广义典型流形的实例构作

陈述了广义典型流形(或拓扑域流形)的概念,并构作了拓扑域流形的若干实例,丰富了这类流形的内容,也给今后研究和运用拓扑域流形提供一些参考.     

Stochastic inertial manifold

A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic mertial manifold is introduced, defined as a random manifold depending on time, which is finite dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as t → ∞. Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.     

Invariant subspaces in higher order jet prolongations of a fibred manifold

We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.     

A two-dimensional intake manifold flow simulation

A computer flow model of an intake manifold of a four cylinder engine has been developed using a computational fluid dynamic code. This code is base on the Conchas-Spray code developed at the Los Alamos National Laboratory. The flow inside the intake manifold is assumed to be two-dimensional, unsteady, compressible, and turbulent. A simple subgrid scale (SGS) turbulence model and hypothetical boundary conditions are employed in the simulation. Atmospheric pressure is specified at the inlet and the velocities are specified at the outlets of the manifold, together with the law of the wall at all the wall boundaries. Numerical results of the simulation are presented in the form of velocity, pressure, density, and temperature fields. The model is designed in such a way that different manifold geometries may be simulated with ease. A simulation of a concept manifold “the loop-manifold” is also presented.     

Equivariant mappings: a new approach in stochastic simulations

Stochastic simulations on manifolds usually are traced back to ⁿ via charts. If a group G is acting on a manifold M and if the respective distribution v is invariant under this group action then in many cases of practical interest there exists a more convenient approach which uses equivariant mappings. The concept of equivariant mappings will be discussed intensively at the instance of the Grassman manifold in which case G equals the orthogonal group. Further advantages of this concept will be demonstrated by applying it to a probabilistic problem from the field of combinatorial geometry.     

非线性Sobolev Galpern方程的近似惯性流形

近似惯性流形概念与耗散偏微分方程的长时间行为研究有关, 该文对非线性Sobolev Galpern方程构造了两个近似惯性流形. 证明了非平滑近似惯性流形Σ和平滑近似惯性流形Σ_0=P_mH对整体吸引子有相同的逼近阶数.     

Integral manifolds of systems of differential equations with a random right-hand side in a Banach space

The concept of an integral manifold of a system of differential equations with a random right-hand side is introduced. The problem of the existence of an integral manifold of a certain class of differential equations in a Banach space and several of its properties are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 17–21, January, 1992.     

Local geometry of even clifford structures on conformal manifolds

We introduce the concept of a Clifford–Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for some “generic” low-dimensional instances, where explicit examples of non-closed Clifford–Weyl structures can be constructed.     

Generalized Cartan Identities on Isoparametric Manifolds

We first introduce the concept of an isoparametric manifold, which is the natural generalization of the concept of an isoparametric hypersurface in a real space form and that of a space-like isoparametric hypersurface in a Lorentzian space form. Finally, we establish generalized Cartan identities for isoparametric manifolds. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the geodesic curvature of an integral curve of a two-dimensional pfaffian manifold inE 4

The concept of generalized geodesic curvature is introduced along with the invariant mean geodesic curvature vector for a two-dimensional Pfaffian manifold inE ₄, and the indicatrix of generalized geodesic curvature of integral curves of this manifold is constructed.It is proved that the ratio of the fourth quadratic form of the manifold to its first fundamental form is equal to the square of the modulus of the generalized geodesic curvature vector, and an analog of a theorem of Euler is deduced.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 22–26.     

Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so-called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painlevé property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painlevé property. We offer six full examples of how our method works for the six equations, which we believe cover all possible cases.     

Thermodynamics in material space-time

The concept of material forces is well established in the continuum theory of defects. These forces are quantities in a three-dimensional material manifold. The manifold can be augmented to a four-dimensional material space-time by a time-like coordinate. The inelastic phenomena caused by moving defects are highly dissipative. So a thermodynamic theory in material space-time seems to be indispensable. Some first steps towards such a theory are developed here. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Remark on the Reduction of Cauchy–Riemann Manifolds

The concept of a moment map for an action of a real Lie group by CR–diffeomorphisms on a CR–manifold M of hypersurface type is introduced. It gives rise to a natural reduction procedure in the sense that we construct a CR–structure on an associated orbit manifold M₀ for the case that the group acts freely and properly on M.     

An Intrinsic Geometric Formulation of the Equilibrium Equations in Continuum Mechanics

The theory of invariant continuum mechanics is based on the concept that forces and stresses are defined as elements of the cotangent bundle of the configuration manifold. While body and physical space are modeled as differentiable manifolds, the infinite dimensional configuration manifold is given by all configurations of the body in the physical space. In this paper a virtual work principle is postulated which leads together with an induced traction stress and Stokes' theorem directly to the local equilibrium equations and the traction boundary conditions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.     

Conjugate and conformally conjugate parallelisms on Finsler manifolds

In this paper we study conjugate parallelisms and their conformal changes on Finsler manifolds. We provide sufficient conditions for a Finsler manifold endowed with two conjugate (resp. conformally conjugate) covering parallelisms to become a Berwald (resp. Wagner) manifold. As an application for Lie groups, we give a new proof for a theorem of Latifi and Razavi about bi-invariant Finsler functions being Berwald. By introducing the concept of a conformal change of a parallelism, we also obtain a conceptual proof of a theorem of Hashiguchi and Ichijyō: the class of generalized Berwald manifolds is closed under conformal change.     

Spiralling invariant manifolds

A definition of the concept of a multidimensional spiralling manifold is studied. Manifolds of this type are shown to occur as invariant manifolds of flows containing hyperbolic restpoints with a pair of complex conjugate eigenvalues. It is shown that spiralling can be a mechanism for producing intersections of invariant manifolds.     

Variational theory of Riemannian subspaces

If a subspace of a Riemannian manifold is varied the tensors associated with the subspace also vary. The appropriate concept of tensorial variation is introduced and the tensorial variations of several tensors are calculated. A general formula for the variation of an integral is given. Applications are made to the mean curvature integrals of hypersurfaces.     

股票市场模型和不变价格流形

本文首次提出股票市场价格流形的概念 ,并对由维纳过程 (布朗运动 )驱动的市场价格动态扩散过程模型 (X)的不变流形作了初步的讨论 ,给出了一个判定一个给定的价格流形M是市场模型 (X)的不变流形的充分条件 .