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A polynomial $p\in\mathbb{R}[z_1,\dots,z_n]$ is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the $z_1z_2\dots z_n$ monomial of a real stable…
We show that the dilogarithm has at most one zero on each branch, that each zero is close to a root of unity, and that they may be found to any precision with Newton's method. This work is motivated by applications to the asymptotics of…
An explanation to the boundness of minimal log discrepancies conjectured by V.V. Shokurov would be that the minimal log discrepancies of a variety in its closed points define a lower semi-continuous function. We check this lower…
We study approximation properties of weighted $L^2$-orthogonal projectors onto the space of polynomials of degree less than or equal to $N$ on the unit disk where the weight is of the generalized Gegenbauer form $x \mapsto…
We propose a single time-scale stochastic subgradient method for constrained optimization of a composition of several nonsmooth and nonconvex functions. The functions are assumed to be locally Lipschitz and differentiable in a generalized…
A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen \cite{AHH}, \cite {HK-2} that a connection exists between discrete Painlev\'e equations and (finite order) Nevanlinna theory is an estimate of the…
We produce uniform and decaying bounds in time for derivatives of the solution to the backwards Kolmogorov equation associated to a stochastic processes governed by a time dependent dynamics. These hold under assumptions over the…
In the present work we consider rational best approximations to the exponential function that minimize a uniform error on a subset of the imaginary axis. Namely, Chebyshev approximation and unitary best approximation where the latter is…
In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine $\F_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate…
A new modulus of smoothness and its equivalent $K$-function are defined on the conic domains in $\mathbb{R}^d$, and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the…
In a recent paper, Bary-Soroker, Koukoulopoulos and Kozma proved that when $A$ is a random monic polynomial of $\mathbb{Z}[X]$ of deterministic degree $n$ with coefficients $a_j$ drawn independently according to measures $\mu_j,$ then $A$…
We study local conservation law multipliers for a generalized fifth-order Kadomtsev--Petviashvili family whose one-dimensional reductions include the Lax, Sawada--Kotera, and Kaup--Kupershmidt equations. Using the direct multiplier method,…
It is well known that for every $f\in C^m$ there exists a polynomial $p_n$ such that $p^{(k)}_n\rightarrow f^{(k)}$, $k=0,\ldots,m$. Here we prove such a result for fractional (non-integer) derivatives. Moreover, a numerical method is…
For a subspace $X$ of functions from $L_2$ we consider the minimal number $m$ of nodes necessary for the exact discretization of the $L_2$-norm of the functions in $X$. We construct a subspace such that for any exact discretization with $m$…
Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where…
In this work, we develop first-order (Hessian-free) and zero-order (derivative-free) implementations of the Cubically regularized Newton method for solving general non-convex optimization problems. For that, we employ finite difference…
We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case…
We develop a new framework for establishing approximate factorization of entropy on arbitrary probability spaces, using a geometric notion known as non-negative sectional curvature. The resulting estimates are equivalent to entropy…
We obtain the decay bounds for Chebyshev series coefficients of functions with finite Vitali variation on the unit square. A generalization of the well known identity, which relates exact and approximated coefficients, obtained using the…
We introduce a derivative-free global optimization algorithm that efficiently computes minima for various classes of one-dimensional functions, including non-convex, and non-smooth functions.This algorithm numerically approximates the…