The Dirac delta function in two settings of Reverse Mathematics

Sanders, Sam; Yokoyama, Keita

doi:10.1007/s00153-011-0256-5

Cited by 19 publications

(14 citation statements)

References 16 publications

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“…Finally, it is shown in [49] that the well-known property of the Dirac delta function (in the weak sense) is equivalent to WWKL 0 . All functions involved are at least continuous, and we believe that a more general theorem involving discontinuous functions yields WHBU.…”

Section: 44mentioning

confidence: 99%

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On the logical and computational properties of the Vitali covering theorem

Normann¹,

Sanders²

2019

Preprint

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The development of measure theory in 'computational' frameworks proceeds by studying the computational properties of countable approximations of measurable objects. At the most basic level, these representations are provided by Littlewood's three principles, and the associated approximation theorems due to e.g. Lusin and Egorov. In light of this fundamental role, it is then a natural question how hard it is to prove the aforementioned theorems (in the sense of the Reverse Mathematics program), and how hard it is to compute the countable approximations therein (in the sense of Kleene's schemes S1-S9). The answer to both questions is 'extremely hard', as follows: one one hand, proofs of these approximation theorems require weak compactness, the measure-theoretical principle underlying e.g. Vitali's covering theorem. In terms of the usual scale of comprehension axioms, weak compactness is only provable using full second-order arithmetic. On the other hand, computing the associated approximations requires a weak fan functional Λ, which is a realiser for weak compactness and is only computable from (a certain comprehension functional for) full second-order arithmetic. Despite this observed hardness, we show that weak compactness, and certain weak fan functionals, behave much better than (Heine-Borel) compactness and the associated class of realisers, called special fan functionals Θ. In particular, we show that the combination of any Λ-functional and the Suslin functional has no more computational power than the latter functional alone, in contrast to Θ-functionals. Finally, we introduce a hierarchy involving Θ-functionals and Heine-Borel compactness.

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Section: 44mentioning

confidence: 99%

“…Note that we need WWKL to guarantee that the integral in MCT net exists, in light of [49,Theorem 10]. Arzelà already studied the monotone convergence theorem (involving sequences) for the Riemann integral in 1885, and this theorem is moreover proved in e.g.…”

Section: Nets and Weak Compactnessmentioning

confidence: 99%

On the logical and computational properties of the Vitali covering theorem

Normann¹,

Sanders²

2019

Preprint

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“…From P [STP ↔ CC ns ], terms t, s can be extracted such that E-PA * (∀Θ 3 ) SFF(Θ) → MU 3 (t(Θ)) ∧ (∀Ξ 3 ) MU 3 (Ξ) → SFF(s(Ξ)) . (6.23) As shown in [62], WWKL 0 is equivalent to the statement that every bounded continuous functional on the unit interval is Riemann integrable. We suspect that adding a boundedness condition to Y 2 ∈ C ns yields an equivalence to LMP.…”

Section: Generalisations Of Weak König's Lemmamentioning

confidence: 99%

Computability Theory, Nonstandard Analysis, and Their Connections

Normann¹,

Sanders²

2019

J. symb. log.

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We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.(T.1) A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $.(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

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“…Thus (p k ) k , v = 2 is a code for the integral of the mollifier. For the proof of this proposition we will need the following notation and theorem from [17]. A partition of [0, 1] is a finite set…”

Section: The Space Of Functions Of Bounded Variationmentioning

confidence: 99%

Bounded variation and the strength of Helly's selection theorem

Kreuzer

2014

Logical Methods in Computer Science

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Abstract. 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 (BV ). For this we utilize a new representation of this space intermediate between L1 and the Sobolev space W 1,1 , compatible with the-so called-weak * topology on BV . We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.In this paper we investigate the space of functions of bounded variation (BV ) and Helly's selection theorem (HST) from the viewpoint of reverse mathematics and computable analysis. Helly's selection theorem is the most important compactness principle on BV . It is used in analysis and optimization, see for instance [1,3].This continues our work in [10] and [12] where (instances of) the Bolzano-Weierstraß principle and the Arzelà-Ascoli theorem were analyzed. There we showed, among others, that an instance of the Arzelà-Ascoli theorem is equivalent to a suitable single instance of the Bolzano-Weierstraß principle (for the unit interval [0, 1]), which, in turn, is equivalent to an instance of WKL for Σ 0 1 -trees. Here, we will show that an instance of Helly's selection theorem is equivalent to a single instance of the Bolzano-Weierstraß principle (and with this to an instance of the other principles mentioned above). It is a priori not clear that this is possible since the proof of HST uses seemingly iterated application of the Arzelà-Ascoli theorem and since there are compactness principles, which are instance-wise strictly stronger than Bolzano-Weierstraß for [0, 1]. (For instance the Bolzano-Weierstraß principle for weak compactness on ℓ 2 has this property, see [11].) A fortori this shows that HST is equivalent to ACA 0 over RCA 0 .

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The Dirac delta function in two settings of Reverse Mathematics

Cited by 19 publications

References 16 publications

On the logical and computational properties of the Vitali covering theorem

On the logical and computational properties of the Vitali covering theorem

Computability Theory, Nonstandard Analysis, and Their Connections

Bounded variation and the strength of Helly's selection theorem

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