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.

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).

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.)

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.

Multiplicative norms on Banach algebras

1951 ◽  
Vol 47 (3) ◽  
pp. 473-474 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Banach Algebras ◽  
Real Field ◽  
Complex Numbers ◽  
Normed Algebra ◽  
Real Numbers ◽  
The Real ◽  
Image Position ◽  
Real Quaternions

1. Mazur(1) has shown that any normed algebra A over the real field in which the norm is multiplicative in the sense thatis equivalent (i.e. algebraically isomorphic and isometric under one and the same mapping) to one of the following algebras: (i) the real numbers, (ii) the complex numbers, (iii) the real quaternions, each of these sets being regarded as normed algebras over the real field. Completeness of A is not assumed by Mazur. A relevant discussion is given also in Lorch (2).

On product sets in a unimodular group

1968 ◽  
Vol 64 (4) ◽  
pp. 1001-1007 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Finite Measure ◽  
Real Numbers ◽  
Locally Compact ◽  
The Real ◽  
Unimodular Group ◽  
Image Position ◽  
Product Sets

Let G be a locally compact Hausdorff topological group, with µ the left Haar measure on G, and µ* the corresponding inner measure. If R denotes the real numbers, ℬ(G) denotes the Borel† subsets of G of finite measure, and VG = {µ(E):E∈ℬ(G)}, then, following Macbeath(4), we define ΦG: VG × VG→ R bywhere AB denotes the product set of A and B in G.

Further results on infinite valued predicate logic

1964 ◽  
Vol 29 (2) ◽  
pp. 69-78 ◽  
Author(s):  
L. P. Belluce
Keyword(s):  
Real Number ◽  
Present Author ◽  
Closed Interval ◽  
Abelian Groups ◽  
Predicate Logic ◽  
Real Numbers ◽  
The Real ◽  
Truth Values ◽  
Recursively Enumerable ◽  
Weakened Form

In a previous paper [1] Chang and the present author presented a system of infinite valued predicate logic, the truth values being the closed interval [0, 1] of real numbers. That paper was the result of an investigation attempting to establish the completeness of the system using the real number 1 as the sole designated value. In fact, we fell short of our mark and proved a weakened form of completeness utilizing positive segments, [0, a], of linearly ordered abelian groups as admissible truth values. A result of Scarpellini [8], however, showing that the set of well-formed formulas of infinite valued logic valid (with respect to the sole designated real number 1) is not recursively enumerable indicates the above mentioned result is the best possible.

Two series inequalities

1979 ◽  
Vol 83 (1-2) ◽  
pp. 109-114 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Finite Difference ◽  
Infinite Series ◽  
Real Function ◽  
Real Numbers ◽  
The Real ◽  
Image Position

SynopsisIn 1932, Hardy and Littlewood proved the inequalitiesThe best possible values of the constants K1 and K2 being 1 and 4 respectively. The object of this paper is to prove analogous results for infinite series in which the derivative of the real function f is replaced by the finite differenceof a sequence {an} of real numbers.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

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