Suppose your favorite brand of cereals is having a competition, where in each box of cereals there is a coupon with a positive integer number between 1 and N, and you win a prize if you collect coupons with all numbers between 1 and N. Moreover, when you buy a new box of cereals, the coupon inside the box independently from […]
Let’s wear our “city-planner’s” hats for this article. Assume that we are building a new city with n neighborhoods and m one-way roads between them. Being city planners, we have a passion for pizza, and we want to place a minimal amount of pizzerias such that from each pizzeria the delivery guy could drive to a certain neighborhood and go back as […]
Complex Analysis Beyond Infinity Analytic Continuation and its Applications in Quantum Physics Complex Analysis The Beautiful World of Complex Functions Exploring Hidden Structure in Higher Dimensions Riemann Hypothesis A Fun Proof of the Riemann Hypothesis Real math about a fictional object. Euler The Most Beautiful Equation in the World And the Geometry of Numbers Complex […]
If you were to go for a little gambling about what it is going to be about then, most likely, you would hit the jackpot. So why not to try? A croupier has opened bets, waiting for your one, spinning a wheel after a while. Over time, when the wheel’s speed decreased to a decent value, the […]
Game Theory Why politicians move to the center In microeconomics and game theory, the median voter theorem states that “In a majority rule voting system, the candidate/party most preferred by the median voter will be elected” In other words, the favorite candidate of the person in the middle of the probability distribution picks the winner of the […]
Today, we will discuss a beautiful result of Paul Erdős (1913–1996) and George Szekeres (1911–2005), proved in 1935, about finite sequences of real numbers. The inspiration of the result comes from Ramsey Theory, which, quoting the Wikipedia article, “Studies the conditions under which order must appear in relation to disorder.” Today’s problem can be very simply stated as the following game. […]
From Newton’s “Lion Claws” to the Modern Solution In June 1696, the renowned mathematician Johann Bernoulli published in the Acta Eruditorum, the first German scientific periodical, the following problem: “Given in a vertical plane two points A and B, assign to the moving [body] M, the path AMB, by means of which — descending by its own weight […]
Physics Beyond Noether’s Theorem: The Physics of Broken Symmetries Back in June, I wrote this article about Noether’s Theorem, the idea that continuous symmetries and conservation laws are intrinsically linked. But, while Noether’s Theorem is one of the most beautiful concepts in physics, symmetries themselves are equally as fundamental, and often similarly beautiful. And further, much of […]
On the day I was introduced to Huber loss by Michal Fabinger, the very first thing that came to my mind was the question: “How did someone joined these two functions in a mathematical way?”. Apart from that, the usage of Huber loss was pretty straightforward to understand when he explained. However, he left me with […]
Each triangle belongs to one of three groups about which membership its angles decide. These are the right, acute, and obtuse triangles. The law of sines holds within them all. Right triangles There is a right-angled triangle with the vertices A, B, C at which angles are α, β, γ, where α is the right […]
In physics, events characterized by a significant difference in size (for example, the behavior of waves in the oceans and of their individual water molecules) have a small influence on one another (they do not communicate). Therefore we can independently study the physical properties corresponding to each order of magnitude. This scale independence is precisely what allows us to […]
Graph Theory City Planning Using Graph Theory Strongly Connected Components Algorithm Pi Approximating Pi Using Needles A simple experiment to approximate π Euler From Derangement To Euler’s Number How e can rise from combinatorics Linear Algebra How Does Compression Work? SVD decomposition and a demo of an image compression. Open Problem The Simplest Open Problem […]
In 1915, mathematicians David Hilbert and Felix Klein came to Emmy Noether with a problem [1]. Einstein had published the field equations for his general relativity earlier that same year, and there seemed to be a worrying hole in the theory. Under certain circumstances, one of the most fundamental principles of physics was violated. Energy […]
Differential calculus was invented independently by Isaac Newton and Gottfried Leibniz and it was understood that the notion of the derivative of nth order, that is, applying the differentiation operation n times in succession, was meaningful. In a 1695 letter, l’Hopital asked Leibniz about the possibility that n could be something other than an integer, […]