Real numbers and functions in the Kleene hierarchy and limits of recursive, rational functions

1969 ◽  
Vol 34 (2) ◽  
pp. 207-214 ◽  
Author(s):  
N. Z. Shapiro
Keyword(s):  
Real Number ◽  
Convergent Sequence ◽  
Rational Functions ◽  
Rational Numbers ◽  
Real Numbers ◽  
Recursive Set

Let ƒ be a real number. It is well known [7] that the set of rational numbers which are less than ƒ is a recursive set if and only if ƒ is representable as the limit of a recursive, recursively convergent sequence of rational numbers. In this paper we replace the condition that the set of rational numbers less than ƒ is recursive by the condition that this set is at various points in the Kleene hierarchy, and we replace the recursive, recursively convergent limit by a variety of other recursive limiting processes.

Have You Read …?

1969 ◽  
Vol 62 (3) ◽  
pp. 220-221
Author(s):  
Philip Peak
Keyword(s):  
Real Number ◽  
Rational Number ◽  
Geometric Approach ◽  
Rational Numbers ◽  
Polynomial Equations ◽  
Real Numbers ◽  
Basic Principles ◽  
New Ideas

One of the basic principles we follow in our teaching is to relate new ideas with old ideas. Dr. Forbes has done just this in his article about extending the concept of rational numbers to real numbers. He points out how this extension cannot follow the same pattern as that of extensions positive to negative integers or from integers to rationals. If we look to a definition for motivating the extension we at best can only say, “Some polynomial equations have no rational number solutions and do have some real number solutions.” We might use least-upperbound idea, or we might try motivating through nonperiodic infinite decimals. However, Dr. Forbes rejects all of these and makes the tie-in through a geometric approach.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals ON THE SPECTRA OF PISOT NUMBERS

2011 ◽  
Vol 54 (1) ◽  
pp. 127-132 ◽  
Author(s):  
TOUFIK ZAIMI
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Pisot Number ◽  
Real Numbers ◽  
Pisot Numbers

AbstractLet θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all$\alpha \in \{ \theta ^{n}\mid n\in \mathbb{N}\} \cup \{ \sum\nolimits_{n=0}^{N}\theta ^{n}\mid \mathit{\}N\in \mathbb{N}\}$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such that$\Vert \lambda \alpha \Vert <\frac{1}{% 3}$for all$\alpha \in \{ \sum\nolimits_{n=0}^{N}a_{n}\theta ^{n}\mid$an ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

Series representing transcendental numbers that are not U-numbers

2015 ◽  
Vol 11 (03) ◽  
pp. 869-892
Author(s):  
Emre Alkan
Keyword(s):  
Rational Functions ◽  
Rational Numbers ◽  
Upper Bounds ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Type Series ◽  
Transcendental Numbers ◽  
Mathematical Constants

Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

scholarly journals Diophantine approximation with mixed powers of primes

2018 ◽  
Vol 14 (07) ◽  
pp. 1903-1918
Author(s):  
Wenxu Ge ◽  
Huake Liu
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers

Let [Formula: see text] be an integer with [Formula: see text], and [Formula: see text] be any real number. Suppose that [Formula: see text] are nonzero real numbers, not all the same sign and [Formula: see text] is irrational. It is proved that the inequality [Formula: see text] has infinitely many solutions in primes [Formula: see text], where [Formula: see text], and [Formula: see text] for [Formula: see text]. This generalizes earlier results. As application, we get that the integer parts of [Formula: see text] are prime infinitely often for primes [Formula: see text].

The Term and Stochastic Ranks of a Matrix

1959 ◽  
Vol 11 ◽  
pp. 269-279 ◽  
Author(s):  
N. S. Mendelsohn ◽  
A. L. Dulmage
Keyword(s):  
Real Number ◽  
Stochastic Matrix ◽  
Real Numbers ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
The Matrix ◽  
Term Rank ◽  
Square Matrix

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

Didactic Features of Studying the Real Numbers in the School Course of Mathematics

2017 ◽  
Vol 6 (1) ◽  
pp. 65-70
Author(s):  
Алексеенко ◽  
A. Alekseenko ◽  
Лихачева ◽  
M. Likhacheva
Keyword(s):  
Real Number ◽  
Complex Numbers ◽  
Algebraic Structures ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Clear Idea

The article is devoted to the study of the peculiarities of real numbers in the discipline &#34;Algebra and analysis&#34; in the secondary school. The theme of &#34;Real numbers&#34; is not easy to understand and often causes difficulties for students. However, the study of this topic is now being given enough attention and time. The consequence is a lack of understanding of students and school-leavers, what constitutes the real numbers, irrational numbers. At the same time the notion of a real number is required for further successful study of mathematics. To improve the efficiency of studying the topic and form a clear idea about the different numbers offered to add significantly to the material of modern textbooks, increase the number of hours in the study of real numbers, as well as to include in the school course of algebra topics &#34;Complex numbers&#34; and &#34;Algebraic structures&#34;.

Sign in / Sign up

Export Citation Format

Share Document