Recursive Integral Method with Cayley Transformation

Recently, a non-classical eigenvalue solver, called RIM, was proposed to compute (all) eigenvalues in a region on the complex plane. Without solving any eigenvalue problem, it tests if a region contains eigenvalues using an approximate spectral projection. Regions that contain eigenvalues are subdivided and tested recursively until eigenvalues are isolated with a specified precision. This makes RIM an eigensolver distinct from all existing methods. Furthermore, it requires no a priori spectral information. In this paper, we propose an improved version of {\bf RIM} for non-Hermitian eigenvalue problems. Using Cayley transformation and Arnoldi's method, the computation cost is reduced significantly. Effectiveness and efficiency of the new method are demonstrated by numerical examples and compared with 'eigs' in Matlab

Confluent primary fields in the conformal field theory

For any complex simple Lie algebra, we generalize primary fileds in the Wess-Zumino-Novikov-Witten conformal field theory with respect to the case of irregular singularities and we construct integral representations of hypergeometric functions of confluent type, as expectation values of products of generalized primary fields. In the case of sl(2), these integral representations coincide with solutions to confluent KZ equations. Computing the operator product expansion of the energy-momentum tensor and the generalized primary field, new differential operators appear in the result. In the case of sl(2), these differential operators are the same as those of the confluent KZ equations.Comment: 15 pages. Corrected typos. Proposition 3.1 rewritten. Other minor changes, title change

Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\binom{2k}k/(k+1)$. We also pose some related conjectures.Comment: 24 pages. Correct few typo

Direct Investigation of Superparamagnetism in Co Nanoparticle Films

A direct probe of superparamagnetism was used to determine the complete anisotropy energy distribution of Co nanoparticle films. The films were composed of self-assembled lattices of uniform Co nanoparticles 3 nm or 5 nm in diameter, and a variable temperature scanning-SQUID microscope was used to measure temperature-induced spontaneous magnetic noise in the samples. Accurate measurements of anisotropy energy distributions of small volume samples will be critical to magnetic optimization of nanoparticle devices and media.Comment: 4 pages, 4 figures. Submitted to Physical Review Letter

Achieving Effective Innovation Based On TRIZ Technological Evolution

Organised by: Cranfield UniversityThis paper outlines the conception of effective innovation and discusses the method to achieve it. Effective Innovation is constrained on the path of technological evolution so that the corresponding path must be detected before conceptual design of the product. The process of products technological evolution is a technical developing process that the products approach to Ideal Final Result (IFR). During the process, the sustaining innovation and disruptive innovation carry on alternately. By researching and forecasting potential techniques using TRIZ technological evolution theory, the effective innovation can be achieved finally.Mori Seiki – The Machine Tool Compan

Tetraphenolate niobium and tantalum complexes for the ring opening polymerization of ε-caprolactone

Reaction of the pro-ligand α,α,α′,α′-tetra(3,5-di-tert-butyl-2-hydroxyphenyl-p-)xylene-para-tetraphenol (p-L¹H₄) with two equivalents of [NbCl₅] in refluxing toluene afforded, after work-up, the complex {[NbCl₃(NCMe)]₂(μ-p-L¹)}·6MeCN (1·6MeCN). When the reaction was conducted in the presence of excess ethanol, the orange complex {[NbCl₂(OEt)(NCMe)]₂(μ-p-L¹)}·3½MeCN·0.614toluene (2·3½MeCN·0.614toluene) was formed. A similar reaction using [TaCl₅] afforded the yellow complex {[TaCl₂(OEt)(NCMe)]₂(μ-p-L¹)}·5MeCN (3·5MeCN). In the case of the meta pro-ligand, namely α,α,α′,α′tetra(3,5-di-tert-butyl-2-hydroxyphenyl-m-)xylene-meta-tetraphenol (m-L²H₄) only the use of [Nb(O)Cl₃(NCMe)₂] led to the isolation of crystalline material, namely the orange bis-chelate complex {[Nb(NCMe)Cl(m-L²H₂)₂]}·3½MeCN (4·3½MeCN) or {[Nb(NCMe)Cl(m-L²H₂)₂]}·5MeCN (4·5MeCN). The molecular structures of 1–4 and the tetraphenols L¹H₄ and m-L²H₄·2MeCN have been determined. Complexes 1–4 have been screened as pre-catalysts for the ring opening polymerization of ε-caprolactone, both with and without benzyl alcohol or solvent present, and at various temperatures; conversion rates were mostly excellent (>96%) with good control either at >100 °C over 20 h (in toluene) or 1 h (neat)

Simulation of spontaneous G protein activation reveals a new intermediate driving GDP unbinding

Activation of heterotrimeric G proteins is a key step in many signaling cascades. However, a complete mechanism for this process, which requires allosteric communication between binding sites that are ~30 Å apart, remains elusive. We construct an atomically detailed model of G protein activation by combining three powerful computational methods: metadynamics, Markov state models (MSMs), and CARDS analysis of correlated motions. We uncover a mechanism that is consistent with a wide variety of structural and biochemical data. Surprisingly, the rate-limiting step for GDP release correlates with tilting rather than translation of the GPCR-binding helix 5. β-Strands 1 - 3 and helix 1 emerge as hubs in the allosteric network that links conformational changes in the GPCR-binding site to disordering of the distal nucleotide-binding site and consequent GDP release. Our approach and insights provide foundations for understanding disease-implicated G protein mutants, illuminating slow events in allosteric networks, and examining unbinding processes with slow off-rates