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 Homomorphisms of near-rings of continuous real-valued functions

1996 ◽  
Vol 53 (3) ◽  
pp. 401-411
Author(s):  
K.D. Magill
Keyword(s):  
Compact Hausdorff Space ◽  
Topological Structure ◽  
Hausdorff Space ◽  
Additive Group ◽  
Continuous Functions ◽  
Endomorphism Semigroup ◽  
Real Numbers ◽  
Extensive Class ◽  
Ring Of Functions

For any topological near-ring (which is not a ring) whose additive group is the additive group of real numbers, we investigate the near-ring of all continuous functions, under the pointwise operations, from a compact Hausdorff space into that near-ring. Specifically, we determine all the homomorphisms from one such near-ring of functions to another and we show that within a rather extensive class of spaces, the endomorphism semigroup of the near-ring of functions completely determines the topological structure of the space.

Positive-moment problems in abstract measure spaces

1975 ◽  
Vol 78 (3) ◽  
pp. 461-469
Author(s):  
H. P. Rogosinski
Keyword(s):  
Moment Problem ◽  
Measurable Function ◽  
Measure Space ◽  
Moment Problems ◽  
Real Numbers ◽  
Index Set ◽  
Almost Everywhere ◽  
Measure Spaces ◽  
Abstract Measure ◽  
Image Position

In this paper we continue the investigation of positive-moment problems, begun in (4). For an arbitrary index set A we consider a family (fα)α ∈ A of measurable real-valued functions on a measure-space (X, µ). We suppose throughout thatwhere (Xm) is an increasing sequence of measurable subsets of X and where, for each α in A and each m, fα is µ-integrable over Xm. Let (сα)α ∈ A be a given family of real numbers. We consider the following restricted positive-moment problem: does there exist a measurable function g on X such that 0 ≤° g ≤° 1 and such thatfor every α in A? (Here the symbol ‘≤°’ indicates that the relation ≤ holds almost everywhere with respect to µ on X. Symbols ‘ = °, <°, …’ are used similarly.) If such a g exists we call (сα)α ∈ A a moment family for the problem:

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.

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.

First order topological structures and theories

1987 ◽  
Vol 52 (3) ◽  
pp. 763-778 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Topological Structure ◽  
Complex Numbers ◽  
Restricted Class ◽  
Real Numbers ◽  
Boolean Combination ◽  
Topological Structures ◽  
First Order ◽  
Definable Set ◽  
Material Basis ◽  
Definable Sets

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis.In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

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.

Uncovering Atherosclerotic Risk Disease Gene Based on Expression and Network Topological Structure*

2010 ◽  
Vol 37 (8) ◽  
pp. 916-922
Author(s):  
Hong WANG ◽  
Xiao-Li QU ◽  
Yan ZHAO ◽  
Jing ZHANG ◽  
Li-Na CHEN
Keyword(s):  
Topological Structure ◽  
Disease Gene

Brief Survey of Biological Network Alignment and a Variant with Incorporation of Functional Annotations

2018 ◽  
Vol 14 (1) ◽  
pp. 4-10
Author(s):  
Fang Jing ◽  
Shao-Wu Zhang ◽  
Shihua Zhang
Keyword(s):  
Gene Ontology ◽  
Topological Structure ◽  
Metabolic Networks ◽  
Biological Network ◽  
Genomic Sequence ◽  
Network Alignment ◽  
Future Directions ◽  
Functional Annotations ◽  
Protein Protein Interaction ◽  
Ppi Networks

Background:Biological network alignment has been widely studied in the context of protein-protein interaction (PPI) networks, metabolic networks and others in bioinformatics. The topological structure of networks and genomic sequence are generally used by existing methods for achieving this task.Objective and Method:Here we briefly survey the methods generally used for this task and introduce a variant with incorporation of functional annotations based on similarity in Gene Ontology (GO). Making full use of GO information is beneficial to provide insights into precise biological network alignment.Results and Conclusion:We analyze the effect of incorporation of GO information to network alignment. Finally, we make a brief summary and discuss future directions about this topic.

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.

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