Hodge cycles on the jacobian variety of the Catalan curve

The jacobian variety of the Catalan curve yq = xp - 1is shown to be nondegenerate. As an application, a 0-1-matrix whose determinantcomputes the relative class number of the pq-th cyclotomic field is constructed.     

Variations of the Catalan numbers from some nonassociative binary operations

We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial objects such as binary trees, plane trees, lattice paths, and permutations.     

On connection sequence for equivalent polynomial sets

The main purpose of this paper is to study the connection sequence between two polynomial sets. We prove that any two equivalent polynomial sets can be connected with a Sheffer sequence. The principal result presents a unification of some useful known results. The inversion, addition and duplication sequences are considered as particular cases. Moreover, some examples are given.     

Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Distribution functions for subsequences of the van der Corput sequence

For an integer b>1$b>1$ let _(?b(n))n≥0${\left({?}_b\left(n\right)\right)}_{n\ge 0}$ denote the base b$b$ van der Corput sequence in [0,1)$[0,1)$. Answering a question of O. Strauch, C. Aisleitner and M. Hofer showed that the distribution function of _{(?b(n),?b(n+1),…,?b(n+s−1))n≥0}${({?}_b\left(n\right),{?}_b(n+1),\dots ,{?}_b(n+s-1))}_{n\ge 0}$ on [0,1)^s${[0,1)}^s$ exists and is a copula. In this note we show that this phenomenon extends to a broad class of subsequences of the van der Corput sequences.     

Extending tests for convergence of number series

Analyzing several classical tests for convergence/divergence of number series, we relax the monotonicity assumption for the sequence of terms of the series. We verify the sharpness of the obtained results on corresponding classes of sequences and functions.     

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

The $k^{\text{th}}$ power of a graph $G=(V,E)$, $G^k$, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of $G^k$ which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.     

The distinguishing number of the augmented cube and hypercube powers

The distinguishing number of a graph G, denoted D(G), is the minimum number of colors such that there exists a coloring of the vertices of G where no nontrivial graph automorphism is color-preserving. In this paper, we answer an open question posed in Bogstad and Cowen [The distinguishing number of the hypercube, Discrete Math. 283 (2004) 29-35] by showing that the distinguishing number of , the pth graph power of the n-dimensional hypercube, is 2 whenever 2<p<n-1. This completes the study of the distinguishing number of hypercube powers. We also compute the distinguishing number of the augmented cube AQ_n, a variant of the hypercube introduced in Choudum and Sunitha [Augmented cubes, Networks 40 (2002) 71-84]. We show that D(AQ₁)=2; D(AQ₂)=4; D(AQ₃)=3; and D(AQ_n)=2 for n?4. The sequence of distinguishing numbers answers a question raised in Albertson and Collins [An introduction to symmetry breaking in graphs, Graph Theory Notes N.Y. 30 (1996) 6-7].     

Strict definiteness of integrals via complete monotonicity of derivatives

Let be a nonnegative integer and let be a function with completely monotone and not constant. If is a signed measure on any euclidean space , with vanishing moments up to order , then the integral is strictly positive whenever it exists. For general no larger class of continuous functions seems to admit the same conclusion. Examples and applications are indicated. A section on 'bilinear integrability' might be of independent interest.

     

On the fractional chromatic number of monotone self-dual Boolean functions

We compute the exact fractional chromatic number for several classes of monotone self-dual Boolean functions. We characterize monotone self-dual Boolean functions in terms of the optimal value of an LP relaxation of a suitable strengthening of the standard IP formulation for the chromatic number. We also show that determining the self-duality of a monotone Boolean function is equivalent to determining the feasibility of a certain point in a polytope defined implicitly.     

Degree sequence conditions for equal edge‐connectivity and minimum degree,depending on the clique number

Using the well‐known Theorem of Turán, we present in this paper degree sequence conditions for the equality of edge‐connectivity and minimum degree, depending on the clique number of a graph. Different examples will show that these conditions are best possible and independent of all the known results in this area. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 234–245, 2003     

On the structure of digital explicit nonlinear and inversive pseudorandom number generators

We analyze the lattice structure and distribution of the digital explicit inversive pseudorandom number generator introduced by Niederreiter and Winterhof as well as of a general digital explicit nonlinear generator. In particular, we extend a lattice test designed for this class of pseudorandom number generators to parts of the period and arbitrary lags and prove that these generators pass this test up to very high dimensions. We also analyze the behavior of digital explicit inversive and nonlinear generators under another very strong lattice test which in its easiest form can be traced back to Marsaglia and provides a complexity measure essentially equivalent to linear complexity.     

On the boundary sequence of an automatic sequence

In this paper, we prove a conjecture of Chen and Wen that the boundary sequence of an automatic sequence is also automatic. In particular, we study the boundary sequences of the generalized Cantor sequences, and give a complete characterization of the periodic boundary sequences. As an application, for a class of automatic sequences, we prove that their abelian complexities are also automatic.     

On the b-chromatic number of Kneser graphs

In this note, we prove that for any integer n≥3 the b-chromatic number of the Kneser graph KG(m,n) is greater than or equal to . This gives an affirmative answer to a conjecture of [6].     

On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice , spanning a Euclidean space V. Let m be a positive integer and be the arrangement of hyperplanes in V of the form for and . It is known that the number of bounded dominant regions of is equal to the number of facets of the positive part of the generalized cluster complex associated to the pair by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of and conjecture that the corresponding refinement of coincides with the $h$-vector of . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1. 2000 Mathematics Subject Classification Primary—20F55; Secondary—05E99, 20H15