N-S方程一般近似惯性流形构造和逼近

1．引言为了用有限维常微分方程来研究Navier七lobes（N七）方程的长时间动力学行为,Foias,Sell和Temaml］引入了耗散系统惯性流形的概念．但惯性流形存在的一个相当苛刻的条件一谱间隔条件一是包括N习方程在内的很多耗散系统无法满足的．因此,Foias,Manlea和TemamZ］随后又提出了近似惯性流形的概念．近似惯性流形也是一个光滑的Lipschitz流形,所有原方程的解在时间充分大时,将被吸引进入该流形的一个三邻域中．因其存在性不需要谱间隔条件来保证,从而可证明包括N－S方程在内的一大类耗散系统存在近似惯性流形．利用近似惯性…     

Noncommutative geometry and classification of elliptic operators

The computation of a stable homotopic classification of elliptic operators is an important problem of elliptic theory. The classical solution of this problem is given by Atiyah and Singer for the case of smooth compact manifolds. It is formulated in terms of K-theory for a cotangent fibering of the given manifold. It cannot be extended for the case of nonsmooth manifolds because their cotangent fiberings do not contain all necessary information. Another Atiyah definition might fit in such a case: it is based on the concept of abstract elliptic operators and is given in term of K-homologies of the manifold itself (instead of its fiberings). Indeed, this theorem is recently extended for manifolds with conic singularities, ribs, and general so-called stratified manifolds: it suffices just to replace the phrase “smooth manifold” by the phrase “stratified manifold” (of the corresponding class). Thus, stratified manifolds is a strange phenomenon in a way: the algebra of symbols of differential (pseudodifferential) operators is quite noncommutative on such manifolds (the symbol components corresponding to strata of positive codimensions are operator-valued functions), but the solution of the classification problem can be found in purely geometric terms. In general, it is impossible for other classes of nonsmooth manifolds. In particular, the authors recently found that, for manifolds with angles, the classification is given by a K-group of a noncommutative C* -algebra and it cannot be reduced to a commutative algebra if normal fiberings of faces of the considered manifold are nontrivial. Note that the proofs are based on noncommutative geometry (more exactly, the K-theory of C* -algebras) even in the case of stratified manifolds though the results are “classical.” In this paper, we provide a review of the abovementioned classification results for elliptic operators on manifolds with singularities and corresponding methods of noncommutative geometry (in particular, the localization principle in C* -algebras).     

VC‐dimension on manifolds: a first approach

The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd.     

Infinitely many nonhomogeneous conditions

A general formula is given for the index of a linear manifold in Banach space. This is expressed in terms of the dimensions of linear manifolds. A fundamental result is established, called a Boundary-Form Formula, for linear manifolds. This is then used to study the existence of the solutions of boundary value problems for linear manifolds subject to finite or infinite nonhomogeneous boundary conditions. An application is made to study the deficiency index of a second-order ordinary differential operator by dividing an interval into infinite subintervals.     

Conformal scalar curvature equations on complete manifolds

This note contains considerations on the existence and non-existence problem of conformal scalar curvature equations on some complete manifolds. We impose two general types of conditions on complete manifolds. The first type is in terms of bounds on curvature and injectivity radius. The second type is in terms of some particular structures on ends of manifolds, for examples, manifolds with cones or cusps and conformally compact manifolds. We obtain non-existence results on both types of conditions. Then we study in more details the existence problem on manifolds with cones, manifolds with cusps and conformally flat manifolds of bounded positive scalar curvature.     

Parameter Estimation for Observed Diffusions in Manifolds

Following the procedure described by Elworthy (1982) and Ikeda& Watanabe (1981) we construct the solution of stochasticdifferential equations (SDEs) in manifolds. We take such SDEsto describe parametrized completely observed stochastic systemsand manifold-valued state processes. The likelihood functionfor the system parameter is computed by two methods: the firstapplies to the case of parallelizable manifolds; the secondapplies to the general case, here the solution of the systemSDE is constructed in the orthonormal frame bundle of the manifold.Two examples are given.     

Singularities in variational calculus

A survey is given of theories of singularities of systems of rays and wave fronts, that is, singularities of systems of extremals of variational problems and solutions of the Hamilton-Jacobi equations near caustics. The problem of passing about an obstacle bounded by a smooth surface of general position is studied in detail. Theorems are proved on the normal forms of Lagrangian manifolds with singularities formed by rays of the system of extremals of a variational problem in the symplectic space of all oriented lines which tear off from the surface of the obstacle as well as theorems on Legendre manifolds with singularities formed by contact elements of a wave front and 1-jets of a solution of the Hamilton-Jacobi equation.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 22, pp. 3–55, 1983.     

Some Global Optimization Problems on Stiefel Manifolds

Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers, and some global optimization methods were considered and tested on Stiefel manifolds. In the paper, test functions are given with known global optimum points and their optimal function values. A restriction, which leads to a discretization of the problem is suggested, which results in a problem equivalent to the well-known assignment problem.     

Two Dimensional Submanifolds in Four Manifolds with Boundary

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环流形上的极值度量——存在性和K-稳定性

本文主要研究环流形上的极值度量的存在性和K-稳定性.本文将Donaldson关于环流形上有关常数量曲率度量的稳定性概念的约化推广到一般的极值度量的情形.通过这个约化,本文证明环流形上极值度量的存在性可以推出流形对于环形变的相对K-稳定性.在不知道是否存在极值度量的情形下,本文还给出环流形相对K-稳定的一个充分性条件.对环曲面的情形,基于Arrezo-Pacard-Singer的工作,本文证明任意一个环曲面上存在含有极值度量的Ka¨hler类,并给出一些环曲面上有不存在极值度量的K¨ahler类的例子.关于一般的环流形上的极值度量的存在性,本文用变分方法研究其弱解,证明在能量泛函逆紧性假设下,存在弱极小化子.     

Classification Problem for Quasitoric Manifolds over a Given Simple Polytope

The classification problem for quasitoric manifolds over a given polytope Pⁿ is considered. The known construction of weights, which was used in the study of the similar problem for toric manifolds, is modified. The construction thus obtained is applied to the solution of the above problem in small dimensions. Classification results are also obtained for polytopes that are products of finitely many simplices of arbitrary dimensions.     

Discrete and sampled-data stochastic control problems with complete and incomplete state information

An unconstrained stochastic optimization problem involving a discrete-time linear process with a normally distributed initial condition and subject to additive gaussian state and measurement noise is formulated in terms of a quite general finite horizon, discrete-time quadratic cost criterion and solved when there is either complete or incomplete state information. It is shown that both the stochastic sampled-data optimal tracker and the stochastic sampled-data optimal regulator are special cases of this problem. A breakdown of the minimum cost for both sampled-data controllers is given.     

基于不变流形的登月轨道设计

研究了基于三体问题的不变流形设计低成本登月轨道的问题．考虑了黄道面和白道面之间夹角不为零的三维情况,将太阳-地球-月亮-卫星组成的四体问题分解成由太阳-地球-卫星和地球-月亮-卫星组成的非共面的两个限制三体问题．给出了这两个三体系统Halo轨道不变流形与两轨道面相交处进行小的变轨来设计低成本探月轨道的一般方法．比较结果表明用该方法设计的轨道比传统的Hohmann变轨节省约20%的燃料．从轨道能量的角度分析了用流形设计轨道比Hohmann变轨节省燃料的原因,并给出了理论表达式．该方法对于深空探测轨道设计的能量分析具有普遍的适用性,可为设计提供一个选择参数的标准．     

The lagrangian approach to stochastic variational principles on curved manifolds

The classical definition of the action functional, for a dynamical system on curved manifolds, can be extended to the case of diffusion processes. For the stochastic action functional so obtained, we introduce variational principles of the type proposed by Morato. In order to generalize the class of process variations, from the flat case originally given by Morato to general curved manifolds, we introduce the notion of stochastic differential systems. These give a synthetic characterization of the process and its variations as a generalized controlled stochastic process on the tangent bundle of the manifold. The resulting programming equations are equivalent to the quantum Schrödinger equation, where the wave function is coupled to an additional vector potential, satisfying a plasma-like equation with a peculiar dissipative behavior.     

Cohomological non-rigidity of generalized real Bott manifolds of height 2

We investigate the following problem: When do two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with ?/2 coefficients and also when are they diffeomorphic? It turns out that in general cohomology rings with ?/2 coefficients do not distinguish those manifolds up to diffeomorphism. This gives a negative answer to the cohomological rigidity problem for real toric manifolds posed earlier by Y. Kamishima and the present author. We also prove that generalized real Bott manifolds of height 2 are diffeomorphic if they are homotopy equivalent.     

Harmonic Maps from Complete Noncompact Manifolds

ln this paper we prove some general existence theorems of harmonic maps from complete noncompact manifolds with tho positive lower bounds of spectrum into convex balls. We solve the Dirichlet problem in classical domains and some special complete noncompact manifolds for harmonic maps into convex balls. We also study the existence of harmonic maps from some special complete noncompact manifolds into complete manifolds with nonpositive sectional curvature which are not simply connected.     

The even dimensional pinching problem and SU(3)/T

A notion of optimal pinching for positively curved manifolds in introduced, and a corresponding rigidity problem is discussed. For the lowest dimensional non-standard homogeneous manifold of positive curvature, SU(3)/T, an estimate is given.     

An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds

In this paper, an inexact proximal point algorithm concerned with the singularity of maximal monotone vector fields is introduced and studied on Hadamard manifolds, in which a relative error tolerance with squared summable error factors is considered. It is proved that the sequence generated by the proposed method is convergent to a solution of the problem. Moreover, an application to the optimization problem on Hadamard manifolds is given. The main results presented in this paper generalize and improve some corresponding known results given in the literature.     

Direct Search Methods on Reductive Homogeneous Spaces

Direct search methods are mainly designed for use in problems with no equality constraints. However, there are many instances where the feasible set is of measure zero in the ambient space and no mesh point lies within it. There are methods for working with feasible sets that are (Riemannian) manifolds, but not all manifolds are created equal. In particular, reductive homogeneous spaces seem to be the most general space that can be conveniently optimized over. The reason is that a ‘law of motion’ over the feasible region is also given. Examples include \(\mathbb {R}^{n}\) and its linear subspaces, Lie groups, and coset manifolds such as Grassmannians and Stiefel manifolds. These are important arenas for optimization, for example, in the areas of image processing and data mining. We demonstrate optimization procedures over general reductive homogeneous spaces utilizing maps from the tangent space to the manifold. A concrete implementation of the probabilistic descent direct search method is shown. This is then extended to a procedure that works solely with the manifold elements, eliminating the need for the use of the tangent space.     

State space models on special manifolds

This paper concerns modeling time series observations in state space forms considered on the Stiefel and Grassmann manifolds. We develop a state space model relating the time series observations to a sequence of unobserved state or parameter matrices assuming the matrix Langevin noise processes on the Stiefel manifolds. We show a Bayes method for estimating the state matrices by the posterior modes. We consider a further extended state space model where two sequences of unobserved state matrices are involved. A simple state space model on the Grassmann manifolds with matrix Langevin noise processes is also investigated.