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Adapted from an appendix of my MS thesis. Markov Chain Monte Carlo Almost as soon as computers were invented, they were used for simulation. Markov chain Monte Carlo (MCMC) was invested as Los Alamos, Metropolis et al (1953) simulated a liquid in equilibrium with its gas phase. Their tour de force was the realization that they did not need to simulate the exact dynamics, they only needed to simulate some Markov chain with the same equilibrium distribution. The Metropolis algorithm was widely used by chemists and physicists, but was not widely known among statisticians until after 1990. Hastings (1970) generalized the Metropolis algorithm, and simulations following his scheme are said to use the Metropolis-Hastings (MH) algorithm [1]. A special case of the MH algorithm was introduced by Geman et al (1984) discussing optimization to find the posterior mode rather than simulation. Algorithms following their scheme are said to use the Gibbs sampler. It took some time for the spatial statistics community to understand that the Gibbs sampler simulated the posterior distribution, thus enabling full Bayesian inference of all kinds. Gelfand et al (1990) made the wider Bayesian community aware of the Gibbs sampler, and then it was rapidly realized that most Bayesian inference could be done using MCMC, whereas very little could be done without MCMC. Green (1995) generalized the MH algorithm as much as it could be generalized [1]. Theoretical Foundations A sequence X 1 ​ , X 2 ​ , … of random elements of some set is a Markov chain if the conditional distribution of X n + 1 ​ given X 1 ​ , … , X n ​ depends on X n ​ only. The set in which the X i ​ take values is called the state space of the Markov chain. A Markov chain has stationary transition probabilities if the conditional distribution of X n + 1 ​ given X n ​ does not depend on n . This is the main kind of Markov chain of interest in MCMC. The joint distribution of a Markov chain is determined by the following [1]. The ma

Pull a hot steel bolt out of a furnace and quench it in oil, and a fair question is: does the bolt cool from the outside in, with a sharp temperature difference between its skin and its core, or does the whole thing drop in temperature more or less together? The answer is not obvious from the part itself. A thin copper washer and a thick ceramic block behave very differently in the same bath, even at the same starting temperature. The Biot number is the small calculation that settles this question before you commit to any heavy analysis. It tells you, in a single dimensionless figure, whether an object can be treated as having one uniform temperature or whether you must resolve a temperature gradient inside it. That distinction changes the math from a one-line exponential decay to a partial differential equation. Why this calculation matters Transient heating and cooling problems show up everywhere: heat-treating metal parts, quenching forgings, cooling electronics, baking or chilling food, warming up an engine block. In every one of these, the engineer wants to know how the temperature changes over time. The hard version of that question requires solving the heat conduction equation across the body, with position and time as variables. The easy version is the lumped-capacitance model, which treats the whole object as a single point at one temperature. It reduces the problem to a simple first-order exponential. The catch is that the lumped model is only valid when internal conduction is fast compared with surface convection. The Biot number is exactly the check that tells you whether that condition holds. Skip the check and apply the lumped model where it does not belong, and you can badly mispredict cooling times, residual stresses, and the risk of cracking from thermal gradients. The core formula The Biot number compares two thermal resistances. One is the resistance to conducting heat through the inside of the solid. The other is the resistance to carrying heat a

Data centers are driving up the company’s use of electricity—and carbon pollution.

Reinforcement learning uses error information to adjust control algorithms.

The Quest shipwreck is in worse shape than expected, but it has turned into a thriving marine ecosystem.

The Earth may not be that massive, but it still distorts space-time.

A recent study tracked hundreds of soccer fans until their favorite team reached the final of a tournament. Their stress levels skyrocketed, and their heart rates jumped too.

This is Optimizer, a weekly newsletter sent from Verge senior reviewer Victoria Song that dissects and discusses the latest gizmos and potions that swear they're going to change your life. Opt in for Optimizer here. Bryan Johnson, best known as the man who wants to live forever, has an incurable autoimmune disease. The internet's most […]

The AC culture wars may be solved by advances in environmentally friendly technology.

English and Norwegian players will face off under extreme and dangerous levels of heat stress, scientists say, thanks to a Wet Bulb Index over over 90°F.

I run a small AI trends site, and three weeks ago I started doing something simple: every day, snapshot the star count of every repo that crosses my GitHub trending scan for AI. No judgment, no curation, just append-only rows in a database. 611 repos and 2,671 data points later (June 19 to July 10), the picture of what's actually growing looks pretty different from what my feeds told me was hot. Here's what the data says. Before publishing this I re-checked every number below against GitHub's live API. Star counts drift by the hour, so treat them as of July 10. The top 10 risers, by raw stars gained Repo Gained Window From → To calesthio/OpenMontage +30,253 21 days 5,899 → 36,152 DeusData/codebase-memory-mcp +20,483 19 days 7,516 → 27,999 mattpocock/skills +19,053 15 days 137,485 → 156,538 obra/superpowers +16,887 20 days 232,908 → 249,795 NousResearch/hermes-agent +14,896 21 days 197,297 → 212,193 Panniantong/Agent-Reach +14,334 14 days 34,780 → 49,114 usestrix/strix +13,243 12 days 26,363 → 39,606 addyosmani/agent-skills +12,685 21 days 63,156 → 75,841 asgeirtj/system_prompts_leaks +11,720 21 days 43,415 → 55,135 msitarzewski/agency-agents +11,055 10 days 118,241 → 129,296 Windows differ because I only hold snapshots for the days a repo appeared in my scan; each row states its own real window. Three things in this data genuinely surprised me. 1. "Skills" are eating agent frameworks Four of the top ten are not agent frameworks. They are collections of packaged expertise that plug into an existing agent: obra/superpowers (still compounding at roughly 840 stars a day on a 250k base), mattpocock/skills, addyosmani/agent-skills, msitarzewski/agency-agents. A year ago this table would have been full of new frameworks. Now the framework layer looks settled and the growth is in what you load INTO the agent. The moat moved from orchestration code to encoded judgment. 2. The sharpest climbs are applications, not infrastructure The steepest sustained climb from a newcomer in

Microsoft may once again be struggling to keep up with its own climate goals, according to its 2026 sustainability report. As reported by GeekWire, the report states that Microsoft's carbon emissions increased 25 percent in 2025, totalling 34 million metric tons "without select interventions." Microsoft says this was "driven primarily by the expansion of our […]

If Homo floresiensis wasn't a fire-using hunter, its origins could be different than we thought.

SpaceX is currently ahead of last year's record-setting pace for Starlink satellite deployments. SpaceX launched 1,589 Starlink satellites into low-Earth orbit in the first half of 2026, according to launch data compiled by Jonathan McDowell's satellite tracker, compared to 1,489 satellites deployed at the same point in 2025. 2025 was already a record year for […]

"The Starship Pez dispenser demonstrates very smart industrial design and scale."

An image of Portugal forward Pedro Neto’s cleats at the World Cup has reignited a practice among some soccer players: modifying their cleats to relieve heel discomfort.

It's too hot. There, we said it. Protect your health and keep your home cool with one of these top-rated air conditioners.

As extreme heat becomes the norm on the continent, the AC culture wars may be solved by advances in environmentally friendly technology.

Picture a power plant burning fuel to spin a turbine. It is tempting to assume that with enough engineering — better seals, smoother bearings, cleaner combustion — the plant could be pushed toward converting nearly all its heat into useful work. It cannot. A large modern thermal power station turns only something like 40 to 45 percent of its fuel energy into electricity, and the missing majority is not lost to sloppy design. It is lost to a law of physics. That law sets a ceiling on every device that turns heat into work, from a car engine to a steam turbine to a jet. The ceiling is called the Carnot efficiency, and the remarkable thing about it is how little it depends on. Not on the working fluid, not on the mechanism, not on the cleverness of the builder — only on two temperatures. This article explains where that limit comes from, how to compute it, and why it reshapes how engineers think about efficiency. Why this calculation matters The Carnot efficiency is the benchmark against which every real engine is judged. When an engineer reports that a gas turbine runs at 38 percent efficiency, that number means little on its own. Compared against the Carnot limit for the same hot and cold temperatures, it suddenly tells you how much room is left — whether the design is already near the physical wall or still has slack worth chasing. It also redirects design effort toward the things that actually matter. Because the Carnot limit depends only on the ratio of cold to hot absolute temperatures, the single most powerful way to raise the ceiling is to raise the temperature at which heat enters the engine, or lower the temperature at which it is rejected. This is why turbine inlet temperatures have climbed for decades, pushing the limits of metallurgy and cooling. Polishing internal friction yields small gains; raising the hot-side temperature raises the ceiling itself. The core formula Sadi Carnot, in 1824, imagined an idealized engine running on a perfectly reversible cyc

It's more of a baa-ter arrangement.