Related papers: On extremal mappings in complex ellipsoids

We develop a constructive process which determines all extreme points of the unit ball of the space of $m$--linear forms, $m\geq1.$ Our method provides a full characterization of the geometry of that space through finitely many elementary…

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…

Considered here is one quite general problem about description of extremal configurations maximizing the product of inner radii non-overlapping domains.

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

We present a collection of results on (weak) $m$-extremals and $m$-geodesics, concerning general properties, the planar case, quasi-balanced pseudoconvex domains, complex ellipsoids, the Euclidean ball and boundary properties. We prove…

In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one…

The aim of this paper is to review how some approximation results in commutative algebra are being used to construct equisingular deformations of singularities. The first example of such an approximation result appeared for the first time…

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichm\"uller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the…

The algorithms given in Karney, J. Geodesy 87, 43-55 (2013), to compute geodesics on terrestrial ellipsoids are extended to apply to ellipsoids of revolution with arbitrary eccentricity. For the direct and inverse geodesic problems, this…

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient…

We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still…

In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Ple\'sniak in the circle of papers (most important joint with W.…

We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of those…

We introduce a method, based on the Poincare-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear…

We consider several algorithmic problems concerning geodesics in finitely generated groups. We show that the three geodesic problems considered by Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other. We study two…

We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and…

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical…

We consider the existence and uniqueness of a minimizer of the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. Meanwhile, this in turn implies a…

Graphical models in extremes have emerged as a diverse and quickly expanding research area in extremal dependence modeling. They allow for parsimonious statistical methodology and are particularly suited for enforcing sparsity in…

We prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack…