Inferability of recursive real-valued functions

On the inductive inference of recursive real-valued functions

This article reports on a study of high-school and of technologist students (prospective engineers and economists) understanding of real numbers. Our study was based on written response to a properly designed questionnaire and on interviews taken from students. The quantitative results of our experiment showed an almost complete failure of the technologist students to deal with processes connected to geometric constructions of incommensurable magnitudes. However, this didn’t prevent them in answering satisfactorily the other questions. The superiority of their correct answers with respect to those of high-school students, although the majority of them correspond to mediocre graduates of the secondary education, was evident in most cases. This is a strong indication that the age and the width of mathematical knowledge of the individual play an important role for the better understanding of the real numbers. The results of our experiment suggest also that the ability to transfer in comfort among several representations of real numbers helps students in obtaining a better understanding of them. A theoretical explanation about this is obtained through the adoption of the conceptual framework of dimensions of knowledge, introduced by Tirosh et al. (1998) for studying the comprehension of rational numbers. Following in part the idea of generic decomposition of the APOS analysis (Weller et al. 2009) we suggest a possible order for development of understanding the real numbers by students when teaching them at school. Some questions open to further research are also mentioned at the end of the paper. Keywords: Real, rational, irrational, algebraic and transcendental numbers, fractions, decimals, representations of real numbers.

A point of Euclidean 1-space R1 has just one coordinate x, and thus it should be denoted by (x), in accordance with the general terminology introduced in I.2.1. However, as a matter of convenience, we shall speak of the point x instead of the point (x). If this terminology is adopted, then the points of R1 are merely the real numbers, and if x′, x″ are two real numbers then their (Euclidean) distance is equal to ∣x″ - x′∣, the absolute value of their difference (see I.2.1). While, in general, there exists a significant conceptual difference between an object x and the set consisting of the single object x, it should be noted that in the present case we could operate directly with the space whose points are the real numbers, the distance of two real numbers x′, x″ being defined as ∣x″ - x′∣. The space so obtained is the real number-line of Analysis. In other words, the notational agreement of writing x instead of (x) has the effect that R1 becomes formally identical with the real number-line. Let now U be a non-empty subset of R1, and consider a transformation T: U → R1. Then T associates with every real number u ∈ U a definite real number x = T(u). Thus T may be thought of as a real-valued function x = T(u) of the real variable u ∈ U.

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well‐known non‐appropriateness of the representation to some basebby proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers.

A real number x is called semi-computable if it is a limit of an increasing or decreasing computable sequence (x n ), n∈ℕ of ratio­nal numbers. In this case, a later member of the sequence is always a better approximation to x in the sense that |x − x n | ≥ |x − x m |, if n ≤ m. As a natural generalization, we call a real number x k-monotone computable (k-mc, for short), for any real number k >0,, if there is a computable sequence (xn),n∈ℕ of rational numbers which converges to x k-monotonically in the sense that k · |x − x n | ≥ |x − x m | for any n ≤ m and x is monotonically computable (mc, for short) if it is k-mc for some k >0. Various properties of k-mc real numbers are discussed in this pa­per. Among others we show that a real number is computable if it is k-mc for some k <1; the 1-mc real numbers are just the semi-computable real numbers and the set of all mc real numbers is contained properly in the set of weakly computable real numbers, where x is weakly computable if there are semi-computable real numbers y, z such that x = y + z. Furthermore, we show also an infinite hierarchy of mc real numbers.

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

Classical Evolutionary Programming (CEP) and Fast Evolutionary Programming (FEP) have been applied to real-valued function optimisation. Both of these techniques directly evolve the real-values that are the arguments of the real-valued function. In this paper we have applied a form of genetic programming called Cartesian Genetic Programming (CGP) to a number of real-valued optimisation benchmark problems. The approach we have taken is to evolve a computer program that controls a writing-head, which moves along and interacts with a finite set of symbols that are interpreted as real numbers, instead of manipulating the real numbers directly. In other studies, CGP has already been shown to benefit from a high degree of neutrality. We hope to exploit this for real-valued function optimisation problems to avoid being trapped on local optima. We have also used an extended form of CGP called Embedded CGP (ECGP) which allows the acquisition, evolution and re-use of modules. The effectiveness of CGP and ECGP are compared and contrasted with CEP and FEP on the benchmark problems. Results show that the new techniques are very effective.

An inequality of the form (1.1) is established, where p , q p,q are real-valued functions on an interval [ a , b ) [a,b) of the real line, with − ∞ &gt; a &gt; b &gt; ∞ , p ( x ) &gt; 0 - \infty &gt; a &gt; b &gt; \infty ,p(x) &gt; 0 on [ a , b ) , μ 0 [a,b),{\mu _0} is a real number and f f is a complex-valued function in a linear manifold so chosen that all three integrals in (1.1) are absolutely convergent. The problem is singular in that while p − 1 ∈ L ( a , b ) {p^{ - 1}} \in L(a,b) we require q q to have a behavior at b b of such a form that q ∉ L ( a , b ) q \notin L(a,b) .

Let w be a real-valued function on R and k be a positive integer. If for every real number x, w(x + 2k) = rw(x + k) + sw(x) for some nonnegative real numbers r and s, then we call such function a second-order linear recurrent function with period k. Similarly, we call a function w : R → R satisfying w(x + 2k) = −rw(x + k) + sw(x) an odd secondorder linear recurrent function with period k. In this work, we present some elementary properties of these type of functions and develop the concept using the notion of f-even and f-odd functions discussed in [9]. We also investigate the products and quotients of these functions and provide in this work a proof of the conjecture of B. Sroysang which he posed in [19]. In fact, we offer here a proof of a more general case of the problem. Consequently, we present findings that confirm recent results in the theory of Fibonacci functions [9] and contribute new results in the development of this topic.

An asymptotic expansion for the total scattering phase in potential scattering is derived under minimal decay and regularity assumptions on the potential. The result is obtained from the representation in stationary scattering theory of the derivative of the total scattering phase with respect to energy proved in [9], combined with a perturbation argument. Let Hi, = denote the free Schrodinger operator in L2(R3). The following assumption on the potential is introduced: Assumption V(;3,m): Let m 2 0 be an integer and B > 0 a real number. Let V E Cm(R 3) be a real-valued function such that for some 8 > 0 and all multi-indices a with lal ::; m

On root arrangements for hyperbolic polynomial-like functions and their derivatives

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability on continuous data types. In the case of the real numbers we illustrate how computability can be expressed in the language of Σ-formulas.KeywordsTuring MachineExpressive PowerEquality TestAtomic FormulaPredicate SymbolThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

n= Zx,+ i=1 The principal tool in the proof is the Hardy-Littlewood circle method, incorporating results and techniques of a powerful new iterative method developed by Vaughan and Wooley ([Va3], [Va4], [VW1], [Wol], [VW2]). In section 2, an algorithm developed in [Fo] for optimizing the parameters in mixed power mean value theorems is generalized and analyzed. Section 3 details a more sophisticated method of generating mixed power mean value theorems, by a limited adaptation of the iterative method itself. The form of these estimates offers many advantages over those of section 2, and provides the key to the elimination of the 16th power from (1.1). These mean value theorems are then applied to the proof of Theorem 1 in section 4. The tools developed here are applicable to a wide range of mixed power representation problems, and we briefly illustrate in section 5 the application to the problem of determining the number of terms required to represent all large n, when the lowest power used is a kth power instead of a square. Throughout, n is a large natural number whose representation as a sum of mixed powers is at issue, and E is an arbitrarily small positive real number. Constants implied by the Landau Oand Vinogradov <<-symbols may depend on E or k. For a real number x, write e(x) for e27iX, [x] for the greatest integer not exceeding x and llxll for the distance from x to the nearest integer. Unless otherwise specified, lowercase Latin letters denote natural numbers and Greek letters denote real numbers. Let A(P, R) denote the set of natural numbers not exceeding P with no

Real numbers are the basic objects in analysis. For most non-mathematicians a real number is an infinite decimal fraction, for example π = 3●14159 .... Mathematicians prefer to define the real numbers axiomatically as follows: (ℝ, +,·,0,1, <) is, up to isomorphism, the only Archimedean ordered field satisfying the axiom of continuity [Die60]. The set of real numbers can also be constructed in various ways, for example by means of Dedekind cuts or by completion of the (metric space of) rational numbers. We will neglect all foundational problems and assume that the real numbers form a well-defined set ℝ with all the properties which are proved in analysis. We will denote the real line topology, that is, the set of all open subsets of ℝ, by τℝ.KeywordsReal NumberRational NumberComputable FunctionComputable SequenceAdmissible RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.