Let F be a union-closed family of subsets of an m-element set A. Let n = |F| ≥ 2. For b ∈ A let w(b) denote the number of sets in F containing b minus the number of sets in F not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over ...

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

Ninety five patients receiving chronic haemodialysis (CHD) were surveyed to determine the prevalence of rheumatic disease and, where possible, its aetiology. At least three distinct rheumatic syndromes were identified--a group of patients with a syndrome consisting of large and medium joint synovial swelling, restricted hips and shoulders, tenosynovitis, carpal tunnel syndrome, and bone cysts d...

1. I am sure that the domain of the creation of the large language corpora, where linguistic annotations are assigned to the input data, still belongs to the interests of the Festschrift's owner. I met Prof. Ferenc Papp for the first time in 1964 in Prague at the Colloquium on Mathematical Linguistics and we immediately have found a common basis of interest: how to store linguistic data about r...

Best and Brouwer [Discrete Math. 17 (1977), 235– 245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2m − 4 and 2m − 3, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computeraided classification of the optimal binary one-erro...

For each v, k and m such that v ≥ k ≥ 2m + 2 ≥ 8, we construct a periodically-cyclic Gale 2m-polytope with v vertices and the period k. For such a polytope, there is a complete description of each of its facets based upon a labelling (total ordering) of the vertices so that every subset of k successive vertices generates a cyclic 2m-polytope. MSC 2000: 52B11, 52B05

Two common features in human immunodeficiency virus infection and acquired immunodeficiency syndrome, rheumatoid arthritis, and hematologic malignancies including multiple myeloma are elevated serum levels of 2microglobulin ( 2M) and activation or inhibition of the immune system. We hypothesized that 2M at high concentrations may have a negative impact on the immune system. In this study, we ex...

Pregnancy-associated plasma protein-A (PAPP-A) has been purified by a combination of methods including antibody-affinity chromatography. The resultant protein, obtained in 16% yield from maternal serum, appeared as a single major component on non-denaturing polyacrylamide and SDS/polyacrylamide gel electrophoresis. The protein showed a single component when analysed by isoelectric focusing unde...