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We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in…
Many data analysis problems can be cast as distance geometry problems in \emph{space forms} -- Euclidean, spherical, or hyperbolic spaces. Often, absolute distance measurements are often unreliable or simply unavailable and only proxies to…
In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as…
Invariant integrals of functions and forms over $q$ - deformed Euclidean space and spheres in $N$ dimensions are defined and shown to be positive definite, compatible with the star - structure and to satisfy a cyclic property involving the…
A simple three-dimensional model of a fluid whose constituent particles interact via a short range attractive and long range repulsive potential is used to model the aggregation into large spherical-like clusters made up of hundreds of…
Spatial classification with limited feature observations has been a challenging problem in machine learning. The problem exists in applications where only a subset of sensors are deployed at certain spots or partial responses are collected…
Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for…
Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely…
Except for crystalline or random structures, an agreed definition of complexity for intermediate and hence interesting cases does not exist. We fill this gap with a notion of complexity that characterises shapes formed by any finite number…
We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are the spheres in the Euclidean spaces, the real, complex and quaternionic…
The scattering of free particles constrained to move on a cylindrically symmetric curved surface is studied. The nontrivial geometry of the space contributes to the scattering cross section through the kinetic as well as a possible scalar…
Distributions over permutations arise in applications ranging from multi-object tracking to ranking of instances. The difficulty of dealing with these distributions is caused by the size of their domain, which is factorial in the number of…
We develop a general framework to study hyperuniformity of various mathematical models of quasicrystals. Using this framework we provide examples of non-hyperuniform quasicrystals which unlike previous examples are not limit-quasiperiodic.…
Position uncertainty (delocalization) measures for a particle on the sphere are proposed and illustrated on several examples of states. The new measures are constructed using suitably the standard multiplication angle operator variances.…
We investigate the formation of cluster crystals with multiply occupied lattice sites on a spherical surface in systems of ultra-soft particles interacting via repulsive, bounded pair potentials. Not all interactions of this kind lead to…
The structure and dynamics of confined suspensions of particles of arbitrary shape is of interest in multiple disciplines, from biology to engineering. Theoretical studies are often limited by the complexity of long-range particle-particle…
We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and non-orientable). We also give improved bounds for the topological complexity of unordered configuration spaces…
Scattering methods are widely used in many research areas to analyze and resolve material structures. Given the importance, a large number of full textbooks are devoted to this topic. However, technical details in experiments and…
We develop a model to describe the properties of random assemblies of polydisperse hard spheres. We show that the key features to describe the system are (i) the dependence between the free volume of a sphere and the various coordination…
Particle tracking is a key to single-particle-level confocal microscopy observation of colloidal suspensions, emulsions, and granular matter. The conventional tracking method has not been able to provide accurate information on the size of…