Related papers: Riesz $s$-equilibrium measures on $d$-rectifiable …

In this note the logarithmic energy problem with external potential $|z|^{2n}+tz^d+\bar{t}\bar{z}^d$ is considered in the complex plane, where $n$ and $d$ are positive integers satisfying $d\leq 2n$. Exploiting the discrete rotational…

We study the topology of the space of probability measures invariant under the geodesic flow, defined on the unit-tangent bundle of a compact Riemannian manifold with non-positive curvature. Building on a previous work by Coud\`ene and…

For a fixed, compactly supported probability measure $\mu$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $\Sigma…

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of…

For certain families of complex maps, we give a formula for the Hausdorff dimension of the equilibrium measure. In particular, given an endomorphism $f$ of $\mathbb C\mathbb P^k$ of algebraic degree $d \ge2$, and given the equilibrium…

We investigate the minimal error in approximating a general probability measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In…

Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and…

Let $\{x\_n\}\_{n\geq 0}$ be a sequence of $[0,1]^d$, $\{\lambda\_n\} \_{n\geq 0}$ a sequence of positive real numbers converging to 0, and $\delta>1$. Let $\mu$ be a positive Borel measure on $[0,1]^d$, $\rho\in (0,1]$ and $\alpha>0$.…

We first interpret Pell's equation satisfied by Chebyshev polynomials for each degree t, as a certain Positivstellensatz, which then yields for each integer t, what we call a generalized Pell's equation, satisfied by reciprocals of…

It is shown that, given a point $x\in\mathbbm{R}^d$, $d\ge 2$, and open sets $U_1,...,U_k$ containing $x$, any convex combination of the harmonic measures for $x$ with respect to $U_n$, $1\le n\le k$, is the limit of a sequence of harmonic…

We consider a class of non-doubling manifolds $\mathcal{M}$ defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$ and the Euclidean dimension $n_i$ are not…

Let $E$ be a Banach space such that $E'$ has the Radon-Nikod\'ym property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(\mu,E)$ to a convex compactness criterion in a weaker…

In a previous work we proved that if a finite Borel measure $\mu$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is…

The family $\mathcal{P}_{d}^{\lambda_{d-1}}$ of all probability measures on $[0,1]^d$ whose $(d-1)$-dimensional marginals are all equal to the Lebesgue measure $\lambda_{d-1}$ on $[0,1]^{d-1}$ contains remarkably pathological elements:…

For a decreasing real valued function $\psi$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\psi$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m <…

In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral…

We study the continuity and the measurability of the solution to Schr\"odinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This…

We prove a conjecture of T. Erd\'{e}lyi and E.B. Saff, concerning the form of the dominant term (as $N\to \infty$) of the $N$-point Riesz $d$-polarization constant for an infinite compact subset $A$ of a $d$-dimensional $C^{1}$-manifold…

For a finite collection $\mathbf A=(A_i)_{i\in I}$ of locally closed sets in $\mathbb R^n$, $n\geqslant3$, with the sign $\pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem…

For a compact $ d $-dimensional rectifiable subset of $ \mathbb{R}^{p} $ we study asymptotic properties as $ N\to\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field…