Related papers: Harmonic Conjugation in Harmonic Matroids
Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…
Various notions of joint majorization are examined in continuous matrix algebras. The relative strengths of these notions are established via proofs and examples. In addition, the closed convex hulls of joint unitary orbits are completely…
We explore a combinatorial theory of linear dependency in complex space, "complex matroids", with foundations analogous to those for oriented matroids. We give multiple equivalent axiomatizations of complex matroids, showing that this…
We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples…
Positive geometries are semialgebraic sets equipped with a canonical differential form whose residues mirror the boundary structure of the geometry. Every full-dimensional projective polytope is a positive geometry. Motivated by the…
Following our previous work, we develop an algorithm to compute a presentation of the fundamental group of certain partial compactifications of the complement of a complex arrangement of lines in the projective plane. It applies, in…
We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the symmetry group consists of rotations and translations. Along the way, we use tropical…
In this work, we introduce a purely combinatorial way to obtain realizable orientations of a gammoid from a total order on the arc set of the digraph representing it, without first obtaining a matrix representing the gammoid over the reals.
We provide a natural definition of an elliptic arrangement, extending the classical framework to an elliptic curve E with complex multiplication. We analyse the intersections of elements of the arrangement and their connected components as…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
Synchronization of identical harmonic oscillators interconnected via position, velocity, and acceleration couplings is studied. How to construct a complex Laplacian matrix representing the overall coupling is presented. It is shown that the…
First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth…
An effective way to design structured coherent wave interference patterns that builds on the theory of coherent lattices, is presented. The technique combines prime number factorization in the complex plane with moir\'e theory to provide a…
In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of…
This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always…
In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define…
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are…
We give a characterization of a matroid to be paving, through its set of hyperplanes and give an algorithm to construct all of them.
In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac…
The existence of a projective plane of order $p\equiv3\pmod{4}$, where $p$ is a prime power, is shown to be equivalent to the existence of a balancedly multi-splittable embeddable $p^2\times p(p+1)$ partial Hadamard matrix.