相关论文: On polarization of spherical codes and designs
Ionic crystals, such as solid electrolytes and complex oxides, are central to modern technologies for energy storage, sensing, actuation, and other functional applications. An important fundamental issue in the atomic and quantum-scale…
Consider the problem of constructing a polar code of block length $N$ for the transmission over a given channel $W$. Typically this requires to compute the reliability of all the $N$ synthetic channels and then to include those that are…
This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is…
We present an analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric, focusing on highly symmetric configurations on the unit sphere $\mathbb S^2$. Three discrete uniform…
We apply the generalized sphere-packing bound to two classes of subblock-constrained codes. A la Fazeli et al. (2015), we made use of automorphism to significantly reduce the number of variables in the associated linear programming problem.…
Due to quantum noise fluctuations, the rate of error achievable in decision problems involving several possible configurations of a scattering system is subject to a fundamental limit known as the Helstrom bound. Here, we present a general…
In this paper we study a mass-constrained free boundary problem modeling cell polarization, in the regime where the mass is small. In the generic case of a signal with nondegenerate maxima, we prove that the solution converges locally to a…
In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This…
We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, We formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the…
A universal and optimal method for the polarimetry of light with arbitrary photon statistics is presented. The method is based on the continuous maximum-likelihood positive operator-valued measure (ML-POVM) for pure polarization states over…
We define three-point bounds for sphere packing that refine the linear programming bound, and we compute these bounds numerically using semidefinite programming by choosing a truncation radius for the three-point function. As a result, we…
The classic lower bound of Kuhn, Moscibroda and Wattenhofer [JACM 2016] states that approximate maximum matching and approximate vertex cover (among other problems) in the LOCAL model require $\Omega(\min\{\sqrt{\frac{\log n}{\log\log n}},…
We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered…
We introduce the design of a set of code sequences $ \{ {\mathscr C}_{n}^{(m)} : n\geq 1, m \geq 1 \}$, with memory order $m$ and code-length $N=O(\phi^n)$, where $ \phi \in (1,2]$ is the largest real root of the polynomial equation…
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for…
Let $\mathcal{C}$ be the family of compact convex subsets $S$ of the hemisphere in $\rn$ with the property that $S$ contains its dual $S^*;$ let $u\in S^*$, and let $ \Phi(S,u)=\frac{2}{\omega_n}\int_{S}\ < \theta, u \ > \,\,…
We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss.…
An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the…
A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically…
We identify a universal structural principle underlying the smoothing of classical divergences: the optimizer of the smoothing problem is a clipped probability vector, independently of the specific divergence. This yields a…