A Komlós Theorem for abstract Banach lattices of measurable functions

Consider a Banach function space X(μ) of (classes of) locally integrable functions over a σ-finite measure space (Ω,Σ,μ) with the weak σ-Fatou property. Day and Lennard (2010) [9] proved that the theorem of Komlós on convergence of Cesàro sums in L¹[0,1] holds also in these spaces; i.e. for every bounded sequence n(fn) in X(μ), there exists a subsequence k(fnk) and a function f∈X(μ) such that for any further subsequence j(hj) of k(fnk), the series converges μ-a.e. to f. In this paper we generalize this result to a more general class of Banach spaces of classes of measurable functions — spaces L¹(ν) of integrable functions with respect to a vector measure ν on a δ-ring — and explore to which point the Fatou property and the Komlós property are equivalent. In particular we prove that this always holds for ideals of spaces L¹(ν) with the weak σ-Fatou property, and provide an example of a Banach lattice of measurable functions that is Fatou but do not satisfy the Komlós Theorem.     

The edge version of Hadwiger’s conjecture

A well known conjecture of Hadwiger asserts that K_n+1 is the only minor minimal graph of chromatic number greater than n. In this paper, all minor minimal graphs of chromatic index greater than n are determined.     

An Extension Theorem for Euler Characteristics of Groups

We give a homological definition of the Euler characteristic (G) of a group G; if N is a normal subgroup of G with quotient group H, and if (H) and (N) are defined, then (G) is defined, and is the product of the other two. Several conjectures and problems are proposed.     

An approximate Herbrand’s theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite‐dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.     

On Seymour’s strengthening of Hadwiger’s conjecture for graphs with certain forbidden subgraphs

Let H be a set of graphs. A graph is called H-free if it does not contain a copy of a member of H as an induced subgraph. If H is a graph then G is called H-free if it is {H}-free. Plummer, Stiebitz, and Toft proved that, for every -free graph H on at most four vertices, every -free graph G has a collection of ⌈|V(G)|/2⌉ many pairwise adjacent vertices and edges (where a vertexvand an edgeeare adjacent if v is disjoint from the set V(e) of endvertices of e and adjacent to some vertex of V(e), and two edgeseandfare adjacent if V(e) and V(f) are disjoint and some vertex of V(e) is adjacent to some vertex of V(f)). Here we generalize this statement to -free graphs H on at most five vertices.     

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M ², let α_x be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σ_x(2π - α_x) = 2πχ(M ²), where χ denotes the Euler characteristic of M ², α_x denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Σ_τ (-1)^dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σ _x∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B:

$$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$

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Theory of Valuations on Manifolds: A Survey

This is a non-technical survey of a recent theory of valuations on manifolds constructed in [A10], [A11], [AF] and [A12], and actually a guide to this series of articles. We also review some recent related results obtained by a number of people. Received: February 2006, Revision: June 2006, Accepted: June 2006     

Cell decomposition and definable functions for weak p‐adic structures

We develop a notion of cell decomposition suitable for studying weak p‐adic structures (reducts of p‐adic fields where addition and multiplication are not (everywhere) definable). As an example, we consider a structure with restricted addition.     

A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

The principal axis theorem for holomorphic functions

An algebraic approach to Rellich's theorem is given which states that any analytic family of matrices which is normal on the real axis can be diagonalized by an analytic family of matrices which is unitary on the real axis. We show that this result is a special version of a purely algebraic theorem on the diagonalization of matrices over fields with henselian valuations.

     

Definetti’s Theorem for Abstract Finite Exchangeable Sequences

We show that a finite collection of exchangeable random variables on an arbitrary measurable space is a signed mixture of i.i.d. random variables. Two applications of this idea are examined, one concerning Bayesian consistency, in which it is established that a sequence of posterior distributions continues to converge to the true value of a parameter θ under much wider assumptions than are ordinarily supposed, the next pertaining to Statistical Physics where it is demonstrated that the quantum statistics of Fermi-Dirac may be derived from the statistics of classical (i.e. independent) particles by means of a signed mixture of multinomial distributions.      

On the Lê–Milnor fibration for real analytic maps

In this paper, we study the topology of real analytic map‐germs with isolated critical value , with . We compare the topology of f with the topology of the compositions , where are the projections , for . As a main result, we give necessary and sufficient conditions for f to have a Lê–Milnor fibration in the tube.     

Linear extension operators for continuous functions on definable sets in the p‐adic context

Let E be a subset of . A linear extension operator is a linear map that sends a function on E to its extension on some superset of E . In this paper, we show that if E is a semi‐algebraic or subanalytic subset of , then there is a linear extension operator such that is semi‐algebraic (subanalytic) whenever f is semi‐algebraic (subanalytic).     

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.     

Quasi-symmetric functions and up–down compositions

Carlitz (1973) [5] and Rawlings (2000) [13] studied two different analogues of up–down permutations for compositions with parts in {1,…,n}. Cristea and Prodinger (2008/2009) [7] studied additional analogues for compositions with unbounded parts. We show that the results of Carlitz, Rawlings, and Cristea and Prodinger on up–down compositions are special cases of four different analogues of generalized Euler numbers for compositions. That is, for any s≥2, we consider classes of compositions that can be divided into an initial set of blocks of size s followed by a block of size j where 0≤j≤s−1. We then consider the classes of such compositions where all the blocks are strictly increasing (weakly increasing) and there are strict (weak) decreases between blocks. We show that the weight generating functions of such compositions w=w₁?w_m, where the weight of w is , are always the quotients of sums of quasi-symmetric functions. Moreover, we give a direct combinatorial proof of our results via simple involutions.     

A Structure Theorem for Homogeneous Spaces

A partial generalization of Sophus Lie’s triangular form for solvable Lie algebras is presented. As application one is led to an iterated fibration for arbitrary homogeneous spaces. In the case of compact homogeneous spaces, one concludes easily that the Euler number is ≥0.     

A Proof of the Relaxation Theorem for Differential Inclusions Based on Euler Approximations

In this note, an alternative proof of the relaxation theorem is presented that is based on a two-level Euler approximation.