On the Computer

Down the rabbit hole on digital calculations. Computers, as programmable tools, trace their origins to the 1800s. Joseph Jacquard’s loom, which used punched cards for pattern storage, indirectly influenced the field (1800s). C. Babbage, inspired by the loom, built the Difference Engine in 1822 for mathematical calculations and later designed the Analytical Engine, the first computer with memory and programmability [2]. Lord Kelvin in the 1870s created an analog computer for tide calculation, notable for its accuracy and long-term use [3]. In some places, still used today for its simplicity and accuracy.

References

  • Jacquard, J. (1800s). Programmable loom development.
  • Babbage, C. (1822). Difference Engine and Analytical Engine.
  • Kelvin, Lord (1870s). Analog computer for tides.

Complex Polynomials

The Fundamental Theorem of Algebra states that the field of complex numbers is algebraically closed, implying that every polynomial equation of degree n has n roots within the complex numbers, with at least one being a solution where the polynomial evaluates to zero. Historically, the theorem’s origin traces back to the conjectures by Albert Girard in 1629 and René Descartes in 1637, though neither provided proofs including complex numbers. A first significant proof attempt was made by Jean d’Alembert (my personal hero) in 1746. The most notable advancements were made by Carl Friedrich Gauss, who, in 1799, offered a proof that, despite its gaps, marked a significant step forward in understanding the theorem. Gauss’s efforts were refined by Jean Robert Argand in 1814, who introduced an existence proof that set the stage for more concrete proofs, including those by Gauss himself in subsequent years using Euler’s earlier work. The theorem’s proof evolution highlights the gradual understanding and acceptance of complex numbers in solving polynomial equations, an understanding put further by Hellmuth Knesser (1940).

References

  • Girard, A. (1629). Invention Nouvelle en l’Algèbre.
  • Descartes, R. (1637). La Géométrie.
  • d’Alembert, J. (1746). Recherches sur le calcul intégral.
  • Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse.
  • Argand, J. R. (1814). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques.

Personal Equation

1796, Maskelyne, Astronomer Royal of Britain, dismissed his assistant for what he believed to be persistent inaccuracies in the timing of observations, specifically delays of approximately half a second. This decision unknowingly set the stage for the development of an important concept in measurement science: the personal equation. Maskelyne’s published their mixed observational results. After Maskelyne’s death, the issue was investigated further by F. Bessel, a German astronomer. Bessel’s research revealed a consistent measurable difference in the timing of observations made by different astronomers. He termed this difference the “personal equation,” highlighting its role in scientific measurements.

The concept of the personal equation stimulated significant research in the newly emerging field of psychology in the late 19th century. Researchers focused on understanding individual differences in reaction times, which could affect observational accuracy. The term personal equation entered general use, applied to any personal influence on objective situations. This illustrates how a specific scientific concept can expand beyond its original context to gain wider societal acceptance.

Important References

  • Astronomische Beobachtungen auf der Koniglichen Universitiits-Sternwarte F. W. Bessel, 8. Abtheilung voe 1. Januar bis 31. December, 1822.

Backus on IBM / Fortran

Much of my work has come from being lazy. I didn’t like writing programs, and so, when I was working on the IBM 701, writing programs for computing missile trajectories, I started work on a programming system to make it easier to write programs.

John Backus, 1979, Interview IBM Think Magazine

Origins of Graph Theory

Graph theory emerged in the 18th century, connecting geometry with fields like topology and set theory. Leonhard Euler formulated graph theory during his time in Königsberg, now Kalingrad. His seminal work began with the 1736 paper, “The Seven Bridges of Königsberg.”

Residents of Königsberg enjoyed crossing the city’s seven bridges in one outing – considered impossible, but not proven. Euler approached this challenge not by traditional geometry or algebra, but through an abstraction where cities and bridges became vertices and edges in a graph. This approach showed that a path crossing each bridge once was impossible if more than two vertices had an odd number of edges.

Euler’s insights created the foundation for graph theory, now used in various applications from computer algorithms for facial recognition to network and process design. Or perhaps a bit of frustration by myself in graduate school studying math in the physics department at Penn State. Crossing all the bridges became possible later on in Kalingrad due to war, as they were partly destroyed.

See

MSDOS 4 Source Code

Find myself going down the rabbit hole as MS posted their MSDOS assembly code on GITHUB. Initials JK appear through source code, which I’m not sure who it belongs to. My assembly programming is at best poor. It’s fun to read the commands like del.asm etc. DOS seemed to be behind the times in 1986, while groups at Bell labs etc. were ahead of the times with parallel or multi-thread programs.

Revisiting \(e\) History

\(e\), defined by the limit \(\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\) and approximating 2.71828, is characterized by its irrational and transcendental nature, indicating it cannot be depicted as a quotient of two integers nor as a solution to any non-trivial polynomial equation with rational coefficients.

John Napier, a Scottish mathematician and theologian, significantly contributed to the field of mathematics through his invention of logarithms, detailed in his seminal work “Mirifici Logarithmorum Canonis Descriptio,” (1614). This development facilitated the transformation of cumbersome multiplicative calculations into simpler additive operations. Napier’s logarithms laid the groundwork for the natural logarithm concept, where \(e\) emerged as the fundamental base that equates the value of the logarithm to unity.

One hundred years later, Jacob Bernoulli, Swiss, explored the concept of compound interest, leading to the discovery of \(e\)’s significance in financial mathematics. His investigations, as detailed in his posthumously published research “Ars Conjectandi,” (1713), demonstrated that continuous compounding yields a growth factor of \(e\), showing \(e\)’s utility in not only financial models but also in natural processes like population dynamics and radioactive decay.

References

  • Napier, John (1614), Mirifici Logarithmorum Canonis Descriptio.
  • Bernoulli, Jakob (1713), Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis.

Removal of Tenure at the University of Florida

A university cannot exist without academic freedom. Academic freedom is the core value under which American universities have operated for nearly the last century. It is outlined in multiple works, including the Chicago Principles and by the American Association of University Professors. Academic freedom is protected by the tenure system. Tenure, which was popularized in America, is the reason why the country today has so many excellent universities. Without tenure, a university is one in name only.

Less than two years ago, I wrote that I earned and was awarded tenure at University of Florida. See https://saemiller.com/2022/07/01/associate-professor-and-tenure/. When I came home last night, I realized that tenure no longer exists at University of Florida, due to the new policies being put in place by all administrators at the university.

Though I have tenure at the University of Florida, I believe it is in name only. It is a facade that covers an Iron Tower, that has transitioned from the Ivory Tower.

References:

Binary’s Origin

Binary numbers were originally used for encryption and communication, a fact recognized as early as the 17th century by Francis Bacon. Bacon used the binary system for encoding the alphabet using strings of binary characters. This laid the framework for subsequent developments in coded communication, such as technologies like the telegraph (Samuel Morse), which relied on a binary tones of ‘dots’ and ‘dashes.’ Binary’s mathematical formulation was created by Gottfried Wilhelm Leibniz, who recognized the system’s elegance. Leibniz’s research created a formal basis for binary arithmetic, outlining methods for converting between binary, decimal, and other number systems.

References:

  • Bacon, F. (1605), “The Advancement of Learning.” Book VI. London: Henrie Tomes.
  • Leibniz, G.W. (1703), “Explication de l’Arithmétique Binaire.” Mémoires de l’Académie Royale des Sciences, pp. 85-89. Paris.