Related papers: Majorization and minimal energy on spheres
We investigate the ground states of a free energy functional on sphere. The energy consists of an entropy and a nonlocal interaction term that are in competition with each other, as they favour spreading and aggregation, respectively.…
Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltr\'an and…
A magnetic monopole is placed at the centre of a 3-ball whose surface, S, is tiled by the symmetry group, G, of a regular solid. The quantum mechanics on the two-dimensional quotient, S/G, is developed and the monopole charge is found to be…
The eigenstate thermalization hypothesis (ETH), which dictates that all diagonal matrix elements within a small energy shell be almost equal, is a major candidate to explain thermalization in isolated quantum systems. According to the…
The quantum Maxwell theory at finite temperature at equilibrium is studied on compact and closed manifolds in both the functional integral- and Hamiltonian formalism. The aim is to shed some light onto the interrelation between the topology…
We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $\Sigma$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone…
We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices.…
The paper deals with minimum energy problems in the presence of external fields on a locally compact space $X$ with respect to a function kernel $\kappa$ satisfying the energy and consistency principles. For quite a general (not necessarily…
We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces…
In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram…
This monograph presents a theoretical background and a broad introduction to the Min-Max Framework for Majorization-Minimization (MM4MM), an algorithmic methodology for solving minimization problems by formulating them as min-max problems…
We consider the minimization problem of $\phi$-divergences between a given probability measure $P$ and subsets $\Omega$ of the vector space $\mathcal{M}_\mathcal{F}$ of all signed finite measures which integrate a given class $\mathcal{F}$…
Consider n points on the unit 2-sphere. The potential of the interaction of two points is a function f(r) of the distance r between the points. The total energy E of n points is the sum of the pairwise energies. The question is how to place…
Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an…
The Minor problem, namely the study of the spectrum of a principal submatrix of a Hermitian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on the use of orbital integrals and on the connection with…
Let $\mathbb{A}$ and $\mathbb{A_{*}}$ be two non-degenerate spherical annuli in $\mathbb{R}^{n}$ equipped with the Euclidean metric and the weighted metric $|y|^{1-n}$, respectively. Let $\mathcal{F}(\mathbb{A},\mathbb{A_{*}})$ denote the…
The lowest radiative correction to the Casimir energy density between two parallel plates is calculated using effective field theory. Since the correlators of the electromagnetic field diverge near the plates, the regularized energy density…
We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two…
We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the…
This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the…