Kronecker Kronecker, God and the Integers Natural numbers were created by God, everything else is the work of men — Kronecker (1823–1891). Number Theory Perfect Numbers An ancient riddle Prime Numbers The Goldbach Conjecture An Equivalent Formulation of an Unsolved Mystery Number Theory The Euler Totient Function A Probabilistic Approach Number Theory Transcendental Numbers Into […]
What happens when logic goes wrong? Let’s see why the Law of the Excluded Middle is wrong (sometimes), discover foundational problems at the heart of Mathematics, before finally trying to resolve what’s going wrong. You’ll also learn how to make sense of the negative root of -1, delve briefly into analysis, as well as proving […]
Computational neuroscience, broadly defined, is the mathematical and physical modeling of neural processes at a chosen scale, from molecular and cellular to systems, for the purpose of understanding how the brain represents and processes information. The ultimate objective is to provide an understanding of how an organism takes in sensory information, how information is integrated […]
Proof Theory Methods of Proof for Every Occasion Three essential proof techniques, followed by many not-so-essential Topology Theorema Egregium and Pizza Eating How can Gauss improve your pizza eating technique? Statistics Maximum Likelihood Estimate Part 2: Mixture Models Probability Theory Maximum Likelihood Estimate Part 1: How to best fit a Gaussian Chaos Theory A Gentle […]
Most people may think that mathematics is about doing hard sums with large numbers. Whilst it is true that you can do seriously hard computations all day long (try computing the integral of the square root of tan), the majority of mathematics is based on proofs. Here I will outline some of the most common […]
Open Problem “Mathematics may not be ready for such problems.” The Collatz Conjecture was posed in 1937 by Lothar Collatz, two years after he received his Ph.D. It remains open to this day. “Mathematics may not be ready for such problems.” — Paul Erdős. “This is an extraordinarily difficult problem, completely out of reach of present day […]
I was recently listening to a podcast that was recommended to me, and during one episode which discussed big-O theory, the Monty Hall problem was mentioned. The commentators talked a little about it, at which point they admitted to not ‘getting it’ and moved on. This motivated me to share my understanding of the problem […]
If you have an elementary knowledge of physics, it is very possible that you are unsatisfied about the metaphoric explanations of the role of the Higgs boson in “giving mass to other particles”. On the other hand we do not explain gravity and electromagnetism in a similar way. We teach them using mathematical equations, whose […]
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 Complex Analysis Beyond Infinity Analytic Continuation and its Applications in Quantum Physics Quantum Physics Nature Isn’t Fuzzy: Quantum Uncertainty, Inconsistency and Paradox … And […]
Calculus Calculus and the Legend of the Founding of Carthage How modern calculus of variations solved an ancient math problem. Optimization The Method of Lagrange Multipliers How to solve constrained optimization problems. Geometry Why Trisecting the Angle is Impossible An accidental introduction to field theory. PDE Laplace’s Equation Flows and fields Relativity Thinking Relativistically Problem-solving […]
Complex Analysis The Beautiful World of Complex Functions Exploring Hidden Structure in Higher Dimensions Fractal (Not Only Frozen) Fractals All Around A Discussion of Fractals Fibonacci How to Find Numbers in Nature Understanding the ubiquitous patterns of the universe Fractal Mandelbulb: Three Dimensional Fractals Searching For The Mandelbrot In Higher Dimensions Fractal Fractal Geometry From […]
Imagine a square of paper lying flat on your desk. I ask you to close your eyes. You hear the paper shift. When you open your eyes the paper doesn’t appear to have changed. What could I have done to it while you weren’t looking? It’s obvious that I haven’t rotated the paper by 30 […]
The American theoretical physicist Richard Feynman is one of the best-known scientists in the world (see this link). Feynman made several contributions to theoretical physics. These include: a new formulation of quantum mechanics using path integrals, the theory of quantum electrodynamics (including the development of his famous pictorial representations known as Feynman diagrams), his contributions to the explanation of the phenomenon of superfluidity of […]
Solving the Basel problem, using an elementary introduction to Fourier series! Part 0: Motivation, some History, and Welcome! Hello and welcome! We are going to be tackling a beautiful problem, the ‘Basel Problem’. It was posed in 1650 by Pietro Mengoli, and it took 84 years to solve it, when the brilliant Euler did so. Euler’s proof was brilliant, but also not […]