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.

Generalized Loss-Sensitive Adversarial Learning with Manifold Margins

Abstract: The classic Generative Adversarial Net and its variants can be roughly categorized into two large families: the unregularized versus regularized GANs. By relaxing the non-parametric assumption on the discriminator in the classic GAN, the regularized GANs have better generalization ability to produce new samples drawn from the real distribution. It is well known that the real data like natural images are not uniformly distributed over the whole data space. Instead, they are often restricted to a low-dimensional manifold of the ambient space. Such a manifold assumption suggests the distance over the manifold should be a better measure to characterize the distinct between real and fake samples. Thus, we define a pullback operator to map samples back to their data manifold, and a manifold margin is defined as the distance between the pullback representations to distinguish between real and fake samples and learn the optimal generators. We justify the effectiveness of the proposed model both theoretically and empirically.

Goursat Flags: Classification of Codimension-One Singularities

Abstract: A distribution D of corank r \ge 2 on a manifold M is Goursat when its Lie square [D, D] is a distribution of constant corank r-1, the Lie square of [D, D] is of constant corank r-2 and so on. Any such D, according to von Weber [21] and E. Cartan [3], behaves in a well-known way at generic points of M: in certain local coordinates it is the chained model (C) given below, a classical object in the control theory. Singularities concealed in Goursat distributions have emerged for the first time in [8]; by now the complete local classification of these objects of coranks not exceeding 7 is known, plus some isolated facts for coranks 8, 9, and 10. In the present paper we deal with the Goursat distributions of any corank r and obtain a complete classification of the first occurring singularities of them, located at points outside a stratified codimension-2 submanifold of M. Off this set there are (on top of (C)) only r-2 non-equivalent singular behaviours possible.

Smooth distributions are finitely generated

Abstract: A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.

Geometric nonlinear thermoelasticity and the time evolution of thermal stresses

Abstract: In this paper we formulate a geometric theory of nonlinear thermoelasticity that can be used to calculate the time evolution of temperature and thermal stress fields in a nonlinear elastic body. In particular, this formulation can be used to calculate residual thermal stresses. In this theory the material manifold (natural stress-free configuration of the body) is a Riemannian manifold with a temperature-dependent metric. Evolution of the geometry of the material manifold is governed by a generalized heat equation. As examples, we consider an infinitely long circular cylindrical bar with a cylindrically symmetric temperature distribution and a spherical ball with a spherically-symmetric temperature distribution. In both cases we assume that the body is made of an arbitrary incompressible isotropic solid. We numerically solve for the evolution of thermal stress fields induced by thermal inclusions in both a cylindrical bar and a spherical ball, and compare the linear and nonlinear solutions for a generaliz...

Generalised connections over a vector bundle map

Abstract: A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is `parametrised' in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.