Strong asymptotics in Lagrange interpolation with equidistant nodes

In this paper we prove three conjectures of Revers on Lagrange interpolation for f_λ(t)=|t|^λ,λ>0, at equidistant nodes. In particular, we describe the rate of divergence of the Lagrange interpolants L_N( f_λ,t) for 0<|t|<1, and discuss their convergence at t=0. We also establish an asymptotic relation for max_|t|1| |t|^λ−L_N( f_λ,t)|. The proofs are based on strong asymptotics for |t|^λ−L_N( f_λ,t), 0|t|<1.     

Markovian network processes: Congestion-dependent routing and processing

A Markovian network process describes the movement of discrete units among a set of nodes that process the units. There is considerable knowledge of such networks, often called queueing networks, in which the nodes operate independently and the routes of the units are independent. The focus of this study, in contrast, is on networks with dependent nodes and routings. Examples of dependencies are parallel processing across several nodes, blocking of transitions because of capacity constraints on nodes, alternate routing of units to avoid congestion, and accelerating or decelerating the processing rate at a node depending on downstream congestion. We introduce a general network process representing the numbers of units at the nodes and derive its equilibrium distribution. This distribution takes the form of a product of functions of vectors in which the arguments of the functions satisfy an interchangeability property. This new type of distribution may apply to other multi-variate processes as well. A basic idea in our approach is a linking of certain micro-level balance properties of the network routing to the processing rates at the nodes. The link is via routing-balance partitions of nodes that are inherent in any network. A byproduct of this approach is a general characterization of blocking of transitions without the restriction that the process is reversible, which had been a standard assumption. We also give necessary and sufficient conditions under which a unit moving in the network sees a time average for the unmoved units (called the MUSTA property). Finally, we discuss when certain flows between nodes in an open network are Poisson processes.This research was sponsored in part by Air Force Office of Scientific Research contract 84-0367.     

Qualitative properties of pulsating fronts for reaction–advection–diffusion equations in periodic excitable media

In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.     

Multicolor particle systems with large threshold and range

In this paper we consider the Greenberg-Hastings and cyclic color models. These models exhibit (at least) three different types of behavior. Depending on the number of colors and the size of two parameters called the threshold and range, the Greenberg-Hastings model either dies out, or has equilibria that consist of debris or fire fronts. The phase diagram for the cyclic color models is more complicated. The main result of this paper, Theorem 1, proves that the debris phase exists for both systems.     

Evolutionary variational inequalities with a volterra term

The mathematical model of the injection molding process is characterized by evolutionary inequalities with an elliptic principal part and a Volterra term. Existence and uniqueness of the solution of these problems are derived by means of an argument based on the Banach fixed point theorem. Regularity results are obtained by using the penalty and Rothe method. The results can be considered as a starting point for the numerical treatment and for the application to injection molding.     

WSANs中基于蜂巢结构的移动容错恢复算法

旨在研究无线传感器与执行器网络(WSANs)中节点失效情况下恢复执行器(actor)节点服务的算法. 首先说明了WSANs中的实时覆盖模型, 证明WSANs覆盖恢复问题是NP难问题, 给出了近似求解方案. 在此基础上, 提出了一种基于六边形蜂巢结构的移动容错算法HMFR用于恢复失效actor节点, HMFR 算法在限制网络初始部署的条件下拥有很好的性能. 通过实验与现有的恢复算法进行比较, 发现HMFR算法在actor覆盖sensor节点数和移动距离方面有更好的性能.     

Controllable Synthesis of Porphyrin‐Based 2D Lanthanide Metal–Organic Frameworks with Thickness‐ and Metal‐Node‐Dependent Photocatalytic Performance

Synthesizing 2D metal–organic frameworks (2D MOFs) in high yields and rational tailoring of the properties in a predictable manner for specific applications is extremely challenging. Now, a series of porphyrin‐based 2D lanthanide MOFs (Ln‐TCPP, Ln=Ce, Sm, Eu, Tb, Yb, TCPP=tetrakis(4‐carboxyphenyl) porphyrin) with different thickness were successfully prepared in a household microwave oven. The as‐prepared 2D Ln‐TCPP nanosheets showed thickness‐dependent photocatalytic performances towards photooxidation of 1,5‐dihydroxynaphthalene (1,5‐DHN) to synthesize juglone. Particularly, the Yb‐TCPP displayed outstanding photodynamic activity to generate O₂^? and ¹O₂. This work not only provides fundamental insights into structure designing and property tailoring of 2D MOFs nanosheets, but also pave a new way to improve the photocatalytic performance.     

Classification of Gauge Orbit Types for SU(n)-Gauge Theories

A method for determining the orbit types of the action of the group of gauge transformations on the space of connections for gauge theories with gauge group SU(n) in spacetime dimension d4 is presented. The method is based on the one-to-one correspondence between orbit types and holonomy-induced reductions of the underlying principal SU(n)-bundle. It is shown that the orbit types are labelled by certain cohomology elements of spacetime satisfying two relations. Thus, for every principal SU(n)-bundle the corresponding stratification of the gauge orbit space can be explicitly determined. As an application, a criterion characterizing kinematical nodes for physical states in Yang–Mills theory with the Chern–Simons term proposed by Asorey et al. is discussed.     

Approximation Constants in Equidistant Lagrange Interpolation

In this paper we present a new limit relation for the equidistant Lagrange interpolation polynomials to |x|, (0, 1] on the internal [–1, 1]. The result extends a well-known result of D. L. Berman and S. M. Losinskii. Furthermore, we briefly discuss on a possible connection of the limit relation to another prominent constant in best uniform polynomial approximation for |x| - the so-called Bernstein constant.