Related papers: An upper bound on Jacobi polynomials

For the $\beta$-Hermite, Laguerre, and Jacobi ensembles of dimension $N$ there exist central limit theorems for the freezing case $\beta\to\infty$ such that the associated means and covariances can be expressed in terms of the associated…

We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…

Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $\Delta$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_\Delta \subset S$ its Stanley--Reisner ideal. We write…

In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of…

In this paper we construct Donoghue $m$-functions for the Jacobi differential operator in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, associated to the differential expression \begin{align*} \begin{split} \tau_{\alpha,\beta} = -…

Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…

Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if…

In this work, we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if…

Let $\mathrm{R}$ be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of $\mathrm{R}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in…

For a class of systems of semi-linear elliptic equations, including \[ -\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k, \] for $p=2$ (variational-type interaction) or $p = 1$ (symmetric-type interaction), we…

We prove certain optimal systolic inequalities for a closed Riemannian manifold (X,g), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves…

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi…

We consider elliptic equations with non-Lipschitz nonlinearity $$ -\Delta u = \lambda |u|^{\beta-1}u-|u|^{\alpha-1}u$$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, with Dirichlet boundary conditions; here…

An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element…

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. Previous works introduced this concept and study when…

In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight $x^{\alpha}(1-x)^{\beta},~x\in[0,1],~\alpha,\beta>0$, the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval…

We describe the asymptotic behavior of the multivariate BC-type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results describe the approximation of the…

Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\mathbb{F}_q$. A general result of our study is that…

Coefficients of super Jacobi polynomials of type $B(1,n)$ are rational functions in three parameters $k,p,q$. At the point $(-1,0,0)$ these coefficient may have poles. Let us set $q=0$ and consider pair $(k,p)$ as a point of $\Bbb A^2$. If…

Let $\mathcal{X}$ be a Banach space. Let $\{\tau_j\}_{j=1}^n, \{\omega_k\}_{k=1}^m\subseteq \mathcal{X}$ and $\{f_j\}_{j=1}^n$, $\{g_k\}_{k=1}^m\subseteq \mathcal{X}^*$ satisfy $ |f_j(\tau_j)|\geq 1$ for all $ 1\leq j \leq n$,…