Related papers: Extremal decomposition problems in the Euclidean s…
We extend classical results by Lavrent'ev and Kufarev concerning the product of the conformal radii of planar non-overlapping domains. We also extend relatively recent results for the case of domains in the $n$-dimensional Euclidean space,…
Considered here is one quite general problem about description of extremal configurations maximizing the product of inner radii non-overlapping domains.
The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved…
In this paper we partly extend the Beauville-Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class…
Consider a finite-dimensional algebra $A$ and any of its moduli spaces $\mathcal{M}(A,\mathbf{d})^{ss}_{\theta}$ of representations. We prove a decomposition theorem which relates any irreducible component of…
We develop the theory of variable exponent Hardy spaces. Analogous to the classical theory, we give equivalent definitions in terms of maximal operators. We also show that distributions in these spaces have an atomic decomposition including…
We consider the moduli space of the extremal K\"ahler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\over2}$-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can…
We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part,…
It is shown that the coincidence isometries of certain modules in Euclidean $n$-space can be decomposed into a product of at most $n$ coincidence reflections defined by their non-zero elements. This generalizes previous results obtained for…
Extremal principles can generally be divided into two rather distinct classes. There are, on the one hand side, formulations based on the Lagrangian or Hamiltonian mechanics, respectively, dealing with time dependent problems, but…
We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the…
We study the duality of moduli of k- and (n-k)-dimensional slices of euclidean n-cubes, and establish the optimal upper bound 1.
New isoperimetric inequalities for lower order eigenvalues of the Laplacian on closed hypersurfaces, of the biharmonic Steklov problems and of the Wentzell-Laplace on bounded domains in a Euclidean space are proven. Some open questions for…
We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and…
We translate notions and results of decomposition and dimension theories for module categories, into the lattice environment. In particular we translate dimension theory in module categories to complete modular upper-continuous lattices.
In this paper we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin, and are normalized to be one at the origin. The conjecture essentially…
The paper contains two natural constructions of extreme hyperspace selections generated by special ordinal decompositions of the underlying space. These constructions are very efficient not only in simplifying arguments but also in…
We prove that for every indecomposable ordinal there exists a (transfinitely valued) Euclidean domain whose minimal Euclidean norm is of that order type. Conversely, any such norm must have indecomposable type, and so we completely…
For the sub-Riemannian problem on the group of motions of Euclidean space we present explicit formulas for extremal controls in a special case, when one of the initial momenta is fixed.
There are significant differences between Helmholtz and Hodge's decomposition theorems, but both share a common flavor. This paper is a first step to bring them together. We here produce Helmholtz theorems for differential 1-forms and…