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2012 | OriginalPaper | Chapter

6. Conclusion: New Opportunities

Author : Mark Burgin

Published in: Hypernumbers and Extrafunctions

Publisher: Springer New York

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Abstract

Many topics and results in the theory of hypernumbers and extrafunctions have been left beyond the scope of this little book as its goal is to give a succinct introduction into this rich and multilayered theory. Here we briefly describe some of these topics and results, articulating open problems and directions for further research.

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Literature
go back to reference Burgin, M.: On the Han-Banach theorem for hyperfunctionals. Not. Nat. Acad. Sci. Ukr. (in Russian and Ukrainian) 7, 9–14 (1991) Burgin, M.: On the Han-Banach theorem for hyperfunctionals. Not. Nat. Acad. Sci. Ukr. (in Russian and Ukrainian) 7, 9–14 (1991)
go back to reference Burgin, M.: Hypernormed spaces and algebras. International Algebraic Conference dedicated to the Memory of Prof. L.M. Gluskin, pp. 97–98. Kharkov, Ukraine, (1997) Burgin, M.: Hypernormed spaces and algebras. International Algebraic Conference dedicated to the Memory of Prof. L.M. Gluskin, pp. 97–98. Kharkov, Ukraine, (1997)
go back to reference Burgin, M.: Extrafunctions, distributions, and nonsmooth analysis. Mathematics Report Series, MRS Report. 1 Feb 2001, p. 47. University of California, Los Angeles (2001) Burgin, M.: Extrafunctions, distributions, and nonsmooth analysis. Mathematics Report Series, MRS Report. 1 Feb 2001, p. 47. University of California, Los Angeles (2001)
go back to reference Burgin, M.: Theory of hypernumbers and extrafunctions: functional spaces and differentiation. Discrete Dyn. Nat. Soc. 7, 201–212 (2002)MathSciNetMATHCrossRef Burgin, M.: Theory of hypernumbers and extrafunctions: functional spaces and differentiation. Discrete Dyn. Nat. Soc. 7, 201–212 (2002)MathSciNetMATHCrossRef
go back to reference Burgin, M.: Hyperfunctionals and generalized distributions. In: Krinik, A.C., Swift, R.J. (eds.) Stochastic processes and functional analysis, a Dekker series of lecture notes in pure and applied mathematics, vol. 238, pp. 81–119. CRC Press, Boca Raton, FL (2004) Burgin, M.: Hyperfunctionals and generalized distributions. In: Krinik, A.C., Swift, R.J. (eds.) Stochastic processes and functional analysis, a Dekker series of lecture notes in pure and applied mathematics, vol. 238, pp. 81–119. CRC Press, Boca Raton, FL (2004)
go back to reference Burgin, M.: Neoclassical analysis: calculus closer to the real world. Nova Science Publishers, Hauppauge, NY (2008a) Burgin, M.: Neoclassical analysis: calculus closer to the real world. Nova Science Publishers, Hauppauge, NY (2008a)
go back to reference Burgin, M.: Inequalities in series and summation in hypernumbers. In: Dragomir, S., Sofo, A. (eds.) Advances in inequalities for series, pp. 89–120. Nova Science Publishers, New York (2008b) Burgin, M.: Inequalities in series and summation in hypernumbers. In: Dragomir, S., Sofo, A. (eds.) Advances in inequalities for series, pp. 89–120. Nova Science Publishers, New York (2008b)
go back to reference Burgin, M.: Hyperintegration approach to the feynman integral. Integr.: Math. Theory. Appl. 1, 59–104 (2008/2009) Burgin, M.: Hyperintegration approach to the feynman integral. Integr.: Math. Theory. Appl. 1, 59–104 (2008/2009)
go back to reference Burgin, M.: Nonlinear partial differential equations in extrafunctions. Integr: Mathl. Theory. Appl. 2, 17–50 (2010)MATH Burgin, M.: Nonlinear partial differential equations in extrafunctions. Integr: Mathl. Theory. Appl. 2, 17–50 (2010)MATH
go back to reference Burgin, M., Krinik, A. C.: Probabilities and hyperprobabilities, 8th Annual International Conference on Statistics, Mathematics and Related Fields, Conference Proceedings, Honolulu, Hawaii, pp. 351–367 (2009) Burgin, M., Krinik, A. C.: Probabilities and hyperprobabilities, 8th Annual International Conference on Statistics, Mathematics and Related Fields, Conference Proceedings, Honolulu, Hawaii, pp. 351–367 (2009)
go back to reference Christov, C.I., Todorov, T.D.: Asymptotic numbers: algebraic operations with them. Serdica, Bulgaricae Math. Publ. 2, 87–102 (1974)MathSciNet Christov, C.I., Todorov, T.D.: Asymptotic numbers: algebraic operations with them. Serdica, Bulgaricae Math. Publ. 2, 87–102 (1974)MathSciNet
go back to reference Clark, F.H.: Optimization and nonsmooth analysis. Willy, New York (1983) Clark, F.H.: Optimization and nonsmooth analysis. Willy, New York (1983)
go back to reference Connor, J.N.L.: Practical methods for the uniform asymptotic evaluation of oscillating integrals with several coalescing saddle points. In: Wong, R. (ed.) Asymptotic and computational analysis, pp. 137–173. Dekker, New York (1990) Connor, J.N.L.: Practical methods for the uniform asymptotic evaluation of oscillating integrals with several coalescing saddle points. In: Wong, R. (ed.) Asymptotic and computational analysis, pp. 137–173. Dekker, New York (1990)
go back to reference Connor, J.N.L., Hobbs, C.A.: Numerical evaluation of cuspoid and bessoid oscillating integrals for applications in chemical physics. Russ. J. Phys. Chem. 23(2), 13–19 (2004) Connor, J.N.L., Hobbs, C.A.: Numerical evaluation of cuspoid and bessoid oscillating integrals for applications in chemical physics. Russ. J. Phys. Chem. 23(2), 13–19 (2004)
go back to reference Döring, A., Isham, C.J.: A topos foundation for theories of physics: I formal languages for physics. J. Math. Phys. 49, 053515 (2008)MathSciNetCrossRef Döring, A., Isham, C.J.: A topos foundation for theories of physics: I formal languages for physics. J. Math. Phys. 49, 053515 (2008)MathSciNetCrossRef
go back to reference Du Bois-Reymond, P.: Sur la grandeur relative des infinis des fonctions. Annali di matematica pura ed applicata 4, 338–353 (1870/1871) Du Bois-Reymond, P.: Sur la grandeur relative des infinis des fonctions. Annali di matematica pura ed applicata 4, 338–353 (1870/1871)
go back to reference Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. McGraw-Hill Companies, New York (1965)MATH Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. McGraw-Hill Companies, New York (1965)MATH
go back to reference Hardy, G.H.: Orders of infinity, the “Infinitärcalcül” of Paul Du Bois-Reymond. Cambridge University Press, Cambridge (1910) Hardy, G.H.: Orders of infinity, the “Infinitärcalcül” of Paul Du Bois-Reymond. Cambridge University Press, Cambridge (1910)
go back to reference Hellman, G.: Randomness and reality. Proc. Biennial Meet. Phil. Sci. Assoc 2, 79–97 (1978) Hellman, G.: Randomness and reality. Proc. Biennial Meet. Phil. Sci. Assoc 2, 79–97 (1978)
go back to reference Johnson, G.W., Lapidus, M.L.: The Feynman integral and Feynman's operational calculus, oxford mathematical monographs. Oxford Univ Press, Oxford (2002) Johnson, G.W., Lapidus, M.L.: The Feynman integral and Feynman's operational calculus, oxford mathematical monographs. Oxford Univ Press, Oxford (2002)
go back to reference Kashiwa, T.: Path integral methods. Oxford Univ Press, Oxford (1997)MATH Kashiwa, T.: Path integral methods. Oxford Univ Press, Oxford (1997)MATH
go back to reference Kauffman, L.: Quantum fields and strings: a course for mathematicians, by Deligne et al. Bull. Amer. Math. Soc. 38, 489–494 (2001)MathSciNetCrossRef Kauffman, L.: Quantum fields and strings: a course for mathematicians, by Deligne et al. Bull. Amer. Math. Soc. 38, 489–494 (2001)MathSciNetCrossRef
go back to reference Luu, D.: Hyperprobabilities of nonstationary Markov Chains. Thesis, California State Polytechnic University, Pomona (2011) Luu, D.: Hyperprobabilities of nonstationary Markov Chains. Thesis, California State Polytechnic University, Pomona (2011)
go back to reference Oberguggenberger, M., Todorov, T.: An embedding of Schwartz distributions in the algebra of asymptotic functions. Int J. Math. Math. Sci. 21, 417–428 (1998)MathSciNetMATHCrossRef Oberguggenberger, M., Todorov, T.: An embedding of Schwartz distributions in the algebra of asymptotic functions. Int J. Math. Math. Sci. 21, 417–428 (1998)MathSciNetMATHCrossRef
go back to reference Sudbury, A.: Scientific laws that are neither deterministic nor probabilistic. Br. J. Philos. Sci. 27, 307–315 (1976)MATHCrossRef Sudbury, A.: Scientific laws that are neither deterministic nor probabilistic. Br. J. Philos. Sci. 27, 307–315 (1976)MATHCrossRef
Metadata
Title
Conclusion: New Opportunities
Author
Mark Burgin
Copyright Year
2012
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4419-9875-0_6

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