Low-energy points on the sphere and the real projective plane

We present a generalization of a family of points on ${\mathbb{S}}^2$, the Diamond ensemble, containing collections of N points on ${\mathbb{S}}^2$ with very small logarithmic energy for all $N\in \mathbb{N}$. We extend this construction to the real projective plane ${\mathbb{RP}}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.     

四川省市州经济发展差异的统计分析

采用因子分析研究四川省21市州之间的相关性,利用主因子计算出各个市州得分,以市州得分作为新变量进行集团排序,再将基于相对位置排序法引入到集团排序中,进而得到最终的市州分类结果,在统计分析的基础上对市州经济发展给出一些建议.     

Assessing the Calibration of High-Dimensional Ensemble Forecasts Using Rank Histograms

Any decision-making process that relies on a probabilistic forecast of future events necessarily requires a calibrated forecast. This article proposes new methods for empirically assessing forecast calibration in a multivariate setting where the probabilistic forecast is given by an ensemble of equally probable forecast scenarios. Multivariate properties are mapped to a single dimension through a prerank function and the calibration is subsequently assessed visually through a histogram of the ranks of the observation’s preranks. Average ranking assigns a prerank based on the average univariate rank while band depth ranking employs the concept of functional band depth where the centrality of the observation within the forecast ensemble is assessed. Several simulation examples and a case study of temperature forecast trajectories at Berlin Tegel Airport in Germany demonstrate that both multivariate ranking methods can successfully detect various sources of miscalibration and scale efficiently to high-dimensional settings. Supplemental material in form of computer code is available online.     

Explicit maximal and minimal curves over finite fields of odd characteristics

In this work we present explicit classes of maximal and minimal Artin–Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1].     

The minimum forcing number of perfect matchings in the hypercube

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim (1998) conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is $2^{n?2}$, for all $n\ge 2$. This was revised by Riddle (2002), who conjectured that it is at least $2^{n?2}$, and proved it for all even $n$. We show that the revised conjecture holds for all $n\ge 2$. The proof is based on simple linear algebra.     

Uniqueness of Solution to Systems of Elliptic Operators and Application to Asymptotic Synchronization of Linear Dissipative Systems II: Case of Multiple Feedback Dampings

In this paper, the authors consider the asymptotic synchronization of a linear dissipative system with multiple feedback dampings. They ?rst show that under the observability of a scalar equation, Kalman’s rank condition is su?cient for the uniqueness of solution to a complex system of elliptic equations with mixedobservations. The authors then establish a general theory on the asymptotic stability and the asymptotic synchronization for the corresponding evolutional system subjected to mixed dampings of various natures. Some classic models are presented to illustrate the ?eld of applications of the abstract theory.     

Characterizing and Generating Bivariate Empirical Rank Distributions Satisfying Certain Positive Dependence Concepts

Abstract

This article introduces an approach for characterizing the classes of empirical distributions that satisfy certain positive dependence notions. Mathematically, this can be expressed as studying certain subsets of the class S_N of permutations of 1, …, N, where each subset corresponds to some positive dependence notions. Explicit techniques for it-eratively characterizing subsets of S_N that satisfy certain positive dependence concepts are obtained and various counting formulas are given. Based on these techniques, graph-theoretic methods are used to introduce new and more efficient algorithms for constructively generating and enumerating the elements of various of these subsets of S_N. For example, the class of positively quadrant dependent permutations in S_N is characterized in this fashion.