Related papers: Kirchberger's Theorem for Complexes of Oriented Ma…

If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T,R) of V(R) such that T is the sum of the topes of Q(T,R). If for…

Let K be an Abstract Elementary Class. Under the asusmptions that K has a nicely behaved forking-like notion, regular types and existence of some prime models we establish a decomposition theorem for such classes. The decomposition implies…

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…

Boij-S\"oderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring $S = k[x_1, \ldots, x_n]$. We posit that a similar combinatorial description can be given for…

Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a partition into circuits. They defined the class of ``finite matching extendable'' matroids and showed in their compactness theorem that those matroids always admit…

We prove that graded K-theory is universal among oriented Borel-Moore homology theories with a multiplicative periodic formal group law.

The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of…

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky's Theorem unifies well-known theorems of Kolchin and Levitzki on…

Main Result: Let $(M,L)$ be a smooth complex polarized threefold. Then the linear system $| K+tL|$ separates any two different points on $M$ for any $t\ge 6$, where $K$ is the canonical bundle of $M$. The argument in the proof is a variant…

Performing both right and left multiplication operations using general regular matrix polynomials, which need not be monic and may possess leading coefficients of arbitrary rank, on a rectangular matrix of measures associated with mixed…

We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a…

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric…

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as…

In this paper we present a combinatorial proof of Selberg's integral formula. We start by giving a bijective proof of a Theorem about the number of topological orders of a certain related directed graph. Selberg's Integral Formula then…

We prove a Kotake-Narasimhan type theorem in general ultradifferentiable classes given by weight matrices. In doing so we simultaneously recover and partially generalize the known results for classes given by weight sequences and weight…

We prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the…

We prove (Theorem 2.4) that the symmetrized deleted join $SymmDelJoin(\mathcal{K})$ of a "balanced family" $\mathcal{K} = \langle K_i\rangle_{i=1}^r$ of collectively $r$-unavoidable subcomplexes of $2^{[m]}$ is $(m-r-1)$-connected. As a…

We conjecture that the class of frame matroids can be characterised by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterisation for the class of bicircular matroids. The proof does not…

We explain a correct proof of the decomposition theorem for direct images of constant Hodge modules by proper K\"ahler morphisms of complex manifolds. We also give some examples showing certain difficulty in the non-constant Hodge module…