A counting proof of the Graham–Pollak Theorem

We give a counting based proof of the Graham–Pollak Theorem.     

A tropical proof of the Brill–Noether Theorem

We produce Brill–Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill–Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.     

A combinatorial proof of Klyachko’s Theorem on Lie representations

Let L be a free Lie algebra of finite rank r over an arbitrary field K of characteristic 0, and let L_n denote the homogeneous component of degree n in L. Viewed as a module for the general linear group GL(r,K), L_n is known to be semisimple with the isomorphism types of the simple summands indexed by partitions of n with at most r parts. Klyachko proved in 1974 that, for n > 6, almost all such partitions are needed here, the exceptions being the partition with just one part, and the partition in which all parts are equal to 1. This paper presents a combinatorial proof based on the Littlewood-Richardson rule. This proof also yields that if the composition multiplicity of a simple summand in L_n is greater than 1, then it is at least .     

On the Adams–Riemann–Roch theorem in positive characteristic

Let p > 0 be a prime number. We give a short proof of the Adams–Riemann–Roch theorem for the p-th Adams operation, when the involved schemes live in characteristic p. We also answer a question of B. Köck.     

Mineyev–Dicks proof of the HN-conjecture and the Euler–Poincaré characteristic

A proof of the Hanna Neumann conjecture (HN-conjecture) based on the ideas of Mineyev andDicks is presented. The new ingredients are theQuillen formula for the Euler–Poincaré characteristic and the “abstract HN-conjecture.”     

A combinatorial proof of Shapiroʼs Catalan convolution

Shapiro proved an elegant convolution formula involving Catalan numbers of even index. This paper gives a combinatorial proof of his formula. In addition, we show that it is equivalent to an alternating convolution formula of central binomial coefficients.     

A simple proof of Veech’s Theorem

We present a simple proof of Veech’s Theorem.     

A proof of the Lyusternik–Graves theorem

In this paper, we give a new proof of the Lyusternik–Graves theorem, based on an intermediate result regarding linear openness inspired by works of Frankowska and Ursescu.     

A Generalization of the Gallai–Roy Theorem

 A well-known and essential result due to Roy ([4], 1967) and independently to Gallai ([3], 1968) is that if D is a digraph with chromatic number χ(D), then D contains a directed path of at least χ(D) vertices. We generalize this result by showing that if ψ(D) is the minimum value of the number of the vertices in a longest directed path starting from a vertex that is connected to every vertex of D, then χ(D) ≤ψ(D). For graphs, we give a positive answer to the following question of Fajtlowicz: if G is a graph with chromatic number χ(G), then for any proper coloring of G of χ(G) colors and for any vertex v∈V(G), there is a path P starting at v which represents all χ(G) colors. Received: May 20, 1999 Final version received: December 24, 1999     

A generalization of the Jaffard–Ohm–Kaplansky Theorem

The well-known Jaffard–Ohm–Kaplansky Theorem states that every abelian ?-group can be realized as the group of divisibility of a commutative Bézout domain. To date there is no realization (except in certain circumstances) of an arbitrary, not necessarily abelian, ?-group as the group of divisibility of an integral domain. We show that using filters on lattices we can construct a nice quantal frame whose “group of divisibility” is the given ?-group. We then show that our construction when applied to an abelian ?-group gives rise to the lattice of ideals of any Prüfer domain assured by the Jaffard–Ohm–Kaplansky Theorem. Thus, we are assured of the appropriate generalization of the Jaffard–Ohm–Kaplansky Theorem.     

A short proof of the Doob–Meyer theorem

Every submartingale S$S$ of class D$D$ has a unique Doob–Meyer decomposition S=M+A$S=M+A$, where M$M$ is a martingale and A$A$ is a predictable increasing process starting at 0.     

A simple proof of the Rogers–Selberg identities

A short and simple proof of the Rogers–Selberg identities is obtained from the work of Andrew Sills on identities of the Rogers–Ramanujan type.     

A Non-Autonomous Version of the Denjoy–Wolff Theorem

The aim of this work is to establish the celebrated Denjoy–Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, ω-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.     

A Generalized Version of the Baker–Pixley Theorem

A famous theorem by Baker and Pixley states that the term functions of a finite algebra with a (d + 1)-ary near-unanimity term are precisely the functions under which all subalgebras of the algebra's dth power are closed. In this paper, we generalize the theorem to a completely abstract level. Indeed, we obtain a version of the theorem that is stated in purely category-theoretic terms, making it applicable in any concrete or abstract category. To motivate this rather abstract result, we also discuss some of its concrete applications.     

A Nonlinear Extension of the Coale–Lopez Theorem

For a sequence (T _n)_nof nonlinear positive operators on a real Banach space, certain monotonicity conditions are specified under which asymptotic equality holds for the inhomogeneous iterates T _n T _n-1 T ₁ T ₀. As a consequence, weak ergodicity or even path stability holds in the norm-topology. Finally, a nonlinear extension of the Coale–Lopez theorem in demography is shown.     

A simple proof of the generalized Craig–Sakamoto theorem

The Craig–Sakamoto theorem establishes the independence of two quadratic forms in normal variates. In this article, we provide a simple proof of a generalized Craig–Sakamoto theorem.     

A new topological proof of the Abhyankar–Moh theorem

We present a new simple proof of the famous theorem of Abhyankar, Moh and Suzuki about rational curves in a plane. This proof relies on the Poincaré–Hopf theorem. This work was supported by Polish KBN Grant No 2 P03A 041 15Mathematics Subject Classification (2002):14E25.     

Frobenius–Ehresmann structures and Cartan geometries in positive characteristic

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogeneous spaces of algebraic groups, which we call Frobenius–Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius–Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann–Weil–Thurston principle for Frobenius–Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius–Ehresmann structure are completely determined by their monodromy crystals.