Related papers: Topological properties of regular generalized func…
In this review article we present regularity properties of generalized functions which are useful in the analysis of non-linear problems. It is shown that Schwartz distributions embedded into our new spaces of generalized functions, with…
A local existence and uniqueness theorem for ODEs in the special algebra of generalized functions is established, as well as versions including parameters and dependence on initial values in the generalized sense. Finally, a Frobenius…
Several general properties, concerning reduction algebras - rings of definition and algorithmic efficiency of the set of ordering relations - are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra…
Generalized topological spaces in the sense of Cs\'{a}sz\'{a}r have two main features which distinguish them from typical topologies. First, these families of subsets are not closed under intersections. Second, we allow for the possibility…
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…
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…
We determine multiplication and convolution topological algebras for classes of $\omega$-ultradifferentiable functions of Beurling type. Hypocontinuity and discontinuity of the multiplication and convolution mappings are also investigated.
We prove locality of superconformal algebras: every pluperfect superconformal algebra is spanned by coefficients of a finite family of mutually local distributions. We also introduce quasi-Poisson algebras and show that they can be used to…
We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In a previous paper, we introduce the notion of formal manifolds and develop the…
We study continuity, H\"older regularity, and $C^{1,1}$-regularity of geodesics between continuous plurisubharmonic functions on bounded domains of $\mathbb{C}^n$. We then derive regularity properties of rooftop envelopes.
We study seminormalization of affine complex varieties. We show that polynomials on the seminormalization correspond to the rational functions which are continuous for the Euclidean topology. We further study this type of functions which…
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…
The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal…
This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of…
We investigate some properties of regularity of homomorphisms of local algebras over positive characteristic fields. We state a result of monomialization of such a homomorphism between algebras of analytic or algebraic power series. From…
We present recent developments concerning Lorentzian geometry in algebras of generalized functions. These have, in particular, raised a new interest in refined regularity theory for the wave equation on singular space-times.
This work presents a collection of useful properties of the Moreau envelope for finite-dimensional, proper, lower semicontinuous, convex functions. In particular, gauge functions and piecewise cubic functions are investigated and their…
Let $\Omega$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $\Omega$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the…
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar…