How e can rise from combinatorics The constant e has a significant role across mathematics from growth problems to compound interest and even eigenvalues problems. e has many representations, some of the most well known are some of many e representations If the above formulation seems unfamiliar or daunting to you, don’t get discouraged since it’s not gonna be relevant […]
Mathematics is often believed to bring people to madness. We hear many stories like those about Gödel, Cantor, Nash, and Grothendieck, describing geniuses haunted by insanity that is developing along with their mathematics. And there is something to it. A certain psychologist said that A paranoid person is irrationally rational. . . . Paranoid thinking is characterized not by […]
In a simple sense, mathematics may be viewed as an extremely useful tool. The English cosmologist, theoretical physicist and mathematician John D. Barrow, however, holds a slightly different view. That is, he posits the view known as Pythagoreanism, summing up the relationship between mathematics and physics in the following way: “By translating the actual into the numerical […]
Feynman “I would like to make a number of remarks on the relation of mathematics and physics” During Richard Feynman’s Messenger Lecture Series on “The Relation of Mathematics & Physics” held at Cornell University in 1965, “The Great Explainer” addressed what he found to be the key differences between mathematics and physics. His thoughts are summarized below. Differences in […]
In mathematics, it is often the case that we derive things through proof or previous results that we don’t fully understand. Allow me to explain in four simple, interesting examples. Part One: Euler’s Identity Euler’s Identity This is popularly known as Euler’s identity. If you ask anyone remotely familiar with the study of mathematics, they’ll […]
Logic Formal Logic vs. Material Logic? In a purely logical argument, even if the premises aren’t in any way (semantically) connected to the conclusion, the argument may still be both valid and sound. Professor Edwin D. Mares displays what he sees as a problem with purely formal logic when he offers us the following example Logic Lewis […]
Stochastic Modelling Calculate Hitting Times, Absorption Probabilities, and more 18 Jan 2021 — 12 min read Image by Pete Linforth from Pixabay. If you’re unfamiliar with Stochastic processes, then I would suggest that you read my introductory article on the topic, as it covers all the background needed to understand this next tutorial. If you’re already familiar with the […]
Number Theory An ancient proof to give insight into an unsolved problem. Photo by Tim Mossholder on Unsplash Perfect numbers have fascinated me for a long time. The concept is simple. Take any number and write out the numbers that divide it (not including itself). For example: 1,2, and 3 all divide 6 evenly. Now […]
In the Deep Learning (DL) age, more and more people have encountered and used (knowingly or not) random matrices. Most of the time this use is limited to the initialization of the networks weights, that can be accomplished with a single line of code in your favorite DL framework. However, Random Matrices have a rich […]
This is a gentle introduction to an important concept in combinatorics that plays a major role in algebra and geometry — root systems. Roots are just vectors in a real vector space with some special properties. A root system is a collection of roots. The special properties have to do with reflections and whether the roots […]
Seeing drones fly always makes you think how it can hold itself stable in the air? Multicopters and their wide-ranging applicability made them to one of the most innovative means of flight and transportation of various kinds. However, in the background run complex mathematical calculations to precisely follow control inputs with respect to environmental influences. […]
Over 2,300 years ago, Euclid proved the Fundamental Theorem of Arithmetic. Now it is our turn. Statement of the Theorem The Fundamental Theorem of Arithmetic states that we can decompose any number uniquely into the product of prime numbers. For example, 350 = 2*7*5², and there is no other way to write 350 as the product of […]
The twin prime conjecture states that: There are infinitely many twin primes A twin prime is a prime that differs from another prime by two. A set of two primes that differ by two are called a twin prime pair. The first twin prime pairs are: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (101, 103), (107, 109), […]
“I am Feynman.”“I am Dirac.” (Silence) Beloved late physicist Richard P. Feynman (1918–1988) first met his hero Paul Dirac (1902–1984) during Princeton University’s Bicentennial Celebration in 1946 and then again at least twice, in 1948 and 1962. Most notably, the two came to heads during the so-called Pocono Conferencewhen Feynman gave a lecture on a nascent “Alternative Formulation of Quantum […]