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.

Normal-Bundle Bootstrap

Abstract: Probabilistic models of data sets often exhibit salient geometric structure. Such a phenomenon is summed up in the manifold distribution hypothesis and can be exploited in probabilistic learning. H...

Crystalline Solids with a Uniform Distribution of Dislocations

Abstract: In this thesis, we investigate the problem of a crystalline solid containing a uniform distribution of elementary dislocations In the first part, we introduce the basic concepts of a material manifold describing the intrinsic properties of a solid and a crystalline structure similar to that of a Lie algebra We define the dislocation density as the continuum limit of a distribution of elementary dislocations in a crystal lattice It is represented by a Burger’s vector In the general case, the dynamics are then described by mappings from space-time to the material manifold In the static case, the Euler-Lagrange equations – a system of elliptic partial differential equations expressing the divergence freeness of the stress tensor – are derived in the most general form We complete the boundary value problem by adding the proper boundary conditions for a mapping from the material manifold into Euclidean space (static spacetime) They reflect the vanishing of the external forces on the boundary of the solid The interchange of the roles of the domain and target space prevents us from solving a free boundary problem Moreover, we state the Legendre-Hadamard conditions to justify the choice of the energy density These conditions are given in terms of the derivatives of the energy with respect to the thermodynamic configuration, a symmetric bilinear form on the crystalline structure The second part ends with a discussion of the equivalences for both the crystalline structures and the mechanics of a solid In the third part, which constitutes the core part of the thesis, we study the special cases of uniform distributions of the two elementary types of dislocations, edge and screw dislocations in two and three space dimensions, respectively It turns out that the material manifold can then be given the structure of a (non-Abelian) Lie group, the corresponding Lie algebra representing the crystalline structure In the first case, upon choosing an isotropic energy density, the problem reduces to the study of energy minimizing mappings from a domain in the hyperbolic plane to Euclidean space We show that the associated boundary value problem has a unique solution up to rotations and translations This result is achieved by an iteration scheme, in which we choose a parameter reflecting the curvature sufficiently small, followed by a scaling argument In the second case, however, the isotropy is broken by the dislocation lines, and we are left with mappings from the Heisenberg group manifold to Euclidean space Using an anisotropic type of energy, we give the strategy to solve the problem, which, in contrast to the two-dimensional case, is not purely Riemannian

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.

Manifold and water distribution system

Abstract: A distribution manifold 10 for domestic water and heating systems comprises a first inlet port 12, at least one first outlet port 16 and a first distribution line 20. The manifold 10 also comprises a second inlet port 14, at least one second outlet port 18 and a second distribution line 22. The first distribution line 20 and second distribution line 22 define separate flowpaths through the manifold 10 and the second distribution line 22 lies between all of the outlet ports 16, 18, and the first distribution line 20. Later embodiments relate to a modular manifold assembly (102, 112, 114 figures 4, 5 & 6) and a manifold pivotable from an operation position to an installation/servicing position (figures 2 & 3).