Related papers: The Dyer-Roeder distance-redshift relation in inho…
The closely related issues of (1) reconstructing density and velocity fields from redshift survey and distance indicator data and (2) determining cosmological parameters from such data sets are discussed. I focus on possible explanations…
We present a unified approach to improved $L^p$ Hardy inequalities in $\R^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where distance is…
Given a convex set $\Omega$ of $\mathbb{R}^n$, we consider the shape optimization problem of finding a convex subset $\omega\subset \Omega$, of a given measure, minimizing the $p$-distance functional $$\mathcal{J}_p(\omega) :=…
We propose and rigorously analyze a finite element method for the approximation of stationary Fokker--Planck--Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients,…
This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet…
We study the possibility of asymmetric dark matter with self-interactions forming compact stable objects. We solve the Tolman-Oppenheimer-Volkoff equation and find the mass-radius relation of such "dark stars", their density profile and…
The inhomogeneous distribution of matter in the non-linear regime of galaxies, clusters of galaxies and voids is described by an exact, spherically symmetric inhomogeneous solution of Einstein's gravitational field equations, corresponding…
This paper focuses on optimal constants and optimizers of the second order Caffarelli-Kohn-Nirenberg inequalities. Firstly, we aim to study optimal constants and optimizers for the following second order Caffarelli-Kohn-Nirenberg inequality…
I present recent progress in theoretical modelling of cosmological density--velocity relations in the weakly nonlinear regime. The relations are local, based on rigorous perturbation theory and include the effects of smoothing of the…
We explore non-canonical scalar field model in the background of non-flat $D$-dimensional fractal Universe with cosmological constant $\Lambda$ on the condition that the matter and scalar field are separately conserved. The potential $V$,…
We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result…
In this work, we have considered a model independent approach to study the nature of the late time cosmic acceleration. We have used the Pade approximation to parametrize the comoving distance. Consequently, from this comoving distance, we…
Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the length (one-dimensional Hausdorff measure $\mathcal H$) at most $l$ that minimizes \[ \max_{y…
$f$-divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability, mathematical statistics and information theory such as Kullback-Leibler…
In the current study, we investigate a scalar field cosmological model with Lyra's geometry to explain the present cosmic expansion in a homogeneous and isotropic flat FRW universe. In Einstein's field equations, we presupposed a variable…
In this study, we present a systematic comparison of various approaches within the constrained random-phase approximation (cRPA) for calculating the Coulomb interaction parameter $U$. While defining the correlated space is straightforward…
Spherically symmetric dust universe models with a positive cosmological constant $\Lambda$, known as $\Lambda$-Lema\^itre-Tolman-Bondi($\Lambda$LTB) models, are considered. We report a method to construct the $\Lambda$LTB model from a given…
We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(\Omega)$ where $\Omega= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $\Gamma=\partial\Omega$ denote the boundary of $\Omega$ and…
Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_{\varphi}(\al):={y\in \R:…
The Hubble relation between distance and redshift is a purely cosmographic relation that depends only on the symmetries of a FLRW spacetime, but does not intrinsically make any dynamical assumptions. This suggests that it should be possible…