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.     

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.     

Spin Dynamics and Dirac Nodes in a Kagome Lattice

Herein, the spin dynamics for various magnetic configurations arranged on a Kagome lattice is investigated. Using a Holstein–Primakoff expansion of the isotropic Heisenberg Hamiltonian with multiple exchange parameters, the development and evolution of magnetic Dirac nodes with both anisotropy and magnetic field are examined. From the classical energies, the phase diagrams for the ferromagnetic (FM), antiferrimagnetic (AfM), and the 120°  phases are shown as functions of J₁, J₂, J₃, and anisotropy. Furthermore, the production of bosonic Dirac and Weyl nodes in the spin-wave spectra is shown. Through frustration of the magnetic geometry, a connection to the asymmetric properties of the Kagome lattice and the various antiferromagnetic configurations is discerned. Most interesting is the 120°  phase, which does not have Dirac nodes when considering only J₁ due to the formation of an analogous antiferromagnetic honeycomb lattice, but gains Dirac symmetry with next-nearest neighbor interactions. Additionally, the presence of flat modes that are characteristic of cluster excitations is shown. Further study of external frustrations from a magnetic field and anisotropy reveals a tunability of the exchange interactions and nodal points.     

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.     

A New Layer W/Cu/S Polymer with Heterobimetallic Nodes and Tridentate Trithiocyanuric Acid

Solvothermal reaction of [WS4]2-with CuCN and trithiocyanuric acid(L1) in orga-nic solvents gave rise to a new W/Cu/S polymer with a 2D anionic network,namely [Et4N]2[WS4Cu3(C3N3S3H1.5)2](1).The anionic layer of 1 is constructed by the T-shaped {WS4Cu3} subunits as nodes and L1 as linkers and features a(4,4) topology.Both 1 and its isomorphous compound [Et4N]2[MoS4Cu3(C3N3S3H1.5)2](2) have been fully characterized by X-ray crystal analysis,IR and microanalyses.Compounds 1 and 2 show optical transitions with band gaps of 2.17 and 1.84 eV,respectively.