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Real-space refinement of atomic models in macromolecular crystallography or in cryo electron microscopy fits a model to a map obtained experimentally. This requires generating model maps of a limited resolution which moreover may vary from…

Skew-symmetric functions are a class of functions defined on a product space $M \times M$ that are antisymmetric with respect to the order of their inputs. In [13], the authors proved that non-deterministic skew-symmetric Gaussian fields…

Isotropic functions of positions $\hat{\bf r}_1, \hat{\bf r}_2,\ldots, \hat{\bf r}_N$, i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch-Gordan…

The electronic Schr\"odinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wavefunctions, depend on 3N variables, three spatial…

We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $\Omega$-deformed sphere, $S^2_\Omega$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the…

Rotational symmetry plays a central role in physics, providing an elegant framework to describe how the properties of 3D objects -- from atoms to the macroscopic scale -- transform under the action of rigid rotations. Equivariant models of…

We present a family of graphical representations for the O($N$) spin model, where $N \ge 1$ represents the spin dimension, and $N=1,2,3$ corresponds to the Ising, XY and Heisenberg models, respectively. With an integer parameter $0 \le \ell…

We consider the asymptotics of $k$-dimensional spherical integrals when $k = o(N)$. We prove that the $o(N)$-dimensional spherical integrals are approximately the products of $1$-dimensional spherical integrals. Our formulas extend the…

We show that generalized spherical harmonics are well suited for representing the space and orientation molecular density in the resolution of the molecular density functional theory. We consider the common system made of a rigid solute of…

The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type…

We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and…

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials…

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. We…

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…

The convolution of a function with an isotropic Gaussian appears in many contexts such as differential equations, computer vision, signal processing, and numerical optimization. Although this convolution does not always have a closed form…

A convolution of the all-sky survey data with a smoothing function is crucial in calculating the smooth surface brightness of the sky survey data. The convolution is usually performed using a spherical version of the convolution theorem.…

We study special functions on euclidean spaces from the viewpoint of riemannian symmetric spaces. Here the euclidean space $E^n = G/K$ where $G$ is the semidirect product $R^n \cdot K$ of the translation group with a closed subgroup $K$ of…

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^N$. Given a continuous plurisubharmonic function $u$ on $\Omega$, we construct a sequence of Gaussian analytic functions $f_n$ on $\Omega$ associated with $u$ such that…

The symmetry properties of a classical N-dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as…

In this work, we introduce new families of nonconforming approximation methods for reconstructing functions on general polygonal meshes. These methods are defined using degrees of freedom based on weighted moments of orthogonal polynomials…