The Mexican-born Israeli-American theoretical physicist J. Bekenstein was the first one to suggest that black holes, a region of spacetime where gravity is so strong that not even light can escape from it, should have a well-defined entropy. According to Bekenstein, one can define black-hole entropy as follows: “Black hole entropy is the amount of entropy that must be assigned to a black […]
«You don’t understand “ordinary people”. To you they are “stupid fools”» Entrepreneur Stephen Wolfram is a unique egg. By age 14, he had written three books on particle physics. He earned his Ph.D. at age 20 and began publishing research papers at the age of 18, some of which have been cited thousands of times. His software package Mathematica is in […]
The Euler totient function plays an important role in the field of number theory so I’ve decided to give a brief introduction to it in this article with some nice applications. Coprime Integers We say that an integer m divides n (or is a divisor of n) if n/m is a whole number and we write m|n. Furthermore, we say that two […]
“The job entailed such onerous duties that only someone with an iron constitution could survive. My constitution, it so happened, was not made by iron. It was made of reeds and bamboo sticks” The year is 1933. A 26 year-old former graduate student at Columbia University, Edgar R. Lorch (1907–1990) has just completed his Ph.D. in mathematics […]
The motion of an incompressible Newtonian fluid with viscosity μ, density ρ, and subject to hydrostatic pressure p and uniform gravity with acceleration g can be described as a velocity vector field V satisfying the Navier-Stokes equations: We use the plural form for this one equation because it represents three equations in vector form. The equations are named for Claude-Louis Navier (1785–1836) and Sir […]
We are analyzing the most hype topic of the modern days, which — to our mind — has created an absolutely unfounded outlook on our way of thinking having no scientific base. To write this article I have read a few books on AI (having tears because of a huge number of logical mistakes) and […]
The Dirichlet integral was first solved by the brilliant Leonhard Euler (1707–1783), it then achieved fame with Peter Gustav Lejeune Dirichlet (1805–1859)’s work on Fourier series. Now it is our turn! Part I: Approach The main difficulty we have with this integral is the pesky 1/x attached to the sin(x). Our strategy will be to ‘remove’ this, but introducing a […]
Last time, we discussed the dual nature of matter, in which particles behave like waves and waves behave like particles. To explain this we introduce the wave function, which describes not the actual position of a particle, but the probability of finding the particle at a given point. Furthermore, when we consider the wave function […]
A beginner’s guide to combinatorics, and using it to solve a rather tricky question. Let’s make sure we understand what the question means! (If you already understand the notation, you can skip to the section called ‘solving’) Factorials A useful bit of mathematical notation is the factorial notation; this is a number followed by an […]
Rainbows, who could remain untouched when run upon it? From ancient to nowadays times, it is an omen for an upcoming fortune, success, happiness, just positives, and no disasters. No surprise, though, because people feel good vibes during beautiful phenomena, they want to link them with heart-warming feelings. Some folk tales and legends go a […]
Optimization, one of the elementary problems of mathematical physics, economics, engineering, and many other areas of applied math, is the problem of finding the maximum or minimum value of a function, called the objective function, as well as the values of the input variables where that optimum value occurs. In elementary single-variable calculus, this is […]
Lord Kelvin wrote of this integral: “A mathematician is one to whom that is as obvious as that twice two makes four is to you.” Enjoy 😉 OK, so I’m going to assume you know some basic integration and differentiation. The following will add some intuition to clever tricks which come later. Don’t worry if […]
Philosophy Mathematics as Fiction: A Common Sense Approach The following essay was written in 2014 when I was an undergraduate student at the University of Florida; it was the winner of the mathematics department’s Robert Long Prize for writing in the history and/or philosophy of mathematics. In the intervening time, my views on ontological/epistemological status […]
This interesting math trick arises from an interesting empirical observation and the Fibonacci sequence. First, we should define the relationship between miles(mi) and kilometers(km): 1 mi = 1.60934 km and 1 km = 0.621371 mi Now, if this number looks familiar, it is because these numbers are extremely close to the golden ratio. By noticing […]