标签：#math

This essay was written with Kasra Rafi, and originally appeared in The Guardian. Earlier this month, about 40 top mathematicians gathered at OpenAI’s offices to discuss the future of their profession. The meeting was off-the-record, but if recent articles by mathematicians are any guide, it was mostly pretty glum. People fear for their jobs, their careers and the work they love. We think the contrary view is more likely, at least in the short-term. AI models are nowhere near as capable as experienced academic mathematicians. This isn’t to say that AIs aren’t producing stunning mathematical results at the level of PhD researchers. In mid-May, OpenAI ...

How a tiny little electronic switch sparked the computer revolution.

How the joule thief circuit “steals” energy from seemingly depleted power cells.

For more than 150 years, the Riemann hypothesis has stood as one of the major unsolved problems in mathematics. Anthropic hasn't solved it — but the company's models made more progress than you might expect.

Zulfi Alam thinks his team doesn’t need to “prove” they engineered a new state of matter in their bid to reinvent computing. Science would disagree.

Never mind the “heat death of the universe.” Entropy is just a bridge between two levels of reality.

Introduction Having laid out some mathematical foundations in the [previous part][1], we will proceed to discuss quantum gates and circuits in depth here. As usual, we will drive home the theories with worked examples from [@Rieffel2011] and check their accuracy with qiskit and/or cirq code. Prerequisite Understanding quantum gates and circuits will be greatly aided by the knowledge of classical gates (AND, OR, NOT, XOR and the universal gates: NAND and NOR). Also, some familiarity with the Python programming language will help you understand qiskit and/or cirq code. Classical & Quantum gates A classical computer, possibly the one you're reading this with, is a sophisticated engineering piece, no doubt. However, the underlying "magic" comprises logic gates. The National Institute of Standards and Technology (NIST) gives this incredible analogical description of classical computers in relation to logic gates [@NISTQGATE2026]: Traditional computers are like microscopic cities. The roads of these cities are wires with electricity coursing through them. These roads have lots of gates, known as logic gates, which enable computers to do their job. Like physical gates that allow or block cars, logic gates allow or block electricity. Electricity that goes through the gates represents a “1” of digital data, and blocked electricity is a “0.” When you pick up a motherboard, for instance, you may not see the said gates as they are made of tiny transistors (in modern systems, MOSFETs) in specific combinations. Also, the $0$ and $1$ referred to in the analogy mean low and high voltage ranges, respectively. Their actual voltage values depend on the hardware. :::note All schematics were written in and rendered by @schemd/core . Check it out. ::: Logic Gates refresher Since we will be realizing some classical circuits using quantum gates, it's necessary to get familiar with common logic gates. NOT ($\neg$) Gate This gate takes a single classical bit and flips it. For instance, if $0

New study confirms 2024 "momentum flux theory" on how angular momentum of water flows drives rotation.

@kungfufk Since the dawn of computing, we have built Artificial Intelligence on a flawed premise: perfect rationality. We brute-force algorithms to find the optimal solution, assuming infinite time and infinite capacity. But humans don't work like that. As Herbert Simon famously coined, we operate on Bounded Rationality. We make decisions based on limited time, limited cognitive capacity, and limited information. What if, instead of forcing AI to be perfectly rational, we created a mathematical equivalent for human processing? What if we modeled human cognition using the laws of physics — wave theory, thermodynamics, and mechanical energy equations — to build a heavy, complex, but highly probabilistic AI engine? Here is a blueprint for a new field of research: Computational Cognitive Mechanics . 1. The Core Equations of Cognitive Processing To model bounded rationality mathematically, we first need to define the relationship between Knowledge ($K$), Cognitive Capacity ($C$), and Processing Time ($T$). Based on human observation, we can establish these foundational proportions: Knowledge vs. Time — The more knowledge you possess, the faster you can generate a decision. $$T \propto \frac{1}{K}$$ Capacity vs. Time — High cognitive capacity (skills, processing power) inversely relates to the time required to solve a problem. $$T \propto \frac{1}{C}$$ Knowledge vs. Capacity — This is the most fascinating limit. Knowledge does not scale linearly with capacity. Gaining true knowledge requires exponential capacity (effort/skill). Therefore, knowledge is roughly proportional to the square root of capacity. $$K \propto \sqrt{C}$$ By integrating these, we can build a baseline processing algorithm for an AI. Instead of giving an AI unlimited time to compute, we cap its computing time based on a synthetic "Knowledge and Capacity" matrix, forcing it to use heuristics — just like a human. 2. Cognitive Wave Theory & FFT: Information as Interference In physics, waves interact throug

Planning a Fourth of July getaway? Use less gas—and cut your emissions—by easing up on the pedal.

VCs remain thirsty to fund AI coding startups. This one, founded by investor Chamath Palihapitiya, is no exception.

The question My son was on the sofa with his iPad, poking at the live "Predict the Bracket" game — the whole 2026 World Cup knockout tree on one screen, every slot already filled with the crowd's favourite for that match. Tap a match, see who most people think goes through, watch the picks flow all the way up to a predicted champion. He frowned at it. "Daddy, how do they know which team plays which team? The teams aren't even decided yet." He'd caught something real. The little cards sitting in those slots were only predictions — the crowd's best hunch — but the shape underneath them, who-plays-who and where, was already locked in. Months before a single match kicks off. Fair question. The 2026 World Cup has 48 teams in 12 groups (A through L). The top two of every group go through — that's 24 teams. Then, to round it up to a nice bracket of 32, they also take the 8 best third-placed teams . Twelve groups, but only eight of their third-place teams get a golden ticket. "So you don't know which eight until the very end," he said. "But the bracket's already sitting right there on the screen." "Right." "That's cheating." It isn't cheating. It's one of the prettiest little bits of planning in all of sport, and by the end of the afternoon he understood it better than most adults do. We did it with the dining chairs. The setup, first Before the chairs, my son needed to know where these kids even come from. So we did the boring-but-important part first. A football group is a handful of teams who all play each other. When it's done, the best go forward, the worst go home, and — this is the bit that matters — there's a kid right on the line: the best of the rest , neither safely through nor clearly out. That borderline kid is the star of this whole story. Call them a wandering kid . To learn the trick, let's make the groups nice and small: two groups, A and B, three kids in each — six kids total. In each group the top kid goes straight through to the next round, the bottom ki

The country was hit hard by a pair of quakes that happened in quick succession and were likely driven by stress being transferred from one part of the fault that runs through the country to another.

As World Cup action kicks off, we look at the physics of the beautiful game.

I tried to explain OpenAI’s solution more clearly than OpenAI did.

AI จะครองโลก เป็นคำที่ได้ยินมานาน เท่าที่ผู้เขียนจำความได้ก็มี Judgement Day ยุคหนัง Terminator แต่หากจะจริงจังขนาดโยงเข้าความเป็นจริงก็ยังไม่มีอะไรชัดเจน แต่วันนี้เรามีหลักฐานพิสูจน์ได้จริงแล้ว ด้วยความ Novelty จาก OpenAI ที่สามารถค้นพบความรู้ใหม่ที่ไม่เคยมีมนุษย์ค้นพบมาก่อน หักล้างความเชื่อที่ว่า AI ทำได้เพียงนำสิ่งที่มนุษย์ค้นพบแล้วมาเรียงต่อกัน ในเดือนพฤษภาคม 2026 reasoning model ภายในของ OpenAI ได้ disprove Erdős Planar Unit Distance Problem ซึ่งเป็นปัญหาและข้อคาดการณ์ทาง Combinatorial geometry ที่ Paul Erdős ตั้งไว้ตั้งแต่ปี 1946 โจทย์ระบุว่า เมื่อวางจุด nn n จุดบนระนาบ จำนวนของคู่จุดที่ห่างกันพอดี 1 หน่วยจะมีได้มากที่สุดเท่าใด Erdős แสดงความเป็นไปได้ผ่านการจัดเรียงแบบ grid ว่าได้จำนวนคู่ที่เติบโตเหนือเส้นตรงเพียงเล็กน้อย และตั้งข้อคาดการณ์ว่าไม่มีโครงสร้างใดทำได้ดีกว่านี้อย่างมีนัยสำคัญ ข้อคาดการณ์นี้ได้รับการยอมรับในวงกว้างตลอด 80 ปีที่ผ่านมา และยังไม่มีข้อคาดการณ์ที่ดีกว่านี้ จนกระทั่ง OpenAI ได้เผยแพร่ Chain of Thought (CoT) เรียบเรียงความยาว 125 หน้า ซึ่งบันทึกลำดับการให้เหตุผลของโมเดลไว้ทั้งกระบวนการว่าโมเดลไปถึงคำตอบอย่างไร กรอบของคำตอบ: lower bound กับ upper bound ก่อนเข้ากระบวนการทำงานของโมเดล ขออธิบาย "กรอบ" ของคำตอบของปัญหานี้ก่อน เพราะคำตอบถูกล้อมไว้ด้วย lower bound จำนวนที่สร้างได้จริงแล้ว อย่างน้อยเท่านี้ และ upper bound เพดานที่พิสูจน์แล้วว่าเกินไม่ได้ ด้าน lower bound นั้น Erdős เอง (1946) ใช้การจัดเรียงแบบ grid แสดงว่าสร้างได้ถึง n1+Ω(1/log⁡log⁡n)n^{1+\Omega(1/\log\log n)} n 1 + Ω ( 1/ l o g l o g n ) ซึ่งมากกว่าเส้นตรงเพียงเล็กน้อย และเข้าใกล้ศูนย์เมื่อ nn n ใหญ่ขึ้น ส่วนด้าน upper bound นั้น Erdős พิสูจน์เพดานแรกไว้ที่ O(n3/2)O(n^{3/2}) O ( n 3/2 ) จากวงกลมหนึ่งหน่วยสองวงตัดกันได้ไม่เกินสองจุด โดยต่อมา Spencer–Szemerédi–Trotter (1984) บีบเพดานนี้ลงมาเป็น O(n4/3)O(n^{4/3}) O ( n 4/3 ) ซึ่งเป็น upper bound ที่ดีที่สุดจนถึงปัจจุบัน และยังคงอยู่หลังการค้นพบของ OpenAI model วิธีเก่าของ Erdős: วงกลมรัศมีเลือกมาลากผ่านจุด grid หลายจุดพร้อมกัน ทำให้ระยะซ้ำมีมาก สิ่งที่ Erdős คาดการณ์คือ คำตอบจริงของ Planar Unit Distance Problem ควรอยู่ชิดด้าน lower

A clever use of CSS to calculate and display a discounted product price by providing a base price and discount amount, featuring modern CSS features like attr() , mod() , and round() . Computing and Displaying Discounted Prices in CSS originally handwritten and published with love on CSS-Tricks . You should really get the newsletter as well.