arXiv math/01042692001-04math.FAmath-phmath.MP

Diffeomorphism invariant Colombeau Algebras. Part I: Local theory

Roland Steinbauer

备注：: Latex, 11 pages, Contribution to Proceedings of the International Conference on Generalized Functions (ICGF 2000, Guadeloupe); typos corrected

英文摘要：

This contribution is the first in a series of three: it reports on the construction of (a fine sheaf of) diffeomorphism invariant Colombeau algebras on open sets of Eucildean space, which completes earlier approaches. Part II and III will show, among others, the way to an intrinsic definition of Colombeau algebras on manifolds which, locally, reproduces the local diffeomorphism invariant theory.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Diffeomorphism invariant Colombeau algebras. Part II: Classification

This contribution presents a comprehensive analysis of Colombeau (-type) algebras in the range between the diffeomorphism invariant algebra introduced in Part I and Colombeau's original algebra. Along the way, it provides several classification results which are indispensable for obtaining an intrinsic description of a (full) Colombeau algebra on a manifold. The latter will be the focus of Part III of this series of contributions.

2001-04-27math-phmath.FAmath.MP

Diffeomorphism invariant Colombeau algebras. Part III: Global theory

We present the construction of an associative, commutative algebra $\hat {\mathcal G}$ of generalized functions on a manifold $X$ satisfying the following optimal set of permanence properties: (i)The space of distributions on $X$ is linearly embedded into $\hat {\mathcal G}$, $f(p)\equiv 1$ is the unity in the algebra. (ii) For every smooth vector field $ξ$ on $X$ there exists a derivation operator $\hat L_ξ: \hat {\mathcal G} \to \hat {\mathcal G}$ which is linear and satisfies the Leibniz rule. (iii) $L_ξ$ restricted to the space of distributions on $X$ is the usual Lie derivative. (iv) Mult…

2001-04-27math-phmath.FAmath.MP

On the foundations of nonlinear generalized functions II

This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra $\mathcal{G}^d = \mathcal{E}_M/\mathcal{N}$ introduced in part I and Colombeau's original algebra $\mathcal{G}^e$. Three main results are established: First, a simple criterion describing membership in $\mathcal{N}$ (applicable to all types of Colombeau algebras) is given. Second, two counterexamples demonstrate that $\mathcal{G}^d$ is not injectively included in $\mathcal{G}^e$. Finally, it is shown that in the range ``between'' $\mat…

1999-12-28math-phmath.FAmath.MP

Point value characterizations and related results in the full Colombeau algebras G^e and G^d

We present a point value characterization for elements of the elementary full Colombeau algebra G^e and the diffeomorphism invariant full Colombeau algebra G^d. Moreover, several results from the special algebra G^s about generalized numbers and invertibility are extended to the elementary full algebra.

2011-04-05math.FA

Full and special Colombeau algebras

We introduce full diffeomorphism-invariant Colombeau algebras with added $\varepsilone$-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.

2016-11-18math.FA

A geometric approach to full Colombeau algebras

We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view on the construction of the intrinsically defined algebra $\hat{\mathcal G}(M)$ on the manifold $M$ given in (Grosser et al., Adv. Math. 166 (2002), 179--206).

2008-09-17math.FA

更多相关论文收起更多论文

The category of Colombeau algebras

In [11], we introduced the notion of asymptotic gauge (AG), and we used it to construct Colombeau AG-algebras. This construction concurrently generalizes that of many different algebras used in Colombeau's theory, e.g. the special one $\mathcal{G}^{\srm}$, the full one $\gse$, the NSA based algebra of asymptotic functions $\hat{\mathcal{G}}$, and the diffeomorphism invariant algebras $\gsd$, $\mathcal{G}^{2}$ and $\hat{\mathcal{G}}$. In this paper we study the categorical properties of the construction of Colombeau AG-algebras with respect to the choice of the AG, and we show their consequence…

2015-07-09math.FA

A new approach to diffeomorphism invariant algebras of generalized functions

We develop the diffeomorphism invariant Colombeau-type algebra of nonlinear generalized functions in a modern and compact way. Using a unifying formalism for the local setting and on manifolds, the construction becomes simpler and more accessible than previously in the literature.

2013-03-13math.FA

Group invariant Colombeau generalized functions

Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of rotations or the Lorentz group. Further, a one-dimensional Colombeau generalized function with two (real) periods is shown to be a generalized constant, when the ratio of the periods is an algebraic nonrational number. Finally, a nonstandard Colombeau generalized function invariant under standard translations is shown to be constant.

2006-11-16math.APmath.FA

Global algebras of nonlinear generalized functions with applications in general relativity

We give an overview of the development of algebras of generalized functions in the sense of Colombeau and recent advances concerning diffeomorphism invariant global algebras of generalized functions and tensor fields. We furthermore provide a survey on possible applications in general relativity in light of the limitations of distribution theory.

2013-10-22math-phmath.FAmath.MP

On the foundations of nonlinear generalized functions I

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given.

1999-12-28math-phmath.FAmath.MP

Group invariants in algebras of generalized functions

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given.

2005-04-04math.FA

Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its C^{1+bv} conjugacy class if and only if it is the time-1 map of a C^1 vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval interval diffeom…

2021-01-04math.DSmath.FAmath.GR

Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations

The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the Chern-Simons path integral. Certain measures of this sort also play the role of states in quantum gravity in Ashtekar's formalism. These measures define link invariants, or more generally multiloop invariants; as noted by Witten, the Chern-Simons path integral gives rise to the Jones polynomial, while in quantum gr…

1993-05-11gr-qchep-thmath.FA

On the extremality of Hofer's metric on the group of Hamiltonian diffeomorphisms

Let M be a closed symplectic manifold, and let | | be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if | | is dominated from above by the L-Infinity-norm, and | | is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume preserving diffeomorphisms. We also prove that if | | is, additionally, not equivalent to the L-Infinity-norm, then the induced Finsler metric on the group of Hamiltonian diffeomorphisms on M vanishes identically.

2005-01-10math.FAmath.SG