Related papers: The Prouhet-Tarry-Escott problem for Gaussian inte…

Given a set $P$ of $n$ points in the plane, its unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ are connected by an edge if their (Euclidean) distance is at most $1$. We consider several classical…

Let $p$ and $q$ be different prime numbers. Using recent results of Ced\'o and Okni\'nski, we describe isomorphism classes of indecomposable set-theoretic solutions to the Yang--Baxter equation of cardinality $pq$.

Chv\'{a}tal and Klincsek (1980) gave an $O(n^3)$-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set $P$ of $n$ points in the plane. This paper examines a generalization of the problem,…

These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is…

The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…

A central question in mathematics and computer science is the question of determining whether a given ideal $I$ is prime, which geometrically corresponds to the zero set of $I$, denoted $Z(I)$, being irreducible. The case of principal…

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates,…

We study the problem of existence of (nontrivial) perfect codes in the discrete $ n $-simplex $ \Delta_{\ell}^n := \left\{ \begin{pmatrix} x_0, \ldots, x_n \end{pmatrix} : x_i \in \mathbb{Z}_{+}, \sum_i x_i = \ell \right\} $ under $ \ell_1…

A $k$-uniform linear path of length $\ell$, denoted by $P^{(k)}_\ell$, is a family of $k$-sets $\{F_1,..., F_\ell\}$ such that $|F_i\cap F_{i+1}|=1$ for each $i$ and $F_i\cap F_j=\emptyset$ whenever $|i-j|>1$. Given a $k$-uniform hypergraph…

A primary pseudoperfect number (PPN) is an integer $K > 1$ such that the reciprocals of $K$ and its prime factors sum to 1. PPNs arise in studying perfectly weighted graphs and singularities of algebraic surfaces, and are related to…

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has…

We study the solutions of the inverse problem \[ g(z)=\int f(y) P_T(z,dy) \] for a given $g$, where $(P_t(\cdot,\cdot))_{t \geq 0}$ is the transition function of a given Markov process, $X$, and $T$ is a fixed deterministic time, which is…

We show that for each finite sequence of algebraic integers $\alpha_1,...,\alpha_n$ and polynomials $P_1(x_1,...,x_n;y_1,...,y_n),..., P_r(x_1,...,x_n;y_1,...,y_n)$ with algebraic integer coefficients, there are a natural number $N$, $n$…

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the…

The decision problem of perfect matchings in uniform hypergraphs is famously an NP-complete problem. It has been shown by Keevash--Knox--Mycroft [STOC, 2013] that for every $\varepsilon>0$, such decision problem restricted to $k$-uniform…

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial…

Text indexing is a classical algorithmic problem that has been studied for over four decades: given a text $T$, pre-process it off-line so that, later, we can quickly count and locate the occurrences of any string (the query pattern) in $T$…

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…

Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(\theta): \theta \in \Theta\}$ and on Gaussian process…

Given a family of subsets $\mathcal S$ over a set of elements~$X$ and two integers~$p$ and~$k$, Max k-Set Cover consists of finding a subfamily~$\mathcal T \subseteq \mathcal S$ of cardinality at most~$k$, covering at least~$p$ elements…