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.

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.

scholarly journals Nonexistence of stable currents in submanifolds of a product of two spheres

1991 ◽  
Vol 44 (2) ◽  
pp. 325-336 ◽  
Author(s):  
Xue-shan Zhang
Keyword(s):  
Calculus Of Variations ◽  
Homology Group ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Vanishing Theorems ◽  
Geometric Measure ◽  
Integral Currents

By using techniques of the calculus of variations in geometric measure theory, we investigate the nonexistence of stable integral currents in Sn1 × Sn2 and its immersed submanifolds. Several vanishing theorems concerning the homology group of these manifolds are established.

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.

Introduction

2014 ◽  
Keyword(s):  
Twentieth Century ◽  
Number Theory ◽  
Algebraic Geometry ◽  
Dynamical Systems ◽  
Complex Analysis ◽  
Measure Theory ◽  
Classical Mechanics ◽  
Holomorphic Dynamics ◽  
Holomorphic Maps

This introductory chapter is an overview into holomorphic dynamics—one of the earliest branches of dynamical systems which is not part of classical mechanics. Holomorphic dynamics studies iterates of holomorphic maps on complex manifolds. As a prominent field in its own right, holomorphic dynamics was founded early in the twentieth century, but was almost completely forgotten for sixty years, only to be revived in the early 1980s partly due to the efforts of John Milnor. The field of holomorphic dynamics is rich in interactions with many branches of mathematics; such as complex analysis, geometry, topology, number theory, algebraic geometry, combinatorics, and measure theory. This chapter briefly explores the extent of such interplay.

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]).

scholarly journals Some Remarks on the Self-Exponential Function: Minimum Value, Inverse Function, and Indefinite Integral

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
J. L. González-Santander ◽  
G. Martín
Keyword(s):  
Closed Form ◽  
Exponential Function ◽  
Real Function ◽  
Inverse Function ◽  
Real Variable ◽  
The Self ◽  
Series Expansions ◽  
Minimum Value ◽  
Indefinite Integral

Considering the function xx as a real function of real variable, what is its minimum value? Surprisingly, the minimum value is reached for a negative value of x. Furthermore, considering the function fβx=x-βx, β∈R and x>0, two different expressions in closed form for the inverse function fβ-1 can be obtained. Also, two different series expansions for the indefinite integral of fβ and fβ-1 are derived. The latter does not seem to be found in the literature.

Hamiltonian Mechanics of Discrete Particle Systems

1994 ◽  
Author(s):  
Antony N. Beris ◽  
Brian J. Edwards
Keyword(s):  
Calculus Of Variations ◽  
Symplectic Structure ◽  
Mathematical Formulation ◽  
Real Variable ◽  
Simple Problem ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Dynamical Equations ◽  
Least Action ◽  
Principle Of Least Action

In the previous chapter, we discussed briefly the fundamental nature of the symplectic structure of theories in optics in order to illustrate the underlying uniformity, physical consistency, and mathematical simplicity inherent to a symplectic mathematical formulation of the governing equations. Hence the main emphasis of chapter 2 was to “discover” the symplectic structure in the physical theories of optics and to see how this structure is interconnected with and implies fundamental theorems in optics, such as Fermat's principle and Hamilton's equations. In the present chapter, we continue our efforts to present a coherent description of symplectic transformations and their applications to physical systems; however, here we switch our emphasis from the underlying symplectic structure of the dynamical equations to the physical integrity of the Poisson bracket and the canonical equations which find their roots in Hamilton's principle of least action and the calculus of variations. Hence we intend to cover ground in this chapter which we neglected in the previous one, and, in so doing, to gradually begin to move towards the applications of the extended bracket formalism at which this book is aimed. In order to apply Hamilton's principle of least action, we first need to study a simple problem of the calculus of variations, following Bedford [1985, §1.1]. Let x be a real variable (x∊ R) on the closed interval x1≤ x≤ x2, denoted [x1 ,x2] .

scholarly journals Análise de Praxeologias em Avaliação Formativa em Ambientes Virtuais na Construção delimites de uma Função Real de uma Variável RealAnalysis of practices in formative evaluation in virtual environments in the construction of the concept of limits of a real function of a real variable

2019 ◽  
Vol 21 (5) ◽  
Author(s):  
Osnildo Andrade Carvalho ◽  
Luiz Marcio Santos Farias ◽  
Itamar Miranda da Silva
Keyword(s):  
Virtual Environments ◽  
Real Function ◽  
Formative Evaluation ◽  
Real Variable

O presente trabalho é um recorte de uma pesquisa em andamento, e tem como objetivo analisar as práticas em avaliação formativa em aulas de cálculo diferencial e integral através de ambientes virtuais. Nosso foco é a incompreensão do conceito de limites, conceito este essencial para o curso de cálculo. Nossa lente teórica está alicerçada nos aportes da Teoria Antropológica do Didático; esta teoria apresenta elementos importantes para o desenvolvimento da pesquisa tais como as praxeologias, os ostensivos e não ostensivos. A nossa questão de investigação é como os estudantes reorganizarão as praxeologias relativas ao conceito de limites, numa organização didática, com a presença sistemática da avaliação formativa, tendo como suporte um ambiente virtual. Temos como hipótese que a atividade institucional somente se completa quando se dá ênfase à exploração dos momentos de trabalho da técnica e do tecnológico-teórico. A nossa abordagem metodológica está pautada na engenharia didática, que procura modelar teoricamente a investigação. E por fim, esperamos, com essa investigação, apresentar contribuições para o aprendizado de cálculo diferencial e integral bem como dialogar com pesquisas relacionadas com o tema.

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