refined Capparelli partitions, and conjecture some combinatorial inequalities. [ABSTRACT FROM AUTHOR]

Two g -regular cyclic difference packings of parameters ( v , 3 , 1 ) (also termed as cyclically relative difference families ( v , g , 3 , 1 ) -CRDFs) A and B are called orbit-disjoint if { A + x : A ∈ A , x ∈ Z v } ∩ { B + x : B ∈ B , x ∈ Z v } = ∅ . In an investigation of multilength optical orthogonal codes (MLOOCs) with weight three, pairwise orbit-disjoint g -regular cyclic difference packings with block size three play an important role. In this paper, we use cyclotomic classes to obtain some pairwise orbit-disjoint ( v , g , 3 , 1 ) -CRDFs. Consequently, some infinite classes of optimal MLOOCs with weight three are obtained. [ABSTRACT FROM AUTHOR]

Disanto, Ferrari, Pinzani and Rinaldi have introduced the concept of Catalan pair , which is a pair of partial orders ( S , R ) satisfying certain axioms. They have shown that Catalan pairs provide a natural description of objects belonging to several classes enumerated by Catalan numbers. In this paper, we first introduce another axiomatic structure ( T , R ) , which we call the Catalan pair of type 2 , which describes certain Catalan objects that do not seem to have an easy interpretation in terms of the original Catalan pairs. We then introduce Fishburn triples , which are relational structures obtained as a direct common generalization of the two types of Catalan pairs. Fishburn triples encode, in a natural way, the structure of objects enumerated by the Fishburn numbers, such as interval orders or Fishburn matrices. This connection between Catalan objects and Fishburn objects allows us to associate known statistics on Catalan objects with analogous statistics of Fishburn objects. As our main result, we then show that several known equidistribution results on Catalan statistics can be generalized to analogous results for Fishburn statistics. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and sufficient conditions for the system of positive numbers M k 1 , M k 2 , … , M k d , 0 ≤ k 1 < … < k d ≤ r , to guarantee the existence of an r -monotone function defined on the negative half-line R − and such that ‖ x ( k i ) ‖ ∞ = M k i , i = 1 , 2 , … , d . We also discuss some applications of the obtained results and connections with other problems. [ABSTRACT FROM AUTHOR]

In this paper we prove that a Cameron–Liebler line class L in PG ( 3 , q ) with parameter x has the property that ( x 2 ) + n ( n − x ) ≡ 0 mod q + 1 for the number n of lines of L in any plane of PG ( 3 , q ) . It follows that the modular equation ( x 2 ) + n ( n − x ) ≡ 0 mod q + 1 has an integer solution in n . This result rules out roughly at least one half of all possible parameters x . As an application of our method, we determine the spectrum of parameters of Cameron–Liebler line classes of PG ( 3 , 5 ) . This includes the construction of a Cameron–Liebler line class with parameter 10 in PG ( 3 , 5 ) and a proof that it is unique up to projectivities and dualities. [ABSTRACT FROM AUTHOR]

Abstract: If is a proper coloring of edges in a graph , then for each vertex it defines the palette of colors of , i.e., the set of colors of edges incident with . In 1997, Burris and Schelp stated the following problem: how many colors do we have to use if we want to distinguish all vertices by their palettes. In general, we may need much more colors than . In this paper we show that if we distinguish the vertices by color walks emanating from them, not just by their palettes, then the number of colors we need is very close to the chromatic index. Actually, not greater than . [Copyright &y& Elsevier]

Abstract: A sequence is said to be an ascent sequence of length if it satisfies and for all , where is the number of ascents in . Recently, Duncan and Steingrímsson proposed the conjecture that 210-avoiding ascent sequences of length are equinumerous with 3-nonnesting set partitions of . In this paper, we confirm this conjecture by showing that 210-avoiding ascent sequences of length are in bijection with 3-nonnesting set partitions of via an intermediate structure of growth diagrams for 01-fillings of Ferrers shapes. [Copyright &y& Elsevier]

Abstract: In this paper, we give some congruences modulo 2 for dimensions of spaces of Siegel cusp forms of degree one or two. First, we review some known results for congruences between dimensions of spaces of cusp forms and class numbers of imaginary quadratic fields. Next, we give our main result which is a congruence relation for the degree two case. It is related to Arthur's conjecture for . [Copyright &y& Elsevier]

Abstract: The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite. [Copyright &y& Elsevier]

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: Consider a graph and, for each vertex , a subset of neighbors of . A -coloring is a coloring of the elements of so that vertices appearing together in some receive pairwise distinct colors. An obvious lower bound for the minimum number of colors in such a coloring is the maximum size of a set , denoted by . In this paper we study graph classes for which there is a function , such that for any graph and any , there is a -coloring using at most colors. It is proved that if such a function exists for a class , then can be taken to be a linear function. It is also shown that such classes are precisely the classes having bounded star chromatic number. We also investigate the list version and the clique version of this problem, and relate the existence of functions bounding those parameters to the recently introduced concepts of classes of bounded expansion and nowhere-dense classes. [Copyright &y& Elsevier]

Abstract: A permutation of a multiset is called Stirling permutation if as soon as and . In our paper we study Stirling polynomials that arise in the generating function for descent statistics on Stirling permutations of any multiset. We develop generalizations of the classical Stirling numbers and present their combinatorial interpretations. Particularly, we apply the theory of -partitions. Using certain specifications we also introduce the Stirling numbers of odd type and generalizations of the central factorial numbers. [Copyright &y& Elsevier]

Abstract: A dominating set of graph is called metric-locating–dominating if it is also locating, that is, if every vertex is uniquely determined by its vector of distances to the vertices in . If moreover, every vertex not in is also uniquely determined by the set of neighbors of belonging to , then it is said to be locating–dominating. Locating, metric-locating–dominating and locating–dominating sets of minimum cardinality are called -codes, -codes and -codes, respectively. A Nordhaus–Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph and its complement . In this paper, we present some Nordhaus–Gaddum bounds for the location number , the metric-location–domination number and the location–domination number . Moreover, in each case, the graph family attaining the corresponding bound is fully characterized. [Copyright &y& Elsevier]

Abstract: G.L. Watson [13,14] introduced a set of transformations, called Watson transformations by most recent authors, in his study of the arithmetic of integral quadratic forms. These transformations change an integral quadratic form to another integral quadratic form with a smaller discriminant, but preserve many arithmetic properties at the same time. In this paper, we study the change of class numbers of positive definite ternary integral quadratic formula along a sequence of Watson transformations, thus providing a new and effective way to compute the class number of positive definite ternary integral quadratic forms. Explicit class number formulae for many genera of positive definite ternary integral quadratic forms are derived as illustrations of our method. [Copyright &y& Elsevier]

Abstract: This article concerns nonconvolutional type operators (also known as Journéʼs type operators) associated with a multiparameter family of dilations given by where and . We are especially interested in the boundedness of such operators on the multiparameter Hardy spaces. This work is motivated by Pipherʼs result on the boundedness of these operators from the multiparameter spaces to spaces for , and Journéʼs counter-example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter Hardy spaces. Journéʼs work shows that there is a sharp difference between the situations for two and three or more parameters. We establish in this paper the necessary and sufficient conditions under which the singular integral operators in Journéʼs class are bounded on multiparameter spaces () with arbitrary number of parameters. [Copyright &y& Elsevier]

Abstract: For a cyclic group G generated by some , i.e. , the atom of a is defined as the set of all elements generating G. Given any two elements a, b of a finite cyclic group G, we study the sumset of the atom of a and the atom of b. It is known that such a sumset is a disjoint union of atoms. The goal of this paper is to offer a deeper understanding of this phenomenon, by determining which atoms make up the sum of two given atoms and by computing the exact number of representations of each element of the sumset. [Copyright &y& Elsevier]

Abstract: Let if , and if . We conjecture that the θ-orbit of every nonnegative rational number ends in 0. A weaker conjecture asserts that there are no positive rational fixed points for any map in the semigroup Λ generated by the maps and . In this paper, we prove that the asymptotic density of the set of maps in Λ that have rational fixed points is zero. Moreover, we prove that certain types of elements in the semigroup Λ cannot have rational fixed points. [Copyright &y& Elsevier]

Abstract: If A is a dense subset of the integers, then contains long arithmetic progressions. This problem has been studied by many people, but results of sparse sets are hard to obtain. In this paper, we prove that if A is a subset of the primes less than n with cardinality , then contains a long arithmetic progression of length at least [Copyright &y& Elsevier]

Abstract: The Isomorphism of Polynomials (IP) is one of the most fundamental problems in multivariate public key cryptography (MPKC). In this paper, we introduce a new framework to study the counting problem associated to IP. Namely, we present tools of finite geometry allowing to investigate the counting problem associated to IP. Precisely, we focus on enumerating or estimating the number of isomorphism equivalence classes of homogeneous quadratic polynomial systems. These problems are equivalent to finding the scale of the key space of a multivariate cryptosystem and the total number of different multivariate cryptographic schemes respectively, which might impact the security and the potential capability of MPKC. We also consider their applications in the analysis of a specific multivariate public key cryptosystem. Our results not only answer how many cryptographic schemes can be derived from monomials and how big the key space is for a fixed scheme, but also show that quite many HFE cryptosystems are equivalent to a Matsumoto–Imai scheme. [Copyright &y& Elsevier]

Abstract: Recently explicit representations of the class of linearized permutation polynomials and the number of such polynomials were given in Zhou (2008) and Yuan and Zeng (2011) . In this paper, we generalize this result to linearized polynomials with kernel of any given dimension, solving an open problem in Charpin and Kyureghyan (2009) . Moreover, more explicit representations of such polynomials are given and several classes of explicit linearized polynomials with kernel of any given dimension are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain an explicit formula for the number of zero-sum k-element subsets in any finite abelian group. [Copyright &y& Elsevier]

Abstract: Consider the problem where Ω is an open bounded subset of (), , λ and s are positive numbers, f is a nonnegative function in some Lebesgue space, is such that which provides a noncoercive operator when . The problem could be seen as a reaction model which produces a saturation effect, that is, the diffusion goes to zero when u goes to infinity. This type of reaction appears as a linearization of the Arrhenius reaction in some solid combustion problems (see for instance Peral and Vazquez (1995) ). The aim of the article is to study existence, nonexistence and regularity of the solutions to problem (P). The main features in the paper are the following. [•] The threshold on s to have existence or nonexistence of solution in some weak sense is . Moreover a bound on λ is needed to have solution in the critical case . [•] The regularity of the solution according to the summability of f. [Copyright &y& Elsevier]

Abstract: Let denote the set of all nonnegative integers. Let be a nonempty subset of . Denote by the set of all finite, nonempty subsets of . Let be the set of all numbers of the form , where . Let be a partition with such that and are infinite. In this paper, we prove that is a minimal asymptotic basis of order 2 if and only if either contains no consecutive integers or contains consecutive integers or both. We resolve three problems on asymptotic bases of order 2 which had been posed by Nathanson. [Copyright &y& Elsevier]

Abstract: In 1934, two kinds of multiplicative relations, the norm and the Davenport–Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport–Hasse relations were the only multiplicative relations connecting Gauss sums over . However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelbergerʼs theorem. [Copyright &y& Elsevier]

Abstract: Let be the kth coefficient of the nth cyclotomic polynomial. Ji, Li and Moree (2009) showed that . In this paper we will determine . [Copyright &y& Elsevier]

Abstract: This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on X are in one-to-one correspondence with -free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with -free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of -free posets on . [Copyright &y& Elsevier]