The Dirac delta function in two settings of Reverse Mathematics
 Author
 Sam Sanders (UGent) and Keita Yokoyama
 Organization
 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 wellknown 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): 171180, 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): 637664, 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 firstorder 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: http://hdl.handle.net/1854/LU1938292
 Chicago
 Sanders, Sam, and Keita Yokoyama. 2012. “The Dirac Delta Function in Two Settings of Reverse Mathematics.” Archive for Mathematical Logic 51 (12): 99–121.
 APA
 Sanders, S., & Yokoyama, K. (2012). The Dirac delta function in two settings of Reverse Mathematics. ARCHIVE FOR MATHEMATICAL LOGIC, 51(12), 99–121.
 Vancouver
 1.Sanders S, Yokoyama K. The Dirac delta function in two settings of Reverse Mathematics. ARCHIVE FOR MATHEMATICAL LOGIC. 2012;51(12):99–121.
 MLA
 Sanders, Sam, and Keita Yokoyama. “The Dirac Delta Function in Two Settings of Reverse Mathematics.” ARCHIVE FOR MATHEMATICAL LOGIC 51.12 (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 wellknown 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): 171180, 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): 637664, 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 firstorder strength.}, author = {Sanders, Sam and Yokoyama, Keita}, issn = {14320665}, journal = {ARCHIVE FOR MATHEMATICAL LOGIC}, keyword = {Nonstandard Analysis,Reverse Mathematics,Dirac Delta function,SET EXISTENCE AXIOMS,ERNA,THEOREM}, language = {eng}, number = {12}, pages = {99121}, title = {The Dirac delta function in two settings of Reverse Mathematics}, url = {http://dx.doi.org/10.1007/s0015301102565}, volume = {51}, year = {2012}, }
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