scholarly journals On Locally Uniformly Differentiable Functions on a Complete Non-Archimedean Ordered Field Extension of the Real Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Khodr Shamseddine ◽  
Todd Sierens
Keyword(s):  
Inverse Function ◽  
Field Extension ◽  
Polynomial Functions ◽  
Inverse Function Theorem ◽  
Real Numbers ◽  
Intermediate Value Theorem ◽  
The Real ◽  
Ordered Field ◽  
Differentiable Functions ◽  
Intermediate Value

We study the properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are C1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

Most Infinitely Differentiable Functions are Nowhere Analytic

1973 ◽  
Vol 16 (4) ◽  
pp. 597-598 ◽  
Author(s):  
R. B. Darst
Keyword(s):  
Open Subset ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
First Category ◽  
Infinitely Differentiable Functions

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

COMPUTING STRENGTH OF STRUCTURES RELATED TO THE FIELD OF REAL NUMBERS

2017 ◽  
Vol 82 (1) ◽  
pp. 137-150 ◽  
Author(s):  
GREGORY IGUSA ◽  
JULIA F. KNIGHT ◽  
NOAH DAVID SCHWEBER
Keyword(s):  
Analytic Function ◽  
Real Numbers ◽  
Computing Power ◽  
The Real ◽  
The Third ◽  
Ordered Field ◽  
Image Position

AbstractIn [8], the third author defined a reducibility $\le _w^{\rm{*}}$ that lets us compare the computing power of structures of any cardinality. In [6], the first two authors showed that the ordered field of reals ${\cal R}$ lies strictly above certain related structures. In the present paper, we show that $\left( {{\cal R},exp} \right) \equiv _w^{\rm{*}}{\cal R}$. More generally, for the weak-looking structure ${\cal R}$ℚ consisting of the real numbers with just the ordering and constants naming the rationals, all o-minimal expansions of ${\cal R}$ℚ are equivalent to ${\cal R}$. Using this, we show that for any analytic function f, $\left( {{\cal R},f} \right) \equiv _w^{\rm{*}}{\cal R}$. (This is so even if $\left( {{\cal R},f} \right)$ is not o-minimal.)

Relative randomness and real closed fields

2005 ◽  
Vol 70 (1) ◽  
pp. 319-330 ◽  
Author(s):  
Alexander Raichev
Keyword(s):  
Real Number ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Closed Fields ◽  
Master's Thesis ◽  
National University

AbstractWe show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

scholarly journals Two constructions of the real numbers via alternating series

1989 ◽  
Vol 12 (3) ◽  
pp. 603-613 ◽  
Author(s):  
Arnold Knopfmacher ◽  
John Knopfmacher
Keyword(s):  
Continued Fractions ◽  
Rational Numbers ◽  
Equivalence Classes ◽  
Real Numbers ◽  
The Real ◽  
New Methods ◽  
Arbitrary Choice ◽  
Ordered Field

Two further new methods are put forward for constructing the complete ordered field of real numbers out of the ordered field of rational numbers. The methods are motivated by some little known results on the representation of real numbers via alternating series of rational numbers. Amongst advantages of the methods are the facts that they do not require an arbitrary choice of "base" or equivalence classes or any similar constructs. The methods bear similarities to a method of construction due to Rieger, which utilises continued fractions.

scholarly journals On the characterization of Jensen m-convex polynomials

2017 ◽  
Vol 3 (2) ◽  
pp. 140-148
Author(s):  
Teodoro Lara ◽  
Nelson Merentes ◽  
Roy Quintero ◽  
Edgar Rosales
Keyword(s):  
Convex Functions ◽  
Polynomial Functions ◽  
Real Polynomial ◽  
Real Numbers ◽  
The Real ◽  
Class Of Functions

AbstractThe main objective of this research is to characterize all the real polynomial functions of degree less than the fourth which are Jensen m-convex on the set of non-negative real numbers. In the first section, it is established for that class of functions what conditions must satisfy a particular polynomial in order to be starshaped on the same set. Finally, both kinds of results are combined in order to find examples of either Jensen m-convex functions which are not starshaped or viceversa.

Multiple Series Manipulations and Generating Functions

1977 ◽  
Vol 20 (1) ◽  
pp. 103-106 ◽  
Author(s):  
R. C. Grimson
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Inverse Function ◽  
Arbitrary Sequence ◽  
Real Numbers ◽  
Multiple Series ◽  
The Real ◽  
Increasing Function

Let γ be an increasing function on the real numbers such that γ(0) = 0 (which, by translation of axes, is no restriction) and suppose that γ(n) is a positive integer if n is a positive integer. Let γ- denote the inverse function of γ. Furthermore, let L(x) be the least integer ≥ x; let [x] be the greatest integer ≤x, and suppose that c0, c1 … is an arbitrary sequence of numbers.

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