A family of log-gamma integrals and associated results

The authors present a systematic investigation of the following log-gamma integral:

     

On the power of a perturbation for testing non-isomorphism of graphs

In this paper we prove an inverted version of A. J. Schwenk's result, which in turn is related to Ulam's reconstruction conjecture. Instead of deleting vertices from an undirected graphG, we add a new vertexv and join it to all other vertices ofG to get a perturbed graphG+v. We derive an expression for the characteristic polynomial of the perturbed graphG+v in terms of the characteristic polynomial of the original graphG. We then show the extent to which the characteristic polynomials of the perturbed graphs can be used in determining whether two graphs are non-isomorphic.This work was supported by the U.S. Army Research Office under Grant DAAG29-82-K-0107.     

The non-isomorphism of H1-spaces in one and several variables

A partial solution is obtained to the problem of the non-isomorphism of the H¹-space on polydiscs. It is shown that H¹(T) and H¹(T, l²) are not isomorphic, which leads to the distinction of H¹ in one and more variables. The proof uses the identification of H¹ with the martingale H¹-space H¹(δ). The main point consists in showing that the range of a so-called order-inverting into-isomorphism of H¹(δ) is non-complemented.     

Common fixed point results for family of mappings under T-weak reciprocal continuity

In this paper we introduce the idea of T-weak reciprocal continuity which is a more general concept than reciprocal continuity. With the help of this concept and the concept of semi-compatibility, we deduce a common fixed point theorem for a converging family of mappings. An existing result of Som (Bull Cal Math Soc 95(4):307–312, 2003) is generalized in some sense. Two illustrative examples are given in support of our result.     

Some results on generalized equilibrium problems involving a family of nonexpansive mappings

In this paper, we introduce an iterative method for finding a common element in the solution set of generalized equilibrium problems, in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.     

A family of polylog-trigonometric integrals

In this paper, we study the sequences

$$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm {Li}_n(\cos \pi x)\mathrm {d}x, \end{aligned}$$

where \(\mathrm {Li}_n\) is the nth polylogarithm function. Among others, we determine their generating functions, asymptotic behaviour and their connection to the well-known log-sine integrals

$$\begin{aligned} S_n=(-1)^n\int _0^1\log ^n(\sin \pi x)\mathrm {d}x. \end{aligned}$$

With the help of the explicit forms of \(I_n\) and \(J_n\), we deduce closed-form evaluations for a number of polylog-trigonometric definite integrals.

     

A family of difference sets

A construction is given for difference sets with parameters $\text{v =}\text{1}\text{2}\text{3}{}^{\mathrm{s+1}}\text{(3}{}^{\mathrm{s+1}}\text{? 1)}$, $\text{k =}\text{1}\text{2}\text{3}{}^{\mathrm{s}}\text{(3}{}^{\mathrm{s+1}}\text{+ 1), λ =}\text{1}\text{2}\text{3}{}^{\mathrm{s}}\text{(3}{}^{\mathrm{s}}\text{+ 1), n = 3}{}^{\mathrm{2s}}$ in certain noncyclic groups of order v. For s = 1 it is shown that the construction yields all possible difference sets with parameters (36, 15, 6, 9) in an abelian group of order 36.     

A family of extension operators

The definition of the degree of a family of extension operators is given. The connection of lower estimates with general properties is established. The problem of approximation of the segment [0, 1] by finite subsets is studied. In the class of homogeneous operators the independence of norms of operators from the degree of the family of extension operators is proved.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 215–219, August, 1977.     

A family of neighborly polytopes

Ad-polytopeP is said to be neighborly provided each [d/2] vertices determine a face ofP. We construct a family ofd-polytopes that are dual to neighborly polytopes by means of facet splitting. We use this family to find a lower bound on the number of combinatorial types of neighborly polytopes. We also show that all members of this family satisfy the famous Hirsch conjecture. Research supported by NSF Grant # MCS-07466.     

A family of nonreconstructible hypergraphs

A k-hypergraph is a hypergraph in which each edge contains k vertices. We describe the construction of an infinite family of finite, nonreconstructible 3-hypergraphs. We also indicate why the same techniques can likely be used to construct nonreconstructible k-hypergraphs for any k ⩾ 3.     

A family of crooked functions

It has been proved in Bierbrauer and Kyureghyan (Des. Codes Cryptogr. 46:269–301, 2008) that a binomial function aX ⁱ  + bX ^j can be crooked only if both exponents i, j have 2-weight  ≤2. In the present paper we give a brief construction for all known examples of crooked binomial functions. These consist of an infinite family and one sporadic example. The construction of the sporadic example uses the properties of an algebraic curve of genus 3. Computer experiments support the conjecture that each crooked binomial is equivalent either to a member of the family or to the sporadic example.      

A family of easy polyhedra

Let E be a finite set and a family of subsets of E such that the symmetric difference of any two members of this family is at least 2. Let be the complement of in , the set of the subsets of E. In this paper we characterize the convex hull of the characteristic vectors of the elements of . We consider also the polar of these polyhedra and study their links with some well known polyhedra. Note from the Editors  This paper was originally submitted directly to a guest editor appointed for a planned special issue dedicated to the memory of Claude Berge. Unfortunately the issue never materialized and had to be canceled. It was only recently discovered that this paper was never processed.     

A family of continued fractions

We investigate a one-parameter family of infinite generalised continued fractions. The fractions converge to rational values which can be explicitly evaluated. The sequences of numerators and denominators are already to be found in the Online Encyclopedia of Integer Sequences with reference to various topics - but not to continued fractions.     

A family of double loops

A family of double loops is constructed and some of their properties are described.     

Some results for a finite family of uniformly L-Lipschitzian mappings in Banach spaces

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend some recent results in Chang [1], Cho et al. [2] Ofoedu [5], Schu [7] and Zeng [8, 9]. The present studies were supported by “the Natural Science Foundation of China (No. 10771141),” the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Scientific Research Foundation from Zhejiang Province Education Committee (20051897).