Search results for "Differential form"
showing 10 items of 28 documents
Wolfe's theorem for weakly differentiable cochains
2014
Abstract A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in R n with the space of flat m -cochains, that is, the dual space of flat chains of dimension m in R n . The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in R n . A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen–Koskela's concept of upper gradient of a function.
High-quality discretizations for microwave simulations
2016
We apply high-quality discretizations to simulate electromagnetic microwaves. Instead of the vector field presentations, we focus on differential forms and discretize the model in the spatial domain using the discrete exterior calculus. At the discrete level, both the Hodge operators and the time discretization are optimized for time-harmonic simulations. Non-uniform spatial and temporal discretization are applied in problems in which the wavelength is highly-variable and geometry contains sub-wavelength structures. peerReviewed
A star product in lattice gauge theory
1993
Abstract We consider a variant of the cup product of simplicial cochains and its applications in discrete formulations of non-abelian gauge theory. The standard geometrical ingredients in the continuum theory all have natural analogues on a simplicial complex when this star product is used to translate the wedge product of differential forms. Although the star product is non-associative, it is graded-commutative, and the coboundary operator acts as a deviation on the star algebra. As such, it is reminiscent of the star product considered in some approaches to closed string field theory, and we discuss applications to the three dimensional non-abelian Chern-Simons theory.
New insights into black bodies
2012
Planck's law describes the radiation of black bodies. The study of its properties is of special interest, as black bodies are a good description for the behavior of many phenomena. In this work a new mathematical study of Planck's law is performed and new properties of this old acquaintance are obtained. As a result, the exact form for the locus in a color-color diagrams has been deduced, and an analytical formula to determine with precision the black body temperature of an object from any pair of measurements has been developed. Thus, using two images of the same field obtained with different filters, one can compute a fast estimation of black body temperatures for every pixel in the image…
Note on the super-extended Moyal formalism and its BBGKY hierarchy
2017
We consider the path integral associated to the Moyal formalism for quantum mechanics extended to contain higher differential forms by means of Grassmann odd fields. After revisiting some properties of the functional integral associated to the (super-extended) Moyal formalism, we give a convenient functional derivation of the BBGKY hierarchy in this framework. In this case the distribution functions depend also on the Grassmann odd fields.
Star calculus on Jacobi manifolds
2002
Abstract We study the Gerstenhaber bracket on differential forms induced by the two main examples of Jacobi manifolds: contact manifolds and l.c.s. manifolds. Moreover, we obtain explicit expressions of the generating operators and the derivations on the algebra of multivector fields. We define star operators for contact manifolds and l.c.s. manifolds and we study some of its properties.
Extensions of Groups of Gauge Transformations
1989
In this chapter we shall discuss the structure of the infinite-dimensional Lie groups associated to the affine Kac-Moody algebras. We shall also construct the group of the current algebra of a gauge field theory in 3+1 space-time dimensions and we shall study the implications of the commutation relations for the spin-statistics relation in 3+1 dimensions.
An Introduction to Hodge Structures
2015
We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kahler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid’s limiting mixed Hodge structure and Steenbrink’s work in the geometric setting. Fin…
Integration on Surfaces
2012
We intend to study the integration of a differential k-form over a regular k-surface of class C 1 in \(\mathbb{R}^n\). To begin with, in Sect. 7.1, we undertake the integration over a portion of the surface that is contained in a coordinate neighborhood. Where possible, we will express the obtained results in terms of integration of vector fields. For example, we study the integral of a vector field on a portion of a regular surface in \(\mathbb{R}^3\) and also the integral over a portion of a hypersurface in \(\mathbb{R}^n\). In Sect. 7.3 we study the integration of differential k-forms on regular k-surfaces admitting a finite atlas.We discuss the need for the surface to be orientable so t…
The algebraic structure of cohomological field theory
1993
Abstract The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature si…