Related papers: The direct sum of $q$-matroids
A positroid is the matroid of a matrix whose maximal minors are all nonnegative. Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an…
Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, and give a complete combinatorial formula. For…
This paper concerns the enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field. This is the same as the classification of commuting tuples of matrices over a finite field up to…
Solving a problem of Diestel and Pott, we construct a large class of infinite matroids. These can be used to provide counterexamples against the natural extension of the Well-quasi-ordering-Conjecture to infinite matroids and to show that…
A graph is chordal if every cycle of length at least four has a chord. In 1961, Dirac characterized chordal graphs as those graphs that can be built from complete graphs by repeated clique-sums. Generalizing this, we consider the class of…
An $n\times n$ real matrix $Q$ is quasi-orthogonal if $Q^{\top}Q=qI_{n}$ for some positive real number $q$. If $M$ is a principal sub-matrix of a quasi-orthogonal matrix $Q$, we say that $Q$ is a quasi-orthogonal extension of $M$. In a…
We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed…
Higher degree forms are homogeneous polynomials of degree $d > 2,$ or equivalently symmetric $d$-linear spaces. This paper is mainly concerned about the algebraic structure of the centers of higher degree forms with applications…
$q$-analogs of sum equals integral relations $\sum_{n\in\mathbb{Z}}f(n)=\int_{-\infty}^\infty f(x)dx$ for sinc functions and binomial coefficients are studied. Such analogs are already known in the context of $q$-hypergeometric series. This…
In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system…
Employing the $q$-Lucas theorem and some known $q$-supercongruences, we give some Dwork-type $q$-congruences, confirming three conjectures in [J. Combin. Theory, Ser. A 178 (2021), Art.~105362]. As conclusions, we obtain the following…
Let $\Phi_n(q)$ be the $n$-th cyclotomic polynomial in $q$. Recently, the author and Zudilin provide a creative microscoping method to prove some $q$-supercongruences mainly modulo $\Phi_n(q)^3$ by introducing an additional parameter $a$.…
We undertake a detailed investigation into the structure of permutations in monotone grid classes whose row-column graphs do not contain components with more than one cycle. Central to this investigation is a new decomposition, called the…
The field diffracted from a one-dimensional, coherently illuminated periodic structure at fractional Talbot distances can be described as a coherent sum of shifted units cells weighted by a set of phases given by quadratic Gauss sums. We…
In this article a new method of generating sums of like powers is presented.
This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly…
We study the distribution of consecutive sums of two squares in arithmetic progressions. If $\{E_n\}_{n \in \mathbb{N}}$ is the sequence of sums of two squares in increasing order, we show that for any modulus $q$ and any congruence classes…
For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…