相关论文: Bounds on Tur{\'a}n determinants

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Tur\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general…

The orthogonal polynomials $p_n$ satisfy Tur\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Tur\'an's…

We show that the normalised ultraspherical polynomials, $G_n^{(\lambda)}(x)=C_n^{(\lambda)}(x)/C_n^{(\lambda)}(1)$, satisfy the following stronger version of Tur\'{a}n inequality, $$|x|^\theta \left(G_n^{(\lambda)}(x)\right)^2…

Originally, Tur\'{a}n's inequality states that if $(P_n(x))_{n\in\mathbb{N}_0}$ is the sequence of Legendre polynomials, then $\Delta_n(x):=P_n^2(x)-P_{n+1}(x)P_{n-1}(x)\geq0$ for all $n\in\mathbb{N}$ and $x\in[-1,1]$. Gasper specified the…

We prove a conjectured determinantal inequality: \frac{\det J}{\prod_{i=1}^nJ_{ii}}\le 2(1-\frac{1}{n-1})^{n-1}, where $J$ is a real $n\times n$ ($n\ge 2$) diagonally balanced symmetric matrix.

We present a short proof that the normalized Tur\'{a}n determinant in the ultraspherical case is convex or concave depending on whether parameter $\lambda$ is positive or negative.

We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator $P_{n}=(-1)^{n} (\partial_x)^{2n}$ on $(0,T)$ with Dirichlet boundary conditions and $n$ a positive integer, and show that it satisfies…

P. Tur\'an was the first to derive lower estimations on the uniform norm of the derivatives of polynomials $p$ of uniform norm $1$ on the interval I:=[-1,1] and the disk D:=$\{z \in C~:~|z| \le 1\}$, under the normalization condition that…

Some 76 years ago P. Tur\'an was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I:=[-1,1] and on the unit disk…

We show how Turan's inequality $P_n(x)^2-P_{n-1}(x)P_{n+1}(x)\geq 0$ for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities.…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x,…

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the…

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…

We prove an upper bound on sums of squares of minors of {+1, -1} matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin (2009), but our proof is simpler. We give several corollaries relevant to minors of…

In 1939 P. Tur\'an started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm 1 on $I:=[-1,1]$ (interval) and $D:=\{z\in\mathbb{C}~:~|z|\le 1\}$ (disk), under the normalization condition that the…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

In 1939 P\'al Tur\'an and J\'anos Er\H{o}d initiated the study of lower estimations of maximum norms of derivatives of polynomials, in terms of the maximum norms of the polynomials themselves, on convex domains of the complex plane. As a…

We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n : Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by \Delta. More precisely, we…

Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove…