Showing papers on "Distribution (differential geometry) published in 2004"

Study of a discrete chaotic system based on tangent-delay for elliptic reflecting cavity and its properties

Abstract: Based on the physical model of ellipse reflecting cavity, the tangent-delay operation is proposed to change the evolution route of the systems , and a new class of discrete chaotic map systems is deduced based on the tangen t-delay operation. Simulation experiments show that the discrete chaotic systems have many special properties such as the maximum Lyapunov exponent is over zero , unchangeable equiprobability distribution and zero correlation in total field, there exists a square chaotic attractor when tangent delays one unit, and becom e ergodic state when tangent delays more units than one. The discrete chaotic sy stems can generate 2 independent pseudo-random sequences together. All of the pr operties suggest that the class of chaos systems possesses the potential applica tion in encryption.

Distribution laws for integrable eigenfunctions

Abstract: We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.

Screen conformal half-lightlike submanifolds

Abstract: We study some properties of a half-lightlike submanifold M, of a semi-Riemannian manifold, whose shape operator is conformal to the shape operator of its screen distribution. We show that any screen distribution S(TM) of M is integrable and the geometry of M has a close relation with the nondegenerate geometry of a leaf of S(TM) . We prove some results on symmetric induced Ricci tensor and null sectional curvature of this class.

Fundamental form and Cartan's tensor of (2,5)-distributions coincide

Abstract: In our previous paper (see this arxiv math.DG/0402171) for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E.Cartan in 1910, using his ``reduction- prolongation'' procedure (we call this form Cartan's tensor). In the present paper we prove that our fundamental form coincides (up to constant factor -35) with Cartan's tensor. This result explains geometric reason for existence of Cartan's tensor (originally this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for Cartan's tensor in terms of structural functions of any frame naturally adapted to the distribution.

Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors

Abstract: Goursat distributions are subbundles, of codimension at least 2, in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing—very slowly—always by 1. The length of a flag thus equals the corank of the underlying distribution.

Nonholonomic Systems and the Geometry of Constraints.

Abstract: In a recent paper [9] we analyze conservation of volume for a series of examples of mechanical systems with linear, affine and non linear constraints aiming to make evident some geometric aspects related with them. Here, we only consider examples with linear constraints (defined by a constant rank distribution), in which we have conservation of volume. Conservation of volume means, equivalently, that the orthogonal distribution (the metric is defined by the kinetic energy) is minimal (see [15]) and so, if it is integrable, the corresponding foliation has minimal leaves. Properties of the falling penny and of the vertical disc rolling on a horizontal plane without slipping are very special. A dynamically symmetric sphere that rolls without slipping on a given surfaceS⊂ℝ3 conserves volume, and the orthogonal distribution is integrable if, and only if,S isparallel to a surface with a fixed constant mean curvature. Semi-simple Lie groups endowed with suitable metrics have foliations with minimal leaves. Geometric questions related with the kinematics of the rolling motion of two surfaces are also considered.

Furnace cooling system and method

Abstract: A cooling system for the distribution and collection of a fluid coolant in a metallurgical vessel used in the processing of molten materials. The cooling system comprises a distribution system including an intake manifold, a plurality of headers attached to the intake manifold, and a plurality of distribution dispensers positioned along each header. A collection system, including a collection manifold, is positioned to collect the fluid coolant. The distribution dispensers are positioned to direct the fluid coolant towards the collection manifold and utilize the majority of the kinetic energy contained within the coolant to direct the coolant towards the collection manifold.

Generic integral manifolds for weight two period domains

Abstract: We define the notion of a generic integral element for the Griffiths distribution on a weight two period domain, draw the analogy with the classical contact distribution, and then show how to explicitly construct an infinite-dimensional family of integral manifolds tangent to a given element.

Coisotropic Submanifolds of a Semi Riemannian Manifold

Abstract: In this paper, we study coisotropic submanifolds of a semi-Riemannian manifold. We investigate the integrability condition of the screen distribution and give a necessary and sucient condition on Ricci tensor of a coisotropic submanifold to be symmetric. Finally, we present some new theorems and results about totally umbilical coisotropic submanifolds of a semi-Riemannian manifold.

Integrals of equivariant forms over non-compact symplectic manifolds

Abstract: This article is a result of the AIM workshop on Moment Maps and Surjectivity in Various Geometries (August 9 - 13, 2004) organized by T.Holm, E.Lerman and S.Tolman. At that workshop I was introduced to the work of T.Hausel and N.Proudfoot on hyperkahler quotients [HP]. One interesting feature of their article is that they consider integrals of equivariant forms over non-compact symplectic manifolds which do not converge, so they formally {\em define} these integrals as sums over the zeroes of vector fields, as in the Berline-Vergne localization formula. In this article we introduce a geometric-analytic regularization technique which makes such integrals converge and utilizes the symplectic structure of the manifold. We also prove that the Berline-Vergne localization formula holds for these integrals as well. The key step here is to redefine the collection of integrals \int_M alpha(X), X \in g, as a distribution on the Lie algebra g. We expect our regularization technique to generalize to non-compact group actions extending the results of [L1,L2].

Correspondence spaces and twistor spaces

Abstract: For a semisimple Lie group G with parabolic subgroups QPG, we associate to a parabolic geometry of type (G, P) on a smooth manifold N the correspondence space CN, which is the total space of a fiber bundle over N with fiber a generalized flag manifold, and construct a canonical parabolic geometry of type (G, Q) on CN. Conversely, for a parabolic geometry of type (G, Q) on a smooth manifold M, we construct a distribution corresponding to P, and find the exact conditions for its integrability. If these conditions are satisfied, then we define the twistor space N as a local leaf space of the corre- sponding foliation. We find equivalent conditions for the existence of a parabolic geometry of type (G, P) on the twistor space N such that M is locally isomorphic to the correspondence space CN, thus obtaining a complete local characterization of correspondence spaces. We show that all these constructions preserve the subclass of normal parabolic geometries (which are determined by some underlying geomet- ric structure) and that in the regular normal case, all characterizations can be expressed in terms of the harmonic curvature of the Cartan con- nection, which is easier to handle. Several examples and applications are discussed.

Constructing a Brownian Sheet with Values in a Compact Riemannian Manifold

Abstract: In this paper, we propose a new method of constructing a two-parameter random field W x (s, t ), x ∈ M , with values in a compact Riemannian manifold M possessing the property that the random processes W x ( · ,t )a ndW x M (s, · ) are Brownian motions on the manifold M with parameters t and s , respectively, issuing from the point x .( By aBrownian motion on a manifold M with parameter t we mean the diffusion process generated by the operator −(t/2)∆M , where ∆M is the Laplace operator on the manifold M.) For the case in which the manifold is a compact Lie group, the two-parameter random field constructed in the paper coincides with the Brownian sheet defined by Malliavin (1) in 1991. (Malliavin called this random field a Brownian motion with values in C((0, 1) ,M ) , which is the set of continuous functions defined on the closed interval (0, 1) and taking values in M.) Nevertheless, for the case in which the manifold is a compact Lie group, the method proposed in the present paper essentially differs from that used in Malliavin's paper. 1. FIRST STEP IN THE CONSTRUCTION OF THE RANDOM FIELD W x Suppose that M is a d-dimensional compact Riemannian manifold without boundary isomet- rically embedded in R m .B y aBrownian sheet with values in R m we mean the family of m independent standard Brownian sheets. Suppose that Wt,s is an n-dimensional Brownian sheet. Consider Wt,s as a process taking values in the space C((0, 1), R m ) . We denote this process by the symbol Wt . We introduce the following notation: if E is a locally convex space, then E t denotes C((0 ,t ) ,E ); if y ∈ C((0, 1), R m ) is a continuous function, then W y denotes the distribution of the process W y = y +Wt .I fψ ∈ C((0, 1), R m ) , then we define the process (W y )t = ψ(t )+ W y . Suppose that W y is the distribution of this process and E y,ψ is the expectation with respect to the measure W y . Further, Ue(M ) denotes the e -neighborhood of the manifold M. We consider W y for functions y and ψ satisfying the conditions: y(0) ∈ M , ψ(0) = 0 . The goal of this section is to prove the existence of a limit (given below) with respect to the family of bounded continu- ous cylindrical functions, where by a cylindrical function C((0, 1) × (0, 1), R m ) → R we mean a

On the Integrability of Orthogonal Distributions in Poisson Manifolds

Abstract: We study conditions for the integrability of the distribution defined on a regular Poisson manifold as the orthogonal complement (with respect to some (pseudo)-Riemannian metric) to the tangent spaces of the leaves of a symplectic foliation. Examples of integrability and non-integrability of this distribution are provided.

Efficient construction of contact coordinates for partial prolongations

Abstract: Let $\CV$ be a vector field distribution on manifold $M$. We give an efficient algorithm for the construction of local coordinates on $M$ such that $\CV$ may be locally expressed as some partial prolongation of the contact distribution $\Cal C^{(1)}_q$, on the first order jet bundle of maps from $\Bbb R$ to $\Bbb R^q$, $q\geq 1$. It is proven that if $\CV$ is locally equivalent to a partial prolongation of $\Cal C^{(1)}_q$ then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on $M$. The number of these first integrals that must be computed satisfies a natural minimality criterion. These results therefore provide a full and constructive generalisation of the classical Goursat normal form from the theory of exterior differential systems.