This story is also available on Kindle! Nash at the Institute for Advanced Study in 2011. (Photo: Serge J-F. Levy) “He was so incredibly himself” — Herta Newman Mathematician John Forbes Nash, Jr (1928-2015). was born in Bluefield, West Virginia in 1928. He died in a car crash in New Jersey on the 23rd of May, […]
In this article, we discuss a beautiful algorithm of Robert S. Boyer and J Strother Moore, published in 1981, that allows to compute the majority vote of a sequence of elements in linear (in the number of elements) time and constant space! Suppose that we have a very long sequence of integer numbers, appearing one by one. Our […]
“The examiner was intelligent enough to quickly quieten Gödel and say ‘Oh god, let’s not go into this’ and broke off the examination at this point, greatly to our relief” — Oskar Morgenstern Kurt Gödel (1906–1978) was the greatest logician who ever lived. At the age of 24, he revolutionized our understanding of the limits of […]
Math can actually be easy! Most of the time, the real problem is not that students aren’t smart enough, but that mathematical topics are taught in highly confusing ways. I have done a lot of math tutoring, and at some point I realized that once students see the bigger picture, things tend to fall into […]
Like many great questions in math, the Cycle Double Cover Conjecture (C.D.C.C.) has the distinction of being relatively easy to state, but quite difficult to prove. It was proposed independently by two mathematicians, George Szekeres [7] and Paul Seymour [6], in the 1970s. As of this writing, it has withstood all attempts at a solution. […]
Let’s prove a beautiful Theorem from complex analysis!! For the determined amateur with some knowledge of 12th grade math and calculus. Some prerequisites Here we give a feel for some of the key concepts underlying the proof. Differentiation of complex functions The Cauchy-Goursat Theorem is about the integration of ‘holomorphic’ functions on triangles. A holomorphic function is […]
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 […]