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1994-1995 | 146 | 1 | 69-84
Tytuł artykułu

A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

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Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, we are led to an integration process over quite general sets $A ⊆ q ℝ^n$ with a regular boundary. The integral enjoys all the usual properties and yields the divergence theorem for vector-valued functions with singularities in a most general form.
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  • Abteilung Mathematik II und V, Universität Ulm, D-89069 Ulm, Germany
  • Abteilung Mathematik II und V, Universität Ulm, D-89069 Ulm, Germany
Bibliografia
  • [Fed] H. Federer, Geometric Measure Theory, Springer, New York, 1969.
  • [Jar-Ku 1] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits $C^1$-transformations, Časopis Pěst. Mat. 109 (1984), 157-167.
  • [Jar-Ku 2] J. Jarník and J. Kurzweil, A non-absolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (110) (1985), 116-139.
  • [Jar-Ku 3] J. Jarník and J. Kurzweil, A new and more powerful concept of the PU integral, ibid. 38 (113) (1988), 8-48.
  • [JKS] J. Jarník, J. Kurzweil and S. Schwabik, On Mawhin's approach to multiple nonabsolutely convergent integral, Časopis Pěst. Mat. 108 (1983), 356-380.
  • [Ju] W. B. Jurkat, The Divergence Theorem and Perron integration with exceptional sets, Czechoslovak Math. J. 43 (118) (1993), 27-45.
  • [Ju-No 1] W. B. Jurkat and D. J. F. Nonnenmacher, An axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, Fund. Math. 145 (1994), 221-242.
  • [Ju-No 2] W. B. Jurkat and D. J. F. Nonnenmacher, A generalized n-dimensional Riemann integral and the Divergence Theorem with singularities, Acta Sci. Math. (Szeged) 59 (1994), 241-256.
  • [Ju-No 3] W. B. Jurkat and D. J. F. Nonnenmacher, The Fundamental Theorem for the $ν_1$-integral on more general sets and a corresponding Divergence Theorem with singularities, Czechoslovak Math. J., to appear.
  • [Maw] J. Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, ibid. 31 (106) (1981), 614-632.
  • [No 1] D. J. F. Nonnenmacher, Sets of finite perimeter and the Gauss-Green Theorem with singularities, J. London Math. Soc., to appear.
  • [No 2] D. J. F. Nonnenmacher, A constructive definition of the n-dimensional ν(S)-integral in terms of Riemann sums, preprint 1992, to appear.
  • [Pf 1] W. F. Pfeffer, The divergence theorem, Trans. Amer. Math. Soc. 295 (1986), 665-685.
  • [Pf 2] W. F. Pfeffer, The Gauss-Green Theorem, Adv. in Math. 87 (1991), 93-147.
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bwmeta1.element.bwnjournal-article-fmv146i1p69bwm
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