scholarly journals The Completeness of the Real Line

2007 ◽  
Vol 39 (117) ◽  
pp. 61-86
Author(s):  
Matthew E. Moore
Keyword(s):  
Real Line ◽  
Geometrical Structure ◽  
Real Numbers ◽  
The Real ◽  
Number Systems ◽  
Lines In Space ◽  
Scientific Hypothesis ◽  
Consensus View

It is widely taken for granted that physical lines are real lines, i.e., that the arithmetical structure of the real numbers uniquely matches the geometrical structure of lines in space; and that other number systems, like Robinson’s hyperreals, accordingly fail to fit the structure of space. Intuitive justifications for the consensus view are considered and rejected. Insofar as it is justified at all, the conviction that physical lines are real lines is a scientific hypothesis which we may one day reject.

scholarly journals 1,2,31,2,3, some inductive real sequences and a beautiful algebraic pattern

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sarsengali Abdygalievich Abdymanapov ◽  
Serik Altynbek ◽  
Anton Begehr ◽  
Heinrich Begehr
Keyword(s):  
Computer Algebra ◽  
Real Line ◽  
Recurrence Relations ◽  
Real Numbers ◽  
The Real

Abstract By rewriting the relation 1 + 2 = 3 {1+2=3} as 1 2 + 2 2 = 3 2 {\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}} , a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of right triangles with 1 2 + 2 2 = 3 2 {\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}} as initial one converging to the segment [ 0 , 1 ] {[0,1]} of the real line. The sequence of their hypotenuses forms a sequence of real numbers which initiates some beautiful algebraic patterns. They are determined through some recurrence relations which are proper for being evaluated with computer algebra.

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

Remarks on Quasi-Hermite-Fejér Interpolation

1964 ◽  
Vol 7 (1) ◽  
pp. 101-119 ◽  
Author(s):  
A. Sharma
Keyword(s):  
Real Line ◽  
Real Numbers ◽  
The Real ◽  
The Moment ◽  
Image Position

Let1be n+2 distinct points on the real line and let us denote the corresponding real numbers, which are at the moment arbitrary, by2The problem of Hermite-Fejér interpolation is to construct the polynomials which take the values (2) at the abscissas (1) and have preassigned derivatives at these points. This idea has recently been exploited in a very interesting manner by P. Szasz [1] who has termed qua si-Hermite-Fejér interpolation to be that process wherein the derivatives are only prescribed at the points x1, x2, …, xn and the points -1, +1 are left out, while the values are prescribed at all the abscissas (1).

Sets and Numbers

2021 ◽  
pp. 3-27
Author(s):  
James Davidson
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Euclidean Distance ◽  
Final Section ◽  
Real Analysis ◽  
Real Numbers ◽  
The Real ◽  
Basic Ideas ◽  
Distance Sets

This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance, sets of the real line, coverings, and compactness.

scholarly journals TOPOLOGICAL FULL GROUPS OF -ALGEBRAS ARISING FROM -EXPANSIONS

2014 ◽  
Vol 97 (2) ◽  
pp. 257-287 ◽  
Author(s):  
KENGO MATSUMOTO ◽  
HIROKI MATUI
Keyword(s):  
Real Line ◽  
Piecewise Linear ◽  
Unit Interval ◽  
Linear Functions ◽  
Discrete Groups ◽  
Piecewise Linear Functions ◽  
Real Numbers ◽  
The Real ◽  
Theoretical Property ◽  
Thompson Groups

AbstractWe introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

Mathematics Developed

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Real Number ◽  
Number System ◽  
Scientific Practices ◽  
Natural Numbers ◽  
Real Numbers ◽  
Sir Isaac Newton ◽  
The Real ◽  
Isaac Newton ◽  
Number Systems ◽  
Real Number System

This chapter considers two crucial shifts in mathematical knowledge: the natural numbers ℕ and the real number system ℝ. ℝ has proved to serve together with the natural numbers ℕ as one of the two core structures of mathematics; together they are what Solomon Feferman described as “the sine qua non of our subject, both pure and applied.” Indeed, nobody can claim to have a basic grasp of mathematics without mastery of the central elements in the theory of both number systems. The chapter examines related theories and conceptions about real numbers, with particular emphasis on the work of J. H. Lambert and Sir Isaac Newton. It also discusses various conceptions of the number continuum, assumptions about simple infinity and arbitrary infinity, and the development of mathematics in relation to the real numbers. Finally, it reflects on the link between mathematical hypotheses and scientific practices.

On the Total Variation of a Function

1981 ◽  
Vol 24 (3) ◽  
pp. 331-340
Author(s):  
B. S. Thomson
Keyword(s):  
Total Variation ◽  
Arbitrary Function ◽  
Real Line ◽  
Real Numbers ◽  
The Real ◽  
Variation Of A Function

There are a number of theories which assign to a function defined on the real line a measure that reflects somehow the variation of that function. The most familiar of these is, of course, the Lebesgue-Stieltjes measure associated with any monotonie function. The problem in general is to provide a construction of a measure from a completely arbitrary function in such a way that the values of this measure provide information about the total variation of the function over sets of real numbers and from which useful inferences can be drawn.

What's Between Q and R

1967 ◽  
Vol 60 (4) ◽  
pp. 308-314
Author(s):  
James Fey
Keyword(s):  
Mathematics Instruction ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Valuable Insight ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Number Systems ◽  
Insight Into ◽  
Structure Properties

Among the objectives of school mathematics instruction, one of the most important is to develop understanding of the structure, properties, and evolution of the number systems. The student who knows the need for, and the technique of, each extension from the natural numbers through the complex numbers has a valuable insight into mathematics. Of the steps in the development, that from the rational numbers to the real numbers is the trickiest.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

AN OSCILLATION FUNCTION ON THE REAL LINE

2000 ◽  
Vol 26 (1) ◽  
pp. 237
Author(s):  
Duszyński
Keyword(s):  
Real Line ◽  
The Real
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