On a conjecture of Herzer concerning linear partitions

We prove: Let G be a finite group with a non-trivial partition (G) and a set (G) of subgroups such that any two components of (G) are contained in at least one W with W (G) and W G; then G is a p-group or a Frobenius group.From this we obtain: A finite group with a linear partition can only be an abelian p-group or a Frobenius group.Dedicated to Prof. H. Lenz on the occasion of his 70^th birthday     

On a conjecture concerning exactly covering systems of congruences

A conjecture of S. Znam [1] is disproved by means of a counterexample.     

On a conjecture of Tutte concerning minimal tree numbers

A counterexample is given to a conjecture by Tutte on the minimum number of spanning trees that a 3-connected planar graph with a prescribed number of edges may have.     

On a conjecture concerning the orientation number of a graph

For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D∈D(G), we denote by d(D) the diameter of D. The orientation number of G is defined by . Let G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs K_p and K_q by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91-106]. Define and . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4.     

On automatic continuity of homomorphisms

Introducing a weaker notion of regularity in a topological algebra, we examine and improve an automatic continuity theorem given by the second author. Examples and applications are given.

     

On a conjecture of Whittaker concerning uniformization of hyperelliptic curves

This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values .

Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality.

Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms.

The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.

     

On a conjecture of Hecke concerning elementary class number formulas

Let K be a totally real algebraic number field of class number h_K and L a totally imaginary quadratic extension of K of class number h_L. Hecke conjectured that there exists an elementary formula for the first factor h_L/h_K of the class number of L. The paper develops a theory which allows computation of h_L/h_K in terms of the periods of certain complex differential forms associated to a manifold defined in a natural way from K. Thus, Hecke's conjecture is reduced to the problem of finding elementary formulas for these periods. The essential idea of the proof consists of establishing a Kronecker limit formula for the non-analytic Eisenstein series for the Hilbert modular group for K.Research supported by NSF Grant GP 20538     

On a conjecture of Thomassen concerning subgraphs of large girth

In 1983 C. Thomassen conjectured that for every k, g∈? there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. Kühn and Osthus [2004] proved the case g = 6. We give another proof for the case g = 6 which is based on a result of Füredi [1983] about hypergraphs. We also show that the analogous conjecture for directed graphs is true. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:316‐331,2011     

On Lam's conjecture concerning signatures of quadratic forms

In this paper, we prove that formally real fields satisfying some explicit conditions on the v-invariant verify a Lam's conjecture concerning signatures of quadratic forms. In particular, it gives a short proof of a Marshall's conjecture for Pythagorean fields.     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.     

On a conjecture of He concerning the spectral reconstruction of matrices

This paper is concerned with a recent conjecture of He (Electron. J. Comb. 14(1), 2007) on the spectral reconstruction of matrices. A counterexample is provided by using Hadamard matrices. We also give some results to the above mentioned conjecture (with slight modifications) in the positive direction.     

On a conjecture of Fink and Jacobson concerning k-domination and k-dependence

A set D of vertices of a graph is k-dependent if every vertex of D is joined to at most k?1 vertices in D. Let β_k(G) be the maximum order of a k-dependent set in G. A set D of vertices of G is k-dominating if every vertex not in D is joined to at least k vertices of D. Let γ_k(G) be the minimum order of a k-dominating set in G. Here we prove the following conjecture of Fink and Jacobson: for any simple graph G and any positive integer k, γ_k(G) ≤ β_k(G).     

On Kemnitz’ conjecture concerning lattice-points in the plane

In 1961, Erdős, Ginzburg and Ziv proved a remarkable theorem stating that each set of 2n−1 integers contains a subset of size n, the sum of whose elements is divisible by n. We will prove a similar result for pairs of integers, i.e. planar lattice-points, usually referred to as Kemnitz’ conjecture. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11B50.     

On strong limit theorems concerning delayed sums of a random sequence

Let (ξ _n ) _n∈N be a sequence of arbitrarily dependent random variables. In this paper, a generalized strong limit theorem of the delayed average of (ξ _n ) _n∈N is investigated, then some limit theorems for arbitrary information sources follow. As corollaries, some known results are generalized.