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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 simplified multidimensional integral

Ágnes M. Backhausz, Vilmos Komornik, Tivadar Szilágyi (2009)

Czechoslovak Mathematical Journal

We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.

Area functionals and Godbillon-Vey cocycles

Takashi Tsuboi (1992)

Annales de l'institut Fourier

We investigate the natural domain of definition of the Godbillon-Vey 2- dimensional cohomology class of the group of diffeomorphisms of the circle. We introduce the notion of area functionals on a space of functions on the circle, we give a sufficiently large space of functions with nontrivial area functional and we give a sufficiently large group of Lipschitz homeomorphisms of the circle where the Godbillon-Vey class is defined.

Continuity of monotone functions

Boris Lavrič (1993)

Archivum Mathematicum

It is shown that a monotone function acting between euclidean spaces R n and R m is continuous almost everywhere with respect to the Lebesgue measure on R n .

Fermat’s method of quadrature

Jaume Paradís, Josep Pla, Pelegrí Viader (2008)

Revue d'histoire des mathématiques

The Treatise on Quadratureof Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, x + m / n d x , or under a higher hyperbola, x - m / n d x —with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of theTreatise is obscure and difficult to read. In it Fermat reduced the quadrature of a great number of algebraic curves in implicit form to the quadrature of known curves: the...

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