关于拟Einstein场方程

In this paper some properties of a symmetric tensor field T(X, Y) = g(A(X), Y) on a Riemannian manifold (M,g) without boundary which satisfies the quasi-Einstein equation Rij-S/2gij=Tij+bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.     

Isolation Phenomena for Einstein Manifolds

Let M be a compact n-dimensional Einstein manifold without boundary. Denoteby R_(ijkl) the Riemannian curvature tensor with respect to a local orthonormal framefield. Tbe Ricci curvature tensor is given by     

Lorentz几何与Einstein方程

本文综述Lorentz几何的一些进展,包括等距嵌入定理、完备稳态真空的Bernstein型定理和局部最优坐标系的构造等.本文的主要目的是发展Lorentz流形上的几何分析并以之来研究Einstein方程.     

可逆扩散过程和Einstein公式

本文由一椭圆型微分式出发,定义了相应的本质自伴算子,从而构造出一可逆扩散过程,给出并严格证明了Einstein公式,这一公式将扩散流的相关特性用算符扩散项的迹通过积分表示出来。     

A Class of Homogeneous Einstein Manifolds

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c·g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.     

Rotating multicomponent Bose–Einstein condensates

In the understanding of the spatial behavior of interacting components of multicomponent Bose–Einstein condensates (BECs), a central problem is to establish whether coexistence of all the components occurs, or the interspecies interaction leads to extinction, that is, configurations where one or more densities are null. In this paper, for the rotating k-mixture BEC, we prove that the interspecies interaction leads to extinction in the Thomas–Fermi approximation.     

Bose–Einstein condensation in satisfiability problems

This paper is concerned with the complex behavior arising in satisfiability problems. We present a new statistical physics-based characterization of the satisfiability problem. Specifically, we design an algorithm that is able to produce graphs starting from a k-SAT instance, in order to analyze them and show whether a Bose–Einstein condensation occurs. We observe that, analogously to complex networks, the networks of k-SAT instances follow Bose statistics and can undergo Bose–Einstein condensation. In particular, k-SAT instances move from a fit-get-rich network to a winner-takes-all network as the ratio of clauses to variables decreases, and the phase transition of k-SAT approximates the critical temperature for the Bose–Einstein condensation. Finally, we employ the fitness-based classification to enhance SAT solvers (e.g., ChainSAT) and obtain the consistently highest performing SAT solver for CNF formulas, and therefore a new class of efficient hardware and software verification tools.     

Einstein’s equations and Clifford algebra

Einstein’s equations of the general theory of relativity are rewritten within a Clifford algebra. This algebra is otherwise isomorphic to a direct product of two quaternion algebras. A multivector calculus is developed within this Clifford algebra which differs from the corresponding complexified algebra used in the standard spacetime algebra approach.     

Bose–Einstein Condensates with Non-classical Vortex

Coupled systems of nonlinear Schrödinger equations have been used extensively to describe Bose–Einstein condensates. In this paper, we study a two-component Bose–Einstein condensate (BEC) with an external driving field in a three-dimensional space. This model gives rise to a new kind of vortex–filaments, with fractional degree and nontrivial core structure. We show that vortex–filaments is 1-rectifiable set, and calculate its mean curvature in the strong coupling (Thomas–Fermi) limit. In particular, we show that large strength of the external driving field causes vortex–filaments for a two-component BEC.     

推广Einstein场方程为随机微分方程

在这篇论文中,我们推广了Einstein场方程成为随机微分方程: 其几何张量和物质张量的分量都被约定为均方连续和均方可微的随机函数。 我们得以建立一些非常深刻的新观点: a.随机Einstein场方程表示随机物质源决定着空-时的随机结构。这一方程的均方解——均方可微的随机度规函数表征着一类随机空-时微分流形。 b.这类随机空-时微分流形可以解释为浸没在R~n空间中的随机超曲面S。在S中任意运动(包括随机运动)的坐标变换下,ds~2是不变量;而且,物理方程也具有协变性,我们称之为随机协变原理。 c.我们解出了这一随机Einstein场方程的一个特殊的均方解(见§4之(18)式)。     

Exact Vacuum Solutions to the Einstein Equation

In this paper,the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation.This procedure of resolution is based on a canonical form of the metric.According to this procedure,the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations, which are much convenient for the resolution.     

Weyl–Einstein structures on K-contact manifolds

We show that a compact K-contact manifold \((M,g,\xi )\) has a closed Weyl–Einstein connection compatible with the conformal structure [g] if and only if it is Sasaki–Einstein.     

Quasi—Einstein Hypersurfaces in a Hyperbolic Space

§1.　IntroductionLetRijbethecomponentsofRiccitensorofann-dimensionalRiemannianmanifoldM.IfRij=Agij Bξiξj,　(i,j=1,2,…,n)(1.1)whereξisanunitvectorfield,thenMiscalledaquasi-EinsteinmanifoldanddenotedbyQE(ξ).Ifξisanisotropicvectorfield,thenMiscalledageneralizedquasi-Einsteinmanifold.Intheequality(1.1),AandBarescalarfunctions.WeknowQE(ξ)manifoldisEinsteinwhenB≡0.Especially,if〈ξ,ξ〉=e=±1,thenQE(ξ)iscalledanormalquasi-Einsteinmani-fold.Itiseasytoknowfrom[1]and[2]:Rij=R-Tn-1…     

4维完备Einstein流形的一些注释

本文给出了在开的定向的4维流形上存在有界曲率的完备Einstein度量的障碍．     

多复变数Einstein空间中的Schwarz引理

<正> 本文主要的目的是来证明定理1.设■域是 n 个复变数■=(z~1,…,z~n)空间中的简单域且为Einstein空间(不失一般性,不妨假设其 Ricci 曲率为-1),其Bergman度量为     

Einstein–Weyl structures on contact metric manifolds

In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ^± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.