Related papers: The Caratheodory-Fej\'er type extremal problem on …

The century old extremal problem, solved by Carath\'eodory and Fej\'er, concerns a nonnegative trigonometric polynomial normalized by a0 = 1, and the quantity to be maximized is the coefficient a1. In the complex exponential form, the…

The following problem originated from a question due to Paul Turan. Suppose $\Omega$ is a convex body in Euclidean space $\RR^d$ or in $\TT^d$, which is symmetric about the origin. Over all positive definite functions supported in $\Omega$,…

Let G be a locally compact abelian group (LCA group) and U be an open, 0-symmetric set. Let F:=F(U) be the set of all real valued continuous functions from G to R which are supported in U and are positive definite. The Turan constant T(U)…

We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carath\'eodory--Fej\'er--Tur\'an problem. The first variation imposes the additional requirement that the function is…

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties.…

Let G be a locally compact Abelian group, and let X, Y be two open sets in G. We investigate the extremal constant C(X,Y) defined to be the supremum of integrals of functions f from the class F(X,Y), where F(X,Y) is the family of positive…

The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open…

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovasz conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the…

We survey the Carath\'eodory extremal problem $\mathrm{Car} \delta$ on the symmetrized bidisc $$ G = \{(z+w,zw):|z|<1, \, |w|<1\} = \{(s,p)\in \mathbb{C}^2: |s-\bar s p| < 1-|p|^2\}. $$ We also give some new results on this topic. We are…

Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szeg\H{o}, where a number of problems were posed, which were partially solved by using orthogonal polynomials. Since then, not too many new…

This paper is concerned with a Delsarte type extremal problem. Denote by $\mathcal{P}(G)$ the set of positive definite continuous functions on a locally compact abelian group $G$. We consider the function class, which was originally…

Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first prove that timelike normal extremals are locally…

We study the following question: Given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Tur\'an extremal problems. A slight variation of the extremal problem is introduced and the…

Tkachenko and Yaschenko [34] characterized the abelian groups G such that all proper unconditionally closed subsets of G are finite, these are precisely the abelian groups G having cofinite Zariski topology (they proved that such a G is…

For the univalent polynomials $F(z) = \sum\limits_{j=1}^{N} a_j z^{2j-1}$ with real coefficients and normalization \(a_1 = 1\) we solve the extremal problem \[ \min_{a_j:\,a_1=1} \left( -iF(i) \right) = \min_{a_j:\,a_1=1}…

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate…

In the class of normalized sine-polynomials $S(t),$ non-negative on $[0,\pi],$ W.Rogosinski and G.Szeg\H{o} 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient…

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$…