Related papers: Rate of convergence of random polarizations
We derive conditions under which random sequences of polarizations (two-point symmetrizations) converge almost surely to the symmetric decreasing rearrangement. The parameters for the polarizations are independent random variables whose…
It was shown by Burchard and Fortier that the expected $L^1$ distance between $f^*$ and $n$ random polarizations of an essentially bounded function $f$ with support in a ball of radius $L$ is bounded by $2dm(B_{2L})||f||_{\infty}n^{-1}$.…
We investigate the rate of convergence in the central limit theorem for convex sets. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the…
A beam of rotating dipolar particles (molecules or clusters) will broaden when passed through an electric or magnetic field gradient region. This broadening, which is a common experimental observable, can be expressed in terms of the…
We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous…
The polarization emerging in the subsequent scattering processes can never exceed $1$ which corresponds to the fully polarized pure state. This property is shown to be provided by the addition rule similar to that for relativistic…
In this paper, convergence for moments of powered normal extremes is considered under an optimal choice of normalizing constants. It is shown that the rates of convergence for normalized powered normal extremes depend on the power index.…
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in…
The standard deviations of the energy and Poynting power densities for an isotropic random field near a perfectly conducting planar boundary are characterized, based on quartic plane-wave expansions. For normal and transverse components,…
The multipole expansion of the polarization interaction between a charged particle and an electrically charged particle has long been known to be asymptotic in nature, i.e. the multiple expansion diverges at any finite distance from the…
It is known that increasing powers of a continuous random variable converge in distribution to Benford's law as the exponent approaches infinity. The rate of convergence has been estimated using Fourier analysis, but we present an…
A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n =…
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean…
We calculate the detectability of the polarization of the cosmic microwave background (CMB) as a function of the sky coverage, angular resolution, and instrumental sensitivity for a hypothetical experiment. We consider the gradient…
The present work contributes to the determination of the $n$-th linear polarization constant $c_n(H)$ of an $n$-dimensional real Hilbert space $H$. We provide some new lower bounds on the value of $\sup_{\|y\|=1}| x_1,y >... x_n,y |$, where…
We calculate the differential cross section of polarized light scattering from a cold and dense atomic ensemble. The regularities in the transformation of the cross section when increasing the size of the atomic ensemble are analyzed…
In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from…
The polarization constant of a Banach space $X$ is defined as $$\mathbf c(X):= \limsup\limits_{k\rightarrow \infty} \mathbf c(k, X)^\frac{1}{k},$$ where $\mathbf c(k, X)$ stands for the best constant $C>0$ such that $ \Vert…
May randomness (real numbers, opinions) evolve into order (regularity) with time? We study some polarization and symmetry properties, which emerge in time evolution of opinions (real numbers) within entries of two and three-dimensional…
Particles scattered off the nuclear target acquire a polarization if the nuclei have a non zero analyzing power. This effect is enhanced when particles traverse a bent crystal. Such enhancement under certain assumptions allows one to…