Graphs of order two less than the Moore bound

The problem of determining the largest order n_d,k of a graph of maximum degree at most d and diameter at most k is well known as the degree/diameter problem. It is known that n_d,k?M_d,k where M_d,k is the Moore bound. For d=4, the current best upper bound for n_4,k is M_4,k-1. In this paper we study properties of graphs of order M_d,k-2 and we give a new upper bound for n_4,k for k?3.     

The cocyclic Hadamard matrices of order less than 40

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.     

Equalities for roots of entire functions of order less than two

Entire functions of order less than two are considered. Relations between the series of absolute values, real parts and imaginary parts of the roots are derived. Moreover, bounds for these series are established in terms of the coefficients of the Taylor series. These results are new even for polynomials. Received: 04 February 2005     

Intersections of maximal subgroups in simple groups of order less than 106

This paper contains a list of intersections of all pairs of maximal subgroups in nonabelian simple groups of order less than 10⁶ excluding PSL(2,q3 for q > 9. To compute possible orders of intersections a computer was used.     

On the Hankel transformation of order zero

In this paper we write the Hankel transform of order zero of a function as a composition of Fourier transforms, and a new proof is given for Hankel's theorem on Hankel transformation of order zero.     

Strongly regular growth of entire functions of order zero

For entire functions of order zero we introduce a new concept of regularity of growth, which is shown to possess properties similar to those which characterize the concept of totally regular growth of entire functions of finite order in the sense of Levin-Pflüger. Translated fromMatematicheskie Zameiki, Vol. 63, No. 2, pp. 196–208, February, 1998. This research was partially supported by the International Science Foundation under grant No. UCR000.     

The mixing advantage is less than 2

Corresponding to n independent non-negative random variables X ₁,...,X _n , are values M ₁,...,M _n , where each M _i is the expected value of the maximum of n independent copies of X _i . We obtain an upper bound for the expected value of the maximum of X ₁,...,X _n in terms of M ₁,...,M _n . This inequality is sharp in the sense that the random variables can be chosen so that the bound is approached arbitrarily closely. We also present related comparison results.      

Solving satisfiability in less than 2 steps

In this paper we describe and analyse an algorithm for solving the satisfiability problem. If E is a boolean formula in conjunctive normal form with n variables and r clauses, then we will show that this algorithm solves the satisfiability problem for formulas with at most k literals per clause in time O(|F|·_kⁿ), where _k is the greatest number satisfying _k = 2−1/_k^k−1 (in the case of 3-satisfiability ₃ = 1,6181).     

The asymptotic behaviour of certain entire functions of order zero

Summary A certain class of entire functionsF(s) of order zero which are asymptotically equal to the sum of just two neighbouring terms of their power series when |s| with |args| < – for any fixed > 0, is investigated. Which two terms one has to take, depends upons. It is shown that these functions have infinitely many negative zeros, and the asymptotic behaviour of the zeros is also determined.