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The Dirac delta function in two settings of Reverse Mathematics

(2012) ARCHIVE FOR MATHEMATICAL LOGIC. 51(1-2). p.99-121
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Abstract
The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property integral(R) f(x)delta(x)dx = f (0) of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak Konig's Lemma (see Yu and Simpson in Arch Math Log 30(3): 171-180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the 'Big' systems of Reverse Mathematics. In the context of ERNA's Reverse Mathematics (Sanders in J Symb Log 76(2): 637-664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA's Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA's Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength.
Keywords
Nonstandard Analysis, Reverse Mathematics, Dirac Delta function, SET EXISTENCE AXIOMS, ERNA, THEOREM

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Citation

Please use this url to cite or link to this publication:

Chicago
Sanders, Sam, and Keita Yokoyama. 2012. “The Dirac Delta Function in Two Settings of Reverse Mathematics.” Archive for Mathematical Logic 51 (1-2): 99–121.
APA
Sanders, S., & Yokoyama, K. (2012). The Dirac delta function in two settings of Reverse Mathematics. ARCHIVE FOR MATHEMATICAL LOGIC, 51(1-2), 99–121.
Vancouver
1.
Sanders S, Yokoyama K. The Dirac delta function in two settings of Reverse Mathematics. ARCHIVE FOR MATHEMATICAL LOGIC. 2012;51(1-2):99–121.
MLA
Sanders, Sam, and Keita Yokoyama. “The Dirac Delta Function in Two Settings of Reverse Mathematics.” ARCHIVE FOR MATHEMATICAL LOGIC 51.1-2 (2012): 99–121. Print.
@article{1938292,
  abstract     = {The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property integral(R) f(x)delta(x)dx = f (0) of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak Konig's Lemma (see Yu and Simpson in Arch Math Log 30(3): 171-180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the 'Big' systems of Reverse Mathematics. In the context of ERNA's Reverse Mathematics (Sanders in J Symb Log 76(2): 637-664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA's Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA's Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength.},
  author       = {Sanders, Sam and Yokoyama, Keita},
  issn         = {1432-0665},
  journal      = {ARCHIVE FOR MATHEMATICAL LOGIC},
  keyword      = {Nonstandard Analysis,Reverse Mathematics,Dirac Delta function,SET EXISTENCE AXIOMS,ERNA,THEOREM},
  language     = {eng},
  number       = {1-2},
  pages        = {99--121},
  title        = {The Dirac delta function in two settings of Reverse Mathematics},
  url          = {http://dx.doi.org/10.1007/s00153-011-0256-5},
  volume       = {51},
  year         = {2012},
}

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