118 research outputs found
Multiplication of Distributions and Nonperturbative Calculations of Transition Probabilities
In a mathematical context in which one can multiply distributions the
"`formal"' nonperturbative canonical Hamiltonian formalism in Quantum Field
Theory makes sense mathematically, which can be understood a priori from the
fact the so called "`infinite quantities"' make sense unambiguously (but are
not classical real numbers). The perturbation series does not make sense. A
novelty appears when one starts to compute the transition probabilities. The
transition probabilities have to be computed in a nonperturbative way which, at
least in simplified mathematical examples (even those looking like
nonrenormalizable series), gives real values between 0 and 1 capable to
represent probabilities. However these calculations should be done numerically
and we have only been able to compute simplified mathematical examples due to
the fact these calculations appear very demanding in the physically significant
situation with an infinite dimensional Fock space and the QFT operators
An algebraic approach to manifold-valued generalized functions
We discuss the nature of structure-preserving maps of varies function
algebras. In particular, we identify isomorphisms between special Colombeau
algebras on manifolds with invertible manifold-valued generalized functions in
the case of smooth parametrization. As a consequence, and to underline the
consistency and validity of this approach, we see that this generalized version
on algebra isomorphisms in turn implies the classical result on algebras of
smooth functions.Comment: 7 page
Hadamard Regularization
Motivated by the problem of the dynamics of point-particles in high
post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a
certain class of functions which are smooth except at some isolated points
around which they admit a power-like singular expansion. We review the concepts
of (i) Hadamard ``partie finie'' of such functions at the location of singular
points, (ii) the partie finie of their divergent integral. We present and
investigate different expressions, useful in applications, for the latter
partie finie. To each singular function, we associate a partie-finie (Pf)
pseudo-function. The multiplication of pseudo-functions is defined by the
ordinary (pointwise) product. We construct a delta-pseudo-function on the class
of singular functions, which reduces to the usual notion of Dirac distribution
when applied on smooth functions with compact support. We introduce and analyse
a new derivative operator acting on pseudo-functions, and generalizing, in this
context, the Schwartz distributional derivative. This operator is uniquely
defined up to an arbitrary numerical constant. Time derivatives and partial
derivatives with respect to the singular points are also investigated. In the
course of the paper, all the formulas needed in the application to the physical
problem are derived.Comment: 50 pages, to appear in Journal of Mathematical Physic
On the completeness of impulsive gravitational wave space-times
We consider a class of impulsive gravitational wave space-times, which
generalize impulsive pp-waves. They are of the form ,
where is a Riemannian manifold of arbitrary dimension and carries
the line element with the line
element of and the Dirac measure. We prove a completeness result
for such space-times with complete Riemannian part .Comment: 13 pages, minor changes suggested by the referee
Regularity properties of distributions through sequences of functions
We give necessary and sufficient criteria for a distribution to be smooth or
uniformly H\"{o}lder continuous in terms of approximation sequences by smooth
functions; in particular, in terms of those arising as regularizations
.Comment: 10 page
On the Geroch-Traschen class of metrics
We compare two approaches to semi-Riemannian metrics of low regularity. The maximally 'reasonable' distributional setting of Geroch and Traschen is shown to be consistently contained in the more general setting of nonlinear distributional geometry in the sense of Colombea
An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms
We offer an axiomatic definition of a differential algebra of generalized
functions over an algebraically closed non-Archimedean field. This algebra is
of Colombeau type in the sense that it contains a copy of the space of Schwartz
distributions. We study the uniqueness of the objects we define and the
consistency of our axioms. Next, we identify an inconsistency in the
conventional Laplace transform theory. As an application we offer a free of
contradictions alternative in the framework of our algebra of generalized
functions. The article is aimed at mathematicians, physicists and engineers who
are interested in the non-linear theory of generalized functions, but who are
not necessarily familiar with the original Colombeau theory. We assume,
however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page
Use of specific Green's functions for solving direct problems involving a heterogeneous rigid frame porous medium slab solicited by acoustic waves
A domain integral method employing a specific Green's function (i.e.,
incorporating some features of the global problem of wave propagation in an
inhomogeneous medium) is developed for solving direct and inverse scattering
problems relative to slab-like macroscopically inhomogeneous porous obstacles.
It is shown how to numerically solve such problems, involving both
spatially-varying density and compressibility, by means of an iterative scheme
initialized with a Born approximation. A numerical solution is obtained for a
canonical problem involving a two-layer slab.Comment: submitted to Math.Meth.Appl.Sc
Singular Modes of the Electromagnetic Field
We show that the mode corresponding to the point of essential spectrum of the
electromagnetic scattering operator is a vector-valued distribution
representing the square root of the three-dimensional Dirac's delta function.
An explicit expression for this singular mode in terms of the Weyl sequence is
provided and analyzed. An essential resonance thus leads to a perfect
localization (confinement) of the electromagnetic field, which in practice,
however, may result in complete absorption.Comment: 14 pages, no figure
Isomorphisms of algebras of Colombeau generalized functions
We show that for smooth manifolds X and Y, any isomorphism between the
special algebra of Colombeau generalized functions on X, resp. Y is given by
composition with a unique Colombeau generalized function from Y to X. We also
identify the multiplicative linear functionals from the special algebra of
Colombeau generalized functions on X to the ring of Colombeau generalized
numbers. Up to multiplication with an idempotent generalized number, they are
given by an evaluation map at a compactly supported generalized point on X.Comment: 10 page
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