Related papers: Complexity of random smooth functions on the high-…

The main purpose of this paper is to compute all irreducible spherical functions on $G=\SU(3)$ of arbitrary type $\delta\in \hat K$, where $K={\mathrm{S}}(\mathrm{U}(2)\times\mathrm{U}(1))\simeq\mathrm{U}(2)$. This is accomplished by…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We find conditions…

The family of Mat\'ern kernels are often used in spatial statistics, function approximation and Gaussian process methods in machine learning. One reason for their popularity is the presence of a smoothness parameter that controls, for…

We show that for a suitable class of functions of finitely-many variables, the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.

We consider steady states of the incompressible Euler equation on two-dimensional domains. For non-radial analytic steady states on bounded simply connected domains, it was shown previously that there must be a global functional…

Let $E = \Bbb C/\Lambda$ be a flat torus and $G$ be its Green function with singularity at $0$. Consider the multiple Green function $G_n$ on $E^n$: $$G_{n}(z_1,\cdots,z_n) := \sum_{i < j} G(z_{i} - z_{j}) - n \sum_{i = 1} ^{n} G(z_{i}).$$…

Most of the analytical studies on spin glasses are performed by using mean-field theory and renormalization group analysis. Analytical studies on finite-dimensional spin glasses are very challenging. In this short note, a possible exten-…

Motivated by the mean field prediction of a Gardner phase transition between a "normal glass" and a "marginally stable glass", we investigate the off-equilibrium dynamics of three-dimensional polydisperse hard spheres, used as a model for…

Reviewing the semiclassical theory for the parametric level density fluctuations, we show that for large parametric changes the density correlation function, after rescaling, becomes universal and coincides with the leading asymptotic term…

For $n\in [-2,2]$ the $O(n)$ model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with $c\in [-\infty,1]$ can be associated. Previously we have…

We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued…

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $\sigma$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in…

We study the energy landscape near the ground state of a model of a single particle in a random potential with trivial topology. More precisely, we find the large dimensional limit of the Hessian spectrum at the global minimum of the…

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $\mathbb{R}^n$, under compactly supported variations. The critical point solves a fourth order…

It was shown by E. Gluskin and V.D. Milman in [GAFA Lecture Notes in Math. 1807, 2003] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to…

By the use of Green's second integral identity we determine the field scattered from a two-dimensional randomly rough isotropic or anisotropic Dirichlet or Neumann surface when it is illuminated by a scalar Gaussian beam. The integral…

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly…

Gaussian processes are the leading class of distributions on random functions, but they suffer from well known issues including difficulty scaling and inflexibility with respect to certain shape constraints (such as nonnegativity). Here we…