Published on: 2025-06-05 01:55:37
Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I“. It was intended to complement the many good available analysis textbooks out there by focusing more on foundational issues, such as the construction of the natural numbers, integers, rational numbers, and reals, as well as providing enough set theory and logic to allow students to develop proofs at high levels of rigor. While some proof assistants such as Coq or Agda were well established when the book was written, fo
Keywords: exercises lean mathlib natural numbers
Find related items on AmazonPublished on: 2025-06-23 00:20:00
If you want to know where AI is headed, this year’s Google I/O has you covered. The company’s annual showcase of next-gen products, which kicked off yesterday, has all of the pomp and pizzazz, the sizzle reels and celebrity walk-ons, that you’d expect from a multimillion dollar marketing event. But it also shows us just how fast this still-experimental technology is being subsumed into a line-up designed to sell phones and subscription tiers. Never before have I seen this thing we call artifi
Keywords: ai gas natural power story
Find related items on AmazonPublished on: 2025-06-25 20:00:00
The AI data center also promises to transform the state’s energy future. Stretching in length for more than a mile, it will be Meta’s largest in the world, and it will have an enormous appetite for electricity, requiring two gigawatts for computation alone (the electricity for cooling and other building needs will add to that). When it’s up and running, it will be the equivalent of suddenly adding a decent-size city to the region’s grid—one that never sleeps and needs a steady, uninterrupted flo
Keywords: data gas natural plants power
Find related items on AmazonPublished on: 2025-06-27 08:40:48
Having shut down its last remaining nuclear plant Saturday, Taiwan is working to secure new imports of natural gas. The Maanshan nuclear plant closed following the expiration of its 40-year operating license, fulfilling a promise by the ruling Democratic Progressive Party of a “nuclear-free” Taiwan. Analysts say the move will make the country more reliant on imported fossil fuels and more exposed to spikes in the price of gas. Already Taiwan is struggling to generate enough electricity, leadin
Keywords: gas natural nuclear taiwan year
Find related items on AmazonPublished on: 2025-07-18 01:00:22
John Wick spinoff Caine will feature another familiar returning face. Eric Kripke teases The Boys‘ long-awaited Supernatural reunion. Plus, get a very tiny look at next week’s Doctor Who. Spoilers, away! Superman In response to a fan at Northern Fancon (via Screen Rant), Sean Gunn confirmed he’s “kinda” in Superman, and may-or-may-not be playing Maxwell Lord, as previously rumored. I can’t believe Pedro Pascal and I keep going after the same roles! [laughs] I can’t answer most of those questi
Keywords: doctor movie really say supernatural
Find related items on AmazonPublished on: 2025-07-29 05:00:58
Mining is the main anthropogenic cause of heavy metal soil contamination. The image shows an aerial view of the Aznalcóllar mine, photographed in 2017. The first 30 centimeters of soil are the foundation of life. This foot-deep slice of the pedosphere is the vital space for most plant roots. When roots go deeper, it’s to anchor the plant, not to nourish it. Within this narrow band, bacteria, fungi, nematodes, and countless other microscopic organisms form the so-called biological crust, which i
Keywords: concentrations human metals natural soil
Find related items on AmazonPublished on: 2025-08-04 05:25:10
is a deputy editor and author of thenewsletter. He has been reporting on the tech industry for more than a decade. Meta has laid off an unspecified number of employees in its Reality Labs division, a company spokesperson confirmed. The cuts affected teams working in Oculus Studios, Meta’s in-house games division for Quest headsets, as well as some employees involved in the company’s hardware efforts, according to people familiar with the matter. Specific titles impacted by the layoffs include
Keywords: company division fitness meta supernatural
Find related items on AmazonPublished on: 2025-08-08 04:00:00
To test this idea, he combined solutions of natural mucin proteins with synthetic mussel-inspired polymers and observed how the resulting mixture solidified and stuck to surfaces over time. “It’s like a two-part epoxy. You combine two liquids together, and chemistry starts to occur so that the liquid solidifies while the substance is simultaneously gluing itself to the surface,” Degen says. The resulting gel strongly adheres even to wet surfaces while preventing the buildup of bacteria. The r
Keywords: gel liquid materials natural resulting
Find related items on AmazonPublished on: 2025-08-09 09:24:53
Surprises in Logic John Baez April 4, 2016 There's a complexity barrier built into the very laws of logic: roughly speaking, while lots of things are more complex than this, we can't prove any specific thing is more complex than this. And this barrier is surprisingly low! Just how low? Read this. And then see what happens when combine this idea with the famous 'surprise examination paradox', also known as the 'unexpected hanging paradox'. Mathematically speaking, these ideas are called Chai
Keywords: natural numbers program standard theorem
Find related items on AmazonPublished on: 2025-08-13 16:30:00
Keeping an eye on your sugar intake isn’t a bad thing -- and neither is treating yourself to something sweet now and then. It’s all about finding the right balance. Instead of cutting out sugar entirely, the smarter move is to pay attention to the kind of sugar you’re consuming. For women and children, studies show that the recommended daily limit is about six teaspoons of sugar. For men, it’s nine. The good news? You don’t have to rely on heavily processed sugars to satisfy your cravings. Natu
Keywords: fruit natural sugar sugars syrup
Find related items on AmazonPublished on: 2025-08-18 05:12:01
Scientists have caught a colossal squid on camera in its natural environment for the very time, according to a report by the BBC . This happened around 100 years after the species was originally discovered, proving just how vast and mysterious the ocean truly is. The squid is a juvenile, so it’s not nearly as large as its namesake suggests. It clocks in at around 11.8-inches long. Adult colossal squid can grow up to 23 feet in length and weigh up to 1,100 pounds. They are the heaviest invertebr
Keywords: colossal juvenile natural ocean squid
Find related items on AmazonPublished on: 2025-08-23 12:00:03
Get your hands on the latest in holographic tech — or if you want to be pedantic about it, volumetric displays. Using a clever system of elastic bands and rapidly projecting images, a team of researchers in Spain have created three-dimensional virtual objects that you can physically manipulate, allowing you to directly poke and prod at them without the need of a boring controller. As detailed in a new paper that's yet to be peer-reviewed and will be presented at the CHI 2025 conference this mo
Keywords: displays elastic images natural volumetric
Find related items on AmazonPublished on: 2025-08-30 05:00:00
Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More For decades, the SQL query language has been the cornerstone of database technology. But what happens when you bring SQL together with modern generative AI? That’s the question that Google Cloud is answering as part of a series of database updates rolling out at the company’s Google Cloud Next conference today. Over the past year, all Google Cloud databases have added
Keywords: database google language natural sql
Find related items on AmazonPublished on: 2025-09-13 07:14:47
01 Intro Conversational interfaces are a bit of a meme. Every couple of years a shiny new AI development emerges and people in tech go “This is it! The next computing paradigm is here! We’ll only use natural language going forward!”. But then nothing actually changes and we continue using computers the way we always have, until the debate resurfaces a few years later. We’ve gone through this cycle a couple of times now: Virtual assistants (Siri), smart speakers (Alexa, Google Home), chatbots (
Keywords: data language like natural transfer
Find related items on AmazonPublished on: 2025-09-19 00:44:37
How does the human brain process natural language during everyday conversations? Theoretically, large language models (LLMs) and symbolic psycholinguistic models of human language provide a fundamentally different computational framework for coding natural language. Large language models do not depend on symbolic parts of speech or syntactic rules. Instead, they utilize simple self-supervised objectives, such as next-word prediction and generation enhanced by reinforcement learning. This allows
Keywords: brain human language models natural
Find related items on AmazonPublished on: 2025-09-24 03:26:05
authored by Premmi and Beguène Previous Topic: An Axiomatic Study of Numbers Introduction Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as 1, 2, 3, etc. with counting, then with what corresponding concepts do we relate numbers like -4, \sqrt{3} \text{ and } \frac{22}{7}? To reason about all kinds of num
Keywords: axioms mathbb natural numbers set
Find related items on AmazonPublished on: 2025-09-25 04:26:05
authored by Premmi and Beguène Previous Topic: An Axiomatic Study of Numbers Introduction Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as 1, 2, 3, etc. with counting, then with what corresponding concepts do we relate numbers like -4, \sqrt{3} \text{ and } \frac{22}{7}? To reason about all kinds of num
Keywords: axioms mathbb natural numbers set
Find related items on AmazonPublished on: 2025-09-28 05:35:33
Why numbering should start at zero To denote the subsequence of natural numbers 2, 3, ..., 12 without the pernicious three dots, four conventions are open to us a) 2 ≤ i < 13 b) 1 < i ≤ 12 c) 2 ≤ i ≤ 12 d) 1 < i < 13 Are there reasons to prefer one convention to the other? Yes, there are. The observation that conventions a) and b) have the advantage that the difference between the bounds as mentioned equals the length of the subsequence is valid. So is the observation that, as a consequen
Keywords: bound convention natural number subscript
Find related items on AmazonPublished on: 2025-09-28 15:35:33
Why numbering should start at zero To denote the subsequence of natural numbers 2, 3, ..., 12 without the pernicious three dots, four conventions are open to us a) 2 ≤ i < 13 b) 1 < i ≤ 12 c) 2 ≤ i ≤ 12 d) 1 < i < 13 Are there reasons to prefer one convention to the other? Yes, there are. The observation that conventions a) and b) have the advantage that the difference between the bounds as mentioned equals the length of the subsequence is valid. So is the observation that, as a consequen
Keywords: bound convention natural number subscript
Find related items on AmazonPublished on: 2025-10-13 08:00:00
Keeping an eye on your sugar intake isn’t a bad thing -- and neither is treating yourself to something sweet now and then. It’s all about finding the right balance. Instead of cutting out sugar entirely, the smarter move is to pay attention to the kind of sugar you’re consuming. For women and children, studies show that the recommended daily limit is about six teaspoons of sugar. For men, it’s nine. The good news? You don’t have to rely on heavily processed sugars to satisfy your cravings. Natu
Keywords: fruit natural sugar sugars syrup
Find related items on AmazonPublished on: 2025-10-22 21:30:39
Characters in horror movies often find themselves in situations involving the urgent need to make or receive phone calls—a simple enough task that’s inevitably complicated by the monster that’s after them. This trope especially thrives in media set during the age of land lines, but it’s followed the genre into the smart-phone era, too. Let’s examine the different types of phone-call horrors in movies. The Scary Call (Non-Supernatural) You’re home alone. The phone rings and there’s a stranger o
Keywords: horror movie phone scary supernatural
Find related items on AmazonPublished on: 2025-11-07 00:00:18
Supernatural fans rejoice: the band’s getting back together on The Boys. The Prime Video series revealed the news on a post shared on X with The Boys‘ Soldier Boy, Jensen Ackles, calling for his old squad to assemble. And of course, the hail was answered by Jared Padalecki and Misha Collins. While we don’t know who they will be playing, it’s an exciting prospect to see them join Ackles on the final season of The Boys. Season 5 just got a bit more supernatural. pic.twitter.com/sbK2ww2EPC — THE B
Keywords: ackles boys collins misha supernatural
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