Related papers: An introduction to generalised functions in period…
We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…
We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the…
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from…
By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal…
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of {\em Colombeau type} in the sense that it contains a copy of the space of Schwartz…
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…
It is proposed the scheme of quantum mechanics, in which a Hilbert space and the linear operators are not primary elements of the theory. Instead of it certain variant of the algebraic approach is considered. The elements of noncommutative…
This is a gentle introduction to Colombeau nonlinear generalized functions, a generalization of the concept of distributions such that distributions can freely be multiplied. It is intended to physicists and applied mathematicians who…
Formulas for the solutions of initial value problems for ordinary differential equations with singular $\delta^{(n)}$-like driving terms are derived in the framework of an algebra of generalized functions (of Colombeau type) over a field of…
The aim of this paper is to prove that the well known non solvable Mizohata type partial differential equations have Colombeau generalized solutions which are distributions if and only if they are solv- able in the space of Schwartz…
The axiomatic formulation of quantum field theory (QFT) of the 1950's in terms of fields defined as operator valued Schwartz distributions is re-examined in the light of subsequent developments. These include, on the physical side, the…
Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a…
The paper addresses linear hyperbolic systems in one space dimension with random field coefficients. In many applications, a low degree of regularity of the paths of the coefficients is required, which is not covered by classical stochastic…
In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz' spaces into the Colombeau algebra G are well known, but for…
This paper is a tutorial that demonstrates various methods from the Colombeau theory of generalized functions in the context of semilinear wave equations. The Colombeau generalized functions constitute differential algebras that contain the…
We investigate homogeneity in the special Colombeau algebra. It is shown that strongly scaling invariant functions on the d-dimensional space are simply the constants. On the pierced space, strongly homogeneous functions admit tempered…
We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with…
We describe a scheme of quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory. As primary structures we consider observables, elements of noncommutative algebra, and the physical…
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics…
A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces…