Cluster sets and topology

2018 ◽  
Vol 68 (6) ◽  
pp. 1465-1476
Author(s):  
Jacek Jędrzejewski ◽  
Stanisław Kowalczyk
Keyword(s):  
Real Function ◽  
Real Variable ◽  
Real Numbers ◽  
And Topology ◽  
Cluster Sets

Abstract Limit numbers (cluster sets) of a real function of a real variable were discussed in the literature by many authors. Generalizations of cluster sets were considered by distinctions of some classes of sets which generated some kind of limit. In general they were close to some topology on the set of real numbers. However not all such classes allowed to define a topology on ℝ in a simple way. We consider some topologies in ℝ generated by those classes of sets. We investigate a connection between limit numbers generated by those classes and limit numbers defined by a topology generated by a class 𝔅.

On manifolds containing a submanifold whose complement is contractible

1961 ◽  
Vol 57 (3) ◽  
pp. 507-515
Author(s):  
G. M. Kelly
Keyword(s):  
Quadratic Form ◽  
Critical Point ◽  
Unit Sphere ◽  
Real Function ◽  
Critical Point Theory ◽  
Compact Manifold ◽  
Point Theory ◽  
Critical Levels ◽  
Real Numbers ◽  
The Real

The problem discussed here arose in the course of some reflections on the critical point theory of Lusternik and Schnirelmann (4). In (4) it is shown how it is possible to associate, with a suitably differentifiable real-valued function f defined on a compact manifold M, a set of real numbers λ1 ≤ λ2 ≤ … λc, which are critical levels of f and which in certain respects are analogous to, and indeed generalizations of, the eigenvalues of a quadratic form. The number c depends on M and is called the category of M. If Rn is Euclidean n-space, Sn the unit sphere of Rn+1, and Pn the real projective n-space obtained from Sn by identifying opposite points, then a quadratic form φ in the (n + 1) coordinates of Rn+1 defines a real function on Sn and, by passage to the quotient, on Pn. Pn has category n + 1, and the numbers λ in this case are just the eigenvalues of the quadratic form.

On the derivation of discontinuous functions

1939 ◽  
Vol 35 (3) ◽  
pp. 373-381
Author(s):  
D. R. Dickinson
Keyword(s):  
Real Function ◽  
Limit Point ◽  
Real Variable ◽  
Positive Direction ◽  
The Real ◽  
Discontinuous Functions ◽  
Set Of Points

Let f(x) be a real function of the real variable x, let P be any point lying on the graph of f(x) and let l be a ray from P making an angle θ (− π < θ ≤ π) with the positive direction of the x-axis. We say that θ is a derivate direction of f(x) at the point P if the ray l meets the graph of f(x) in a set of points having a limit point at P.

Almost Automorphic Integrals of Almost Automorphic Functions

1972 ◽  
Vol 15 (3) ◽  
pp. 433-436 ◽  
Author(s):  
M. Zaki
Keyword(s):  
Infinite Sequence ◽  
Real Variable ◽  
List Type ◽  
Real Numbers ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Automorphic Functions ◽  
Automorphic Integrals

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that(i) exits for every real t but no kind of uniformity of convergence is stipulated;(ii) exits for every t;(iii) for every t.

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

The law of the iterated logarithm for certain power series and generalized Nörlund methods

1996 ◽  
Vol 120 (4) ◽  
pp. 735-753 ◽  
Author(s):  
Rüdiger Kiesel
Keyword(s):  
Power Series ◽  
Random Variables ◽  
Real Numbers ◽  
Iterated Logarithm ◽  
Cluster Sets ◽  
The Law

AbstractLet (pn) be a sequence of real numbers with pn ~ R(n), R(.) a regulary varying function with index greater than −1/2. We prove the Hartman–Wintner law of the iterated logarithm for the corresponding (Jp) power series transform and generalized Nörlund transforms (Nβp) of sequences (Xn) of i.i.d. random variables with mean-zero and variance 1. We also identify the cluster sets.

scholarly journals Development of the Theory of the Functions of Real Variables in the First Decades of the Twentieth Century

2020 ◽  
Author(s):  
Loredana Biacino
Keyword(s):  
Twentieth Century ◽  
Calculus Of Variations ◽  
Real Function ◽  
Measure Theory ◽  
Real Variable ◽  
Italian Mathematicians ◽  
Lebesgue Theorem

In (Biacino 2018) the evolution of the concept of a real function of a real variable at the beginning of the twentieth century is outlined, reporting the discussions and the polemics, in which some young French mathematicians of those years as Baire, Borel and Lebesgue were involved, about what had to be considered a genuine real function. In this paper a technical survey of the arising function and measure theory is given with a particular regard to the contribution of the Italian mathematicians Vitali, Beppo Levi, Fubini, Severini, Tonelli etc … and also with the purpose of exposing the intermediate steps before the final formulation of Radom-Nicodym-Lebesgue Theorem and the Italian method of calculus of variations.

A Functional Equation for the Cosine

1968 ◽  
Vol 11 (3) ◽  
pp. 495-498 ◽  
Author(s):  
PL Kannappan
Keyword(s):  
Functional Equation ◽  
Measurable Function ◽  
Trivial Solution ◽  
Real Variable ◽  
Real Constant ◽  
Real Numbers ◽  
Measurable Solution ◽  
Image Position ◽  
Complex Valued ◽  
Cosine Functions

It is known [3], [5] that, the complex-valued solutions of(B)(apart from the trivial solution f(x)≡0) are of the form(C)(D)In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],(E)In this paper, the functional equation(P)where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.

scholarly journals Solovay reducibility and continuity

2020 ◽  
Vol 12 ◽  
Author(s):  
Masahiro Kumabe ◽  
Kenshi Miyabe ◽  
Yuki Mizusawa ◽  
Toshio Suzuki
Keyword(s):  
Real Function ◽  
Computable Analysis ◽  
Real Numbers ◽  
Lipschitz Continuous ◽  
Complete Sets ◽  
Partial Randomness ◽  
Turing Reduction

The objective of this study is a better understandingof the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether thereexists a reducibility concept that corresponds to H¨older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H¨older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.

scholarly journals On extensions of inequalities of Kolmogoroff and others and some applications to almost periodic functions

1972 ◽  
Vol 13 (1) ◽  
pp. 1-16 ◽  
Author(s):  
C. J. F. Upton
Keyword(s):  
Real Function ◽  
Real Line ◽  
Complex Function ◽  
Real Variable ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Almost Everywhere ◽  
Image Position

Let f(x) be a complex function of a real variable, defined over the whole real line, which possesses n derivatives (the nth at least almost everywhere) and is such that . Then, if k is any integer for which 0< k < n, Kolmogoroff's inequality may be written as,or, by putting ,The constant K=K (k, n) known explicitly and is the best possible, i.e., there is a (real) function for which equality holds (see Bang [1]).

The Duality Theorem for Curves of Order n in n-Space

1951 ◽  
Vol 3 ◽  
pp. 159-163 ◽  
Author(s):  
Douglas Derry
Keyword(s):  
Duality Theorem ◽  
Real Variable ◽  
Projective Line ◽  
Real Numbers

Let Cn be a curve in real projective n-space which is a continuous 1— 1 image of either the projective line or one of its closed segments. Consequently its points depend continuously on a real variable s for which , with the understanding that s = 0 and s = 1 represent the same curve point in the case that Cn is the image of the complete projective line. The points of Cn will be described by their corresponding real numbers s.

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