On the Definition of Computable Function of a Real Variable

1976 ◽  
Vol 22 (1) ◽  
pp. 391-402 ◽  
Author(s):  
J. C. Shepherdson
Keyword(s):  
Real Variable ◽  
Computable Function ◽  
Definition Of

scholarly journals Fractional Difference Equations with Real Variable

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Jin-Fa Cheng ◽  
Yu-Ming Chu
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Real Variable ◽  
Fractional Difference ◽  
Basic Properties ◽  
Fractional Difference Equations ◽  
Fractional Difference Equation ◽  
Definition Of

We independently propose a new kind of the definition of fractional difference, fractional sum, and fractional difference equation, give some basic properties of fractional difference and fractional sum, and give some examples to demonstrate several methods of how to solve certain fractional difference equations.

Computability and λ-definability

1937 ◽  
Vol 2 (4) ◽  
pp. 153-163 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Computable Function ◽  
General Recursive Function ◽  
Technical Details ◽  
Intuitive Idea ◽  
Short Space ◽  
Definition Of ◽  
Definable Functions ◽  
Positive Integers ◽  
Computable Functions ◽  
Definable Function

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ-K-definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ-K-definable function is computable; that every λ-definable function is λ-K-definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ-K-definability.A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

α‎-c.a. functions

2020 ◽  
pp. 23-54
Author(s):  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Computable Function ◽  
Order Type ◽  
Truth Table ◽  
Main Issue ◽  
Weak Truth Table Degrees ◽  
Definition Of

This chapter discusses the notion of α‎-c.a. functions. The main issue is to properly define what is meant by a computable function o from N to α‎, which is required for the definition of α‎-computable approximations. Naturally, to deal with an ordinal α‎ computably, one needs a notation for this ordinal, or more generally, a computable well-ordering of order-type α‎. To form the basis of a solid hierarchy, the notion of α‎-c.a. should not depend on which well-ordering one takes, rather it should only depend on its order-type. Thus, one cannot consider all computable copies of α‎. Rather, one restricts one's self to a class of particularly well-behaved well-orderings, in a way that ensures that they are all computably isomorphic. Having defined α‎-c.a. functions, the chapter relates these functions to iterations of the bounded jump (the jump inside the weak truth-table degrees).

Tangential properties of Fréchet surfaces

1958 ◽  
Vol 54 (2) ◽  
pp. 187-196 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Real Variable ◽  
Parametric Surfaces ◽  
Tangential Plane ◽  
The Subject ◽  
Definition Of ◽  
Variable Problem

The first two papers of Reifenberg ((4), (5)) under the general heading ‘Parametric Surfaces’ contain a detailed and profound study of the tangential properties of these surfaces. Since their publication one of the fundamental problems in the subject, that of obtaining a convenient representation for the surface, has been solved by Cesari (1). In this paper we obtain direct proofs of Reifenberg's results from Cesari's theorem. Whereas Reifenberg had to contend with both topological and real-variable problems combined the effect of Cesari's theorem is to remove the topological difficulties and to leave a straightforward real variable problem. The definition of approximate tangential plane used here is not the same as either of the two employed by Reifenberg, but the differences between it and one of Reifenberg's definitions ((5), definition 2) are not very important.

Computing and Deciding

2019 ◽  
pp. 21-30
Author(s):  
Giuseppe Primiero
Keyword(s):  
Decision Problem ◽  
Computable Function ◽  
Computability Theory ◽  
Definition Of

This chapter illustrates the basic tools of computability theory, essential to the formulation of the decision problem and the definition of the notion of computable function.

A Note on Measures Determined by Continuous Functions

1972 ◽  
Vol 15 (2) ◽  
pp. 289-291 ◽  
Author(s):  
A. M. Bruckner
Keyword(s):  
Continuous Function ◽  
Total Variation ◽  
Bounded Variation ◽  
Real Variable ◽  
Continuous Functions ◽  
Residual Subset ◽  
Zero Measure ◽  
Borel Measures ◽  
Left And Right ◽  
Definition Of

Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.

Some properties of convex functions

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 3015-3021
Author(s):  
Miloljub Albijanic
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Real Variable ◽  
First Derivative ◽  
Upper Limit ◽  
Definition Of

We treat two problems on convex functions of one real variable. The first one is concerned with properties of tangent lines to the graph of a convex function and essentially is related to the questions on the first derivative (if it exists). The second problem is related to Schwarz?s derivative, in fact its upper limit modification. It gives an interesting characterization of convex functions. Let us recall the definition of a convex functions.

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