Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

Pit your wits against young minds!

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Theory of Non-Equilibrium Stationary States as a Theory of Resonances

We study a small quantum system (e.g., a simplified model for an atom or molecule) interacting with two bosonic or fermionic reservoirs (say, photon or phonon fields). We show that the combined system has a family of stationary states parametrized by two numbers, T ₁ and T ₂ (‘reservoir temperatures’). If T ₁ ≠ T ₂, then these states are non-equilibrium stationary states (NESS). In the latter case we show that they have nonvanishing heat fluxes and positive entropy production and are dynamically asymptotically stable. The latter means that the evolution with an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures T ₁ and T ₂, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound min (T ₁,T ₂) > g ^{2 + α}, where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions. Submitted: March 20, 2006. Revised: March 19, 2007. Accepted: May 11, 2007. Marco Merkli: Partly supported by an NSERC PDF, the Institute of Theoretical Physics of ETH Zürich, Switzerland, the Departments of Mathematics of McGill University and the University of Toronto, Canada. Matthias Mück: Supported by DAAD under grant HSP III. Israel Michael Sigal: Supported by NSERC under grant NA7901.     

The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L ₁-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C ₀-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.     

STABILITY NUMBER IN SUBCLASSES OF P5-FREE GRAPHS

Two new hereditary classes of P5-free graphs where the stability number can be found in polynomial time are proposed. They generalize several known results.     

非园断面通道层流摩擦阻力与换热理论研究的新进展

本文介绍作者近些年来对椭圆、矩形、扇形、任意三角形、正多边形、环形等各种非圆断面通道层流摩擦阻力与换热理论的研究成果，给出大量新的速度与温度场表达式，以及摩擦阻力与换热的新的计算公式。本文对传统的层流阻力与换热理论进行了一次系统的补充与完善，同时考虑到工程需要，对所有繁杂的理论公式都进行了化简，给出形式简单而且精度相当高的近似公式。     

Studies on self-pumped phase conjugation in Cu-doped (K0.5Na0.5)0.2(Sr0.61Ba0.39)0.9Nb2O6 crystals

The use of Cu-doped (K_0.5Na_0.5)_0.2(Sr_0.61Ba_0.39)_0.9Nb₂O₆ as self-pumped phase conjugators using internal reflection is reported. Reflectivities as high as 63% at 5145.5 nm and 60% at 632.8 nm were realized. It was found that the photorefractive response changes significantly at different doping concentration. The responses of the crystals to the laser wavelength, incident angle and dopant concentration are studied.     

Con-A诱发的淋巴细胞化学发光

本文以刀豆素 A(ConA)为诱发剂,研究了绵羊肺淋巴液中的淋巴细胞鲁米诺依赖的化学发光现象.结果表明,ConA诱发的淋巴细胞化学发光测定具有迅速、简便和敏感等优点,可作为淋巴细胞活化的检测方法.淋巴细胞悬液(106～107个细胞/ml)1ml、ConA溶液(2mg/ml)0.1ml、鲁米诺溶液(10-3M)50μl为最佳测定体系.多形核白细胞(PMN)混杂可明显增强发光体系的发光强度.因此,测定外周血中的淋巴细胞化学发光,细胞的纯化是一个不可忽视的问题.消炎痛可抑制其发光,说明发光可能与花生四烯酸代谢有关.