Related papers: An upper bound on the smallest singular value of d…

Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) =…

We consider the least singular value of $M = R^* X T + U^* YV$, where $R,T,U,V$ are independent Haar-distributed unitary matrices and $X, Y$ are deterministic diagonal matrices. Under weak conditions on $X$ and $Y$, we show that the…

We perturb a real matrix $A$ of full column rank, and derive lower bounds for the smallest singular values of the perturbed matrix, in terms of normwise absolute perturbations. Our bounds, which extend existing lower-order expressions,…

We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds…

We derive tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components.

Let $K_n$ denote the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the number sequence $$ c_n=\min\{\lambda\mid\lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad…

Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\times n matrix almost surely has a constant…

For any real number $p > 0$, we nearly completely characterize the space complexity of estimating $\|A\|_p^p = \sum_{i=1}^n \sigma_i^p$ for $n \times n$ matrices $A$ in which each row and each column has $O(1)$ non-zero entries and whose…

We study the singularity probability of random integer matrices. Concretely, the probability that a random $n \times n$ matrix, with integer entries chosen uniformly from $\{-m,\ldots,m\}$, is singular. This problem has been well studied in…

For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…

Let $K_n$ be the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the numbers $$ c_n={\rm min}\{\lambda\mid \lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad…

The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$…

We approximate intersection numbers $\big\langle \psi_1^{d_1}\cdots \psi_n^{d_n}\big\rangle_{g,n}$ on Deligne-Mumford's moduli space $\overline{\mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain…

Let A be an n x n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and…

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer $d$ there is a constant $c_d > 0$ such that whenever $X_1,..., X_{d+1}$ are $n$-element subsets of $\mathbb{R}^d$, then we…

Let $M_{n}$ denote a random symmetric $n\times n$ matrix, whose entries on and above the diagonal are i.i.d. Rademacher random variables (taking values $\pm 1$ with probability $1/2$ each). Resolving a conjecture of Vu, we prove that the…

In this paper we determine the threshold for collapsibility in the probabilistic model $X_d(n,p)$ of $d$-dimensional simplicial complexes. A lower bound for this threshold $p=\frac{c_d}{n}$ was established in \cite{ALLM}. Here we show that…

We study the deviation inequality for the spectral norm of structured random matrices with non-gaussian entries. In particular, we establish an optimal bound for the $p$-th moment of the spectral norm by transfering the spectral norm into…

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following…

Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal…