Abstract:
Let $M$ be a Riemannian manifold and ${\mathcal P}M$ be the space of all smooth paths on $M$. We describe geodesics on path space ${\mathcal P}M$. Normal neighbourhood structure on ${\mathcal P}M$ has been discussed. We identify paths on $M$ under "back-track" equivalence. Under this identification we show that if $M$ is complete, then geodesics on path space yield a double category.We gave a physical interpretation of this double category.

Abstract:
We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category. 1. Introduction and Geometric Background The interaction of point particles through a gauge field can be encoded by means of Feynman diagrams, with nodes representing particles and directed edges carrying an element of the gauge group representing parallel transport along that edge. If the point particles are replaced by extended one-dimensional string-like objects, then the interaction between such objects can be encoded through diagrams of the form (1) where the labels and describe classical parallel transport and , which may take values in a different gauge group, describes parallel transport over a space of paths. We will now give a rapid account of some of the geometric background. We refer to our previous work [1] for further details. This material is not logically necessary for reading the rest of this paper but is presented to indicate the context and motivation for some of the ideas of this paper. Consider a principal -bundle , where is a smooth finite dimensional manifold and a Lie group, and a connection on this bundle. In the physical context, may be spacetime, and describes a gauge field. Now consider the set of piecewise smooth paths on , equipped with a suitable smooth structure. Then, the space of -horizontal paths in forms a principal -bundle over . We also use a second gauge group (that governs parallel transport over path space), which is a Lie group along with a fixed smooth homomorphism and a smooth map such that each is an automorphism of , such that for all and . We denote the derivative by , viewed as a map , and denote by , to avoid notational complexity. Given also a second connection form on and a smooth -equivariant vertical -valued -form on , it is possible to construct a connection form on the bundle where is the -valued -form on specified by which is a Chen integral. Consider a path of paths in specified through a smooth map where each is -horizontal and the path is -horizontal. Let . The bi-holonomy is specified as follows: parallel translate along by , then up the path by , back along -reversed by and then down by , then the resulting point is The following result is proved in [1]. Theorem 1. Suppose that is smooth, with each being -horizontal and the path being -horizontal. Then, the parallel

Abstract:
Compressed sensing is a thriving research field covering a class of problems where a large sparse signal is reconstructed from a few random measurements. In the presence of several sensor nodes measuring correlated sparse signals, improvements in terms of recovery quality or the requirement for a fewer number of local measurements can be expected if the nodes cooperate. In this paper, we provide an overview of the current literature regarding distributed compressed sensing; in particular, we discuss aspects of network topologies, signal models and recovery algorithms.

Abstract:
We develop parallel transport on path spaces from a differential geometric approach, whose integral version connects with the category theoretic approach. In the framework of 2-connections, our approach leads to further development of higher gauge theory, where end points of the path need not be fixed.

Abstract:
We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces.

Abstract:
The generalized vector is defined on an $n$ dimensional manifold. Interior product, Lie derivative acting on generalized $p$-forms, $-1\le p\le n$ are introduced. Generalized commutator of two generalized vectors are defined. Adding a correction term to Cartan's formula the generalized Lie derivative's action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application.

Abstract:
We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework.

Abstract:
We study a type of connection forms, given by Chen integrals, over pathspaces by placing such forms within a category-theoretic framework of principal bundles and connections. We introduce a notion of 'decorated' principal bundles, develop parallel transport on such bundles, and explore specific examples in the context of pathspaces.

Abstract:
We consider the problem of recovering an $N$-dimensional sparse vector $\vm{x}$ from its linear transformation $\vm{y}=\vm{D} \vm{x}$ of $M(< N)$ dimension. Minimizing the $l_{1}$-norm of $\vm{x}$ under the constraint $\vm{y} = \vm{D} \vm{x}$ is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful $l_1$-recovery is universal over a variety of randomly constructed matrices $\vm{D}$. For examining the extent of the universality, we focus on the case in which $\vm{D}$ is provided by concatenating $\nb=N/M$ matrices $\vm{O}_{1}, \vm{O}_{2},..., \vm{O}_\nb$ drawn uniformly according to the Haar measure on the $M \times M$ orthogonal matrices. By using the replica method in conjunction with the development of an integral formula for handling the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than what the universality predicts when the density of non-zero signals is not uniform among the $\nb$ matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.