线性流形上矩阵方程B^TXB=D的加权最小二乘对称解和解的最佳逼近

本文利用矩阵的奇异值分解（SVD）,给出了在一流形上矩阵方程B^TXB=D的加权最小二乘对称解的通解表达式,并解决了加权最小二乘对称解的最佳逼近问题。     

辛流形上的拟方向导数

设M为有限维微分流形,V(M)与∧'(M)分别表示M上的向量场和1-形之全体,本文引入了V(M)与∧'(M)间三类线性映射沿向量场X∈V(M)的拟方向导数的概念。以此为工具,有限维辛流形上Hamilton 算子的经典定义取得了与无限维情形相一致的等价形式,从而为近年来在无限维情形发展起来的卓有成效的双Hamilton 结构方法应用于有限维辛流形之研究提供了一条途径。作为应用的例子,我们给出了有限维辛流形上遗传算子的一个重要性质的一个新的证明。     

Note on Left Invariant Almost Cosymplectic Structures

Periodica Mathematica Hungarica -     

SUBMANIFOLDS OF PRODUCT RIEMANNIAN MANIFOLD

1IntroductionLetUandVbeRiemannianmanffolds,withthedimensionn1andn2respectively.UxVistheRiemannianproductofUandV.WedenotebyPandQtheprojectionmappingsofT(UxV)toTUaildTVrespectively.ThenwehaveWeputJ=P-Q.ItiseasytoseethatJ~=I.WedefineaRiemannianmetricofUxVbyg(X,Y)==g1(PX,PY) g2(QX,QY),wllereg1andg2areRiemannia11metricofUandVrespectively.ItfollowsthatBy7wedellotetheg'sLevi-Civitaconnection.ThenwecaneasilyseethatInfact,Frollltlledefillitiollofg,wecangetthatUalldVareallgeodesicsub…     

A CSP and tabulation-based adaptive chemistry model

We demonstrate the feasibility of a new strategy for the construction of an adaptive chemistry model that is based on an explicit integrator stabilized by an approximation of the Computational Singular Perturbation (CSP)-slow-manifold projector. We examine the effectiveness and accuracy of this technique first using a model problem with variable stiffness. We assess the effect of using an approximation of the CSP-slow-manifold by either reusing the CSP vectors calculated in previous steps or from a pre-built tabulation. We find that while accuracy is preserved, the associated CPU cost was reduced substantially by this method. We used two ignition simulations – hydrogen–air and heptane–air mixtures – to demonstrate the feasibility of using the new method to handle realistic kinetic mechanisms. We test the effect of utilizing an approximation of the CSP-slow-manifold and find that its use preserves the order of the explicit integrator, produces no degradation in accuracy, and results in a scheme that is competitive with traditional implicit integration. Further analysis on the performance data demonstrates that the tabulation of the CSP-slow-manifold provides an increasing level of efficiency as the size of the mechanism increases. From the software engineering perspective, all the machinery developed is Common Component Architecture compliant, giving the software a distinct advantage in the ease of maintainability and flexibility in its utilization. Extension of this algorithm is underway to implement an automated tabulation of the CSP-slow-manifold for a detailed chemical kinetic system either off-line, or on-line with a reactive flow simulation code.     

On the Elliptic Equation △u ＋ K（x）e^2u -0 with K（x） Positive Somewhere

This paper considers the existence problem of an elliptic equation.which is equivalent to the prescribing conformal Gaussian curvature problem on R2.An existence result is pried.In particular,K(x)is allowed to be unbounded above.     

Generalized Bergman kernels on symplectic manifolds

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As a consequence, we calculate the density of states function of the Bochner-Laplacian and establish a symplectic version of the convergence of the induced Fubini-Study metric. We also discuss generalizations of the asymptotic expansion for non-compact or singular manifolds as well as their applications. Our approach is inspired by the analytic localization techniques of Bismut and Lebeau.     

Existence and Concentration of Ground States of Coupled Nonlinear Schr(o)dinger Equations with Bounded Potentials

A 2-coupled nonlinear Schr(o)dinger equations with bounded varying potentials and strongly attractive interactions is considered.When the attractive interaction is strong enough,the existence of a ground state for sufficiently small Planck constant is proved.As the Planck constant approaches zero,it is proved that one of the components concentrates at a minimum point of the ground state energy function which is defined in Section 4.     

The Three Gap Theorem and Riemannian geometry

The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.