Related papers: Kirchberger's Theorem for Complexes of Oriented Ma…
We define united KK-theory for real C*-algebras A and B such that A is separable and B is sigma-unital, extending united K-theory in the sense that KK\crt(\R, B) = K\crt(B). United KK-theory contains real, complex, and self-conjugate…
Tropical oriented matroids were defined by Ardila and Develin in 2007 in analogy to (classical) oriented matroids. In this paper we present one tropical analogue for the Topological Representation Theorem.
Let $x\geqslant 3$, for $1\leqslant n \leqslant x$ an integer, let $\omega(n)$ be its number of distinct prime factors. We show that, among the values $n\leqslant x$ with $\omega(n)=k$ where $1\leqslant k \ll \log_2 x$, $\omega(n-1)$…
We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that…
Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial.…
In this paper, we present a constructive proof of Herschfeld's Convergence Theorem. Our formulation differs from Herschfeld's in a few ways: We consider radicals that nest transfinitely many times, as these are essential to the proof;…
We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic…
Using the Hilbert-Schmidt theorem, we reformulate the R-matrix theory in terms of a uniformly and absolutely convergent expansion. Term by term differentiation is possible with this expansion in the neighborhood of the surface. Methods for…
An affine oriented matroid is a combinatorial abstraction of an affine hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a squarefree monomial ideal in a polynomial ring, called an oriented matroid ideal, and got…
The theorem of Mather on generic projections of smooth algebraic varieties is also proved for the singular ones.
Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.
Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus…
A Euclidean oriented matroid program yields a partial ordering of the cocircuits of its cocircuit graph. We show that every linear extension of that ordering yields a topological sweep and induces a recursive atom-ordering (a shelling of…
One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures…
We argue that the chirotope concept of oriented matroid theory may be found in different scenarios of physics, including classical mechanics, quantum mechanics, gauge field theory, p-branes formalism, two time physics and Matrix theory. Our…
A weak bialgebra is known to be a special case of a bialgebroid. In this paper we study the relationship of this fact with the Tannaka theory of bialgebroids as developed in [4]. We obtain a Tannaka representation theorem with respect to a…
We show that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension $d$ admit a proper labeled sample compression scheme of size $d$. This considerably extends results of Moran and Warmuth on ample classes, of…
This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over GF(2), originally alluded to by Bland and Jensen in their seminal paper on weak…
A fundamental theorem of matroid theory establishes that a transversal matroid is representable over fields of any characteristic. It was proved in 1970 by Piff and Welsh: their proof is elegant and concise and, moveover, constructive.…
White's conjecture asserts that any two tuples of matroid bases that have the same multi-set union can be transformed from one to another by symmetric exchanges; it also implies that the toric ideals of matroids are generated by the…