On a refinement of a theorem of Landau on Koebe domains

We state and prove a refinement of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions introduced by A. W. Goodman. Our geometric approach in this article enables us to derive several statements of interest, which would not be produced via the methods in Goodman's paper, as immediate corollaries of the proof of the main theorem.     

Yet another proof of the addition formula for Jacobi polynomials

Short proofs of the addition formulas for Gegenbauer polynomials and for Jacobi polynomials are given. The properties of certain special orthogonal polynomials in two, respectively three, variables are used.     

Interdependence of a theorem of Koebe and a theorem of caratheodory

We determine the widest class of topological mappings for which a correspondence of boundaries is describable in terms of prime ends in the sense of Caratheodory. Relying on a concept of relative distance, we explain why the class so determined is the widest possible, and using a characteristic property of mappings of this class we prove a generalized theorem of Koebe on correspondence of accessible points and we establish its logical equivalence to a fundamental theorem of the Caratheodory theory.Translated from Matematicheskie Zametki, Vol. 10, No. 4, 399–406, October, 1971.     

Yet another single law for lattices

In this note we show that the equational theory of all lattices is defined by the single absorption law This identity of length 29 with 8 variables is shorter than previously known such equations defining lattices.     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

The chaos game revisited: Yet another, but a trivial proof of the algorithm’s correctness

The aim of this work is to explain why the most popular algorithm for approximating IFS fractals, the chaos game, works. Although there are a few proofs of the algorithm’s correctness in the relevant literature, the majority of them utilize notions and theorems of measure and ergodic theories. As a result, paradoxically, although the rules of the chaos game are very simple, the logic underlying the algorithm seems to be hard to comprehend for non-mathematicians. In contrast, the proof presented in this work uses only fundamentals of probability and can be understood by anyone interested in fractals.     

Yet another optimal algorithm for 3-edge-connectivity

An optimal algorithm for 3-edge-connectivity is presented. The algorithm performs only one pass over the given graph to determine a set of cut-pairs whose removal leads to the 3-edge-connected components. An additional pass determines all the 3-edge-connected components of the given graph. The algorithm is simple, easy to implement and runs in linear time and space. Experimental results show that it outperforms all the previously known linear-time algorithms for 3-edge-connectivity in determining if a given graph is 3-edge-connected and in determining cut-pairs. Its performance is also among the best in determining the 3-edge-connected components.     

A proof of a partition theorem for

In this note we give a proof of Devlin's theorem via Milliken's theorem about weakly embedded subtrees of the complete binary tree . Unlike the original proof which is (still unpublished) long and uses the language of category theory, our proof is short and uses direct combinatorial reasoning.

     

A short proof for a generalization of Vizing's theorem

For a simple graph of maximum degree Δ, it is always possible to color the edges with Δ + 1 colors (Vizing); furthermore, if the set of vertices of maximum degree is independent, Δ colors suffice (Fournier). In this article, we give a short constructive proof of an extension of these results to multigraphs. Instead of considering several color interchanges along alternating chains (Vizing, Gupta), using counting arguments (Ehrenfeucht, Faber, Kierstead), or improving nonvalid colorings with Fournier's Lemma, the method of proof consists of using one single easy transformation, called “sequential recoloring”, to augment a partial k-coloring of the edges.     

New proof of a theorem of Szekeres

A Hadamard matrix H of order 16t² is constructed for all t for which there is a Hadamard matrix of order 4t, in such a way that each row of H contains exactly 8t² + 2t ones. As a consequence a new method of constructing the symmetric block designs with parameters (16t², 8t² + 2t, 4t² + 2t) for all t for which there is a Hadamard matrix of order 4t is given.     

Elementary proof of a theorem of Blackburn

For a positive integer n, a finite p-group G is called an ℳ _n -group, if all subgroups of index p ⁿ of G are metacyclic, but there is at least one subgroup of index p ⁿ⁻¹ that is not. A classical result in p-group theory is the classification of ℳ₁-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result.     

A short proof of a theorem on Hamiltonian graphs

In this note, we give a short proof of a stronger version of the following theorem: Let G be a 2-connected graph of order n such that $ d(u)+d(v)+d(w) \geq n + \mid N(u) \cap N(v) \cap N(w) \mid $ for any independent set {u, v, w}, then G is hamiltonian. © 1996 John Wiley & Sons Inc.