Related papers: Stolarsky principle and energy optimization on the…
We give sharp limiting case Hardy inequalities on the sphere $\mathbb{S}^{2}$ and show that their optimal constants are unattainable by any $f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}$. The singularity of the problem is related to…
We propose a notion of distance between two parametrized planar curves, called their discrepancy, and defined intuitively as the minimal amount of deformation needed to deform the source curve into the target curve. A precise definition of…
The effect of a stochastic background of cosmological perturbations on the luminosity-redshift relation is computed to second order through a recently proposed covariant and gauge-invariant light-cone averaging procedure. The resulting…
Stochastic monotonicity is a well known partial order relation between probability measures defined on the same partially ordered set. Strassen Theorem establishes equivalence between stochastic monotonicity and the existence of a coupling…
The boundary-value problem for the perturbation of an electric potential by a homogeneous anisotropic dielectric sphere in vacuum was formulated. The total potential in the exterior region was expanded in series of radial polynomials and…
We consider the minimum energy problem on the unit sphere $\mathbb S^{d-1}$ in the Euclidean space $\mathbb R^d$, $d\geq 3$, in the presence of an external field $Q$, where the charges are assumed to interact according to Newtonian…
The Euclidean space, obtained by the analytical continuation of time, to an imaginary time, is used to model thermal systems. In this work, it is taken a step further to systems with spatial thermal variation, by developing an equivalence…
We study the Casimir interaction between perfectly conducting sphere and plate in the classical limit of high temperatures. By taking the small-distance expansion of the exact scattering formula, we compute the leading correction to the…
The classical Onofri inequality in the two-dimensional sphere assumes a natural form in the plane when transformed via stereographic projection. We establish an optimal version of a generalization of this inequality in the d-dimensional…
We introduce (integro-differential) harmonic maps into spheres, which are defined as critical points of the Besov-Slobodeckij energy $\int\limits_{\Omega}\int\limits_{\Omega} \frac{|v(x)-v(y)|^{p_s}}{|x-y|^{n+sp_s}}\ dx\ dy$. For $p_s = 2$…
In this paper, we study forms of the uncertainty principle suggested by problems in control theory. We obtain a version of the classical Paneah-Logvinenko-Sereda theorem for the annulus. More precisely, we show that a function with spectrum…
I briefly outline a new physical interpretation to the average cosmological parameters for an inhomogeneous universe with backreaction. The variance in local geometry and gravitational energy between ideal isotropic observers in bound…
In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict…
We study the path of light rays passing near a massive object, in the context of the scale invariant equation of the geodesics first obtained by Dirac (1973). Using the exterior Schwarzschild solution for the metric, we derive the complete…
I investigate spacetime singularities from the point of view of the wavefunction of the universe. In order to extend the classical notion of geodesic incompleteness one has to include the proper time of an observer as a degree of freedom in…
The classical result of concentration of the Gaussian measure on the sphere in the limit of large dimension induces a natural duality between Gaussian and spherical models of spin glass. We analyse the Legendre variational structure linking…
The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian…
The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic…
This paper deals with different image theorems, i.e., Love's equivalence principle, the induction equivalence principle and the physical optics equivalence principle, in the spherical geometry. The deviation of image theorem approximation…
We study a finite-element based space-time discretisation for the 2D stochastic Navier-Stokes equations in a bounded domain supplemented with no-slip boundary conditions. We prove optimal convergence rates in the energy norm with respect to…