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Manifold-valued generalized functions in full Colombeau spaces

Michael Kunzinger, Eduard Nigsch (2011)

Commentationes Mathematicae Universitatis Carolinae

We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

Means and generalized means.

Toader, Gheorghe, Toader, Silvia (2007)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Mean-value theorem for vector-valued functions

Janusz Matkowski (2012)

Mathematica Bohemica

For a differentiable function 𝐟 : I k , where I is a real interval and k , a counterpart of the Lagrange mean-value theorem is presented. Necessary and sufficient conditions for the existence of a mean M : I 2 I such that 𝐟 ( x ) - 𝐟 ( y ) = ( x - y ) 𝐟 ' ( M ( x , y ) ) , x , y I , are given. Similar considerations for a theorem accompanying the Lagrange mean-value theorem are presented.

Monotonicity of generalized weighted mean values

Alfred Witkowski (2004)

Colloquium Mathematicae

The author gives a new simple proof of monotonicity of the generalized extended mean values M ( r , s ) = ( ( f s d μ ) / ( f r d μ ) ) 1 / ( s - r ) introduced by F. Qi.

Multifunctions of two variables: examples and counterexamples

Jürgen Appell (1996)

Banach Center Publications

A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.

Multiplication, distributivity and fuzzy-integral. I

Wolfgang Sander, Jens Siedekum (2005)

Kybernetika

The main purpose is the introduction of an integral which covers most of the recent integrals which use fuzzy measures instead of measures. Before we give our framework for a fuzzy integral we motivate and present in a first part structure- and representation theorems for generalized additions and generalized multiplications which are connected by a strong and a weak distributivity law, respectively.

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