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A metric probability space $M$ admits thresholds if the random geometric graph on $M$ has a threshold for every monotone graph property. We connect the existence of thresholds to the uniform expansion of $M$ and prove that all standard…
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions…
A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…
Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability…
If the minimal supersymmetric standard model is the solution to the hierarchy problem, the scalar top quark (stop) and the Higgsino should weigh around the electroweak scale such as 200 GeV. A low messenger scale, which results in a light…
We show that, for vector spaces in which distance measurement is performed using a gauge, the existence of best coapproximations in $1$-codimensional closed linear subspaces implies in dimensions $\geq 2$ that the gauge is a norm, and in…
Gromov--Wasserstein (GW) distances compare graphs, shapes, and point clouds through internal distances, without requiring a common coordinate system. This invariance is powerful, but discrete GW is a nonconvex quadratic optimal transport…
We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler's measure arises in connection with extremal problems for Bergman spaces. It…
The perimeter of a measurable subset of $\mathbb R^N$ is the total variation of its characteristic function. We generalize this notion to a subset $E$ of a closed Riemannian manifold. We show that the perimeter of $E$ is the limit of the…
We study linear spectral statistics of $N \times N$ Wigner random matrices $\mathcal{H}$ on mesoscopic scales. Under mild assumptions on the matrix entries of $\mathcal{H}$, we prove that after centering and normalizing, the trace of the…
We introduce a new class of $\varkappa$-metrizable spaces, namely countably $\varkappa$-metrizable spaces. We show that the class of all $\varkappa$-metrizable spaces is a proper subclass of counably $\varkappa$-metrizable spaces. On the…
We propose a variational approach to approximate measures with measures uniformly distributed over a 1 dimentional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of…
We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a…
In this article, we provide some necessary and sufficient coefficients conditions for a harmonic mapping to be hereditarily spirallike. Also, we give growth estimate for certain harmonic hereditarily spirallike mappings. Moreover, we…
In this work we study three topologies defined over the same set: the hedgehog. As the name suggests, the hedgehog can be described as a set of spines identified at a single point. Among others, we give a proof of the Kowalsky hedgehog…
Classical Hamming graphs are Cartesian products of complete graphs, and two vertices are adjacent if they differ in exactly one coordinate. Motivated by connections to unitary Cayley graphs, we consider a generalization where two vertices…
The marginal likelihood, or Bayesian evidence, is a crucial quantity for Bayesian model comparison but its computation can be challenging for complex models, even in parameters space of moderate dimension. The learned harmonic mean…
We consider elliptic transmission problems in several space dimensions near an interface which is $C^{1,1}$ diffeomorphic to an axisymmetric reference-interface with a singular point of cusp type. We establish the regularity of the gradient…
New realizations of finite W algebras are constructed by relaxing the usual constraint conditions. Then, finite W algebras are recognized in the Heisenberg quantization recently proposed by Leinaas and Myrheim, for a system of two identical…
In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal…