Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Duality respecting representations and compatible complexity measures for gammoids

We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums.     

Kinetics of carbon steel dissolution in ammonium chloride solution containing sodium thiosulfate

The dissolution behavior of carbon steel in ammonium chloride (NH₄Cl) solution containing sodium thiosulfate (Na₂S₂O₃) of various concentrations (0.01 and 0.1 M) was investigated using electrochemical impedance spectroscopy (EIS) and other nonelectrochemical techniques. The weight loss and polarization measurements indicate a significant increase in the NH₄Cl corrosion rate of carbon steel on addition of Na₂S₂O_3. The EIS measurements exhibited two capacitive loops at multiple direct current (dc) potentials for both the concentrations. Electrical equivalent circuit (EEC) and reaction mechanism analysis (RMA) were employed to analyze the impedance data. A four-step mechanism with two intermediate adsorbate species of same charge was proposed to explain the dissolution behavior of carbon steel in the given system. The surface coverage values enumerated that the surface was entirely covered with adsorbed species unlike in the pure NH₄Cl system. Charge transfer resistance and polarization resistance values estimated from RMA parameters indicate the increase in a dissolution rate with dc potential. The surface morphology was inspected via field emission scanning electron microscopy, and the corrosion products including surface state of carbon steel electrode were analyzed using energy dispersive X-ray spectroscopy and X-ray photoelectron spectroscopy.     

Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

QK乘子代数上的Corona定理和Wolff定理

本文给出对数K-Carleson测度的一个新特征,并以此为工具研究Q_K空间的乘子代数M(Q_K),给出乘子代数M(Q_K)的某些特征描述.利用对数K-Carleson测度及Q_K空间的一个新特征,建立乘子代数M(Q_K)上的Corona定理和Wolff定理.     

基于等效介质原理的宽角超材料吸波体的理论分析

目前,很少有文章就如何实现宽角度吸波材料进行详细的理论分析和设计指导,设计宽角度吸波材料仍然是一件很困难的事情.本文基于等效介质理论对带有反射地板的单层介质超材料吸波体进行较为详细的理论分析.从基础电磁理论出发,推导TE波(横电波,电场方向与入射面垂直的平面电磁波)和TM波(横磁波,磁场方向与入射面垂直的平面电磁波)照射下吸波体的反射系数,分析实现宽角度吸波效果所需的等效电磁参数,为宽角度超材料吸波体的设计提供了理论基础.此外,论文还理论分析了实现宽带宽角吸波等效电磁参数所要满足的条件,并做了计算检验.结果表明,当介质等效电磁参数按照特殊曲线随频率发生变化时,理论上能实现宽带宽角的吸波效果.