Related papers: Polarization, sign sequences and isotropic vector …

Quantifying the amount of polarization is crucial for understanding and studying political polarization in political and social systems. Several methods are used commonly to measure polarization in social networks by purely inspecting their…

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{\alpha,\beta}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$…

The spectral $p$-norm and nuclear $p$-norm of matrices and tensors appear in various applications albeit both are NP-hard to compute. The former sets a foundation of $\ell_p$-sphere constrained polynomial optimization problems and the…

Let $d\ge 3$ be a fixed integer. Let $y:= y(p)$ be the probability that the root of an infinite $d$-regular tree belongs to an infinite cluster after $p$-bond-percolation. We show that for every constants $b,\alpha>0$ and $1<\lambda< d-1$,…

Let $M_{G}$ be the centered Hardy-Littlewood maximal operator on a finite graph $G$. We find $\underset{p\to \infty}{\lim}\|M_{G}\|_{p}^{p }$ when $G$ is the start graph ($S_n$) and the complete graph ($K_n$), and we fully describe…

In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result…

We show that the VC-density of any partitioned formula in a pair of ordered vector spaces is bounded above by twice the number of parameter variables. We also show that this bound is optimal and, as a by-product, we prove that no dense pair…

We develop a nonperturbative approach to the bulk polarization of crystalline electric insulators in $d\geq1$ dimensions. Formally, we define polarization via the response to background fluxes of both charge and lattice translation…

We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space $\dot{W}^{s,p}({\mathbb R}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise…

We propose a formal resource-theoretic approach to quantify the degree of polarization of two and three-dimensional random electromagnetic fields. This endows the space of spectral polarization matrices with the orders induced by…

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\frac{\lambda}{(1-u)^{p}} & \{in}\ \ B, 0<u\leq 1 & \{in}\ \ B, u=\frac{\partial u}{\partial n} =0 & \{on}\ \ \partial B. {array}…

R. B. Kusner [R. Guy, Amer. Math. Monthly 90 (1983), 196--199] asked whether a set of vectors in a d-dimensional real vector space such that the l-p distance between any pair is 1, has cardinality at most d+1. We show that this is true for…

We consider the PDE $-\Delta_pu=\rho$, where $\rho$ is a signed Borel measure on $\mathbb{R}^n$. For each $p>n$, we characterize solutions as extremals of a generalized Morrey inequality determined by $\rho$.

Let $\Omega$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $\omega$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{\Lambda_{p}%…

Let $G$ be a finite group having a normal $p$-subgroup $N$ that contains its centralizer $\text{C}_{G}(N)$, and let $R$ be a $p$-adic ring. It is shown that any finite $p$-group of units of augmentation one in $RG$ which normalizes $N$ is…

Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the…

We investigate two kind of polarization of localized optical waves in nonlinear Kerr type media, linear and combination of linear and circular. In the first case of linear polarized components we obtained the vector version of 3D+1…

We consider the rate of convergence of the expected loss of empirically optimal vector quantizers. Earlier results show that the mean-squared expected distortion for any fixed distribution supported on a bounded set and satisfying some…

New degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the…

After n random polarizations of Borel set on a sphere, its expected symmetric difference from a polar cap is bounded by C/n, where the constant depends on the dimension [arXiv:1104.4103]. We show here that this power law is best possible,…