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In this paper, a description of the small-h limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers. The considered potentials are those which provide us with the simple turning points only. Three types of Stokes graphs (SG's) associated with the potentials are investigated - the general non-critical ones, the general critical ones but with only single internal Stokes line (SL), and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized h. In particular due to the fact that the small-h limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines. [ABSTRACT FROM AUTHOR]

Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow. [ABSTRACT FROM AUTHOR]

It was shown by Griggs and Wu that a graph of minimal degree 4 on n vertices has a spanning tree with at least $$ \frac{2}{5} $$ n leaves, which is asymptomatically the best possible bound for this class of graphs. In this paper, we present a polynomial time algorithm that finds in any graph with k vertices of degree greater than or equal to 4 and k′ vertices of degree 3 a spanning tree with $$ \left[ {\frac{2}{5} \cdot k + \frac{2}{{15}} \cdot k'} \right] $$ leaves. [ABSTRACT FROM AUTHOR]

We deal with the functional equation motivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λ were supposed to be rational or the equation in question was solved only for n=2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F. [ABSTRACT FROM AUTHOR]

We consider a model of recommendation systems, where each member from a given set of players has a binary preference to each element in a given set of objects: intuitively, each player either likes or dislikes each object. However, the players do not know their preferences. To find his preference of an object, a player may probe it, but each probe incurs unit cost. The goal of the players is to learn their complete preference vector (approximately) while incurring minimal cost. This is possible if many players have similar preference vectors: such a set of players with similar “taste” may split the cost of probing all objects among them, and share the results of their probes by posting them on a public billboard. The problem is that players do not know a priori whose taste is close to theirs. In this paper we present a distributed randomized peer-to-peer algorithm in which each player outputs a vector which is close to the best possible approximation of the player’s real preference vector after a polylogarithmic number of rounds. The algorithm works under adversarial preferences. Previous algorithms either made severely limiting assumptions on the structure of the preference vectors, or had polynomial overhead. [ABSTRACT FROM AUTHOR]

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]

Let T be a self-affine tile in n defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in 2, for any integral expanding matrix A, there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given. [ABSTRACT FROM AUTHOR]

In this paper, we find coefficient estimates by a new method making use of the Faber polynomial expansions for a comprehensive subclass of analytic bi-univalent functions, which is defined by subordinations in the open unit disk. The coefficient bounds presented in this paper would generalize and improve some recent works appeared in the literature. [ABSTRACT FROM AUTHOR]

The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete $$r$$ -partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour et al. (Ars Comb, ). [ABSTRACT FROM AUTHOR]

In this paper, we discuss α-type polynomial sets and also generalized α -type polynomial sets of type zero in two variables. Some properties of certain polynomials have also been shown, in support of α-type zero in two variables. [ABSTRACT FROM PUBLISHER]

In this paper, we establish a novel duality relationship between node-capacitated multiflows and tree-shaped facility locations. We prove that the maximum value of a tree-distance-weighted maximum node-capacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree, and the maximum is attained by a half-integral multiflow. Utilizing this duality, we show that a half-integral optimal multiflow and an optimal location can be found in strongly polynomial time. These extend previously known results in the maximum free multiflow problems. We also show that the set of tree-distance weights is the only class having bounded fractionality in maximum node-capacitated multiflow problems. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical, and at any level of the poset, all the blocks have the same dimension. When the poset-block admits the MacWilliams-type identity, we explicitly state the relation between the weight enumerators of a code and its dual. [ABSTRACT FROM AUTHOR]

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups). [ABSTRACT FROM AUTHOR]

Let f(n) denote the number of relatively prime sets in {1, … , n}. This is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. Motivated by a paper of Ayad and Kihel [1], we show that there are at most finitely many positive integers n such that f(n) is a perfect power of exponent > 1 of some other integer. We also show that the sequence {f(n)} n≥1 is not holonomic; that is, it satisfies no recurrence relation of finite order with polynomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper the notion of Coxeter length for a subset of a Coxeter group, as introduced in [9], is investigated for various subsets of a Coxeter group. Mostly cosets of various subgroups are examined as well as the associated idea of X-posets, which is a vast generalization of the Bruhat order. [ABSTRACT FROM AUTHOR]

The main object of this paper is to introduce and investigate two new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials. In particular,we obtain a new addition formula for the new class of the generalized Apostol-Euler polynomials. We also give an extension and some analogues of the Srivastava-Pintér addition theorem obtained in the works by Srivastava and Pintér (2004) and R. Tremblay, S. Gaboury, B.-J. Fugère, and Tremblay et al. (2011). for both classes. [ABSTRACT FROM AUTHOR]

The paper is devoted to the composition method of constructing families of irreducible polynomials over finite fields. [ABSTRACT FROM AUTHOR]

Abstract: Došlić and Måløy (2010) obtained the extremal 6-cactus chains with respect to the number of matchings and of independent sets. Motivated by the prior paper, in this paper we give recurrences for matching polynomials of ortho-chains and meta-chains, and show that they are the -cactus chains with the most matchings. [Copyright &y& Elsevier]

In this paper, we consider the problem of finding the set {A, B} C {0, 1}m that maximizes, among all 2-subsets of {0, 1}m, the Shannon capacity, H(A, B), of a constrained system of binary sequences that do not contain A or B as a contiguous subsequence. This problem is motivated by the problem of finding a pair of length-m binary sequences, called markers, that achieves the maximum rate, R(2, m, n), of a (2, m, n) periodic prefix-synchronized (PPS) code. A (2, m, n) PPS code is a binary block code with two length-m markers, A, B, and codewords of length n that inserts A and B alternately at regular intervals in the encoded bitstream, with the additional constraint that A and B may not appear anywhere in the encoded bitstream other than where inserted. We show that for any m ≥ 2, limn → ∞R(2, m, n) = max{H(A,B) : {A,B} ⊂ {0,1}m} = log2 ρm - 1, where ρm-1 is the largest-magnitude zero of the polynomial zm-1 - zm-2 - … - 1. Moreover, we completely characterize the sequences A and B that achieve maxH(A, B), as well as those that achieve R(2, m, n) for all sufficiently large n. [ABSTRACT FROM AUTHOR]

The article presents a research paper which investigates the relationship between nondeterministic polynomial time (NP) and exponential linear time (EL). The study determines the incomparability of the said classes, the existence or nonexistence of immune sets, and sparse sets. Findings reveal 24 different cases for their relationship, wherein the 16 cases are impossible in any relativized world, while the other eight cases are realizable in relativized world.

In this paper, we study certain polynomial invariants of links (singular or non-singular) that are related to the Homfly polynomial and Vassiliev's invariants. The Homfly polynomial HL [3] (also known as the Flypmoth polynomial) satisfies the well-known skein relation xHL+ +yHL +zHL0=0. The Vassiliev invariants [1, 2] (of order 1) satisfy the relations VL× =VL+ VL and VL××=0. The invariants that we study satisfy the skein relationsformula here [ABSTRACT FROM AUTHOR]

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset E in a regular variety satisfies | E | ≫ q d − 1 2 + 1 k − 1 , then Δ k , F ( E ) : = { F ( x 1 + ⋯ + x k ) : x i ∈ E , 1 ≤ i ≤ k } ⊇ F q ∖ { 0 } , for some certain families of polynomials F ( x ) ∈ F q [ x 1 , … , x d ] . [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q . In this paper, we present four new classes of permutation trinomials of the form x r h ( x 2 m − 1 ) over F 2 2 m . [ABSTRACT FROM AUTHOR]

We investigate the effect of Ito noise on the stability of stationary solutions to a class of semilinear degenerate parabolic equations with the nonlinearity satisfying an arbitrary polynomial growth condition. We will show that an Ito noise of sufficient intensity will stabilize the unstable stationary solution. [ABSTRACT FROM AUTHOR]

This paper contains two new projection theorems in the plane. First, let K ⊂ B ( 0 , 1 ) ⊂ R 2 be a set with H ∞ 1 ( K ) ∼ 1 , and write π e ( K ) for the orthogonal projection of K into the line spanned by e ∈ S 1 . For 1 / 2 ≤ s < 1 , write E s : = { e : N ( π e ( K ) , δ ) ≤ δ − s } , where N ( A , r ) is the r -covering number of the set A . It is well-known – and essentially due to R. Kaufman – that N ( E s , δ ) ⪅ δ − s . Using the polynomial method, I prove that N ( E s , r ) ⪅ min ⁡ { δ − s ( δ r ) 1 / 2 , r − 1 } , δ ≤ r ≤ 1 . I construct examples showing that the exponents in the bound are sharp for δ ≤ r ≤ δ s . The second theorem concerns projections of 1-Ahlfors–David regular sets. Let A ≥ 1 and 1 / 2 ≤ s < 1 be given. I prove that, for p = p ( A , s ) ∈ N large enough, the finite set of unit vectors S p : = { e 2 π i k / p : 0 ≤ k < p } has the following property. If K ⊂ B ( 0 , 1 ) is non-empty and 1-Ahlfors–David regular with regularity constant at most A , then 1 p ∑ e ∈ S p N ( π e ( K ) , δ ) ≥ δ − s for all small enough δ > 0 . In particular, dim ‾ B π e ( K ) ≥ s for some e ∈ S p . [ABSTRACT FROM AUTHOR]

A polyomino graph P is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. A dimer covering of P corresponds to a perfect matching. Different dimer coverings can interact via an alternating cycle (or square) with respect to them. A set of disjoint squares of P is a resonant set if P has a perfect matching M so that each one of those squares is M-alternating. In this paper, we show that if K is a maximum resonant set of P, then P - K has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to the cardinality of a maximum resonant set. This confirms a conjecture of Xu et al. [26]. We also show that if K is a maximal alternating set of P, then P - K has a unique perfect matching. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. Firstly we investigate the distribution, simplicity and monotonicity of the zeros around the unit circle and real line of the real self-reciprocal polynomials R n ( λ ) ( z ) = 1 + λ ( z + z 2 + ⋯ + z n − 1 ) + z n , n ≥ 2 and λ ∈ R . Secondly, as an application of the first results we give necessary and sufficient conditions to guarantee that all zeros of the self-reciprocal polynomials S n ( λ ) ( z ) = ∑ k = 0 n s n , k ( λ ) z k , n ≥ 2 , with s n , 0 ( λ ) = s n , n ( λ ) = 1 , s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , ⌊ n / 2 ⌋ when n is odd, and s n , n − k ( λ ) = s n , k ( λ ) = 1 + k λ , k = 1 , 2 , … , n / 2 − 1 , s n , n / 2 ( λ ) = ( n / 2 ) λ when n is even, lie on the unit circle, solving then an open problem given by Kim and Park in 2008. [ABSTRACT FROM AUTHOR]

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , ⋯ p − 1 } , where p is a prime and F = { F 1 , F 2 , … , F m } be a family of subsets of [ n ] such that | F i | (mod p ) ∈ K for all F i ∈ F and | F i ∩ F j | (mod p ) ∈ L for i ≠ j . In 1991 Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | F | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

A hyperbinary expansion of a positive integer n is a partition of n into powers of 2 in which each part appears at most twice. In this paper, we consider a generalization of this concept and a certain statistic on the corresponding set of expansions of n. We then define q-generalized m-ary trees whose vertices are labeled by ratios of two consecutive terms within the sequence of distribution polynomials for the aforementioned statistic. When m = 2, we obtain a variant of a previously considered q-Calkin-Wilf tree. [ABSTRACT FROM AUTHOR]

Abstract: The δ-annulus of a polygon P is the closed region containing all points in the plane at distance at most δ from the boundary of P. An inner (resp., outer) δ-offset polygon is the polygon defined by the inner (resp., outer) boundary of its δ-annulus. In this paper we address three major problems of covering a given point set S by an offset version or a polygonal annulus of a polygon P. First, the Maximum Cover objective is, given a value of δ, to cover as many points from S as possible by the δ-offset (or by the δ-annulus) of P, allowing translation and rotation. Second, the Containment problem is to minimize the value of δ such that there is a rigid transformation of the δ-offset (or the δ-annulus) of P that covers all points from S. Third, in the Partial Containment problem we seek the minimum offset of P covering points. These problems arise in many applications where one needs to match a given polygonal figure (a known model) to a set of points (usually, obtained measures). We address several variants of these problems, including convex and simple polygons, as well as polygons with holes and sets of polygons, and obtain algorithms with low-degree polynomial running times in all cases. [Copyright &y& Elsevier]

Abstract: In Huang et al. [17] it was proved that for any Nil d Bohr0-set , there are a minimal system and a non-empty open subset of with , and for any minimal system and any open non-empty , the set is an almost Nil d Bohr0-set. The polynomial form of this problem is considered in this paper. It is shown that the latter is still true in the polynomial case, while the former is not in general. We also consider the special case when the system is a nilsystem. [Copyright &y& Elsevier]

Abstract: In Suzuki (1998) [7] Suzuki gave a classification of association schemes with multiple Q-polynomial structures, allowing for one exceptional case which has five classes. In this paper, we rule out the existence of this case. Hence Suzukiʼs theorem mirrors exactly the well-known counterpart for association schemes with multiple P-polynomial structures, a result due to Eiichi Bannai and Etsuko Bannai in 1980. [Copyright &y& Elsevier]

Abstract: In this paper, three classes of monomials and one class of trinomials over finite fields of even characteristic are proposed. They are proved to be complete permutation polynomials. [Copyright &y& Elsevier]

Abstract: Let q be a prime power and , where . It was recently conjectured that f is a permutation polynomial of if and only if one of the following holds: (i) , ; (ii) , ; (iii) , . We confirm this conjecture in the present paper. [Copyright &y& Elsevier]

Abstract: This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the “approximate”, or “almost” zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the “approximate decoupling polynomials”. The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of “approximate decoupling polynomials”. [Copyright &y& Elsevier]

In this paper, we introduce the notion of groupoids belongs to fuzzy subalgebras, and show that the induced poset of a groupoid belongs to fuzzy subalgebras is an ordinal sum of anti-chains, and obtain several properties of induced posets by fuzzy sets. [ABSTRACT FROM AUTHOR]

Abstract: A class of permutation polynomials with given form over finite fields is investigated in this paper, which is a further study on a recent work of Zha and Hu. Based on some particular techniques over finite fields, two results obtained by Zha and Hu are improved and new permutation polynomials are also obtained. [Copyright &y& Elsevier]

Abstract: Let be positive integers. A -edge-colored graph is -e.c. or -existentially closed if for any disjoint sets of vertices with , there is a vertex not in such that all edges from this vertex to the set are colored by the -th color. In this paper, we give an explicit construction of a -e.c. graph of polynomial order. [Copyright &y& Elsevier]

Abstract: Crouzeix’s conjecture is that for any square matrix A and any polynomial p we havewhere is the field of values of A and denotes the spectral norm. In this paper, we show that the conjecture holds for the matrices of the formwhere and . [Copyright &y& Elsevier]

Abstract: Ritt–Wuʼs algorithm of characteristic sets is the most representative for triangularizing sets of multivariate polynomials. Pseudo-division is the main operation used in this algorithm. In this paper we present a new algorithmic scheme for computing generalized characteristic sets by introducing other admissible reductions than pseudo-division. A concrete subalgorithm is designed to triangularize polynomial sets using selected admissible reductions and several effective elimination strategies and to replace the algorithm of basic sets (used in Ritt–Wuʼs algorithm). The proposed algorithm has been implemented and experimental results show that it performs better than Ritt–Wuʼs algorithm in terms of computing time and simplicity of output for a number of non-trivial test examples. [Copyright &y& Elsevier]

For a ring R, we denote by R[C] the free R-module spanned by the isotopy classes of singular links in S3. Given two invertible elements x,t ∊ R, the HOMFLY-PT skein module of singular links in S3 (relative to the triple (R,t,x)) is the quotient of R[C] by local relations, called skein relations, that involve t and x. We compute the HOMFLY-PT skein module of singular links for any R such that (f-1 --t + x ) and (t-1 -t-x) are invertible. In particular, we deduce the Conway skein module of singular links. [ABSTRACT FROM AUTHOR]

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

Abstract: Let G σ be an oriented graph obtained by assigning an orientation σ to the edge set of a simple undirected graph G such that G σ becomes a directed graph. Let S(G σ ) be the skew adjacency matrix of G σ . The skew energy of G σ is defined as the sum of the absolute values of all eigenvalues of S(G σ ). In this paper, we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first largest skew energies for . [Copyright &y& Elsevier]

The authors mainly concern the set U of c ∈ ℂ such that the power deformation $$ z(\frac{{f(z)}} {z})^c $$ is univalent in the unit disk | z| < 1 for a given analytic univalent function f( z) = z + a z + ... in the unit disk. It is shown that U is a compact, polynomially convex subset of the complex plane ℂ unless f is the identity function. In particular, the interior of U is simply connected. This fact enables us to apply various versions of the λ-lemma for the holomorphic family $$ z(\frac{{f(z)}} {z})^c $$ of injections parametrized over the interior of U. The necessary or sufficient conditions for U to contain 0 or 1 as an interior point are also given. [ABSTRACT FROM AUTHOR]

Abstract: In 2008, Marshall (2010) settled a long-standing open problem by showing that if is a polynomial that is non-negative on the strip , then there exist sums of squares such that . In this paper, we generalize Marshall’s result to various strips and half-strips in the plane. Our results give many new examples of non-compact semialgebraic sets in for which one can characterize all polynomials which are non-negative on the set. For example, we show that if is a compact subset of the real line and a specific set of generators for as a semialgebraic set, then whenever is non-negative on , there are sums of squares such that . [Copyright &y& Elsevier]

Abstract: Let be the maximum possible minimum Hamming distance of a linear [] code over . Tables of best known linear codes exist for small fields. In this paper, linear codes over are constructed for k up to 6. The codes constructed are from the class of quasi-cyclic codes. The number of circulant matrices over is enumerated. In addition, the minimum distance of the extended quadratic residue code of length 44 is determined. [Copyright &y& Elsevier]

By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials. [ABSTRACT FROM AUTHOR]

Abstract: The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems like Dominating Set and Independent Set. This approach is used in this paper to obtain a faster exact algorithm for Dominating Set. We obtain this algorithm by considering a series of branch and reduce algorithms. This series is the result of an iterative process in which a mathematical analysis of an algorithm in the series with measure and conquer results in a convex or quasiconvex programming problem. The solution, by means of a computer, to this problem not only gives a bound on the running time of the algorithm, but can also give an indication on where to look for a new reduction rule, often giving a new, possibly faster algorithm. As a result, we obtain an time and polynomial space algorithm. [Copyright &y& Elsevier]

Abstract: Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that does not vanish, where is the Dirichlet polynomial associated to the group G. [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]