### The Dirac delta function in two settings of Reverse Mathematics

(2012) ARCHIVE FOR MATHEMATICAL LOGIC. 51(1-2). p.99-121- 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.

Please use this url to cite or link to this publication:
http://hdl.handle.net/1854/LU-1938292

- author
- Sam Sanders and Keita Yokoyama
- organization
- year
- 2012
- type
- journalArticle (original)
- publication status
- published
- subject
- keyword
- Nonstandard Analysis, Reverse Mathematics, Dirac Delta function, SET EXISTENCE AXIOMS, ERNA, THEOREM
- journal title
- ARCHIVE FOR MATHEMATICAL LOGIC
- Arch. Math. Log.
- volume
- 51
- issue
- 1-2
- pages
- 99 - 121
- Web of Science type
- Article
- Web of Science id
- 000306331000007
- JCR category
- LOGIC
- JCR impact factor
- 0.281 (2012)
- JCR rank
- 17/20 (2012)
- JCR quartile
- 4 (2012)
- ISSN
- 1432-0665
- DOI
- 10.1007/s00153-011-0256-5
- language
- English
- UGent publication?
- yes
- classification
- A1
- copyright statement
*I have transferred the copyright for this publication to the publisher*- id
- 1938292
- handle
- http://hdl.handle.net/1854/LU-1938292
- date created
- 2011-11-01 13:09:53
- date last changed
- 2016-12-19 15:42:00

@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}, }

- 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.