Related papers: $2$-Modular Matrices

We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we…

A theory of single-element extensions of integer polymatroids analogous to that of matroids is developed. We present an algorithm to generate a catalog of $2$-polymatroids, up to isomorphism. When we implemented this algorithm on a…

We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right $R$-module $M$ is non-singular if $xI$ is nonzero for every nonzero $x \in M$ and every essential…

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask if…

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2k+1$ is either a power of a…

Let $A$ be a symbolic (or an extended symbolic) Rees algebra (need not be Noetherian) of dimension $d$. Let $P$ be a finitely generated projective $A$-module of rank $\geq$ $d$. Then P has a unimodular element. This improves the classical…

Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ tadpole Cartan matrix is modular, and they provided a proof for $r=2$. The $r=3$ case was recently resolved by Milas and Wang. We…

Let A be a k-vector space of dimension a. A subvector space M of End(A) is said to be of rank r if every non-zero f in M has rank r. The problem considered in this paper is to determine l(r;a) the maximal dimension of a rank r subspace of…

We prove that a complete family of linearly non-degenerate rational curves of degree $e > 2$ in $\mathbb{P}^n$ has at most $n-1$ moduli. For $e = 2$ we prove that such a family has at most $n$ moduli. It is unknown whether or not this is…

A transversal matroid $M$ of rank $r$ on $[n]$ can be associated to a family of binary matrices corresponding to different presentations of $M$. We describe those matrices which arise from unique maximal presentations of size $r$- giving a…

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…

We consider the problem of computing the rank of an m x n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in \~O(|A| + r^\omega) field operations, where |A| denotes the…

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more…

Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group $SL_2(\mathbb{Z})$. In this paper, we show that there is a one-to-one correspondence between…

Let $A$ be a tubular algebra and let $r$ be a positive irrational. Let ${\mathcal D}_r$ be the definable subcategory of $A$-modules of slope $r$. Then the width of the lattice of pp formulas for ${\mathcal D}_r$ is $\infty$. It follows that…

We consider 4-block $n$-fold integer programming, which can be written as $\max\{w\cdot x: H x=b, l\le x\le u, x\in \mathbb{Z}^{N} \}$ where the constraint matrix $H$ is composed of small submatrices $A,B,C,D$ such that the first row of $H$…

Let $R$ be a ring with unity. The graph $\Gamma(R)$ is a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let $\Gamma_2(R)$ is the subgraph of $\Gamma(R)$ induced by the…

It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D\neq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of…

We classify degeneration patterns of Verma modules over the N=2 superconformal algebra in two dimensions. Explicit formulae are given for singular vectors that generate maximal submodules in each of the degenerate cases. The mappings…