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We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$ on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if $m>…
We use N-body simulations to investigate the structure of dark halos in the standard Cold Dark Matter cosmogony. Halos are excised from simulations of cosmologically representative regions and are resimulated individually at high…
Kinematic studies have produced accurate measurements of the total dark matter mass and mean dark matter density within the optical extent of galaxies, for large samples of objects. Here we consider theoretical predictions for the latter…
The universe, with large-scale homogeneity, is locally inhomogeneous, clustering into stars, galaxies and larger structures. Such property is described by the smoothness parameter $\alpha$ which is defined as the proportion of matter in the…
Based upon the intrinsic symmetries approach to inhomogeneous cosmologies, we propose an exact solution to Einstein's field equations where the spatial sections are flat and the source is a non-perfect fluid such that the dissipative terms…
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity.…
In this paper, we study the normalised characteristic scale of transition to cosmic homogeneity, $\mathcal{R}_H/d_V$, as a cosmological probe. We use a compilation of SDSS galaxy samples, comprising more than $10^6$ galaxies in the redshift…
The Etherington reciprocity theorem, or distance duality relation (DDR), relates the mutual scaling of cosmic distances in any metric theory of gravity where photons are massless and propagate on null geodesics. In this paper, we make use…
We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$…
We relate the structure factor $S(\mathbf{k} \to \mathbf{0})$ in a system of jammed hard spheres of number density $\rho$ to its complexity per particle $\Sigma(\rho)$ by the formula $S(\mathbf{k} \to \mathbf{0})=-1/…
The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the…
For all integers $n \geq k > d \geq 1$, let $m_{d}(k,n)$ be the minimum integer $D \geq 0$ such that every $k$-uniform $n$-vertex hypergraph $\mathcal H$ with minimum $d$-degree $\delta_{d}(\mathcal H)$ at least $D$ has an optimal matching.…
The concept of hyperuniformity has been a useful tool in the study of large-scale density fluctuations in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to…
Halo bias links the statistical properties of the spatial distribution of dark matter halos to those of the underlying dark matter field, providing insights into clustering properties in both general relativity (GR) and modified-gravity…
Using the framework of supersymmetric non-linear $\sigma$-model we develop a general non-perturbative characterisation of universal features of the density $\rho(\Gamma)$ of the imaginary parts (``width'') for $S$-matrix poles…
We show that the hybrid resonances of a DMR backed by a cavity are meta-resonances, in that they can be made as perfect as possible by fine tuning the structural parameters but without the requirements of extreme materials properties, such…
The $\gamma_2$ norm of a real $m\times n$ matrix $A$ is the minimum number $t$ such that the column vectors of $A$ are contained in a $0$-centered ellipsoid $E\subseteq\mathbb{R}^m$ which in turn is contained in the hypercube $[-t, t]^m$.…
The theory of Big Bang nucleosynthesis, coupled with an estimate of the primordial deuterium abundance (D/H)_pr, offers insights into the baryon density of the Universe. Independently, the baryon density can be constrained during a…
We identify a universal relation between the radial density slope \alpha (r) and the velocity anisotropy \beta (r) for equilibrated structures. This relation holds for a variety of systems, including disk galaxy mergers, spherical…
A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on…