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.)

COMPARING TWO VERSIONS OF THE REALS

2016 ◽  
Vol 81 (3) ◽  
pp. 1115-1123
Author(s):  
G. IGUSA ◽  
J. F. KNIGHT
Keyword(s):  
Residue Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Computing Power ◽  
Ordered Field ◽  
Image Position ◽  
Recursively Saturated

AbstractSchweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$. We prove that ${\cal R}$ is not reducible to ${\cal W}$. As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

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

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.

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).

A theorem about infinite-valued sentential logic

1951 ◽  
Vol 16 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Robert McNaughton
Keyword(s):  
Logical Formula ◽  
Real Numbers ◽  
The Real ◽  
Truth Values ◽  
Sentential Logic ◽  
Usual Manner ◽  
Image Position

In this paper we shall use a logic with truth values ranging over all the real numbers x such that 0 ≦ x ≦ 1.1 will be “complete truth” and 0 will be “complete falsity.” The primitive sentential connectives are ‘⊃’ and ‘∼’; other connectives are ‘∨’ and ‘·’. Assume that ‘p’ and ‘q’ are sentential variables, whose truth values are respectively x and y. Then1.1. ‘p ⊃ q’ has the value min(1 − x + y, 1),1.2. ‘∼p’ has the value 1 − x,1.3. ‘p∨q’ has the value max(x, y), and1.4. ‘p·q’ has the value min (x, y).‘∨’ and ‘·’ can be defined as follows:It is the purpose of this paper to prove a theorem which will be stated in the next section. The following symbolism and convention will be used throughout the paper:S is a logical formula.ν (S) is the value of S.‘p’, ‘pi1, ’p2, …, ‘q’, are sentential variables.ν(p) = x and ν(x1) = x1, etc.ν(S) = σ and ν(S1) = σ1, etc.If S contains the sentential variables ‘p1’, ‘p2’, …, then we write for S, S(p1, P2, …). Also ν{S(p1, p2, …)) = σ(x1, x2, …).A logical formula is defined in the usual manner. 1. A sentential variable is a logical formula; 2. if S is a logical formula then ·S is a logical formula; and 3. if S and S′ are logical formulae then (S ⊃ S′) is a logical formula.

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 Universality probability of a prefix-free machine

2012 ◽  
Vol 370 (1971) ◽  
pp. 3488-3511 ◽  
Author(s):  
George Barmpalias ◽  
David L. Dowe
Keyword(s):  
Turing Degree ◽  
Real Numbers ◽  
Arithmetical Hierarchy ◽  
Halting Problem ◽  
The Real ◽  
The Third ◽  
Free Machine

We study the notion of universality probability of a universal prefix-free machine, as introduced by C. S. Wallace. We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers that are universality probabilities of universal prefix-free machines.

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 Hausdorff dimension of certain solenoids

1995 ◽  
Vol 15 (3) ◽  
pp. 449-474 ◽  
Author(s):  
H. G. Bothe
Keyword(s):  
Hausdorff Dimension ◽  
Dense Subset ◽  
Solid Torus ◽  
Cantor Set ◽  
Real Numbers ◽  
The Real ◽  
Mapping Degree ◽  
Image Position ◽  
Expanding Mapping

AbstractFor the solid torus V = S1 × and a C1 embedding f: V → V given by with dϕ/dt > 1, 0 < λi(t) < 1 the attractor Λ = ∩i = 0∞fi(V) is a solenoid, and for each disk D(t) = {t} × (t ∈ S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by where the real numbers pi are characterized by the condition that the pressure of the function log : S1 → ℝ with respect to the expanding mapping ϕ: S1 → S1 becomes zero. (There are two further characterizations of these numbers.)It is proved that (0.1) holds provided λ1, λ2 are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the C1 space of all embeddings f with sup λi > Θ−2 (Θ the mapping degree of ϕ: S1 → S1) all those f which have an intrinsically transverse attractor Λ form an open and dense subset.

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