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This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space $mathcal{G}[X,Y]$ of Colombeau generalized functions defined on a manifold $X$ and taking values in a manifold $Y$. This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings $mathcal{C}(X,Y)$ into $mathcal{G}[X,Y]$ and study the sheaf properties of $mathcal{G}[X,Y]$. Similar results are obtained for spaces of generalized vector bundle homomorphisms. Based on these constructions we propose the definition of a space $mathcal{D}[X,Y]$ of distributions on $X$ taking values in $Y$. $mathcal{D}[X,Y]$ is realized as a quotient of a certain subspace of $mathcal{G}[X,Y]$.
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordi
In this note we define and study a Hilbert space-valued stochastic integral of operator-valued functions with respect to Hilbert space-valued measures. We show that this integral generalizes the classical Ito stochastic integral of adapted processes
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k
Characterizing the appearance of real-world surfaces is a fundamental problem in multidimensional reflectometry, computer vision and computer graphics. For many applications, appearance is sufficiently well characterized by the bidirectional reflecta
New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established