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• Post #4520182

  Lobachevsky's integral formula \[\displaystyle \begin{aligned} \int_{0}^{\infty} \frac{\sin^{2}x}{x^{2}}\,f(x)\,dx \\[0.5em] = \int_{0}^{\infty} \frac{\sin x}{x}\,f(x)\,dx \\[0.5em] = \int_{0}^{\pi/2} f(x)\,dx \end{aligned} \]

• Post #4520181

  Analysis of the xG (expected goals) created and conceded per game for each R16 team, along with Germany and the Netherlands (R32). The statistics suggest that 🇪🇸 Spain's World Cup victory was driven primarily by defensive dominance rather than overwhelming attacking output. Clearly, they had the best balance between attacking output and defensive solidity (both efficient and defensive) throughout the tournament. They were technically the most complete team and thoroughly deserved to be c...

• Post #4520180

  Fields Medals by country, from 1936 to 2026 🥇🏅 China 🇨🇳 enters the top tier with 2 new medalists in 2026! #FieldsMedal #FieldsMedalists #Medalists #Medal #Medalist #Math #Mathematics #EastAsia #China

• Post #4520179

  This is unironically the mid-term future of research math. The next generations of models will have taste and the ability to invent new theories. Then math will be fully automated.

• Post #4520178

  Mathematics is the canary in the coal mine of any education system: It's a delicate pyramid that crumbles if the base is not strong. Using IMO results, here are the countries that improved most from 2006 to 2026 and those that declined the most. Using rank percentile within each year’s field: Most improved 🇸🇦 Saudi Arabia: 2.2% → 69.8% (+67.6pp) 🇲🇾 Malaysia: 22.5% → 83.6% (+61.1pp) 🇰🇬 Kyrgyzstan: 14.6% → 70.7% (+56.1pp) 🇧🇩 Bangladesh: 11.2% → 67.2% (+56.0pp) 🇮🇳 India: 61.8% → 94.8% (+33.0pp...

• Post #4520177

  🌪️ Turbulence: The Greatest Unsolved Problem in Classical Physics Despite centuries of research, turbulence remains one of the deepest mysteries in physics. From the chaotic wake behind an aircraft to swirling hurricanes, ocean currents, combustion, blood flow, and even the birth of stars, turbulent motion is everywhere, yet predicting it precisely remains extraordinarily difficult. This challenge has fascinated some of history's greatest scientific minds. Horace Lamb famously remarked...

• Post #4520176

  Some scientists whose lives ended before the age of 40: 🇫🇷 Évariste Galois — 20 🇳🇴 Niels Henrik Abel — 26 🇮🇳 Srinivasa Ramanujan — 32 🇬🇧 William Kingdon Clifford — 33 🇫🇷 Blaise Pascal — 39 🇩🇪 Bernhard Riemann — 39 🇫🇷 Sadi Carnot — 36 🇬🇧 Ada Lovelace — 36 🇩🇪 Heinrich Hertz — 36 🇷🇺 Alexander Friedmann — 37 🇫🇷 Augustin-Jean Fresnel — 39 🇬🇧 Rosalind Franklin — 37 🇮🇹 Ettore Majorana — 31

• Post #4520175

  Neural networks train by constructing a computational graph in the forward pass, chaining basic operations such as x × y × z into composite functions of the inputs. Derivatives of the output with respect to every input are then obtained in the backward pass by applying the chain rule at each node, where local gradients multiply: ∂f/∂y = ∂f/∂(x × y) × ∂(x × y)/∂y and ∇ₓz = ∇ₓy · ∇ᵧz, with intermediate values stored along the graph. This process updates the parameters of convolutional networks t...

• Post #4520174

  Green's Function ✍️ It explains how mathematicians and physicists solve complex equations by first understanding the response to the simplest possible disturbance. Imagine tapping the surface of a calm pond with a single pebble. Ripples spread outward in every direction, carrying information about how the water reacts to that one tiny push. Once this basic response is known, the effect of any larger or more complicated disturbance can be built by combining countless such ripples. A Gree...

• Post #4520173

  Brilliant Minds Born in Budapest 🇭🇺 Budapest 🇭🇺 is home to barely two million people today, yet its intellectual legacy has shaped the modern world far beyond its size. So many extraordinary scientists emerged from Budapest in the early twentieth century that fellow physicists jokingly invented the "Martians" hypothesis: the only explanation, they said, was that these brilliant Hungarians were actually visitors from Mars, hiding in plain sight, speaking a strange language (Hung...

• Post #4520172

  On May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model, one not available to the public, had come up with a counterexample to the “unit distance” problem, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician. Erdős posed thousands of questions, but this one was special: It was both simple to explain and mathematically deep. It was the first historically significant proof to come from an AI model. Though the model’s...

• Post #4520171

  Erdős problems resolved by year, 1930–2026.

• Post #4520170

  Niels Abel was born 224 years ago, on August 5, 1802. At just 21, he proved that no general algebraic formula can solve equations of degree 5, a problem that had challenged mathematicians for roughly 250 years. Short on money, Abel paid the printer himself and condensed his proof into just six pages to keep costs down. He sent a copy to Gauss, but never received a response. Abel died of tuberculosis at just 26. Two days later, a letter arrived announcing that he had been appointed professor in...

• Post #4520169

  Hilbert wakes up 1,000 years later. “Has the Riemann hypothesis been proven?” “Yes. But there’s something else you should know…” It wasn’t proven by a human.

• Post #4493397

  This is the part of 2FA/TOTP that many people don’t realize: Your phone isn’t receiving the 6-digit code from the server. Instead, your authenticator app acts like a specialized cryptographic calculator. 🧮🔐 It takes a shared secret key, combines it with the current time, and applies the TOTP algorithm to generate a temporary 6-digit code. At the same time, the server independently performs the same calculation using its copy of the secret key and the same time counter. Same secret + same ti...

• Post #4118527

  John Tukey the Fast Fourier Transform (FFT) inventor, who coined both "bit" and "software" died exactly 26 years ago today. The Fast Fourier Transform (FFT), one of the most important algorithms in signal processing and data analysis, was introduced by Tukey & Cooley in 1965. In 1805, Gauss - studying the orbits of asteroids Pallas and Juno - came up with a method to interpolate their trajectories from discrete samples. What he came up with was mathematically very close...

• Post #2061622

  The hierarchy of mathematical spaces These diagrams illustrate the hierarchical relationships and nesting of various mathematical spaces used in functional analysis and geometry. They demonstrate how specific structures, such as Hilbert and Banach spaces, are specialized subsets of broader categories like normed linear spaces and metric spaces. The visuals highlight the essential properties required for each classification, ranging from basic topological sets to complex systems involving inner...

• Post #1035197

  Mathematics behind Artemis II

• Post #1035196

  Noether’s Theorem ✍️ This equation reveals that every continuous symmetry in nature, a change you can make to a system without affecting its physical laws, brings about a conservation law. In simple terms, if the universe does not react to a certain change in perspective, it must keep a related physical quantity constant. For example, since the laws of physics remain unchanged no matter when you are (Time Symmetry), energy is conserved. Since the laws are the same regardless of where you are (Sp...

• Post #715765

  I wasn't ready for it, lol. The book: Introduction to Linear Algebra by Gilbert Strang, the sixth edition