In a footnote to the previous post, I said that Python’s math library can calculate the logarithm of extremely large numbers but not the cosine. This post will expand on that comment. In this post I’ll use n = 200! as my example rather than 1000! nbecause this value of N is larger than the largest representable floating point number but small enough to be more convenient to work with. Suppose someone calculates 200! for you: 78865786736479050355236321393218506229513597768717326329474253324435\…

NASA has released a new 3D model of the Earth’s gravitational field. “The geoid is an equipotential surface that can be thought of as the shape an ocean surface would take due to the Earth’s… More

This is a Blaugust 2026 blog post. I’m now in my third term of music theory. Back in the first term, I created a handy reference sheet for major and minor scales. I just added a new ref sheet for what’s called applied (or secondary) chords. A Bit of Music Theory (If you know basic music theory, go ahead and skip this part.) First, listen to this music snippet: Here is the music score for what you just heard: If you enumerate the C major scale – C-D-E-F-G-A-B – you see that C is the first note.…

In this post I mentioned two series which I denoted t(n) and c(n). The former is the number of unlabeled rooted trees with n nodes. The latter is the cumulative sum of the former, i.e. The sequence c(n) is also the number of constrains on an n-step Runge-Kutta method; that’s how I became interested in it. Now the t(n) sequence has been cataloged as OEIS A000081 and OEIS gives the asymptotic estimate of t(n) for large n as where C = 0.4399… and α = 2.9557…. The cumulative sum of t(n), what I’ve…

Yesterday I wrote that a lot of the special functions that pop up in mathematical physics are solutions to second order linear differential equations with polynomial coefficients. More generally, holonomic functions are defined to be those functions that are the solutions to linear differential equations, of any order, with polynomial coefficients. Most special functions are holonomic. To quantify that statement, I went through the special functions covered in Abramowitz and Stegun. The large…

ICM2026 in Philadelphia was, for me, a celebration of mathematics and people. I met many friends and colleagues, as well as several mathematical ideas and results. The nature of these meetings is that they are abrupt and random. (Some people don’t like it, but I do.) For me, mathematics and personal stories are often entangled. I tried to attend most of the plenary lectures and most of the combinatorics and computer science sectional lectures. Of course, given the vast wealth of mathematics…

by Gertrud U. Rey This article marks my 100th contribution to Virology Blog since I published my first post on March 1, 2018. Above all, I have learned that nature does not recognize the boundaries humans use to organize scientific disciplines. My host Vincent Racaniello and the treasure trove of information on the MicrobeTV network […]

Maxwell’s equations describe the behaviour of electric fields and magnetic fields; a more detailed introduction appears in post 25.12. Equations (1)-(4) (highlighted above) are Maxwell’s equations in vector notation. Here E is an electric field, B is a magnetic field, ∇ is the vector operator del, q is charge, ε0 is the permittivity of free space, ∂/∂t denotes partial differentiation with respect to time, μ0 is the permeability of free space and J is current density. Equations (1) and (2) are…

前回の話はこちらにあるので見てほしい 今、このブログではBekko a25mで関連記事が出るようになっている。そこに至るまでの経緯を書いていこう。 最新版のレポートはこちら レポートを別タブで開く Opus 5が設計する代理指標の罠 正解ラベルのない推薦タスクなため、タグ・カテゴリ一致率という指標をOpus 5が提案した。この代理指標の重大な問題にOpus 5が気づいたため追加調査を行った。 一致率の分母 |P| というのは、推薦された記事に依存して変わるのである。 Bekko a25mは、タグ・カテゴリのついた記事を推薦しやすいという傾向が見られた。なので、再現率が上がり、分母が増えるから一致率が下がるという現象が発生していた。しかも、これをずっとOpus 5はHTMLに補足として書いていたのである！（人もLLMも真面目に読んでいなかった！） Fable 5への切り替えとアノテーションツール これを改善するために、Fable 5に切り替え指標を提案してもらった。今回は、分母を固定する固定プール precision@k…

Double Maths First Thing is square in hex AND decimal! Hello! My name is Colin and I am a mathematician on a mission to share the joy and love in mathematics, to show that it’s more than the mindless memorisation of arbitrary rules. Guess who nearly got sucked into a 6 ÷ 2 × 3 debate on Mathstodon? I note that “the ISO 80000-2 standard for mathematical notation recommends only the solidus / or fraction bar for division.” So there. In more edifying news, this is issue #64 of Double Maths First…

I've seen pegboard backdrops and kitchen-style surrounds for a number of years, but an open benchtop with lidless tills and wells works best for instancy in the zone. That's because I do believe lidded boxes and tills and drawers are very practical for storing things like tools and equipment at the end of a tidy-up, but storing them after each use is mostly inefficient and wasted time. Clutter is only clutter if the person working is careless, and by that I mean that they care less than they…

このサイトでは、 している。prelims-cliはfrontmatterに色々と情報を付加できる をCLIで動くようにしたものである。 実装した当初、embedding modelを使った類似記事推薦をやってみたいと思ったが、当時使ったembedding modelではTF-IDFベースの手法に比べてイマイチな結果しか出なかったため諦めていた。 時は流れ、 に「今どきはCPUでもサクサク動く軽量embedding modelがありますよ」と教えられてGitHub Actionsで動く のであった。 ruri-v3とgraniteが軽量だが性能が良いと言うので試してみたら、TF-IDFと比較してもレコメンド結果をざっと目視で見ても悪くない結果だったので採用した。なにより、sudachiの辞書をGHAからダウンロードするよりもembedding modelをダウンロードする方が速いのである。 つい先週、seconさんが という小型モデルをリリースしたという記事を読んだ。これは、ワンチャン日英大統一できないかなー、日英以外もサポートできないかなーと思って、Claude任せに実験をしてみた。…

A characteristic of emergent phenomena is how novel and complex properties can emerge from apparently simple laws. Quantum ChromoDynamics (QCD) describes the interaction of quarks and gluons. The classical Lagrangian is remarkably simple.It has an SU(3) local gauge symmetry and the A_v^mu are the associated gauge fields (gluons).The quarks are fermions with fractional electrical charge. The only parameters in the theory are the coupling constant g_s, which describes the self-interaction of the…

Tenho curtido muito o som da banda norueguesa Slomosa. Foi uma dica do meu grande amigo e sócio Felippe Mercurio, que tem um baita bom gosto pra rock'n roll. Nota engraçada: ele me mandou mais ou menos na época do jogo do Brasil e da Noruega (um pouco antes) pela Copa do Mundo de 2026. https://www.youtube.com/watch?v=xRlUn6GpxPQ&t=1670s

Eu gosto muito do Pagecord, e agora colocaram comentários. Aos meus 7 leitores e leitoras, podem comentar.

The next issue of the Carnival of Mathematics, rounding up blog posts from the months of June and July 2026, is now online courtesy of Robin Whitty at Theorem of the Day. The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

SENSURROUND ON X: RADIATION AND YOU! Hello there, boys and girls! Today we’re going to learn about one of nature’s most fascinating phenomena! Radiation! I’m sorry. Several of you just started panicking. Read the whole thing.

Over the years, physicists have become increasingly enamored with “quantum weirdness,” the many strange features of quantum mechanics that are extremely hard to explain. In a new paper, a group of researchers have tried to cut through the weirdness by analyzing how causal order really works. By doing so, they hope to set up a deeper framework that can combine quantum mechanics with gravity. Let’s

Superforecasting is the art/science/sport of predicting the future - for example, who will win elections, which countries will fight wars, when key technologies will be discovered. Over the past few years, it went from an obscure academic subfield to a multibillion dollar industry in the form of prediction markets. More recently, AI superforecasters have come close to the accuracy of top humans, and their performance is rising rapidly. In a year or two, we’ll see one of the following…

It wasn’t until 1959 that any being on Earth saw our Moon’s farside for the first time. Because the Moon is tidally locked to Earth, we only ever see its familiar nearside face. In the early days of space exploration, the Soviet Luna 3 and spacecraft succeeding it transmitted the first lunar farside images. While these captured images at low resolutions by today’s standards, the photos nevertheless revealed a far more cratered hemisphere than the Moon’s nearside. It looked more akin to Mercury…

Welcome to my 201st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.1. Volumes of RevolutionMy excellent A level student Samarth made a website volumesofrevolution.co.uk. He has very kindly shared it so that A level Further Maths teachers can use it in lessons when teaching volumes of revolution. The website is very easy to use. You can

Topology is the field of mathematics describing the properties of geometric objects that do not change when they are smoothly deformed. These properties only change in steps by cutting or gluing. Concepts in topology can be illustrated with everyday objects such as balls, doughnuts, coffee cups, and pretzels. For example, a doughnut can be gradually and smoothly deformed into the shape of a coffee mug (Figure 33). No ripping or cutting is required. In contrast, it is impossible to turn a ball…

Yesterday, OpenAI announced the solution to ten open problems in mathematics, all discovered by a yet-unreleased model. There’s only one, the coding theory one, where I know enough to say “huh, that’s important”, but according to the mathematicians I trust this is important. Inevitably, people will spin some sophistry to cope, to convince themselves nothing will change. And that’s fine. People need to cope. But we can’t put our heads in the sand forever while the world is transformed around us.…

Versão EN Essa história já tem um tempinho, mas como eu recebi atualizações recentemente eu decide trazer aqui. Bem, tudo começou em junho, no aniversário da minha mãe. Ela não é muito de fazer festa nem nada grande, mas a família estava passando por um momento difícil e ela queria muito se reunir com quem ela podia. Nisso vem meu tio (um querido) e a mulher dele (não tão querida). Eu sendo como sou, evitei a festa o quanto pude. Cumprimentei geral, mas comi escondida no meu quarto. Até ai tudo…

Second order linear differential equations with polynomial coefficients form their own area of study. This seems like a narrow class of equations, but it’s very important in applications. This class of equations seems like a mathematically natural topic, but why is it so important in applications? I did a PhD in differential equations without ever learning why. The theory of second order linear equations with polynomial coefficients is too complicated for undergraduate courses and too…

The textbook version of the Runge-Kutta method for solving differential equations has 4 stages and has 4th order error. For lower order versions of RK the number of stages s also matches the order of the error p. But in order to achieve error on the order of p ≥ 5, you need more than p stages. This is known as the Butcher barrier. Before going any further, let’s back up and say what we mean by stages and by order. Stages The number of stages in an RK method to solve the equation is the number…

Here’s a short round-up of maths news from this month. The 2026 Fields medals have been awarded to Yu Deng, John Pardon, Jacob Tsimerman and Hong Wang. The Fields Medal is awarded to recognize outstanding mathematical achievement for existing work and for the promise of future achievement, and this year’s winners work across PDEs, symplectic geometry, algebraic geometry, harmonic analysis and measure theory. Hannah Fry has been awarded the IMU’s Leelavati prize for increasing public awareness…