scholarly journals Solovay reducibility and continuity

2020 ◽  
Vol 12 ◽  
Author(s):  
Masahiro Kumabe ◽  
Kenshi Miyabe ◽  
Yuki Mizusawa ◽  
Toshio Suzuki
Keyword(s):  
Real Function ◽  
Computable Analysis ◽  
Real Numbers ◽  
Lipschitz Continuous ◽  
Complete Sets ◽  
Partial Randomness ◽  
Turing Reduction

The objective of this study is a better understandingof the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether thereexists a reducibility concept that corresponds to H¨older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H¨older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.

On manifolds containing a submanifold whose complement is contractible

1961 ◽  
Vol 57 (3) ◽  
pp. 507-515
Author(s):  
G. M. Kelly
Keyword(s):  
Quadratic Form ◽  
Critical Point ◽  
Unit Sphere ◽  
Real Function ◽  
Critical Point Theory ◽  
Compact Manifold ◽  
Point Theory ◽  
Critical Levels ◽  
Real Numbers ◽  
The Real

The problem discussed here arose in the course of some reflections on the critical point theory of Lusternik and Schnirelmann (4). In (4) it is shown how it is possible to associate, with a suitably differentifiable real-valued function f defined on a compact manifold M, a set of real numbers λ1 ≤ λ2 ≤ … λc, which are critical levels of f and which in certain respects are analogous to, and indeed generalizations of, the eigenvalues of a quadratic form. The number c depends on M and is called the category of M. If Rn is Euclidean n-space, Sn the unit sphere of Rn+1, and Pn the real projective n-space obtained from Sn by identifying opposite points, then a quadratic form φ in the (n + 1) coordinates of Rn+1 defines a real function on Sn and, by passage to the quotient, on Pn. Pn has category n + 1, and the numbers λ in this case are just the eigenvalues of the quadratic form.

Absolutely non-computable predicates and functions in analysis

2009 ◽  
Vol 19 (1) ◽  
pp. 59-71
Author(s):  
KLAUS WEIHRAUCH ◽  
YONGCHENG WU ◽  
DECHENG DING
Keyword(s):  
Topological Structure ◽  
Computable Analysis ◽  
Real Numbers ◽  
Simple Lemma ◽  
Measure Spaces ◽  
Computable Measure

In the representation approach (TTE) to computable analysis, the representations of an algebraic or topological structure for which the basic predicates and functions become computable are of particular interest. There are, however, many predicates (like equality of real numbers) and functions that are absolutely non-computable, that is, not computable for any representation. Many of these results can be deduced from a simple lemma. In this article we prove this lemma for multi-representations and apply it to a number of examples. As applications, we show that various predicates and functions on computable measure spaces are absolutely non-computable. Since all the arguments are topological, we prove that the predicates are not relatively open and the functions are not relatively continuous for any multi-representation.

scholarly journals Extensional constructive real analysis via locators

2020 ◽  
pp. 1-25
Author(s):  
Auke B. Booij
Keyword(s):  
Cauchy Sequence ◽  
Computable Analysis ◽  
Real Analysis ◽  
Additional Structure ◽  
Real Numbers ◽  
Decimal Expansion ◽  
Discrete Information

Abstract Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known. We overcome this by considering real numbers equipped with additional structure, which we call a locator. With this structure, it is possible, for instance, to construct a signed-digit representation or a Cauchy sequence, and conversely, these intensional representations give rise to a locator. Although the constructions are reminiscent of computable analysis, instead of working with a notion of computability, we simply work constructively to extract observable information, and instead of working with representations, we consider a certain locatedness structure on real numbers.

Cluster sets and topology

2018 ◽  
Vol 68 (6) ◽  
pp. 1465-1476
Author(s):  
Jacek Jędrzejewski ◽  
Stanisław Kowalczyk
Keyword(s):  
Real Function ◽  
Real Variable ◽  
Real Numbers ◽  
And Topology ◽  
Cluster Sets

Abstract Limit numbers (cluster sets) of a real function of a real variable were discussed in the literature by many authors. Generalizations of cluster sets were considered by distinctions of some classes of sets which generated some kind of limit. In general they were close to some topology on the set of real numbers. However not all such classes allowed to define a topology on ℝ in a simple way. We consider some topologies in ℝ generated by those classes of sets. We investigate a connection between limit numbers generated by those classes and limit numbers defined by a topology generated by a class 𝔅.

Two series inequalities

1979 ◽  
Vol 83 (1-2) ◽  
pp. 109-114 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Finite Difference ◽  
Infinite Series ◽  
Real Function ◽  
Real Numbers ◽  
The Real ◽  
Image Position

SynopsisIn 1932, Hardy and Littlewood proved the inequalitiesThe best possible values of the constants K1 and K2 being 1 and 4 respectively. The object of this paper is to prove analogous results for infinite series in which the derivative of the real function f is replaced by the finite differenceof a sequence {an} of real numbers.

Alan Turing and the Foundations of Computable Analysis

2011 ◽  
Vol 17 (3) ◽  
pp. 394-430 ◽  
Author(s):  
Guido Gherardi
Keyword(s):  
Computability Theory ◽  
Computable Analysis ◽  
Real Numbers ◽  
Machine Model ◽  
Alan Turing ◽  
Real Functions ◽  
The Subject

AbstractWe investigate Turing's contributions to computability theory for real numbers and real functions presented in [22, 24, 26]. In particular, it is shown how two fundamental approaches to computable analysis, the so-called ‘Type-2 Theory of Effectivity’ (TTE) and the ‘realRAM machine’ model, have their foundations in Turing's work, in spite of the two incompatible notions of computability they involve. It is also shown, by contrast, how the modern conceptual tools provided by these two paradigms allow a systematic interpretation of Turing's pioneering work in the subject.

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Dynamics of a family of orbitally continuous mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3507-3517
Author(s):  
Abhijit Pant ◽  
R.P. Pant ◽  
Kuldeep Prakash
Keyword(s):  
Fixed Points ◽  
Julia Sets ◽  
Real Numbers ◽  
Continuous Mappings ◽  
Non Linear

The aim of the present paper is to study the dynamics of a class of orbitally continuous non-linear mappings defined on the set of real numbers and to apply the results on dynamics of functions to obtain tests of divisibility. We show that this class of mappings contains chaotic mappings. We also draw Julia sets of certain iterations related to multiple lowering mappings and employ the variations in the complexity of Julia sets to illustrate the results on the quotient and remainder. The notion of orbital continuity was introduced by Lj. B. Ciric and is an important tool in establishing existence of fixed points.

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