Related papers: Multidimensional Pad\'e approximation of binomial …

We investigate generalized Laurent multiple orthogonal polynomials on the unit circle satisfying simultaneous orthogonality conditions with respect to $r$ probability measures or linear functionals on the unit circle. We show that these…

Let $ f_0 $ and $ f_\infty $ be formal power series at the origin and infinity, and $ P_n/Q_n $, with $ \mathrm{deg}(P_n),\mathrm{deg}(Q_n)\leq n $, be a rational function that simultaneously interpolates $ f_0 $ at the origin with order $…

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$,…

It is known from the work of Shearer (1985) (and also Scott and Sokal (2005)) that the independence polynomial $Z_G(\lambda)$ of a graph $G$ of maximum degree at most $d+1$ does not vanish provided that $\vert{\lambda}\vert \leq…

In this article, we use Pad\'{e} approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Pad\'{e} approximants with a simple…

We study the computational limits of the following general hypothesis testing problem. Let H=H_n be an \emph{arbitrary} undirected graph on n vertices. We study the detection task between a ``null'' Erd\H{o}s-R\'{e}nyi random graph G(n,p)…

Rigid graph theory is an active area with many open problems, especially regarding embeddings in $\mathbb{R}^d$ or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number…

Let $m,r\in\mathbb{Z}$ and $\omega\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $\omega\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) =…

Given a polynomial $g$ of positive degree over a finite field, we show that the proportion of polynomials of degree $n$, which can be written as $h+g^k$, where $h$ is an irreducible polynomial of degree $n$ and $k$ is a nonnegative integer,…

We present a family of high-order finite element approximation spaces on a pyramid, and associated unisolvent degrees of freedom. These spaces consist of rational basis functions. We establish conforming, exactness and polynomial…

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…

For various Hilbert spaces of analytic functions on the unit disk, we characterize when a function $f$ has optimal polynomial approximants given by truncations of a single power series. We also introduce a generalized notion of optimal…

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the…

For any real numbers $b,c\in\mathbb{R}$, we form the sequence of polynomials $\left\{ H_{m}(z)\right\} _{m=0}^{\infty}$ satisfying the four-term recurrence \[ H_{m}(z)+cH_{m-1}(z)+bH_{m-2}(z)+zH_{m-3}(z)=0,\qquad m\ge3, \] with the initial…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

From a sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ whose coefficients are numbers in this sequence. We…

In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube $[0, 1]^d$ and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its…

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions…

Let $X$ be an inner product space, let $G$ be a group of orthogonal transformations of $X$, and let $R$ be a bounded $G$-stable subset of $X$. We define very weak and very strong regularity for such pairs $(R,G)$ (in the sense of…

We introduce new vanishing subspaces of the homogeneous H\"{o}lder space $\dot{C}^{0,\omega}(X,Y)$ in the generality of a doubling modulus $\omega$ and normed spaces $X$ and $Y.$ For many couples $X,Y,$ we show these vanishing subspaces to…