Related papers: Notes on $\{a,b,c\}$-Modular Matrices
Quadratic functions have applications in cryptography. In this paper, we investigate the modular quadratic equation $$ ax^2+bx+c=0 \quad (mod \,\, 2^n), $$ and provide a complete analysis of it. More precisely, we determine when this…
Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of $O(n^3 \log^2 n . E +n^4 {\log}^{O(1)}…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…
Given an ordered triple of positive integers $(n,r,b)$, where $1\leq b\leq\binom{n}{r}$, does there exist a matrix of size $r\times n$ with exactly $b$ invertible submatrices of size $r\times r$? Such a matrix is called an $(n,r,b)$-matrix.…
Let $r\geq 1$ be a positive integer, $A$ a real positive semi-definite symmetric $r\times r$ rational matrix, $B$ a rational vector of length $r$, and $C$ a rational scalar. Nahm's problem is to find all triples $(A,B,C)$ such that the…
In this paper, we present a modular strategy which describes key properties of the absolute primary decomposition of an equidimensional polynomial ideal defined by polynomials with rational coefficients. The algorithm we design is based on…
We convert, within polynomial-time and sequential processing, NP-Complete Problems into a problem of deciding feasibility of a given system S of linear equations with constants and coefficients of binary-variables that are 0, 1, or -1. S is…
Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether all convex combinations of k matrices in R^{n x n} are stable. In particular, we are interested whether there exist…
We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex $n\times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not…
Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily…
Given a field $F$, an integer $n\geq 1$, and a matrix $A\in M_n(F)$, are there polynomials $f,g\in F[X]$, with $f$ monic of degree $n$, such that $A$ is similar to $g(C_f)$, where $C_f$ is the companion matrix of $f$? For infinite fields…
In this article, we study the classification of some natural numbers related to the combinatorics of congruence subgroups of the modular group. More precisely, we will focus here on the notion of minimal monomial solutions. These are the…
The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been…
It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying…
Given any positive integer $r$, Nahm's problem is to determine all $r\times r$ rational positive definite matrix $A$, $r$-dimensional rational vector $B$ and rational scalar $C$ such that the rank $r$ Nahm sum associated with $(A,B,C)$ is…
When is a $q$-series modular? This is an interesting open question in mathematics that has deep connections to conformal field theory. In this paper we define a particular $r$-fold $q$-hypergeometric series $f_{A,B,C}$, with data given by a…
A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer…
In this paper, we introduce the notion of (strictly) semimonotone matrices of exact order $k$, where $0\leq k\leq n$, and explore their properties. We fully characterize the $3 \times 3$ (strictly) semimonotone matrices of exact order $2$,…
Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by…