A note on a conjecture of Ryser

A special case of a conjecture of Ryser states that if a 3-partite 3-uniform hypergraph has at mostv pairwise disjoint edges then there is a set of vertices of cardinality at most 2v meeting all edges of the hypergraph. The best known upper bound for the size of such a set is (8/3)v, given by Tuza [7]. In this note we improve this to (5/2)v.     

Classification of Ryser graphs

The object of study is (2, r ₁, r ₂)-regular graphs in which the union of the neighborhoods of two different vertices u and w contains r ₁ or r ₂ vertices, depending on whether u and w are adjacent. It is proved that such graphs either are strongly regular or decompose into the direct sum of a complete multipartite graph and a clique. Earlier, the case r ₁ = r ₂ was studied by other authors.     

On a minimax theorem

The aim of this note is to prove the following statement. It gives an extension of the theorem proved by O. Y. Borenshtein and V. S. Shul'man in [1].     

ON A MIN-MAX THEOREM

In this paper we prove a nonvariational form of a rain-max problem in [l] and give its application to a periodical problem.     

On a comparison theorem

This paper gives a generalization of the Sturm comparison theorem for differential equations (p): y″ = p(t)y, (q): y″ = q(t)y under the assumption that the function p ? q changes its sign exactly once on [a, b] or ∝_t^bp ? q, ∝_a^tp ? q maintain the sign on [a, b]. The results are used for investigating the distributions of zeros of solutions and the derivative of solutions of (p), (q).     

On a density theorem

Translated from Matematichskie Zametki, Vol. 51, No. 6, pp. 32–40, June, 1992.     

On a fixed-point theorem

Institute of Mathematics, Hanoi. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 30, No. 3, pp. 93–94, July–September, 1996.     

On a Kleinecke-Shirokov theorem

Czechoslovak Mathematical Journal - We prove that for normal operators N1, $${N_2} \in {\cal L}({\cal H})$$ , the generalized commutator [N1, N2; X] approaches zero when [N1, N2; [N1, N2; X]] tends...     

On a theorem of Karlin

Starting from an old result of S. Karlin, we demonstrate the usefulness of couplings within the theory of random systems with complete connections. We also give a short exposé of some limit results for the state sequences associated to random systems with complete connections.     

On a theorem of Ismagilov

If A and B are bounded selfadjoint operators and AB is trace class, then the absolutely continuous part of A + B is unitarily equivalent to the direct sum of the absolutely continuous parts of A and B.     

On a theorem of Kisin

This note provides a short proof of a theorem of Kisin on crystalline representations.     

On a theorem of Picard

We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous -order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determined. Connections with completely integrable systems are mentioned.

     

On a theorem of Fatou

We prove a result on the backward dynamics of a rational function nearby a point not contained in the -limit set of a recurrent critical point. As a corollary we show that a compact invariant subset of the Julia set, not containing critical or parabolic points, and not intersecting the -limit set ofrecurrent critical points, is expanding, thus extending a classical criteria of Fatou. We also prove that the boundary of a Siegel disk is always contained in the -limit set of arecurrent critical point.     

On a theorem of Ganley

LetD be a relative difference with parameters (n, n, n, 1) in an abelian groupG of even ordern ². By a result of Ganley [3],n is necessarily a power of 2 andG is a direct sum of copies of ₄. We present a simple (and much shorter) alternative proof of this result, based on a geometric argument and a simple characterisation of the groups in question.