Ordinarily the sine of x radians and the sine of x degrees are very different numbers. Having your calculator in radian mode when it should be in degree mode, or vice versa, results in a major error. But sometimes it doesn’t matter. A trivial example is when x = 0. A more interesting example is x = 180π/(180 + π) = 3.08770208…. For that vale of x, sin(x) = sin(x°). In this article I’ll use the common convention of using radians by default and denoting degrees with ° as above. Note that x° =…

Back in the 90s when I was a young professor at the University of Chicago, we would have a Complexity Class of the Week where I would take some interesting complexity class, write down on a white board everything we knew about it with some open problems and students and faculty would muse over it. When I started the blog in 2002, I took the concept online. My first Complexity Class of the Week post covered the class \(S_2^P\). Recently Rahul Santhanam said to me "\(L_2^P\) is the new…

I don’t usually talk about particle physics here. I have a whole series of articles about octonions and the Standard Model on my other blog. But I’m kind of excited about this new paper, so I’ll talk about it here too: • John Baez, Endre Bokor and Latham Boyle, Jordan pair quantum theory and the Standard Model. Jordan algebras were introduced by Jordan, von Neumann and Wigner in 1934 in an attempt to formalize algebras of observables in quantum theory. They come in 4 infinite series—but there’s…

For over half a century, people taking the psychedelic drug DMT have reported meeting alien beings. Now a scientist says these aliens might be real, and that he has the mathematics to prove it. Forget string theory and extra dimensions; here come aliens and psychedelics. Let's take a look.

Double Maths First Thing doesn’t fear the REAPER Hello! My name is Colin and I am a mathematician on a mission to spread mathematical joy and surprise. This week is full of leavers’ assemblies and general end-of-term commitments, so you’ll forgive me for keeping it brief. Links Let’s start at EMF, which (for various good reasons) is not my cup of tea, but might well be yours – especially if you revel in things like Ayliean spending five hours doing 14 levels of a Tower of Hanoi. It always…

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be…

Poststratification uses population data on X to estimate E(Y) via E(E(Y | X, R = 1)), where R = 1 are survey respondents who provide Y and X. When the inner expectation “E” is estimated via Multilevel Regression, this is called MRP. The outer “E” needs p(X), population data on X. Sometimes we have to estimate the population distributions. We’ve seen a few examples: “2 flavors of calibration”: Say we have p(X), but we also need p(Z | X), the population distribution of another variable Z. We can…

Reference Reading A Chemical Hunger, Part VII and Interludes C, G, and H U.S. Geological Survey, Public Water Supplies of the 100 Largest Cities in the United States, 1962 U.S. Geological Survey, Lithium in U.S. Groundwater, 2021 Ferensztajn-Rochowiak, E., & Rybakowski, J. K. (2023). Long-term lithium therapy: side effects and interactions. Pharmaceuticals, 16(1), 74. Background Information In clinical settings, lithium is usually prescribed as lithium carbonate, and doses are given in…

FAREWELL AND THANK YOU: SpaceX booster B1069 to be expended after its 32nd flight (scheduled this evening) on the Northrop Grumman MRV-MEP Mission, after an ISS resupply flight, three commercial satellite launches, and 27 Starlink deployments. Normally recovered and reused, tonight’s flight requires so much energy from the booster that there won’t be enough fuel to safely land it.

This year, I have the honor of serving as the program chair for the Haskell Symposium, co-located with the 2026 International Conference on Functional Programming, or ICFP to its friends1, on August 28-29 in Indianapolis, Indiana! The Haskell Symposium has existed in some form or another since 1995, and I’m proud to be part of this long tradition, now in its thirtieth iteration this year (wow!). It’s also a homecoming for me, since I did my PhD in Indiana back in my callow youth. I’m excited to…

In July 1925, a small courthouse in Dayton, Tennessee, became the unlikely stage for one of the strangest legal spectacles in American history. A 24 year old football coach was put on trial for teaching Charles Darwin's theory of evolution to his students, and within days the case had ballooned into a full blown circus, complete with chimpanzees in fedoras, hot dog stands, and a national radio audience hanging on every word. The State of Tennessee v. John Thomas Scopes, better known as the...

Joshua Brooks writes: I know you’ve posted on the topic more generally but don’t recall if you’ve discussed this in particular. Given the timing in relation to cuts in food assistance, It seems a particularly egregious example of the politicization of data. The news article, published in late 2025 by an organization called Food Tank (“The Think Tank for Food”) is titled, USDA Ends Key Food Security Report, Leaving Advocates in the Dark, and it begins: The U.S. Department of Agriculture (USDA)…

There is a new website, hyperelliptic.ncag.info, on the classification of hyperelliptic (or generalized hyperelliptic) varieties in complex dimensions 2, 3 and 4. For now it lives as a subdomain of ncag.info. A hyperelliptic variety is a quotient $X = T/G$ of a complex torus $T$ by a finite group $G$ acting freely and without translations. In dimension 1 these are the elliptic curves, in dimension 2 the seven bielliptic surfaces of Bagnera and De Franchis; the word has nothing to do with…

Read: Misinterpretion by Ledis Xhoga ★★★ 📚 An Albanian interpreter gets into a lot of communication difficulties. Set in New York & Albania Constantly surprising narrative as the narrator bounces from one complex situation to another.

Rudolph Marcus died last week. He was 102. There is a nice obituary in The New York Times. He was best known for his theory of electron transfer, for which he was the sole recipient of the Nobel Prize in Chemistry in 1992.Although the theory was proposed for electron transfer in a polar solvent it applies to a wide range of other systems, where two quantum states are coupled to one another and to an environment. One example is for Forster transfer of excitons between molecules, which is central…

MARTINELLI, Carlo. Debord, la foto rubata. [Faenza]: All’Insegna del Babbomorto, February 2026. Second edition. 16 p.; 15 x 21 cm.; stapled wrappers, cream stock, title and monogram printed in dark red; cover illustration reproducing, in half-tone, the photograph of Guy Debord published in Paris Match of 6 April 1984. A short prose piece devoted to a single episode: the appearance, in the 6 April 1984 issue of Paris Match, of a grainy, evidently clandestine photograph of Guy Debord, captioned…

网上看到很多人说到 EP4，也就是出题篇结束，读者就已经有了所有的线索，虽然大部分人都还得往下再看一两集才能自己解开。我当然是解不开的啦，但还有些想法和问题吧。主要是问题。想着先写下来，到时看完了后还能再回来看看自己说没说对，哈哈哈。 当然，本文有《海猫鸣泣之时》1 - 4 集的剧透！没看过的真的不要读啊！去看原作！可好看了！ 我是真觉得，至少在这个阶段，能“解开”谜题也是瞎猜歪打正着。要考虑到凶手有共犯，可能有很多个共犯，而且每一轮游戏的凶手和共犯都不一定一样，同一轮游戏内甚至都可能有不同的凶手（虽然明明看到有别人死了你还自顾自地执行自己的杀人计划还是有点骇人的但不能否定这个可能性），唯一能确定的是战人肯定永远不是凶手或共犯，不然没法玩了。而且八成也能假定没人死于自杀或意外，不然就没意思了。唯一能相信的只有红字和战人亲眼看到的东西，剩下的都能理解为贝阿朵在糊弄你。虽然这些魔法激光剑东西肯定有自己的意义，但我压根不知道有什么意义啊啊啊啊。 首先…

In today's video I talk about some commonly misunderstood points in the double slit experiment.

The volume of a sphere of radius r is V = 4πr³ / 3 and the surface area is A = 4πr² and so the ratio of volume to area is V / A = r / 3. Surprisingly, the same ratio holds for all regular solids if r is the radius of the largest sphere that can be inscribed inside the regular solid. For example, if the edge of a cube is a, then r = a/2. The volume is 8r³, the area is 24r², and the ratio is r/3. The relationship between edge length and radius, and between radius and volume, is more complicated…

In 2021, the United Nations proclaimed July 20th as International Moon Day, via the resolution 76/76 on “international cooperation in the peaceful uses of outer space”. Some key goals of the day are to promote awareness about our Moon’s unique scientific and cultural importance as well as how countries worldwide are exploring it. To that end, I present a curated webpage of linked, free resources for people from all backgrounds to learn about our Moon as well as dive into specific exploration…

Jonathan Pinchera is an artist based in Brooklyn.Support our publishing & projects:In the above image, rapper Skrilla wears “The Mask of Bobobja” which I designed and embroidered for the Philadelphia-based textile collective Chalk.Press in October 2022. After sitting largely untouched for nearly three years the mask was sold to a close associate of Skrilla, who later gifted it to him in the middle of 2025. Skrilla began wearing the mask incessantly in music videos, interviews, and instagram…

In this week’s issue of The Savvy Diabetic: Islet Cells Still Working Without Immune-Suppressing Drugs No Touchy! When Your Diabetes Device and Your Skin Just Can’t Get Along Continuous Ketone Monitors with Dr. David Ahn/@AhnCall, Insulet, Calm join forces for diabetes care offerings Finerenone Improves Kidney Function in Type 1 Diabetes Dual-Hormone Pumps: A Solution in Search of a Problem What’s Happening at ADA?

[ in progress... ]

By David Tuller, DrPH The Journal of Psychosomatic Research has often served as a house organ of sorts for the “biopsychosocial” ideological brigades. That’s to be expected, given that folks like Per Fink, Michael Sharpe, and Jon Stone are all members of the journal’s editorial board. It should therefore not be surprising that the journal … Trial By Error: Yet Another CBT Trial for Long Covid with Poor Results Read More »

Maxwell’s equations are invariant under Lorentz transformations. The usual equations of fluid flow are not! Like the rest of Newtonian mechanics, they’re invariant under Galilean transformations like So, if we simply slap these two theories together, we get a mess! How can we study electrically conductive fluids—like plasma—without bringing special relativity into the game? We can use a limiting case of Maxwell’s equations where we ignore terms that become tiny when all the particles are moving…

I came across this webpage by Sheeva Azma entitled, “Here’s every scientist I have found in the Epstein Files so far.” She’s missing a few big fish: Dan Ariely (professor at MIT and Duke, Ted talk star, and teller of a story about a possibly nonexistent paper shredder) Donald Rubin (professor at Harvard and one of the most influential statisticians of the twentieth century) Stephen Hawking (late physicist and culture hero) Henry Rosovsky (professor and dean at Harvard; ok, he’s just an…

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 six papers, bringing the number in Volume 9 (2026) to 151 and the total so far published by OJAp up to 599. I will continue to include the posts made on our Mastodon account (on Fediscience); these announcements also show the DOI for each paper. The first paper to report this week, published on Tuesday 14th July, is “On combining estimated and…

“Progress” is a word that often generates both a lot of positive vibes, and also a lot of pushback? Progress toward what? At what cost? So it’s perhaps useful to specify that when I refer to “progress,” I’m referring to an overall movement toward good things of life: in no particular order, longer and healthier lives; a rising standard of living, especially for the poor; more education for more people; environmental protection; leisure time; personal safety, resources of time and money for…

In Einstein's theory of general relativity, gravity is always attractive. But according to a new paper, quantum mechanics offers a loophole that can create antigravity. And not only this, if it was measurable that would prove that gravity has quantum properties. How does this work and how can repulsive gravity be compatible with the laws of physics that we know and like? Let's take a look.

I really, really, like Saturday Morning Breakfast Cereal, because Zach clearly gets the joke. Like this comic: It’s great not only because it got an initial laugh. It also has some additional embedded assumptions that are just wild, y’all. The black swan fallacy. The argument above only works if all sims are porn. They’re not, so this proves nothing. People don’t actually like it when they get everything handed to them, either in narrative or in real life. On the other hand, there’s also lots…