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

In the general setting of a locally compact Abelian group $G$, the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions $f: G \to \mathbb{R}$ satisfying $f(0) = 1$ and…

A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal…

We show how realization theory can be used to find the solutions of the Carath\'eodory extremal problem on the symmetrized bidisc \[ G \stackrel{\rm{def}}{=} \{(z+w,zw):|z|<1, \, |w|<1\}. \] We show that, generically, solutions are unique…

Fix an abelian variety $A$ of dimension $g\geq 1$ defined over a number field $K$. For each prime $\ell$, the Galois action on the $\ell$-power torsion points of $A$ induces a representation $\rho_{A,\ell}\colon Gal_K \to…

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an…

The Delsarte extremal problem for positive definite functions, originally introduced by Delsarte in coding theory to bound the size of error-correcting codes, has since found applications in diverse areas such as sphere packing, Fuglede's…

We study the extremal solution for the problem $(-\Delta)^s u=\lambda f(u)$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$, where $\lambda>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when…

Consider the semilinear elliptic equation $-\Delta u=\lambda f(u)$ in the unit ball $B_1\subset \mathbb{R}^N$, with Dirichlet data $u|_{\partial B_1}=0$, where $\lambda\geq 0$ is a real parameter and $f$ is a $C^1$ positive, nondecreasing…

Let V be a finite set of points in Euclidean d-space (d >= 2). The intersection of all unit balls B(v,1) centered at v, where v ranges over V, henceforth denoted by B(V) is the ball polytope associated with V. Note that B(V) is non-empty…

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the…

We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials,…

This paper studies the existence of extremal problems for the Hardy-Littlewood-Sobolev inequalities on compact manifolds without boundary via Concentration-Compactness principle.

We show uniqueness for overdetermined elliptic problems defined on topological disks $\Omega$ with $C^2$ boundary, i.e., positive solutions $u$ to $\Delta u + f(u)=0$ in $\Omega \subset (M^2,g)$ so that $u = 0$ and $\frac{\partial…

The approximate Carath\'eodory theorem states that given a compact convex set $\mathcal{C}\subset\mathbb{R}^n$ and $p\in\left[2,+\infty\right[$, each point $x^*\in\mathcal{C}$ can be approximated to $\epsilon$-accuracy in the $\ell_p$-norm…

The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of Hermitian matrices given the eigenvalues of the summands. This is a problem about the Lie algebra of the maximal compact subgroup of $G=\operatorname{SL}(n)$ .…

A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this…

A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover,…

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier…

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology $\tau_{ac}$. We show that, under mild conditions on a…

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…