Euler, the clothoid and

2016 ◽  
Vol 100 (548) ◽  
pp. 266-273 ◽  
Author(s):  
Nick Lord
Keyword(s):  
History Of Mathematics ◽  
Complex Function ◽  
Real Variable ◽  
Complex Function Theory ◽  
Subsequent Work ◽  
Definite Integrals ◽  
History Of ◽  
The Many ◽  
Image Position

One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title,On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recentGazettearticle [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

Pandora’s Blocks

2020 ◽  
pp. 11-30
Author(s):  
Jack Parkin
Keyword(s):  
Short History ◽  
Key Stakeholders ◽  
Political Economies ◽  
History Of ◽  
Technical Functions ◽  
Set Up ◽  
The Many

Chapter 1 opens the lid on Bitcoin so that all of its attributes, problems, and connotations come spilling out. At the same time, it pulls these disparate strands back into focus by outlining the many discrepancies examined in subsequent chapters. So while in some ways the chapter acts like a primer for cryptocurrencies, blockchains, and their political economies, the material laid out works to set up the book’s underlying argument: asymmetric concentrations of power inevitably form though processes of algorithmic decentralisation. In the process, a short history of Bitcoin introduces some of its key stakeholders as well as some of its core technical functions.

Painless Drilling Not Your Dentist, but the History of Mathematics

1980 ◽  
Vol 27 (8) ◽  
pp. 40-42
Author(s):  
Stephen Krulik
Keyword(s):  
History Of Mathematics ◽  
Theoretical Base ◽  
Body Of Knowledge ◽  
Practical Necessity ◽  
History Of ◽  
Points Of View ◽  
The Many ◽  
Drill And Practice

Many of our students are far from receptive to the many problems of drill materials with which some teachers provide them. And yet, most students need some drill and practice before they can successfully master a new concept or skill. The history of mathematics can play an important role in making these apparently contradictory points of view compatible. Many of the concepts and ideas in the history of mathematics were developed from practical necessity rather than from a theoretical base; it is these same ideas that offer a great deal of practice material for our students. This ancient body of knowledge can provide drill that is interesting, satisfying, challenging. And, at the same time, it offers the necessary drill to achieve competence in fundamental skills and concepts.

scholarly journals The Myth of Perfect Fidelity

2021 ◽  
pp. 16-43
Author(s):  
Melle Jan Kromhout
Keyword(s):  
Twentieth Century ◽  
Noise Reduction ◽  
Random Noise ◽  
History Of ◽  
Late Twentieth ◽  
Late Nineteenth ◽  
Sound Media ◽  
The Many ◽  
Late Twentieth Century

Chapter 1 gives a brief history of the noise of sound media from the late nineteenth to the late twentieth century, tracing the development of different concepts of noise in dialogue with and reaction to ever more complex and sophisticated technologies. It explores the many ways in which inventors, engineers, producers, and musicians have sought to prevent, reduce, and eliminate this noise. The chapter thereby draws the contours of a myth of perfect fidelity or the idea that reproduced sound can in principle be separated from the noise of the medium and complete similitude between original and copy can be achieved. This myth is illustrated by two examples of noise-related technologies: Dolby analog noise reduction, which actively reduces the noise of sound media, and the counterintuitive practice of “dithering” in digital recording, by means of which small amounts of random noise are introduced to reduce digitization errors.

Mathematics and Politics: Thomas Pynchon, against the Day

2018 ◽  
pp. 24-58
Author(s):  
Nina Engelhardt
Keyword(s):  
Thomas Pynchon ◽  
History Of Mathematics ◽  
First Century ◽  
Complex Numbers ◽  
Mathematical Concepts ◽  
Crisis Of Representation ◽  
History Of ◽  
Crisis Of Modernity ◽  
Concrete Concepts

Chapter 1 on Pynchon’s Against the Day focuses on interrelations between mathematics and politics as domains that are both shaken by crises of fundamental beliefs. It examines how Pynchon’s novel draws on the history of mathematics and on concrete concepts to explore the crisis of representation, the transformation of anarchism from political to artistic expression, and the possibilities inherent in imaginary domains. Main mathematical concepts and metaphors include imaginary and complex numbers and the ‘foundational crisis of mathematics’, which Against the Day establishes as producing a mathematics that is ‘an-archistic’ in terms of its loss of foundations and that forms part of the exploration of anarchism across the twentieth century. This chapter demonstrates the centrality of mathematics to Against the Day’s renegotiations of possibilities and responsibilities of the political and the literary, and it shows how the novel’s reimagining of modernism illustrates the relevance of mathematics in developing twenty-first-century responses to the crisis of modernity.

Historically Speaking—: Identification of Napier's Inequalities

1970 ◽  
Vol 63 (1) ◽  
pp. 67-71
Author(s):  
Sidney G. Hacker
Keyword(s):  
Mathematical Knowledge ◽  
History Of Mathematics ◽  
Real Variable ◽  
Numerical Resolution ◽  
Wide Range ◽  
History Of

The discovery of logarithms by John Napier (1550-1617) is a well known facet in the history of mathematics. His singular accomplishment in defining the logarith mic function of a real variable by providing a numerical description of it, over a wide range of its argument, at small intervals and to several (decimal) places, antedated by many yeara the development of funda mental concepts which the modern stu dent regards as necessary to achieve even the same limited goals. Napier success fully bridged, solely in regard to this function, these lacunae in the mathematical knowledge of bis day. It has long been of interest to identify the concepts which he intuitively invoked. This is not done, it should be clearly said, with any idea of assigning to him some kind of priority for them, but merely in the interests of a elearer appreciation of the ingenuity he displayed and the power of his methods. Two inequalities that he obtained are the key to his numerical resolution of the problem and his consequent table of logarithms. The analytical identification of these inequalities appears to have been overlooked. Before exhibiting this identi-fication we shall speak of the fundamental role that these inequalities played. In the interests of intelligibility we first recollect a few familiar facts regarding Napier's formulation of the problem.

The Many Uses of Algebraic Variables

1992 ◽  
Vol 85 (7) ◽  
pp. 557-561 ◽  
Author(s):  
Randolph A. Philipp
Keyword(s):  
Elementary School ◽  
Turning Point ◽  
History Of Mathematics ◽  
History Of ◽  
The Many ◽  
Experience Difficulty

The concept of variable is one of the most fundamental ideas in mathematics from elementary school through college (Davis 1964; Hirsch and Lappan 1989). This concept is so important that its invention constituted a turning point in the history of mathematics (Rajaratnam 1957). However, research indicates that students experience difficulty with the concept of variable, a difficulty that might partially be explained by the fact that within mathematics, variables can be used in many different ways (Rosnick 1981; Schoenfeld and Arcavi 1988; Wagner 1983).

The Fekete–Szegö coefficient functional problems for q-starlike and q-convex functions related with lemniscate of Bernoulli

2019 ◽  
Vol 12 (02) ◽  
pp. 1950019
Author(s):  
S. Sivasubramanian ◽  
P. Gurusamy
Keyword(s):  
Convex Functions ◽  
Complex Function ◽  
Univalent Functions ◽  
Complex Function Theory ◽  
Lemniscate Of Bernoulli ◽  
Definite Integrals ◽  
Near Future ◽  
Appl Math ◽  
Functional Problems ◽  
Mathematics And Physics

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].

Fancy

2020 ◽  
pp. 19-45
Author(s):  
Alexandra J. Finley
Keyword(s):  
Reproductive Labor ◽  
The Past ◽  
Enslaved Women ◽  
Boarding House ◽  
History Of ◽  
The Many

Chapter 1 tells the history of Corinna Hinton, an enslaved woman who had children with the man who enslaved her, slave trader Silas Omohundro, in Richmond, Virginia, to consider the lives of women caught in the so-called fancy trade, or the sale of enslaved women for sex. Omohundro paid Hinton for clothing the enslaved people he jailed; Hinton also ran the boarding house connected to his slave jail, cooking, sewing, doing laundry, and running errands for boarders. Hinton's experiences highlight the many forms that women's socially reproductive labor could take under capitalism: enslaved and free, waged and unwaged. The chapter draws primarily on financial records to uncover Hinton's life, highlighting the ways in which the archive limits the stories historians can and cannot write about the past.

The Hum of Progress: Motorcars and the Modernization of West Africa

2017 ◽  
Author(s):  
Lindsey B. Green-Simms
Keyword(s):  
World War Ii ◽  
West Africa ◽  
World War ◽  
History Of ◽  
The Many

Chapter 1 focuses on the history of motorization from the colonial moment to the decolonizing decades following World War II. Examining various events or episodes alongside key literary and cinematic texts, this chapter explores the many ambivalences and conflicts present in the process of motorization. The chapter also discuss how African entrepreneurs took the lead in importing automobiles and establishing a system of transportation while Europeans were often ambivalent or even hostile to the idea of motorizing Africa.

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