Related papers: The corank of a rectangular random integer matrix

We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c <…

A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The…

We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer $n$, which is not a prime power, is an elliptic Carmichael number for a random curve…

In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n\times n$ sparse symmetric random matrix $A$ with $A_{ij} = \delta_{ij} \xi_{ij}$ is invertible with high probability. Here, $\delta_{ij}$s,…

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the…

We study the cover time $\tau_{\mathrm{cov}}$ by (continuous-time) random walk on the 2D box of side length $n$ with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as $n$ increases,…

We consider a uniform distribution on the set $\mathcal{M}_k$ of moments of order $k \in \mathbb{N}$ corresponding to probability measures on the interval $[0,1]$. To each (random) vector of moments in $\mathcal{M}_{2n-1}$ we consider the…

We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with independent and identically distributed subgaussian entries, the smallest singular value…

In this paper we show the following conjecture of Noga Alon. Fix a positive integer d>2 and real epsilon > 0; consider the probability that a random d-regular graph on n vertices has the second eigenvalue of its adjacency matrix greater…

Given some binary matrix $M$, suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, are we able to recover the unique original orderings and matrix? We present an…

Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$, \begin{align*} {\mathbb…

We give a self-contained proof that for all positive integers $r$ and all $\epsilon > 0$, there is an integer $N = N(r, \epsilon)$ such that for all $n \ge N$ any regular multigraph of order $2n$ with multiplicity at most $r$ and degree at…

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar{A}$ with entries in $\{0,1\}$ such that their Cuntz--Krieger algebras ${\mathcal{O}}_A$…

In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv\{X_1,...,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\R^n$. We…

Using the $det^{S^2}$ map from [5], we introduce the notion of $S^2$-rank of a matrix of type $d\times \frac{s(s-1)}{2}$. As an application, we show that the conditional probability matrix associated to two random variables has the…

Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors…

A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently…

We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice…