CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size

Let $\mathbf{A}=\frac{1}{\sqrt{np}}(\mathbf{X}^T\mathbf{X}-p\mathbf {I}_n)$ where $\mathbf{X}$ is a $p\times n$ matrix, consisting of independent and identically distributed (i.i.d.) real random variables $X_{ij}$ with mean zero and variance one. When $p/n\to\infty$, under fourth moment conditions a central limit theorem (CLT) for linear spectral statistics (LSS) of $\mathbf{A}$ defined by the eigenvalues is established. We also explore its applications in testing whether a population covariance matrix is an identity matrix.Comment: Published at http://dx.doi.org/10.3150/14-BEJ599 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of $G$, denoted by $\chi'(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $$\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},$$ where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ is a lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G), \, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjectured that $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $\chi'(G)$, so an analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $\chi'(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $\Gamma(G)>\Delta(G)$; third, every multigraph $G$ satisfies $\chi'(G)-\chi^*(G) \le 1$, so FECP has a fascinating integer rounding property

Product and process design optimization by quality engineering /

This research is concerned with product and process design optimization by quality engineering based on the work of Dr. Taguchi, with emphasis on the optimization of dynamic systems and tolerance design. Various quality loss functions are presented in this thesis which can be used for quality evaluation. The goal of robust design for dynamic systems is to reduce the deviations of quality characteristics for the real system from an ideal target which can change based on the requirements of the customer. The optimization can be simplified by decomposing the selection of control factors Z and the adjustment of scaling/leveling factors R into a two-step procedure. The first step is selecting levels for factors Z to maximize the signal-to-noise (SN) ratio that is supposed to be independent of the adjustment of factors R. The second step is used to adjust the real system to a desired model. A systematic approach to optimization is provided for dynamic systems. The motivation of the SN ratio is given and the validity of the SN ratio is examined for various systems. However, for the specific models where the use of the SN ratio is questionable, the necessary modification is suggested. In addition, discrete dynamic characteristics are discussed. The objective of tolerance design is to balance quality loss due to variations and cost increase due to control of variations. Based on the variation transmission equation developed in this thesis, the best tolerance levels are specified for components and subsystems. The tolerance design approach is presented for quality characteristics which may deteriorate over time. Also, a method is presented to develop the tolerances for lower-level quality characteristics based on the tolerances for higher-level quality characteristics, to reflect the voice of the customer. Illustrations are given to demonstrate the efficiency of the tolerance design methodology

Real-Time Illegal Parking Detection System Based on Deep Learning

The increasing illegal parking has become more and more serious. Nowadays the methods of detecting illegally parked vehicles are based on background segmentation. However, this method is weakly robust and sensitive to environment. Benefitting from deep learning, this paper proposes a novel illegal vehicle parking detection system. Illegal vehicles captured by camera are firstly located and classified by the famous Single Shot MultiBox Detector (SSD) algorithm. To improve the performance, we propose to optimize SSD by adjusting the aspect ratio of default box to accommodate with our dataset better. After that, a tracking and analysis of movement is adopted to judge the illegal vehicles in the region of interest (ROI). Experiments show that the system can achieve a 99% accuracy and real-time (25FPS) detection with strong robustness in complex environments.Comment: 5pages,6figure

Experimental Study of Explosion Limits of Refrigerants and Lubricants’ Mixture

The explosion limits of refrigerants and lubricants’ mixture were studied. The refrigerants like R161, R1234yf and R152a are combustible. Lubricants, to a certain extent, are combustion-supporting. In many actual conditions, lubricants and refrigerants are mixed together. In this paper, a test device which can be run automatically was established according to ASTM E681-09, and the explosive experimental of refrigerants and lubricants’ mixture in some ratio was studied. By altering the proportions of refrigerants and lubricants, we got curve and scope of explosions. In some certain ratio, refrigerants and lubricants’ mixture has different explosion limits compared to refrigerants with no lubricants in it