On a combinatorial problem related to permanents

Some numerical results will be presented concerning the largest and smallest values of square permanents containing the integers 1, 2, 3, ...,n ² n=1(1)8.Dedicated to Peter Naur on the occasion of his 60th birthday     

Positivity of Szegö's rational function

We consider the problem of deciding whether a given rational function has a power series expansion with all its coefficients positive. Introducing an elementary transformation that preserves such positivity we are able to provide an elementary proof for the positivity of Szegö's function

which has been at the historical root of this subject starting with Szegö. We then demonstrate how to apply the transformation to prove a 4-dimensional generalization of the above function, and close with discussing the set of parameters (a,b) such that

has positive coefficients.     

On a theorem of G. Szegö

It is shown that the precise result of G. Szegö on the asymptotic behavior of the Toeplitz determinants D_n(f), generated by the nonnegative summable function off() holds if Inf L₁ (–, ).Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 693–702, June, 1968.     

A combinatorial proof of a combinatorial theorem

Acta Mathematica Hungarica -     

A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications

In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.

     

A combinatorial proof of Andrews' partition functions related to Schur's partition theorem

We construct an involution to show equality between partition functions related to Schur's second partition theorem.

     

On a functional equation related to a problem of G. Derfel

Aequationes mathematicae - Two years ago, during the 21st European Conference on Iteration Theory, Gregory Derfel asked: “Does there exist a non-trivial bounded continuous solution of the...     

On an extremal problem related to a theorem of Whitney

A well known theorem of Whitney states that for any graph G,

κ(G)?κ^′(G)?δ(G).     

On the exactness of a theorem of G.A. Fomin

, . . [1] - . .     

Problems related to a de Bruijn-Erdös theorem

De Bruijn and Erdös proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on related extremal combinatorial problems.     

A combinatorial analog of a theorem of F.J. Dyson

Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem and the case $n=2$ was proposed by Tucker in 1945. Numerous generalizations and applications of the lemma have appeared since then. In 2006 Meunier proved the lemma in its full generality in his PhD thesis. There are generalizations and extensions of the Borsuk–Ulam theorem that do not yet have combinatorial analogs. In this note, we give a combinatorial analog of a result of Freeman J. Dyson and show that our result is equivalent to Dyson's theorem. As with Tucker's lemma, we hope that this will lead to generalizations and applications and ultimately a combinatorial analog of Yang's theorem of which both Borsuk–Ulam and Dyson are special cases.     

The 6-property for simplicial complexes and a combinatorial Cartan-Hadamard theorem for manifolds

The 6-property for 2-dimensional simplicial complexes is the condition that every nontrivial circuit in the link of a vertex has length greater than or equal to six. If a compact -manifold has a 2-dimensional spine with the 6-property, then we show that the interior of is covered by euclidean -space. In dimension , we show further that such a 3-manifold is Haken.

     

A combinatorial proof of a theorem of Freund

In 1989, Robert W. Freund published an article about generalizations of the Sperner lemma for triangulations of n-dimensional polytopes, when the vertices of the triangulations are labeled with points of Rn. For y∈Rn, the generalizations ensure, under various conditions, that there is at least one simplex containing y in the convex hull of its labels. Moreover, if y is generic, there is generally a parity assertion, which states that there is actually an odd number of such simplices.For one of these generalizations, contrary to the others, neither a combinatorial proof, nor the parity assertion were established. Freund asked whether a corresponding parity assertion could be true and proved combinatorially.The aim of this paper is to give a positive answer, using a technique which can be applied successfully to prove several results of this type in a very simple way. We prove actually a more general version of this theorem. This more general version was published by van der Laan, Talman and Yang in 2001, who proved it in a non-combinatorial way, without the parity assertion.     

Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .