Our Moon as seen by the Galileo spacecraft when it flew past on December 7, 1992. Image: NASAWhere did our Moon come from? The origin of Earth’s cosmic companion is a fundamental question in planetary science. From Galileo’s first telescopic observations of the Moon to humans walking on its surface, our understanding of its origins has come a long way. Yet it’s far from complete.Over the decades, scientists have proposed multiple hypotheses to attempt explaining how the Moon came to be. In this…

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A and use the Cauchy-Schwarz inequality. Detailed proof Now let’s unpack that. Any real symmetric matrix A is similar to a matrix D with the eigenvalues of A along…

Concise and Consistent Naming Florian Deißenb¨ock and Markus Pizka Proceedings of the 13th International Workshop on Program Comprehension (IWPC’05)

After a very busy week, it’s time once more for another Saturday update of activity at the Open Journal of Astrophysics. Since the last update we have published a further five papers, bringing the number in Volume 9 (2026) to 194 and the total so far published by OJAp up to 642. As explained at the end of this post, one of the five publications is a correction of an earlier paper. I continue to include the posts made on our Mastodon account (on Fediscience); these announcements also show the…

Today came the news that the Fermats' Last Theorem has been formalized in Lean by the AI agent Claude.

Richard Stearns (right) and Juris Hartmanis in May 1963. The main theorem from their seminal paper is on the blackboard. Stearns sent Lance this picture to help celebrate the 50th anniversary of their paper.Richard Stearns died on August 29, 2026. Readers of this blog probably know him from the Hartmanis-Stearns paper On the computational complexity of algorithms which appeared in Transactions of the American Mathematical Society in 1965, and his paper with F. C. Hennie that gave the still…

My collaborators and I recently posted a preprintExact mapping from short-ranged harmonic elastic models of spin crossover materials to Ising models with interactions at all length scalesNadeem Natt, Gian Ruzzi, Jace Cruddas, Ross H. McKenzie, Ben J. PowellSpin crossover (SCO) materials are reversible molecular switches found in a wide range of transition metal complexes and metal organic frameworks (MOFs). They exhibit diverse spin state orderings and transitions between them. Here we present…

International Gemini Observatory/NOIRLab/NSF/AURA Image of a centaur object looking very comet-like. Centaurs are small solar system bodies that orbit the Sun between Jupiter and Neptune. Because they move among the gas giants, their orbits are generally unstable. Gravitationally nudged to and fro until they either tumble toward the inner solar system or are tossed away to the cold of deep space. Centaurs also straddle the gap between asteroids and comets. They aren’t as rocky as the warm…

If you're reading this, you've probably been hitting your head against a brick wall trying to parse and decipher the new HTTP Signatures sent by Mastodon and other Fediverse servers. This is a basic and somewhat incomplete guide to accepting these signatures. I'm sure there are various gotchas, but it works with the signatures I've seen in the wild. Shut Up And Show Me The Code! OK, wow, no need to be a dick about it! Here's how I validated a real signature that my server received. $verified =…

The θ(p꜀) = 0 conjecture is solved in all dimension. In 2024 Gady Kozma and Shahaf Nitzan showed a derivation of the dying percolation conjecture from a proposed conjecture about percolation for general graphs. (I briefly discussed it in this post. Their conjecture was so general that many of us expected a counterexample to be discovered not before long.) The dying percolation conjecture asserts that for percolation in at the critical probability, with probability one, there is no infinite…

Visualization by Arpit Arora and Adrian Price-Whelan. Milky Way image credit: Stefan Payne-Wardenaar. Simulated stellar streams arc around the Milky Way. Ah, that ever-elusive dark matter. Although there has been some tantalizing evidence suggesting the detection of a dark matter particle, the data isn’t strong enough to be conclusive. So astronomers continue to rely upon indirect evidence and computer simulations to help us understand the detailed properties of dark matter. A new study in The…

In this post, I am returning to the subject of relative motion, that I first described in posts 16.4 and 16.12. But we shall see that this description is complicated by what I wrote in post 26.6. According to post 26.6, the speed, c, of an electromagnetic wave (for example, light), in a vacuum, is given by where ε0 is the permittivity of free space and μ0 is the permeability of free space. Since we consider ε0 and μ0 to be fundamental constants, we would then expect c to be a fundamental…

I'm not a maths guy. The way I approach maths is with pure, brute strength.503 words · 1 image

As described in our previous post (.htm), back in 2006 ariely@mit.edu sent three Excel spreadsheets, each “last saved” by user “Dan Ariely”, to then-PhD-student-but-now-full-professor Kyle Hyndman. Those spreadsheets contained the data required to reproduce the results reported in Ariely and Wertenbroch’s (2002) influential article on deadlines and procrastination. In our previous post, we reported data...

Im so incredibly proud of my boy!

After returning from holidays earlier this summer, I organized all the stuff on my desk a bit, and I ran into various half-finished computations and notes. Whilst some might eventually become (short) papers, most of them will (at least initially) just be fun blog posts. The first topic I want to tackle is the Brauer group of Fano 3-folds. In the wonderful paper Fano varieties with torsion in the third cohomology group by John Christian Ottem and Jørgen Vold Rennemo, they construct…

Akiva Kadari, Pablo Soberon and the cascade conjecture The cascade conjecture is discussed in this post. I proposed the conjecture back in 1974, inspired by work by Meir Katchalski (though the name “Cascade Conjecture” only came into use over the last decade or two). Akiva Kadari, a master’s student of Micha Perles, proved the planar case in his M.Sc. thesis. Although the proof was ready in the early-to-mid 1980s, writing the thesis was delayed until 1990, when Micha was on sabbatical and I…

I discovered a neat way to derive Stirling’s approximation for the factorial in a succinct manner. I came across this neat derivation in Information Theory, Inference and Learning Algorithms by David J.C. MacKay (though I’ve also just noticed that it’s listed on the Wikipedia page too). This is an addendum of sorts to my Factorial Overflow post. Consider an indexed family of Poisson random variables $N_{\lambda }\sim \operatorname{Pn}(\lambda), \lambda \in \mathbb N$ with probability mass…

Conference: 'XI Partial Differential Equations, Optimization, Numerics, and Machine Learning'

tag(s): #random-thoughts #español (For reasons unknown to me, this post has to be in Spanish, and I know the one about my second tatoo will be in English. Yup, despite both of them being irreconcilably linked. I don't know why. That's just how it feels.) La decisión de hacerme un tatuaje cuando cumpliera 40 años, existío desde un tiempo antes de esa fecha. María me ayudó a elegir el diseño. Lo procrastiné durante 3 años y medio. Digo que lo dilaté, porque decido, estaba. El diseño que quería es…

My favorite statistical podcast is actually about virology. The hosts of This Week in Virology (TWiV) are superb data communicators, and they often highlight statistical virtues or shortcomings of the virology and infectious disease papers they discuss. In TWiV 1352, recorded on August 26, 2026, Dr. Daniel Griffin used a statistical term that was new to me, saying that the Centers for Disease Control and Prevention (CDC) had “bluewashed” COVID-19 heat maps. Heat maps indicate activity level by…

Um dos melhores adágios corporativos que eu já escutei vai mais ou menos assim: Quem diz que uma coisa é fácil de fazer, em geral não é quem tem que fazer essa coisa. Eu não faço ideia da origem disso, mas que é impecável, é.