8 research outputs found
The Dirac delta function in two settings of Reverse Mathematics
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
Bounded variation and the strength of Helly's selection theorem
We analyze the strength of Helly's selection theorem HST, which is the most
important compactness theorem on the space of functions of bounded variation.
For this we utilize a new representation of this space intermediate between
and the Sobolev space W1,1, compatible with the, so called, weak*
topology. We obtain that HST is instance-wise equivalent to the
Bolzano-Weierstra\ss\ principle over RCA0. With this HST is equivalent to ACA0
over RCA0. A similar classification is obtained in the Weihrauch lattice
Grilliot's trick in Nonstandard Analysis
The technique known as Grilliot's trick constitutes a template for explicitly
defining the Turing jump functional in terms of a given
effectively discontinuous type two functional. In this paper, we discuss the
standard extensionality trick: a technique similar to Grilliot's trick in
Nonstandard Analysis. This nonstandard trick proceeds by deriving from the
existence of certain nonstandard discontinuous functionals, the Transfer
principle from Nonstandard analysis limited to -formulas; from this
(generally ineffective) implication, we obtain an effective implication
expressing the Turing jump functional in terms of a discontinuous functional
(and no longer involving Nonstandard Analysis). The advantage of our
nonstandard approach is that one obtains effective content without paying
attention to effective content. We also discuss a new class of functionals
which all seem to fall outside the established categories. These functionals
directly derive from the Standard Part axiom of Nonstandard Analysis.Comment: 21 page