Related papers: An upper bound on Jacobi polynomials

We use precise asymptotic expansions for Jacobi functions $\phi^{(\alpha,\beta)}_\lambda$ parameters $\alpha$, $\beta$ satisfying $\alpha>1/2$, $\alpha>\beta>-1/2$, to generalizing classical H\"ormander-type multiplier theorem for the…

t is proved in this paper that there is a fine correlation between the values of $|\zeta(1/2+i\varphi(t)/2)|^4$ and $|\zeta(1/2+it)|^2$ which correspond to two segments with gigantic distance each from other. This new asymptotic formula…

We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial $C_{n}^{(\lambda)}$ that is greater than $1$ when $-3/2 < \lambda < -1/2.$ Our first approach uses mixed…

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}(k[\Delta])$ of the Stanley-Reisner ring $k[\Delta]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the…

In this paper, we investigate an upper bound of the polar derivative of a polynomial of degree $n$ $$p(z)=(z-z_m)^{t_m} (z-z_{m-1})^{t_{m-1}}\cdots (z-z_0)^{t_0}(a_0+\sum\limits_{\nu=\mu} ^{n-(t_m+\cdots+t_0)} a_{\nu}z^\nu)$$ where zeros…

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…

Let $P(z)$ be a polynomial of degree $n,$ then it is known that for $\alpha\in\mathbb{C}$ with $|\alpha|\leq \frac{n}{2},$ \begin{align*} \underset{|z|=1}{\max}|\left|zP^{\prime}(z)-\alpha P(z)\right|\leq…

In this paper we consider the polynomial sequence $(P_{n}^{\alpha,q}(x))$ that is orthogonal on $[-1,1]$ with respect to the weight function $x^{2q+1}(1-x^{2})^{\alpha}(1-x), \alpha>-1, q\in \mathbb N$; we obtain the coefficients of the…

We improve on Jarn\'{\i}k's inequality between uniform Diophantine exponent $\alpha $ and ordinary Diophantine exponent $\beta$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'{\i}k (1949, 1954) proved that…

We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our…

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s} q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1, \ldots, \beta_t$ bounded by a given constant.…

Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=\delta$ where $\delta$ is an…

We prove an operator level limit for the circular Jacobi $\beta$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the…

We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an…

In this paper we return to the study of the Watson kernel for the Abel summabilty of Jacobi polynomial series. These estimates have been studied for over more than 30 years. The main innovations are in the techniques used to get the…

We prove that for sets $A, B, C \subset \mathbb{F}_p$ with $|A|=|B|=|C| \leq \sqrt{p}$ and a fixed $0 \neq d \in \mathbb{F}_p$ holds $$ \max(|AB|, |(A+d)C|) \gg|A|^{1+1/26}. $$ In particular, $$ |A(A+1)| \gg |A|^{1 + 1/26} $$ and $$…

Finding a good bound on the maximal edge diameter $\Delta(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known,…

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^\alpha(1+x)^\beta$ for functions in the Jacobi weighted $L_p[-1,1]$, $0<p\le \infty$, spaces. These moduli are used to characterize the…

We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(\delta V)^{d/2}$ bound for perturbations…

We combine our version of the resonance method with certain convolution formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$ result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval $1 \le t \le…