SAR @RMHCASE

New book chapter: "How to Weigh a Cell." This chapter explains how scientists have weighed cells throughout history, often using simple equipment and back-of-the-envelope calculations. It has lots of interactives, so you can "repeat" the experiments directly in your browser.

Most engineers have seen this formula. P(A|B) = P(B|A) × P(A) / P(B) Almost none can explain what it actually does. Here's Bayes' Theorem in plain English, and where it's hiding inside systems you use every day. The core idea in one sentence: Bayes' Theorem updates your belief about something after seeing new evidence. That's it. Four terms: Prior → what you believed before the evidence Likelihood → how probable the evidence is, given your hypothesis Evidence → how common the evidence is overall Posterior → your updated belief after seeing the evidence A concrete example: Say 40% of all emails are spam (your prior). You see a new email containing the word "lottery." 10% of spam emails contain "lottery." Only 1% of legitimate emails do. Plug into Bayes: P(spam | "lottery") = (0.10 × 0.40) / P("lottery") ≈ 87% The word "lottery" updated your belief from 40% → 87%. That's Bayes in action. Prior belief + new evidence = updated belief. Where it lives in AI: 1/ Spam filters The Naive Bayes classifier, the algorithm behind most spam filters - applies this exact calculation word by word across an entire email. Each word shifts the probability up or down. It's called "naive" because it assumes each word is independent of the others, which isn't realistic, but works remarkably well in practice. 2/ Medical diagnosis AI A patient has symptom X. What's the probability of disease Y? Bayes updates the base rate (how common the disease is) with the likelihood of seeing that symptom in patients who have it. Same formula, different domain. 3/ Your LLM's uncertainty Modern language models don't just predict the next token, they assign a probability to every possible token. The sampling process (temperature, top-p) is directly working with those probability distributions. Bayesian reasoning is embedded in every response your model generates. The insight most engineers miss: Bayes doesn't give you certainty. It gives you a rational way to update uncertainty. That's exactly why it's foundational to AI - real-world systems are never certain. They're always working with incomplete, noisy, probabilistic information. Every model that learns from data is, at its core, doing some version of this: Start with a belief. See evidence. Update the belief. That's Bayes. That's machine learning.

What if an AI could update its belief... every time it sees new evidence? That concept behind it is called Bayes’ Theorem. At first, it may look like just another probability formula. But in reality, it is one of the most important ideas behind intelligent decision-making. Don't get scared. It's easy to understand... Bayes’ Theorem helps us answer a simple question: “How should we change our prediction when new information arrives?” Imagine a medical test. A person tests positive for a disease. Does that definitely mean the person is sick? Not necessarily. Bayes’ Theorem Works With: - prior belief - new evidence - probability of correctness to calculate a more realistic answer. That is what makes it powerful. It does not blindly trust new data. It updates belief step-by-step. You can think of it as learning from evidence. And that idea appears everywhere in AI & machine learning. For example: - Spam filters use Bayesian probability - Recommendation systems - Self-driving systems continuously update - environmental understanding - Medical AI models revise diagnosis probabilities - LLMs also work through probabilistic prediction mechanisms underneath The interesting part is this: Intelligence is often not about being 100% certain. It is about updating beliefs correctly when new information appears. Step-by-step intuition: - Start with an initial belief - Observe new evidence - Measure how likely that evidence is - Update the belief mathematically - Repeat continuously as more data arrives Thomas Bayes introduced this idea centuries ago. But today, modern AI systems silently depend on this way of thinking every second. Sometimes the smartest systems are not the ones that “know” everything. They are the ones that continuously learn from uncertainty.

@TheMathFlow I want upload this vidéo please !!

What's really inside a root canal?

Anthropic engineer showed how one person can run 5 AI agents, that code, test, review, and deploy at the same time. In 30 minutes they built the whole thing live in one session. Here's what they cover: > when to use one agent vs a full team > how to split work so agents don't step on each other > the exact framework for deciding what each agent handles that's exactly why, I put together a guide on building agent teams that actually work. full guide in the article below 👇

rody @0x_rody

2 weeks ago

x.com/i/article/2058…

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@TheMathFlow @beyoumf Please let us to upload your videos😁😁

Bac Maths 2026 – Amérique du Nord Le sujet vient de sortir et il est disponible sur Math93. 📌 Le corrigé détaillé est en cours de réalisation et sera ajouté progressivement. 👉 math93.com/index.php/anna… #Bac2026 #Maths #SpécialitéMaths #Math93

Converts academic PDFs to Markdown with LaTeX math and tables github.com/facebookresear…

@TheMathFlow Je suis souscripteur, puis-je télécharger la vidéo ??

What warm water does to your body when you start your day.

Wisdom teeth are weird 🦷😨

Hillel Furstenberg is one of the most influential mathematicians in ergodic theory, dynamical systems, and combinatorics. Born in Germany in 1935 and later moving to the United States and Israel, Furstenberg transformed modern mathematics by connecting probability, dynamics, and number theory in unexpected ways. His most famous achievement was introducing ergodic-theoretic methods into combinatorics. In 1977, he gave a groundbreaking ergodic proof of Szemerédi's Theorem, showing that sufficiently dense subsets of integers contain arbitrarily long arithmetic progressions. This created a deep bridge between dynamical systems and number theory. Furstenberg also made foundational contributions to random walks on groups, stationary processes, and boundary theory. His work influenced probability theory, statistical mechanics, information theory, and modern data science. Many ideas in stochastic systems, reinforcement learning, and sequential decision processes echo ergodic principles developed in his research. In 2020, Furstenberg received the Abel Prize for pioneering the use of probability and dynamics in number theory and combinatorics.

This Putnam Integral Has a Ridiculously Simple Answer youtu.be/HSn9V9ok7bw?si… via @YouTube

@dment37 @geogebra @bancoche Dans Euler, il faut diviser par 2i pour sinx !!

ブラウザで動作するLaTeX数式ビューア＆画像出力ソフト「TeXportLab」をリリース！ 以下のURLで誰でも利用できる状態になりました。 texportlab.com ・インストール不要 ・LaTeXコードで書かれた数式をリアルタイムレンダリング ・豊富なサンプルでLaTeXに詳しくなくても使える ・静的ブラウザアプリのため外部へデータ送信なし ・透過背景.pngを出力可能 ・透過背景.svgとしてベクター画像も出力可能 ・Copy PNGボタンでクリップボードにコピー 　→MS Office製品へ直接ペースト等が可能 ・スマホ対応 ・モダンなデザイン ・完全無料（OSSとして公開）

How can a falling chain fountain upward?✍️ h2/h1 = α / (1 - α - β) This is the Mould Effect. Because the chain links are rigid, pulling one end up causes the other to pivot and push off the stack. This downward kick creates the upward force that makes the chain leap out of the jar! The higher the drop, the higher the fountain.

What if a simple rule hides an unsolved mystery? The Collatz Conjecture says every number eventually reaches 1… but no one can prove it. Have you tried it? #Math #Mathematics #NumberTheory #CollatzConjecture #MathType #STEM

Geometry challenge: two tangent circles of radius 1 with a square fitted between them. Can you find the side of the square? Share your answer in the comments 👇 #Math #Geometry #MathPuzzle #STEM