Related papers: An upper bound for permanents of nonnegative matri…

Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\em $\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\leq \alpha |u|$. Our main result is the $O(\alpha n)$ bound on the…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

It is a well-known conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a…

We represent the number of mxn non-negative integer matrices (contingency tables) with prescribed row sums and column sums as the expected value of the permanent of a non-negative random matrix with exponentially distributed entries. We…

In the paper "The best m-term approximation and greedy algorithms" (V. N. Temlyakov), an error bound for a near best m-term approximation of a function g in L^p([0,1]^d) is provided, using a basis L^p-equivalent to the Haar system, where p…

We study the standard-form ILP problem $\max\{ c^\top x \colon A x = b,\; x \in Z_{\geq 0}^n \}$, where $A\in Z^{k\times n}$ has full row rank. We obtain refined FPT algorithms parameterized by $k$ and $\Delta$, the maximum absolute value…

Let $\{u_n\}_n$ be a non-degenerate linear recurrence sequence of integers with Binet's formula given by $u_n= \sum_{i=1}^{m} P_i(n)\alpha_i^n.$ Assume $\max_i \vert \alpha_i \vert >1$. In 1977, Loxton and Van der Poorten conjectured that…

Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a strong submultiplicative upper bound on the rank of a three-tensor obtained as an iterated Kronecker product of a constant-size base tensor. The conjecture, if true,…

Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully…

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect…

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

We study the low rank approximation problem of any given matrix $A$ over $\mathbb{R}^{n\times m}$ and $\mathbb{C}^{n\times m}$ in entry-wise $\ell_p$ loss, that is, finding a rank-$k$ matrix $X$ such that $\|A-X\|_p$ is minimized. Unlike…

In this paper, we study the behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^n$ for large $n$. The main result of this paper is that the $L_p$-bounds ($p>1$) can be taken to be independent of $n$, which is a…

Let A be an n-by-n matrix of real numbers which are weakly decreasing down each column, Z_n = diag(z_1,..., z_n) a diagonal matrix of indeterminates, and J_n the n-by-n matrix of all ones. We prove that per(J_nZ_n+A) is stable in the z_i,…

The unconditional, i.e. without assuming validity of RH, sharp limit relationship (as p tends to infinity) is found between the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals and maximal values of…

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a…

Let $\xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(\xi)$ denote an $n\times n$ random matrix with entries that are independent copies of $\xi$. For $\xi$ which is not uniform on its support, we show…

We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let v be a vector field defined on the unit square such that v(x,y) = (1,u(x)) for some measurable u from [0,1] to [0,1].…

We show that the capacity of the Ising perceptron is with high probability upper bounded by the constant $\alpha_\star \approx 0.833$ conjectured by Krauth and M\'ezard, under the condition that an explicit two-variable function…

New upper and lower bounds for the $\ell_p (1<p<\infty)$ norms of Cauchy-Toeplitz matrices in the form $T_n=[2/(1+2(i-j))]_{i,j=1}^n$ are derived. Moreover, we give a complete answer to a conjecture proposed by D. Bozkurt.