Related papers: Discrepancy of minimal Riesz energy points

In this paper we find asymptotic equalities for the discrete logarithmic energy of sequences of well separated spherical $t$-designs on the unit sphere ${\mathbb{S}^{d}\subset\mathbb{R}^{d+1}}$, $d\geq2$. Also we establish exact order…

We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $A\subset \mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $s\in [p-2, p-1)$, we…

It is shown that the supports of measures minimizing weakly repulsive energies on Riemannian manifolds with sectional curvature bounded below do not have concentration points. This extends the results of Bj\"orck and Carrillo, Figalli, and…

We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite dimensional optimization problems in this…

Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey…

The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels $|x-y|^{\alpha-n}$, $0<\alpha<n$, on $\mathbb R^n$, $n\geqslant2$. For quite a general (not necessarily lower semicontinuous)…

In this paper, we obtain new upper bounds for the Lieb-Thirring inequality on the spheres of any dimension greater than $2$. As far as we have checked, our results improve previous results found in the literature for all dimensions greater…

We extend the celebrated theorem of Kellogg for conformal mappings to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimiser of Dirichlet energy of Sobolev mappings between double connected domains $D$ and…

Fix $d\geq 2$, and $s\in (d-1,d)$. We characterize the non-negative locally finite non-atomic Borel measures $\mu$ in $\mathbb{R}^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\mu)$ in terms of the Wolff energy. This…

We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss.…

In this short note, we employ well-known results to improve the lower bound for the constant associated with the linear term in the asymptotic expansion of the minimal logarithmic energy on the sphere.

We show an alternative proof of the sharpest known lower bound for the logarithmic energy on the unit sphere $\mathbb{S}^2$. We then generalize this proof to get new lower bounds for the Green energy on the unit $n$-sphere $\mathbb{S}^n$.

Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a…

We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with Bent Fuglede (Potential Anal. 51 (2019), 197--217). Having now added to the analysis constraint and external…

We study the global Lipschitz character of minimisers of the Dirichlet energy of diffeomorphisms between doubly connected domains with smooth boundaries from Riemann surfaces. The key point of the proof is the fact that minimisers are…

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schr\"odinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ d\rho \propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x -…

For compact, isometrically embedded Riemannian manifolds $ N \hookrightarrow \mathbb{R}^L$, we introduce a fourth-order version of the wave map equation. By energy estimates, we prove an $\textit{a priori}$ estimate for smooth local…

We introduce and study the $\rho$-Loewner energy, a variant of the Loewner energy with a force point on the boundary of the domain. We prove a large deviation principle for SLE$_\kappa(\rho)$, as $\kappa \to 0+$ and $\rho>-2$ is fixed, with…

We prove that the information complexity (i.e., the inverse) of the classical spherical cap $L_2$ discrepancy on the $d$-dimensional sphere $\mathbb{S}^d$ decreases with dimension $d$, indicating a ``blessing of dimensionality'' for the…

We give a new lower bound for the first gap $\lambda_2 - \lambda_1$ of the Dirichlet eigenvalues of the Schr{\"o}dinger operator on a bounded convex domain $\Omega$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new…