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.

Nonstandard Student Conceptions About Infinitesimals

2010 ◽  
Vol 41 (2) ◽  
pp. 117-146 ◽  
Author(s):  
Robert Ely
Keyword(s):  
Real Number ◽  
Nonstandard Analysis ◽  
Number Line ◽  
Real Numbers ◽  
Student Conceptions ◽  
The Real ◽  
New Perspective ◽  
Real Number Line ◽  
The Relationship

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

scholarly journals Collaborative calculation and application of interreal and real interval relations to protect privacy

PLoS ONE ◽  
2021 ◽  
Vol 16 (12) ◽  
pp. e0261213
Author(s):  
Shaofeng Lu ◽  
Yuefeng Lu ◽  
Ying Sun
Keyword(s):  
Real Number ◽  
Data Privacy ◽  
Basic Problem ◽  
Homomorphic Encryption ◽  
Real Interval ◽  
Real Numbers ◽  
Sign Function ◽  
The Real ◽  
The Relationship

The determination of the relation between a number and a numerical interval is one of the core problems in the scientific calculation of privacy protection. The calculation of the relationship between two numbers and a numerical interval to protect privacy is also the basic problem of collaborative computing. It is widely used in data queries, location search and other fields. At present, most of the solutions are still fundamentally limited to the integer level, and there are few solutions at the real number level. To solve these problems, this paper first uses Bernoulli inequality generalization and a monotonic function property to extend the solution to the real number level and designs two new protocols based on the homomorphic encryption scheme, which can not only protect the data privacy of both parties involved in the calculation, but also extend the number domain to real numbers. In addition, this paper designs a solution to the confidential cooperative determination problem between real numbers by using the sign function and homomorphism multiplication. Theoretical analysis shows that the proposed solution is safe and efficient. Finally, some extension applications based on this protocol are given.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

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

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

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;.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

Criteria of constructibility for real numbers

1953 ◽  
Vol 18 (1) ◽  
pp. 7-10 ◽  
Author(s):  
John Myhill
Keyword(s):  
Real Number ◽  
Lower Half ◽  
Real Numbers ◽  
The Real

The purpose of this paper is to prove two theorems and a conjecture (Conjecture II) announced in section 15 an earlier paper of the author's (cited as “CT”), and to compare them briefly with related results of Specker. Familiarity with both papers is assumed; the terminology of the former is used throughout. On two points however clarification of the usage of CT is in order, and to this chore we must first proceed.A half-section is the lower half of a Dedekind cut; if the cut is rational, the half section is to include the rational corresponding to the real defined by the cut. A whole-section is the relation which holds between any member of the lower and any member of the upper half of some Dedekind cut. If the cut is rational the corresponding rational is to be a member of both halves.A real number α is said to be approximate in K to any required number of decimal places if it is possible to define the predicates ‘x < α’, ‘x ≤ α’, ‘x > α’, and ‘x ≥ α’ (x rational) in K. In view of section 7 of CT this will mean that every true inequation between α and a terminating decimal will be provable in K.

scholarly journals The discrepancy of some real sequences

2003 ◽  
Vol 93 (2) ◽  
pp. 268
Author(s):  
H. Kamarul Haili ◽  
R. Nair
Keyword(s):  
Real Number ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Real Numbers ◽  
The Real ◽  
Almost Everywhere

Let $(\lambda_n)_{n\geq 0}$ be a sequence of real numbers such that there exists $\delta > 0$ such that $|\lambda_{n+1} - \lambda_n| \geq \delta , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ \lambda _0 x \}, \cdots , \{ \lambda _{N-1} x \}$. We show that given $\varepsilon > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + \varepsilon})9774 almost everywhere with respect to Lebesgue measure.

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