Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Nullity and generalized characteristic classes of differential manifolds

Abstract: Using the Kamber-Tondeur construction of characteristic classes for foliated bundes, the author has given a method for constructing generalized characteristic classes for a differentiable manifold M without imposing conditions on M. In particular a vanishing theorem on the manifold M is obtained. The construction is particularly useful if the ordinary characteristic ring Pont*(M) of the manifold M vanishes much below the dimension of M. 1. Introduction. The theory of generalized (ordinary and secondary) characteristic classes of a manifold M which is associated with a certain geometric structure is a recent achievement in differential geometry. In contrast to the ordinary characteristic classes, the newly constructed secondary classes carry with them underlying geometric structures. The construction of secondary classes began in a series of papers by Bott (1), Bott and Haefliger (2), Chern and Simons (5), Godbillon and Vey (6), Kamber and Tondeur (7), (8) and others. The idea of the construction is as follows, namely, if the characteristic form f(K) is zero, then the transgression Tf(K) of f(K) defines a secondary class in the principal fibre bundle (5); here / is an invariant polynomial and K is the curvature form. Kamber and Tondeur found the usefulness of the Weil algebra W{G) of a Lie group G in this context. They showed that for a foliated bundle (P, E, {«}) (7), Bott's vanishing theorem (1), when interpreted in the Weil homomorphism (3), k(w): W(G)->A(P), gave rise to a new generalized characteristic homomorphism A,: H(W(G, H)q) -» H(M, R) where E denotes a foliation on the manifold M of codimension q, {co} is the family of adapted connections and W(G, H)q denotes the truncated relative Weil algebra (7), (8). We are able to define generalized characteristic classes on an affine (or Riemannian) manifold M using the Cartan-Kamber-Tondeur map At. We can do this by observing that the affine {or Riemannian) manifold is already foliated by its nullity distribution (§2). This construction is particularly useful if

Distribution of Characteristic Roots and Exponential Stability of Singular Neutral Differential Delay Systems

Abstract: This paper firstly discussed the distribution of characteristic roots of two types of representative linear autonomy neutral delay differential systems,then given the fitting curve of appromate characteristic roots about the characteristic roots.Finally,we focused on the distribution of characteristic roots of singular neutral differential delay systems.Under a condition,we can obtain the conditions of exponential stability of singular neutral differential differential delay systems.

Rolling on Affine Tangent Planes: Parallel Transport and the Associated Sub-Riemannian Problems

Abstract: This paper addresses the natural optimal control problem associated with the kinematic equations generated by the rollings of symmetric Riemannian spaces on their affine tangent spaces at a fixed point. This optimal problem can be viewed as a certain left-invariant sub-Riemannian problem on a Lie group G associated with a three-step distribution \(\mathcal {F}\) on the Lie algebra \(\mathfrak {g}\) of G. We will use the Maximum Principle of optimality to obtain the appropriate Hamiltonian, and then we will show that the corresponding Hamiltonian system of equations admits an isospectral representation, and hence, is completely integrable. As a byproduct, this representation reveals intriguing connections with mechanical tops, while at the same time, it sheds additional light on the discovery in [7] that the elastic curves, elasticate in Euler’s terminology, can be obtained entirely by the rolling sphere problems on spaces of constant curvature.

Constructing the External Contour of the Frankl Nozzle Using S-Shaped Curves with Quadratic Distribution of the Curvature

Abstract: A mathematical model, an algorithm, and a software are developed for the problem of constructing an S-shaped curve passing through two given points with specified tangent inclination angles and providing a tangent inclination angle at a point with a given abscissa. To control the inflection point of the S-shaped curve with quadratic distribution of curvature in natural parameterization, the tangent inclination angle at the point with the known abscissa is used. The algorithm is based on the modification of the method with space dilation in the direction of the difference of two successive generalized gradients. Computational experiments have shown the efficiency of the developed algorithm for constructing the external contour of a Frankl-type nozzle.

Adaptive domain selection algorithm for manifold learning based on curvature prediction

Abstract: The invention discloses an adaptive domain selection algorithm for manifold learning based on curvature prediction, relates to the adaptive domain selection algorithm applied to manifold learning, and solves the problems of poor adaptability, poor low-dimensional embedding quality and higher algorithm complexity in the application of the conventional domain selection algorithm to manifold learning. The adaptive domain selection algorithm comprises the following concrete steps: 1, calculating the curvature of a high-dimensional discrete data point; and 2, carrying out adaptive domain selection. The adaptive domain selection algorithm can be widely applied to the conventional manifold learning algorithm, can be used for selecting a proper domain size according to different curvatures of data set distribution, and has the effects of effectively lowering the complexity of the manifold learning algorithm and finding the optimized domain size for achieving the optimal low-dimensional embedding with good quality. The adaptive domain selection algorithm is applied to the manifold learning algorithm.