Related papers: Small-scale equidistribution for random spherical …
Scattering resonances play a central role in collision processes in physics and chemistry. They help building an intuitive understanding of the collision dynamics due to the spatial localization of the scattering wavefunctions. For…
We present new methods for uniformly sampling the solid angle subtended by a disk. To achieve this, we devise two novel area-preserving mappings from the unit square $[0,1]^2$ to a spherical ellipse (i.e. the projection of the disk onto the…
We study the oscillation frequencies and radiative decay rates of surface plasmon modes of a simple-metal sphere as a function of sphere radius without any assumptions concerning the sphere size. We re-examine within the framework of…
Finite Sample Smeariness (FSS) has been recently discovered. It means that the distribution of sample Fr\'echet means of underlying rather unsuspicious random variables can behave as if it were smeary for quite large regimes of finite…
This chapter reviews methods for linear shrinkage of the sample covariance matrix (SCM) and matrices (SCM-s) under elliptical distributions in single and multiple populations settings, respectively. In the single sample setting a popular…
The orthonormal set of Spherical Harmonics provides a natural way of expanding whole sky redshift and peculiar velocity surveys.
We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected…
Compared with the traditional spherical harmonics, the spherical needlets are a new generation of spherical wavelets that possess several attractive properties. Their double localization in both spatial and frequency domains empowers them…
We consider scattering of a free quantum particle on a singular potential with rather arbitrary shape of the support of the potential. In the classical limit $\hbar=0$ this problem reduces to the well known problem of chaotic scattering.…
In their seminal 1981 article, Sacks-Uhlenbeck famously proved the existence of non-trivial harmonic 2-spheres in every closed Riemannian manifold with non-zero second homotopy group. Their arguments heavily rely on PDE techniques. The…
We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: i.e., the length of the zero set $\mathcal{Z}_{\ell,r_\ell} :=…
We compute the minimum number of critical points of a small codimension smooth map between two manifolds. We give as well some partial results for the case of higher codimension when the manifolds are spheres.
We present a Gaussian ensemble of random cyclic matrices on the real field and study their spectral fluctuations. These cyclic matrices are shown to be pseudo-symmetric with respect to generalized parity. We calculate the joint probability…
The study of physics at the Planck scale has garnered significant attention due to its implications for understanding the fundamental nature of the universe. At the Planck scale, quantum fluctuations challenge the classical notion of…
We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=\alpha$, the empirical spectral…
We investigate the reliability of mass estimators based on the observable velocity dispersion and half-light radius $R_\mathrm{h}$ for dispersion-supported galaxies. We show how to extend them to flattened systems and provide simple…
Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence…
The general relationship between an arbitrary frequency distribution and the expectation value of the frequency distributions of its samples is discussed. A wide set of measurable quantities ("invariant moments") whose expectation value…
Numerical studies in random systems are plagued with strong finite-size effects and boundary effects. We introduce a window-measurement method as a practical solution to these difficulties. We observe physical quantities only within a…
We address the problem of estimating the spherical-harmonic power spectrum of a statistically isotropic scalar signal from noise-contaminated data on a region of the unit sphere. Three different methods of spectral estimation are…