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We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address…
Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…
We study a model elliptic pseudo-differential equation and simplest boundary value problems for a half-space and a special cone in Sobolev--Slobodetskii spaces which have different smoothness with respect to separate variables. Sufficient…
We discuss a rational version of a conjecture of Matiyasevich, Davis, and Putnam on the relative decidability of the finiteness problem for Diophantine equations with respect to the existence problem. We formulate a suspicion that for…
In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2…
The problem of ensuring constraints satisfaction on the output of machine learning models is critical for many applications, especially in safety-critical domains. Modern approaches rely on penalty-based methods at training time, which do…
We study the Cauchy problem for Schr\"odinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy…
Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…
We conjecture that if a system S \subseteq {x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \leq…
We analyze the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in $\Bbb C^n$. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDdegree) of a…
We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives…
We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.
We study the computational complexity of deciding whether a given set of term equalities and inequalities has a solution in an $\omega$-categorical algebra $\mathfrak{A}$. There are $\omega$-categorical groups where this problem is…
In an earlier paper, Tatong and Suvarnamani explores the Diophantine equation $p^x + p^y = z^2$ for a prime number $p$. In that paper they find some solutions to the equation for $p=2, 3$. In this paper, we look at a general version of this…
We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by…
This article discusses the question - how to estimate the number of solutions of algebraic Diophantine equations with natural coefficients using Circular method developed by Hardy and Littlewood. This paper considers the estimate of the…
Query evaluation over probabilistic databases is known to be intractable in many cases, even in data complexity, i.e., when the query is fixed. Although some restrictions of the queries [19] and instances [4] have been proposed to lower the…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
We prove $L_p$ estimates of solutions to a conormal derivative problem for divergence form complex-valued higher-order elliptic systems on a half space and on a Reifenberg flat domain. The leading coefficients are assumed to be merely…
This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are…