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A concept of absolute continuity and a Riemann type integral

B. Bongiorno, Washek Frank Pfeffer (1992)

Commentationes Mathematicae Universitatis Carolinae

We present a descriptive definition of a multidimensional generalized Riemann integral based on a concept of generalized absolute continuity for additive functions of sets of bounded variation.

A full descriptive definition of the BV-integral

B. Bongiorno, Luisa Di Piazza, Washek Frank Pfeffer (1995)

Commentationes Mathematicae Universitatis Carolinae

We present a Cauchy test for the almost derivability of additive functions of bounded BV sets. The test yields a full descriptive definition of a coordinate free Riemann type integral.

a functional analysis model for natural images permitting structured compression

Jacques Froment (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper describes a compact perceptual image model intended for morphological representation of the visual information contained in natural images. We explain why the total variation can be a criterion to split the information between the two main visual structures, which are the sketch and the microtextures. We deduce a morphological decomposition scheme, based on a segmentation where the borders of the regions correspond to the location of the topological singularities of a topographic map. This...

Absolutely continuous functions of several variables and diffeomorphisms

Stanislav Hencl, Jan Malý (2003)

Open Mathematics

In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.

Change of variables formula under minimal assumptions

Piotr Hajłasz (1993)

Colloquium Mathematicae

In the previous papers concerning the change of variables formula (in the form involving the Banach indicatrix) various assumptions were made about the corresponding transformation (see e.g. [BI], [GR], [F], [RR]). The full treatment of the case of continuous transformation is given in [RR]. In [BI] the transformation was assumed to be continuous, a.e. differentiable and with locally integrable Jacobian. In this paper we show that none of these assumptions is necessary (Theorem 2). We only need...

Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation

Dariusz Idczak (1998)

Czechoslovak Mathematical Journal

We give characterizations of the distributional derivatives D 1 , 1 , D 1 , 0 , D 0 , 1 of functions of two variables of locally finite variation. Then we use these results to prove the existence theorem for the hyperbolic equation with a nonhomogeneous term containing the distributional derivative determined by an additive function of an interval of finite variation. An application of the above theorem to a hyperbolic equation with an impulse effect is also given.

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