Analysis of Weinberg's algebraic relations

Implications of Weinberg's algebraic relations when only p-wave pions are taken into account are investigated. It is shown that decay rates and mass spectra of hadrons assigned to certain classes of unitary irreducible representations of theSU(4) group are in a strong disagreement with experimental data.     

Electron induced fission of the 238U, 237Np, 239Pu and 243Am nuclei in the energy region 100–1000 MeV

Electrofission cross sections for ²³⁸U, ²³⁷Np, ²³⁹Pu and ²⁴³Am have been measured over the energy range 100–1000 MeV. Relative flssilities are evaluated. Analysis in terms of the virtual photon spectra technique involving a nuclear size effect is made. Contributions of various electronuclear excitation mechanisms in the large-energy region are discussed.     

The Number of Edge Colorings with no Monochromatic Cliques

Let F(n,r,k) denote the maximum possible number of distinctedge-colorings of a simple graph on n vertices with r colorswhich contain no monochromatic copy of K_k. It is shown thatfor every fixed k and all n>n₀(k), and , where t_k–1(n)is the maximum possible number of edges of a graph on n verticeswith no K_k (determined by Turán's theorem). The caser=2 settles an old conjecture of Erds and Rothschild, whichwas also independently raised later by Yuster. On the otherhand, for every fixed r>3 and k>2, the function F(n,r,k)is exponentially bigger than . The proofs are based on Szemerédi's regularity lemmatogether with some additional tools in extremal graph theory,and provide one of the rare examples of a precise result provedby applying this lemma.     

A Coding Theory Bound and Zero-Sum Square Matrices

For a code C=C(n,M) the level k code of C, denoted C _k , is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive integer divisible by 8, and n=k, M=k2k then there is a codeword in C _k whose weight is either 0 or at most . In particular, if <(4–2)²/48 then there is a codeword in C _k whose weight is n/2–(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k,p) denote the minimum integer guaranteeing that in any square matrix over Z _p , of order f(k,p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f(k,2)/k<3.836. For general p we obtain, using a different approach, that f(k,p)p( ^{k / ln k )(1+ o} _k ⁽¹⁾⁾.     

Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K_t, we have c₁mt²/ln t ≤ f(m, K_t) ≤ c₂mt², and cmt³/ln t ≤ g(m, K_t) ≤ cmt³/ln t, where c₁, c₂, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

The longest cycle of a graph with a large minimal degree

We show that every graph G on n vertices with minimal degree at least n/k contains a cycle of length at least [n/(k ? 1)]. This verifies a conjecture of Katchalski. When k = 2 our result reduces to the classical theorem of Dirac that asserts that if all degrees are at least 1/2n then G is Hamiltonian.     

On the maximum number of Hamiltonian paths in tournaments

By using the probabilistic method, we show that the maximum number of directed Hamiltonian paths in a complete directed graph with n vertices is at least (e?o(1)) (n!/2^n?1). © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 291–296, 2001