Albert Einstein’s brain was studied long after the physicist’s death. The conclusions reached by researchers who studied his brain would be that the lower part of the parietal lobes was 15% more developed than normal. This “growth” could, therefore, have been related to Einstein’s ability to make equations. In fact, when asked about his intuitions […]
Let’s take a look at an unusual proof of the infinity of prime numbers. By the Fundamental Theorem of Arithmetic, we can write any number as the product of primes. For example, 45 = 5*3², and 100 = 2²5² A variation of this is that any number can be written as the product of a square-free number s and a square, r², […]
“I had never seen anything in the least like [it] before” — G.H. Hardy On or about the 31st of January 1913, mathematician G.H. Hardy (1877-1947) of Trinity College at Cambridge University received a parcel of papers from Madras, India. The package included a cover letter where a young clerk by the name of Srinivasa Ramanujan (1887–1920) provided an introduction of himself […]
We interact with the number π (‘pi’) everyday. Due to its fundamental involvement with circles, π has found applications in virtually every industry, ranging from sports to designing power lines and everything in between. It would, then, seem absurd that a number that is seemingly so specific and universal can be estimated using just random […]
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 of mathematical objects has evolved somewhat. I reproduce […]
Introduction to the most famous equation in finance The Black–Scholes model is a mathematical model simulating the dynamics of a financial market containing derivative financial instruments. Since its introduction in 1973 and refinement in the 1970s and 80s, the model has become the de-facto standard for estimating the price of stock options. The key idea behind the model is to hedge the options […]
History has moments defining the future for decades or even centuries. The collectiveness of many people triggers a countdown for the upcoming revolution of the world. Last was between the second and first half of the 19th and 20th centuries, at least in science. It might seem like our civilization has made progress since then, […]
Statistics Do Masses of Fundamental Particles Follow Benford’s Law? How a statistical test used to uncover tax frauds may help us find an explanation to a long-standing puzzle in particle physics Statistics Huber Loss: Why Is It, Like How It Is? On the day I was introduced to Huber loss by Michal Fabinger, the very first […]
When I tell people that I use mathematics and engineering to study and understand how the brain works, the reaction I get is oftentimes one of confusion. Despite the seemingly head scratching connection between the two, the reality is that we will never be able to understand how the brain works as a system without the […]
Mathematical intuitionism (which is a subset of mathematical constructivism) has it (or had it) that mathematics is purely the result of the mental activities of human beings; not the discovery of mathematical entities which exist in an “objective” (or Platonic) realm. In terms of L. E. J. Brouwer’s original intuitionism (i.e., of the first three decades […]
Symmetry Subgroups and Lagrange’s Theorem The nature of subsymmetries Group Theory Group Theory The nature of symmetry and the symmetry of nature Symmetry The Many Symmetries of Tessellations ft. Harry’s Tiles Tessellations Euler From Derangement To Euler’s Number How e can rise from combinatorics Fractal Fractal Geometry From Gaston Julia to Benoit Mandelbrot Group Theory […]
In the context of economics and game theory, envy-freeness is a criterion of fair division where every person feels that in the division of some resource, their share is at least as good as the share of any other person — thus they feel no envy. For n=2 people, the protocol proceeds by the so-called divide and choose procedure: The Envy-Free Cake-Cutting procedure […]
“No one will drive us from the paradise which Cantor created for us” — David Hilbert What better way to spend in isolation than to ponder the infinite? Let’s prove perhaps the simplest and most elegant proof in mathematics: Cantor’s Theorem. I said simple and elegant, not easy though! Part I: Stating the problem Cantor’s theorem […]
Mathematical Modelling The Time Evolution of a Particle’s Position To determine the position of a particle can be a very challenging task. The obvious way is to look at it and see where it is at that time. However, as one can expect, measuring the position of a particle at every instance of time is […]