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The aim of this paper is to give a new description of the geometry appearing in the multi-specialization along a general family of submanifolds of a real analytic manifold (including some important cases as clean intersection or a…
For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [20]. This result…
For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct non-large vector subspaces in the rational homology of finitely generated groups. The functorial properties of this…
Modelling of singularities given by discontinuous functions or distributions by means of generalized functions has proved useful in many problems posed by physical phenomena. We introduce in a systematic way generalized functions of…
As a measure for the centrality of a point in a set of multivariate data, statistical depth functions play important roles in multivariate analysis, because one may conveniently construct descriptive as well as inferential procedures…
We extend calculus from smooth manifolds to topological manifolds making use of a theory of generalized functions developed for this aim. Actually such extension fits into a boarder context: the universal construction of a site containing…
In this paper, we introduce the concept of nearly convex set-valued mappings and investigate fundamental properties of these mappings. Additionally, we establish a geometric approach for generalized differentiation of nearly convex…
We extend the functional analytic approach to Colombeau-type spaces of nonlinear generalized functions in order to study algebras of tempered generalized functions. We obtain a definition of Fourier transform of nonlinear generalized…
We propose a new framework for the study of homological properties for (compactly generated) triangulated categories such as regularity, finiteness of global or finitistic dimension, gorensteinness or injective generation and the relation…
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…
Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in…
Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical…
This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which…
In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the…
We compute bilinear integrals involving Macdonald and Gegenbauer functions. These integrals are convergent only for a limited range of parameters. However, when one uses generalized integrals they can be computed essentially without…
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…
This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric…
In this paper, we generalize the principle of the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Namely, we introduce endofunctors over a functor category that encodes…
Gel'fand triples of test and generalized functionals in Gaussian spaces are constructed and characterized.
We survey foundational principles of Grothendieck's generalized spaces, including a critical glossary of the various, and often conflicting, terminological usages. Known results using generalized points support a fully pointwise notation…