Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Derivation and Integration
  • Cited by 18
Publisher:
Cambridge University Press
Online publication date:
July 2014
Print publication year:
2001
Online ISBN:
9780511574764
DOI:
https://doi.org/10.1017/CBO9780511574764
Subjects:
Mathematics (general), Mathematics, Abstract Analysis, Real and Complex Analysis
Series:
Cambridge Tracts in Mathematics (140)
Information

Book description

This 2001 book is devoted to an invariant multidimensional process of recovering a function from its derivative. It considers additive functions defined on the family of all bounded BV sets that are continuous with respect to a suitable topology. A typical example is the flux of a continuous vector field. A very general Gauss-Green theorem follows from the sufficient conditions for the derivability of the flux. Since the setting is invariant with respect to local lipeomorphisms, a standard argument extends the Gauss-Green theorem to the Stokes theorem on Lipschitz manifolds. In addition, the author proves the Stokes theorem for a class of top-dimensional normal currents - a first step towards solving a difficult open problem of derivation and integration in middle dimensions. The book contains complete and detailed proofs and will provide valuable information to research mathematicians and advanced graduate students interested in geometric integration and related areas.

Reviews

Review of the hardback:‘…warmly recommended to researchers and advanced graduate students …’.

József Németh Source: Acta Sci. Math.

Review of the hardback:‘…I warmly recommended this book …’.

Thierry de Pauw Source: Bulletin of the Belgian Mathematical Society

Review of the hardback:' … written by one of the leading specialists in this field.'

Source: EMS

Review of the hardback:'Readers with a good background in analysis will find this an illuminating account.'

Source: Mathematika

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
Bibliography
[1] S.I., Ahmed and W.F., Pfeffer, A Riemann integral in a locally compact Hausdorff space, J. Austral. Math. Soc. 41, Series A (1986), 115–137.
[2] L., Ambrosio, A compact theorem for a new class of functions of bounded variation, Boll. Un. Mat. Ital. 3-B (1989), 857–881.
[3] T., Bagby and W.P., Ziemer, Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129–148.
[4] H., Bauer, Der Perronsche Integralbegriff und seine Beziehung zum Lebesgue-shen, Monatshefte Math. Phys. 26 (1915), 153–198.
[5] A.S., Besicovitch, On sufficient conditions for a function to be analytic, and behaviour of analytic functions in the neighbourhood of non-isolated singular points, Proc. London Math. Soc. 32 (1931), 1–9.
[6] B., Bongiorno, Essential variation, Measure Theory Oberwolfach, Lecture Notes in Math. no. 945, Springer-Verlag, New York, 1981, pp. 187-193.
[7] B., Bongiorno, U., Darji, and W.F., Pfeffer, On indefinite BV-integrals, Comment. Math. Univ. Carolinae 41 (2000), 843–853.
[8] B., Bongiorno, M., Giertz, and W.F., Pfeffer, Some nonabsolutely convergent integrals in the real line, Boll. Un. Mat. Ital. 6-B (1992), 371–402.
[9] B., Bongiorno, L. Di, Piazza, and D., Preiss, Infinite variations and derivatives in ℝm, J. Math. Anal. Appl. 224 (1998), 22–33.
[10] B., Bongiorno, L. Di, Piazza, and D., Preiss, A constructive minimal integral which includes Lebesgue integrable functions and derivatives, J. London Math. Soc. 62 (2000), 117–126.
[11] B., Bongiorno and P., Vetro, Su un teorema di F. Riesz, Atti Acc. Sei. Lettere Arti Palermo (IV) 37 (1979), 3–13.
[12] Z., Buczolich, Functions with all singular sets of Hausdorff dimension bigger than one, Real Anal. Exchange 15(1) (1989–1990), 299-306.
[13] Z., Buczolich, Density points and bi-Lipschitz finctions in ℝm, Proc. American Math. Soc. 116 (1992), 53–56.
[14] Z., Buczolich, A v-integrable function which is not Lebesgue integrable on any portion of the unit square, Acta Math. Hung. 59 (1992), 383–393.
[15] Z., Buczolich, The g-integral is not rotation invariant, Real Anal. Exchange 18(2) (1992–1993), 437-447.
[16] Z., Buczolich, Lipeomorphisms, sets of bounded variation and integrals, Acta Math. Hung. 87 (2000), 243–265.
[17] Z., Buczolich, T. De, Pauw, and W.F., Pfeffer, Charges, BV functions, and multipliers for generalized Riemann integrals, Indiana Univ. Math. J. 48 (1999), 1471–1511.
[18] Z., Buczolich and W.F., Pfeffer, Variations of additive functions, Czechoslovak Math. J. 47 (1997), 525–555.
[19] Z., Buczolich and W.F., Pfeffer, On absolute continuity, J. Math. Anal. Appi. 222 (1998), 64–78.
[20] G., Congedo and I., Tamanini, Note sulla regolarità dei minimi di funzionali del tipo dell'area, Rend. Acad. Sci. XL, Mem. Mat. 106 (XII, 17) (1988), 239–257.
[21] R., Engelking, General Topology, PWN, Warsaw, 1977.
[22] L.C., Evans and R.F., Gariepy, Measure Theory and Fine Properties of Functions, CRP Press, Boca Raton, 1992.
[23] K.J., Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge, 1985.
[24] H., Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
[25] H., Federer and W.H., Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458–520.
[26] E., Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel, 1984.
[27] C., Goffman, T., Nishiura, and D., Waterman, Homeomorphisms in Analysis, Amer. Math. Soc., Providence, 1997.
[28] R.A., Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Amer. Math. Soc, Providence, 1994.
[29] P.R., Halmos, Measure Theory, Van Nostrand, New York, 1950.
[30] R., Henstock, Definitions of Riemann type of the variational integrals, Proc. London Math. Soc. 11 (1961), 79–87.
[31] I.N., Herstein, Topics in Algebra, Blaisdell, London, 1964.
[32] J., Holec and J., Mařík, Continuous additive mappings, Czechoslovak Math. J. 14 (1965), 237–243.
[33] J., Horváth, Topological Vector Spaces and Distributions, vol. 1, Addison-Wesley, London, 1966.
[34] E.J., Howard, Analyticity of almost everywhere differentiable functions, Proc. American Math. Soc. 110 (1990), 745–753.
[35] J., Jarník and J., Kurzweil, A nonabsolutely convergent integral which admits transformation and can be used for integration on manifolds, Czechoslovak Math. J. 35 (1986), 116–139.
[36] T., Jech, Set Theory, Academic Press, New York, 1978.
[37] R.L., Jeffery, The Theory of Functions of a Real Variable, Dover, New York, 1985.
[38] W.B., Jurkat, The divergence theorem and Perron integration with exceptional sets, Czechoslovak Math. J. 43 (1993), 27–45.
[39] 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.
[40] K., Karták and J., Mařík, A non-absolutely convergent integral in Em and the theorem of Gauss, Czechoslovak Math. J. 15 (1965), 253–260.
[41] L., Kuipers and H., Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974.
[42] J., Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 82 (1957), 418–446.
[43] J., Kurzweil, Nichtabsolut Konvergente Integrale, Taubinger, Leipzig, 1980.
[44] J., Kurzweil, J., Mawhin, and W.F., Pfeffer, An integral defined by approximating BV partitions of unity, Czechoslovak Math. J. 41 (1991), 695–712.
[45] U., Massari and M., Miranda, Minimal Surfaces of Codimension One, North-Holland, Amsterdam, 1984.
[46] P., Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, Cambridge, 1995.
[47] J., Mařík, Extensions of additive mappings, Czechoslovak Math. J. 15 (1965), 244–252.
[48] J., Mafik and J., Matyska, On a generalization of the Lebesgue integral in Em, Czechoslovak Math. J. 15 (1965), 261–269.
[49] J., Mawhin, Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields, Czechoslovak J. Math. 31 (1981), 614–632.
[50] J., Mawhin, Generalized Riemann integrals and the divergence theorem for differentiate vector fields, E.B. Christoffel (Basel), Birkhäuser, 1981, pp. 704-714.
[51] V.G., Maz'ja, Sobolev Spaces, Springer-Verlag, New York, 1985.
[52] R.M., McLeod, The Generalized Riemann Integral, Math. Asso. Amer., Washington, D.C., 1980.
[53] E.J., McShane, A Riemann-type Integral that Includes Lebesgue-Stieltjes, Boch-ner and Stochastic Integrals, Mem. Amer. Math. Soc., 88, Providence, 1969.
[54] E.J., McShane, A unified theory of integration, Amer. Math. Monthly 80 (1973), 349–359.
[55] E.J., McShane, Unified Integration, Academic Press, New York, 1983.
[56] F., Morgan, Geometric Measure Theory, Academic Press, New York, 1988.
[57] D.J.F., Nonnenmacher, Sets of finite perimeter and the Gauss-Green theorem, J. London Math. Soc. 52 (1995), 335–344.
[58] J.C., Oxtoby, Measure and Category, Springer-Verlag, New York, 1971.
[59] T. De, Pauw, Topologies for the space of BV-integrable functions in ℝm, J. Func. Anal. 144 (1997), 190–231.
[60] W.F., Pfeffer, Integrals and measures, Marcel Dekker, New York, 1977.
[61] W.F., Pfeffer, The divergence theorem, Trans. Amer. Math. Soc. 295 (1986), 665–685.
[62] W.F., Pfeffer, The multidimensional fundamental theorem of calculus, J. Australian Math. Soc. 43 (1987), 143–170.
[63] W.F., Pfeffer, A descriptive definition of a variational integral and applications, Indiana Univ. Math. J. 40 (1991), 259–270.
[64] W.F., Pfeffer, The Gauss-Green theorem, Adv. Math. 87 (1991), 93–147.
[65] W.F., Pfeffer, The Riemann Approach to Integration, Cambridge Univ. Press, New York, 1993.
[66] W.F., Pfeffer, The generalized Riemann-Stielijes integral, Real Anal. Exchange 21(2) (1995–1996), 521-547.
[67] W.F., Pfeffer, The Lebesgue and Denjoy-Perron integrals from a descriptive point of view, Ricerche Mat. 48 (1999), 211–223.
[68] W.F., Pfeffer and B.S., Thomson, Measures defined by gages, Canadian J. Math. 44 (1992), 1306–1316.
[69] W.F., Pfeffer and Wei-Chi, Yang, The multidimensional variational integral and its extensions, Real Anal. Exchange 15(1) (1989–1990), 111-169.
[70] K.P.S. Bhaskara, Rao and M. Bhaskara, Rao, Theory of Charges, Acad. Press, New York, 1983.
[71] C.A., Rogers, Hausdorff measures, Cambridge Univ. Press, Cambridge, 1970.
[72] P., Romanovski, Intégrale de Denjoy dans l'espace á n dimensions, Math. Sbornik 51 (1941), 281–307.
[73] W., Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.
[74] W., Rudin, Functional Analysis, McGraw-Hill, New York, 1991.
[75] S., Saks, Theory of the Integral, Dover, New York, 1964.
[76] V.L., Shapiro, The divergence theorem for discontinuous vector fields, Ann. Math. 68 (1958), 604–624.
[77] L., Simon, Lectures on Geometric Measure Theory, Proc. CM.A. 3, Australian Natl. Univ., Cambera, 1983.
[78] E.H., Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
[79] M., Spivak, Calculus on Manifolds, Benjamin, London, 1965.
[80] E.M., Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
[81] I., Tamanini and C., Giacomelli, Approximation of Caccioppoli sets, with applications to problems in image segmentation, Ann. Univ. Ferrara VII (N.S.) 35 (1989), 187–213.
[82] I., Tamanini and C., Giacomelli, Un tipo di approssimazione “dall'interno” degli insiemi di perimetro finito, Rend. Mat. Acc. Lincei (9) 1 (1990), 181–187.
[83] B.S., Thomson, Spaces of conditionally integrable functions, J. London Math. Soc. 2 (1970), 358–360.
[84] B.S., Thomson, Derivatives of Interval Functions, Mem. Amer. Math. Soc., 452, Providence, 1991.
[85] B.S., Thomson, σ-finite Borel measures on the real line, Real Anal. Exch. 23 (1997–1998), 185-192.
[86] A.I., Volpert, The spaces BV and quasilinear equations, Math. USSR-Sbornik 2 (1967), 225–267.
[87] H., Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, 1957.
[88] W.P., Ziemer, Weakly Differentiate Functions, Springer-Verlag, New York, 1989.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.