Relative constructivity

1998 ◽  
Vol 63 (4) ◽  
pp. 1218-1238 ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Numerical Analysis ◽  
Approximation Theory ◽  
Free Choice ◽  
Functional Interpretation ◽  
Uniform Bounds ◽  
Free Variables ◽  
Image Position ◽  
Definable Functions ◽  
Classical Arithmetic

In a previous paper [13] we introduced a hierarchy (GnAω)n∈ℕ of subsystems of classical arithmetic in all finite types where the growth of definable functions of GnAω corresponds to the well-known Grzegorczyk hierarchy. Let AC-qf denote the schema of quantifier-free choice.[11], [13], [8] and [7] study various analytical principles Γ in the context of the theories GnAω + AC-qf (mainly for n = 2) and use proof-theoretic tools like, e.g., monotone functional interpretation (which was introduced in [12]) to determine their impact on the growth of uniform bounds Φ such thatwhich are extractable from given proofs (based on these principles Γ) of sentencesHere A0(u, k, v, w) is quantifier-free and contains only u, k, v, w as free variables; t is a closed term and ≤p is defined pointwise. The term ‘uniform bound’ refers to the fact that Φ does not depend on v ≤ptuk (see [12] for the relevance of such uniform bounds in numerical analysis and for concrete applications to approximation theory).

scholarly journals Effective bounds from ineffective proofs in analysis: An application of functional interpretation and majorization

1992 ◽  
Vol 57 (4) ◽  
pp. 1239-1273 ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Approximation Theory ◽  
Classical Analysis ◽  
Functional Interpretation ◽  
Image Position ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractWe show how to extract effective bounds Φ for ⋀u1⋀v ≤ytu⋁wηG0-sentences which depend on u only (i.e. ⋀u⋀v ≤y, tu⋁w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form(ϒ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ ≤ 2, (*) can be weakened toThis is used to establish new conservation results about weak Konig's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers.

An extension of Ackermann's set theory

1972 ◽  
Vol 37 (4) ◽  
pp. 703-704
Author(s):  
Donald Perlis
Keyword(s):  
Set Theory ◽  
Free Variables ◽  
Image Position

Ackermann's set theory [1], called here A, involves a schemawhere φ is an ∈-formula with free variables among y1, …, yn and w does not appear in φ. Variables are thought of as ranging over classes and V is intended as the class of all sets.S is a kind of comprehension principle, perhaps most simply motivated by the following idea: The familiar paradoxes seem to arise when the class CP of all P-sets is claimed to be a set, while there exists some P-object x not in CP such that x would have to be a set if CP were. Clearly this cannot happen if all P-objects are sets.Now, Levy [2] and Reinhardt [3] showed that A* (A with regularity) is in some sense equivalent to ZF. But the strong replacement axiom of Gödel-Bernays set theory intuitively ought to be a theorem of A* although in fact it is not (Levy's work shows this). Strong replacement can be formulated asThis lack of A* can be remedied by replacing S above bywhere ψ and φ are ∈-formulas and x is not in ψ and w is not in φ. ψv is ψ with quantifiers relativized to V, and y and z stand for y1, …, yn and z1, …, zm.

scholarly journals Implicational complexity in intuitionistic arithmetic

1981 ◽  
Vol 46 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Normal Form ◽  
Form Theorem ◽  
Natural Measure ◽  
Normal Form Theorem ◽  
Propositional Constants ◽  
Image Position ◽  
Classical Arithmetic ◽  
Intuitionistic Arithmetic

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].

Numerical analysis of different methods to calculate residual stresses in thin films based on instrumented indentation data

2012 ◽  
Vol 27 (13) ◽  
pp. 1732-1741 ◽  
Author(s):  
Carlos Eduardo Keutenedjian Mady ◽  
Sara Aida Rodriguez ◽  
Adriana Gómez Gómez ◽  
Roberto Martins Souza
Keyword(s):  
Thin Films ◽  
Numerical Analysis ◽  
Residual Stresses ◽  
Instrumented Indentation ◽  
Image Position

Abstract

Relative consistency of an extension of Ackermann's set theory

1976 ◽  
Vol 41 (2) ◽  
pp. 465-466
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Relative Strength ◽  
Axiom Of Choice ◽  
Relative Consistency ◽  
Free Variables ◽  
Image Position ◽  
The Axiom Of Choice

The set theory AFC was introduced by Perlis in [2] and he noted that it both includes and is stronger than Ackermann's set theory. We shall give a relative consistency result for AFC.AFC is obtained from Ackermann's set theory (see [2]) by replacing Ackermann's set existence schema with the schema(where ϕ, ψ, are ∈-formulae, x is not in ψ, w is not in ϕ, y is y1, …, yn, z is z1, …, zm and all free variables are shown) and adding the axiom of choice for sets. Following [1], we say that λ is invisible in Rκ if λ < κ and we haveholding for every ∈-formula θ which has exactly two free variables and does not involve u or υ. The existence of a Ramsey cardinal implies the existence of cardinals λ and κ with λ invisible in Rκ, and Theorem 1.13 of [1] gives some further indications about the relative strength of the notion of invisibility.Theorem. If there are cardinals λ and κ with λ invisible in Rκ, then AFC is consistent.Proof. Suppose that λ is invisible in Rκ and we will show that 〈Rκ, Rλ, ∈〉 ⊧ AFC (Rλ being the interpretation of V, of course).

The λ-calculus is ω-incomplete

1974 ◽  
Vol 39 (2) ◽  
pp. 313-317 ◽  
Author(s):  
G. D. Plotkin
Keyword(s):  
The Other ◽  
Free Variables ◽  
Syntactic Identity ◽  
Image Position

The ω-rule in the λ-calculus (or, more exactly, the λK-β, η calculus) isIn [1] it was shown that this rule is consistent with the other rules of the λ-calculus. We will show the rule cannot be derived from the other rules; that is, we will give closed terms M and N such that MZ = NZ can be proved without using the ω-rule, for each closed term Z, but M = N cannot be so proved. This strengthens a result in [4] and answers a question of Barendregt.The language of the λ-calculus has an alphabet containing denumerably many variables a, b, c, … (which have a standard listing e1, e2, …), improper symbolsλ, ( , ) and a single predicate symbol = for equality.Terms are defined inductively by the following:(1) A variable is a term.(2) If M and N are terms, so is (MN); it is called a combination.(3) If M is a term and x is a variable, (λx M) is a term; it is called an abstraction.We use ≡ for syntactic identity of terms.If M and N are terms, M = N is a formula.BV(M), the set of bound variables in M, and FV(M), its free variables, are defined inductively byA term M is closed iff FV(M) = ∅.

Numerical analysis of the mean structure of gaseous detonation with dilute water spray

2020 ◽  
Vol 887 ◽  
Author(s):  
Hiroaki Watanabe ◽  
Akiko Matsuo ◽  
Ashwin Chinnayya ◽  
Ken Matsuoka ◽  
Akira Kawasaki ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Gaseous Detonation ◽  
Water Spray ◽  
Mean Structure ◽  
The Mean ◽  
Image Position

scholarly journals The ruin probability in a special case

1971 ◽  
Vol 6 (1) ◽  
pp. 66-68 ◽  
Author(s):  
H. Bohman
Keyword(s):  
Numerical Analysis ◽  
Laplace Transform ◽  
Ruin Probability ◽  
Claim Amount ◽  
The Laplace Transform ◽  
Desk Calculator ◽  
Numerical Tools ◽  
The Mean ◽  
Image Position ◽  
Special Case

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expressionwhere c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we havewhereCramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.

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