AN APPLICATION OF RECURSION THEORY TO ANALYSIS

2020 ◽  
Vol 26 (1) ◽  
pp. 15-25
Author(s):  
LIANG YU
Keyword(s):  
Approximation Property ◽  
Theoretical Method ◽  
Recursion Theory ◽  
Borel Set ◽  
Analytic Set ◽  
Null Set

AbstractMauldin [15] proved that there is an analytic set, which cannot be represented by $B\cup X$ for some Borel set B and a subset X of a $\boldsymbol{\Sigma }^0_2$-null set, answering a question by Johnson [10]. We reprove Mauldin’s answer by a recursion-theoretical method. We also give a characterization of the Borel generated $\sigma $-ideals having approximation property under the assumption that every real is constructible, answering Mauldin’s question raised in [15].

scholarly journals Corrigendum: amenability of ultrapowers of Banach algebras

2010 ◽  
Vol 53 (3) ◽  
pp. 633-637 ◽  
Author(s):  
Matthew Daws
Keyword(s):  
Approximation Property ◽  
Banach Algebras ◽  
The Other

AbstractSome of the results of § 5 of the cited paper are incorrect: in particular, the characterization of when an algebra is ultra-amenable, in terms of a diagonal like construction, is not proved; and Theorem 5.7 is stated wrongly. The rest of the paper is unaffected. We shall show in this corrigendum that Theorem 5.7 can be corrected and that the other results of § 5 are true if the algebra in question has a certain approximation property.

scholarly journals NOTES ON EXTREMAL AND TAME VALUED FIELDS

2016 ◽  
Vol 81 (2) ◽  
pp. 400-416
Author(s):  
SYLVY ANSCOMBE ◽  
FRANZ-VIKTOR KUHLMANN
Keyword(s):  
Approximation Property ◽  
Elementary Theory ◽  
Residue Field ◽  
Axiom System ◽  
Complete Characterization ◽  
Valued Fields ◽  
Mixed Characteristic ◽  
Valued Field ◽  
Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

scholarly journals A new characterization of the bounded approximation property

2016 ◽  
Vol 7 (4) ◽  
pp. 672-677
Author(s):  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Approximation Property

Every analytic set is Ramsey

1970 ◽  
Vol 35 (1) ◽  
pp. 60-64 ◽  
Author(s):  
Jack Silver
Keyword(s):  
Measurable Cardinal ◽  
Infinite Subset ◽  
The Other ◽  
Natural Numbers ◽  
Ramsey Property ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Principal Theorem ◽  
Countably Infinite

If X is a set, [Χ]ω will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω]ω, we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ]ω ⊆ S or [Χ]ω ∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.The principal theorem of this paper is: Every Σ11 (i.e., analytic) subset of [ω]ω is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ21 Π21 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ21 ∩ Π21, rather directly gives a Σ21 ∩ Π21 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ21 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.

scholarly journals Fifty years of the spectrum problem: survey and new results

2012 ◽  
Vol 18 (4) ◽  
pp. 505-553 ◽  
Author(s):  
Arnaud Durand ◽  
Neil D. Jones ◽  
Johann A. Makowsky ◽  
Malika More
Keyword(s):  
Complexity Theory ◽  
Recursion Theory ◽  
Finite Model ◽  
Chronological Order ◽  
Descriptive Complexity ◽  
First Order ◽  
Structural Graph ◽  
Theoretic Problem ◽  
Bounded Complexity

AbstractIn 1952, Heinrich Scholz published a question in The Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser in turn asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached by means of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally by means of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open.

Recursion theory on orderings. II

1980 ◽  
Vol 45 (2) ◽  
pp. 317-333 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Large Class ◽  
General Model ◽  
Recursion Theory ◽  
Second Order ◽  
Splitting Theorem ◽  
Natural Numbers ◽  
Partial Orderings

In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q, ≤ 〉, the rationals under the usual ordering, and a large class of n-dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r-maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.

A characterization of jump operators

1988 ◽  
Vol 53 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Characterization Theorem ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Descriptive Set Theory ◽  
Jump Operator ◽  
General Study ◽  
General Topic ◽  
Descriptive Set ◽  
Increasing Function

The topic of this paper is jump operators, a subject which originated with some questions of Martin and a partial answer to them obtained by Steel [18]. The topic of jump operators is a part of the general study of the structure of the Turing degrees, but it is concerned with an aspect of that structure which is different from the usual concerns of classical recursion theory. Specifically, it is concerned with studying functions on the degrees, such as the Turing jump operator, the hyperjump operator, and the sharp operator.Roughly speaking, a jump operator is a definable ≤T-increasing function on the Turing degrees. The purpose of this paper is to characterize the jump operators, in terms of concepts from descriptive set theory. Again roughly speaking, the main theorem states that all jump operators (other than the identity function) are obtained from pointclasses by the same process by which the hyperjump operator is obtained from the pointclass Π11; that is, if Γ is the pointclass, then the operator maps the real x to the universal Γ(x) subset of ω. This characterization theorem has some corollaries, one of which answers a question of Steel [18]. In §1 we give a brief introduction to this general topic, followed by a brief (and still somewhat imprecise) description of the results contained in this paper.

scholarly journals Finite element modeling of nanoindentation on an elastic-plastic microsphere

2020 ◽  
Author(s):  
Jialian Chen ◽  
Hongzhou Li
Keyword(s):  
Finite Element ◽  
Mechanical Behavior ◽  
Computational Study ◽  
Theoretical Method ◽  
Finite Element Analyses ◽  
Structured Materials ◽  
Elastic Plastic ◽  
Element Simulation ◽  
Insight Into

Abstract The understanding of the mechanical indentation on a curved specimen (e.g., microspheres and microfibers) is of paramount importance in the characterization of curved micro-structured materials, but there has been no reliable theoretical method to evaluate the mechanical behavior of nanoindentation on a microsphere. This article reports a computational study on the instrumented nanoindentation of elastic-plastic microsphere materials via finite element simulation. The finite element analyses indicate that all loading curves are parabolic curves and the loading curve for different materials can be calculated from one single indentation. The results demonstrate that the Oliver-Pharr formula is unsuitable for calculating the elastic modulus of nanoindentation involving cured surfaces. The surface of the test specimen of a microsphere requires prepolishing to achieve accurate results of indentation on a micro-spherical material. This study provides new insight into the establishment of nanoindentation models that can effectively be used to simulate the mechanical behavior of a microsphere.

scholarly journals Alternating Turing machines and the analytical hierarchy

10.29007/t77g ◽  
2018 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Recursion Theory ◽  
Turing Machines ◽  
Arithmetical Hierarchy ◽  
Definition Of ◽  
Alternating Turing Machines ◽  
Insight Into ◽  
Computably Enumerable

We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy..The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

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