Bartosz Milewski

Physicist, programmer (Haskell, C++), mathematician, category theorist. Author of Category Theory for Programmers and The Dao of Functional Programming

blog: https://bartoszmilewski.com/
github: https://github.com/BartoszMilewski
book: https://github.com/BartoszMilewski/DaoFP/blob/master/DaoFP.pdf

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Last Notes

2026-04-16 07:48:37 UTC

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Unfortunately, this is offset by Citizens United, which gives rights to corporations

2026-04-15 18:40:45 UTC

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On the other hand, humanity is a transient problem. We are nearing peak humanity, after which there will be exponential decay. As life gets more and more comfortable, the prospect of raising children becomes less and less atractive.
https://media.mathstodon.xyz/media_attachments/files/116/410/181/878/983/425/original/c20f4a92ceb5bbfa.png

2026-04-14 18:03:23 UTC

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By coincidence, yesterday I asked Claude about the most common form of life.
By Number of Individuals: Bacteria (domain Prokaryota) dwarf everything else on Earth. Estimates put the total number of bacterial cells at around 10³⁰ (one nonillion) — that's more than the number of stars in the observable universe.
But viruses outnumber bacterial cells 10 to 1.

2026-04-14 15:52:18 UTC

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I think we are all hallucinating

2026-04-14 08:35:53 UTC

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A good source of information is ISW. https://understandingwar.org/research/russia-ukraine/russian-offensive-campaign-assessment-april-13-2026/

2026-04-14 08:34:01 UTC

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This is how Claude explains the sudden change: The fall of Avdiivka triggered a phase where Russia, emboldened and pressing forward across open ground, ran headlong into a Ukraine that had massively scaled up cheap, lethal FPV drone warfare to compensate for its artillery shell shortage. The combination of Russian infantry-wave tactics and Ukrainian drone density proved catastrophically costly for Moscow.

2026-04-12 21:08:47 UTC

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I think Vance was sent to Hungary and to Pakistan as a punishment for his opposition to the war in Iran. He's a despicable guy, but his IQ is slightly higher than Trump's, and that's a fatal flaw in Trump's eyes

2026-04-10 18:18:14 UTC

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You can also find microtonal influences in the Amercian blues. On a guitar you can do all kinds of bends, which gives the blues this emotional, longing quality.
Microtonality can be also heard in singing (before autotune). I think that's what gives me a lump in the throat every time I listen to Billie Holiday's Strange Fruit. Her voice matches perfectly the imagery.

2026-04-10 07:30:33 UTC

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I love their music. I was wondering when you'd mention them

2026-04-07 18:43:09 UTC

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Maybe some republicans could be persuaded to join in the vote for invoking the Twenty-fifth Amendment

2026-03-25 22:12:55 UTC

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The most surprising thing is that these virtual particles are able to bend spacetime. 98% of the mass of any star is virtual quarks and that's what generates their gravitational field.

2026-03-06 23:50:07 UTC

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That reminds me of the lemur aye-aye that also has very long fingers.
https://media.mathstodon.xyz/media_attachments/files/116/184/904/615/630/855/original/9ff41d450570a5f4.png

2026-02-22 00:15:23 UTC

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Unless the perfect fifth is a cultural artifact

2026-02-20 19:44:18 UTC

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I can confirm the Polish version:
- jej psa (accusative case of her dog)
- swojego psa (same but reflexive)

2026-02-17 09:06:44 UTC

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It made me wonder what other physical phenomena exhibit this kind of resonant behavior, and I thought of the Titius-Bode law of planets. Wouldn't it be cool if it was explained using the zeta function? (You also mentioned the rings of Saturn, which is a similar phenomenon.)
https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law

2026-01-28 13:48:40 UTC

The piñata effect: At the final moment of a black hole evaporation through Hawking radiation, all objects that were swallowed by it suddenly come back, thus restoring the conservation of information required by quantum mechanics.

2026-01-28 11:01:11 UTC

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This is even more obvious in constructive mathematics, where things are constructed rather than discovered. The ideas that are effective are kept alive by natural selection. Wigner was wrong-- the effectiveness of mathematics is very reasonable.

2026-01-15 11:41:45 UTC

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Maybe we are moving with the speed of light in the fourth dimension and it shrinks to zero.

2026-01-11 11:50:15 UTC

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In other words, the radius of curvature is a little more than one AU, the distance between the Sun and the Earth

2026-01-07 14:02:14 UTC

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Omnipotent within limits

2025-12-24 10:24:40 UTC

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I can imagine Stephen Wolfram's dilemma: Is the universe a cellular automaton? Or was I wrong all along, and it's a billiard ball?

2025-12-22 13:02:45 UTC

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The dark archive reminds me of the Second Foundation as the defense against the Mule

2025-12-22 09:55:35 UTC

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For scale: light travels 1.5 km (or about 1 mile) during that time

2025-12-20 17:27:30 UTC

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The US has a huge infrastructure debt. All those overhead power lines that snap every time the wind starts blowing. Not to mention the esthetic nightmare of tangles of sky-obscuring wires in every residential area. Most Americans don't even notice it.

2025-12-16 20:36:32 UTC

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I whould divide the two numbers and see if the result is less than one. The advantage is that you can cancel the factor of 2⁴

2025-12-06 12:06:39 UTC

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What's fascinating about burds is that they have to be very thrifty with their brains, or they wouldn't be able to lift them into the air. And their brains have to deal with 3-d navigation and superior vision. Compared to birds, human brains are extremely wasteful.

2025-11-29 08:39:01 UTC

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I'm imagining a taffy machine running for billions of years, with an emerging Taffian civilization discovering Taffian physics, when suddenly the movement is reversed.

2025-11-27 22:19:24 UTC

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I wonder if it could be undone.
https://www.youtube.com/watch?v=UpJ-kGII074

2025-11-06 14:03:47 UTC

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The centrifugal acceleration is the rate of change of velocity. Velocity vector just rotates with the angular velocity \(\omega\). So its rate of change is the length \(v\) times the angular velocity: \(a = v \omega \). Since \(v = r \omega\), we get
\[a = r \omega^2 = v^2/r \]

2025-11-06 13:46:00 UTC

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Naively, the centrifugal force increases with distance, \(F = m \omega^2 r\), but that is only true if you keep the angular velocity constant. In your case, you are keeping the angular momentum, \(L = m r^2 \omega \) constant. Thus \(\omega = \frac{L}{m r^2}\) and \(F=\frac{L^2}{m r^3}\)

2025-11-03 17:04:04 UTC

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Although gravity is not a Y-M field, you can formulate it as a gauge theory of the local Poincare group. The "gauge potentials" for translations are vierbeins (tetrads) and spin connections correspond to local Lorentz transformations.
We used this trick when working with supergravity. The regular Y-M fields were just the components of gravitational fields in compactified dimensions, a la Kaluza-Klein.

2025-10-30 13:09:52 UTC

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I think its the Kolmogorov-Arnold theorem
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold_representation_theorem

2025-10-22 19:45:56 UTC

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I know the pain. You want to simplify things but you don't want to lie.

2025-10-22 14:06:16 UTC

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Since you mention that the simple Feynman diagram for electron/positron annihilation is not physical, the same is true about the two others. You cannot satisfy the conservation of energy-momentum on-shell for these diagrams. And that's lucky, otherwise electrons in vacuum would spontaneously emit photons.

2025-10-09 10:01:51 UTC

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I voted for the bubble, but the technology behind AI will have many useful applications. So it's just like the dotcom bubble: many companies went bankrupt, but the internet survived. It's a good bubble for the technology, bad for the investors.

2025-10-06 20:25:40 UTC

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Don't get me wrong, I don't doubt the reality of the AB effect. I'm just sceptical about the arguments that it somehow proves that the magnetic potential is "real," and not a clever computational trick.

2025-10-06 18:42:23 UTC

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Since we're on the subject, I'm growing a bit sceptical of the AB effect.
AFAIK, in the actual experiment we have an "infinite" string of electrons whose spins are aligned in one direction, so their magnetic fields cancel each other. We replace these electrons with a classical magnetic potential, and it looks like the passing electron can feel this potential, even though the (classical) field strength vanishes.
At what point can stacking N electrons on top of each other be described by a classical potential? It's not a phase transition. For any finite N, the system is just a bunch of electrons interacting by bouncing photons back and forth.

2025-10-06 16:37:51 UTC

In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken.
In HoTT, the value of a mapping out of identity type does not depend on the path, only on its endpoints.
The tricky part is when the path is closed. The total work then is zero. In HoTT there are non-trivial paths that cannot be shrunk to a point. This reminds me of the Aharonov-Bohm effect. I wonder how far one can stretch this physics analogy.

2025-10-06 08:35:18 UTC

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On the bright side, they don't burn scientists at the stake any more.

2025-10-05 19:16:53 UTC

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Great video! I think this is not just a physics problem, but a more general societal problem. We are living in times of unprecedented inequality. This is obvious when you look at the ownership of resources, with all these billionaires monopolizing financial wealth. But there is a similar divergence in the distribution of knowledge. Only the top one percent have the means, the dedication, and the ability to get basic understanding of the current knowledge in any given area of science. The physics-bashing or anti-vax propaganda is just a reflection of the rise of populism in politics. Uneducated people are easier to manipulate.

2025-10-04 11:05:12 UTC

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"True" chemical bond is the result of a (crude) approximation, and as such it has its limitations.
For instance, we assume that the outer electron shell of the carbon atom has four lobes forming a tetrahedron (that's what the black blobs with four holes represent). It's a result of the hybridization of the S and P orbitals, which is an approximate solution of the Schroedinger equation.
Next we assume that when atoms form compounds, they essentially keep the shapes of their orbitals. This is the basis of stick models. Again, it's a crude approximation.
Theoretically, we could just solve the Schroedinger equation for all the electrons and nuclei and be done with it--no orbitals, no hybridization, no sticks. Just one glorious n-electron wave function. Since we can't do it, we use the crutches of valence, stick models, Hartree-Fock approximations, etc.

2025-10-02 11:45:41 UTC

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It makes sense. For a moment I thought that it would change chirality, but it doesn't.

2025-10-02 10:53:28 UTC

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There is one place in QFT where mass enters linearly: in the Lagrangian and the propagator for fermions. I'm not sure if it only influences polarization, or dynamics as well.

2025-10-01 11:43:53 UTC

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I read the introduction, but unfortunately it doesn't talk about the mechanism for the production of heavy elements. You're right that beta decay is slow, but I can imagine that the merger could produce neutron rich isotopes of, say, mercury, which would then slowly decay into gold. Either way, a neutron star has an abundance of neutrons. Even the outer crust contains neutron-heavy isotopes. It's the protons that are hard to come by.

2025-09-30 23:09:43 UTC

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I'm trying to understand why neutrino oscillations would help gold creation. Neutron stars are all neutrons, so in order to get gold you'd need to produce protons. These could come from beta decay, which also produces electron anti-neutrinos. These could oscillate into muon anti-neutrinos, which would prevent them from turning protons back to neutrons + positrons. Is that what's happening?

2025-09-26 15:22:44 UTC

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Make sure you have a good microphone. Sound quality is even more important than video quality.

2025-09-18 22:17:40 UTC

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So how exactly would you argue that the Universe doesn't have an equator? Just curious.

2025-09-11 15:09:47 UTC

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My guess would be Compton scattering

2025-09-11 13:10:28 UTC

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That makes sense. Inside a star, the accretion would be more uniform, thus no fluctuations caused by chunky matter found in galaxies. And the outer layes of the star could absorb the resulting X-rays.

2025-09-11 12:22:42 UTC

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One argument against black holes is that LRDs don't emit gamma rays, and don't fluctuate.

2025-09-11 11:15:14 UTC

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Here's another explanation of red dots by Nandal and Loeb.
https://arxiv.org/abs/2507.12618

2025-09-04 11:26:04 UTC

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I looked at the original paper and I'm confused. They are saying that the actual equations depend on the squares of masses, but the non-relativistic approximation becomes linear in the masses. That doesn't make sense.
https://media.mathstodon.xyz/media_attachments/files/115/145/770/367/890/352/original/c786aa870198f4d1.png

2025-09-04 11:15:12 UTC

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If gravitational effects only depend on the square of the masses, I wonder why they are saying: "More gravitational lensing is also what you’d expect from neutrinos with negative masses."

2025-09-04 11:06:22 UTC

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"the Higgs boson of the Standard Model of particle physics, which has an imaginary mass in its uncondensed phase". That's what makes it unstable, though.

2025-08-09 16:14:32 UTC

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In the modern world you need a PhD to realize how insignificant the chances of you advancing science really are.

2025-08-04 10:52:46 UTC

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Next he's going to install a garbage disposal in his bathtub.
https://www.youtube.com/watch?v=rqkMo3nThHk

2025-07-19 21:55:50 UTC

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I experienced it on myself, and my life got so much less stressful.

2025-07-18 12:02:15 UTC

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No AC/DC ?
Men at Work "Down Under"

2025-07-02 15:48:42 UTC

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More precisely, take an infinite number of drawers, each containing a pair of socks. The AoC says that it's possible to pick one sock from each drawer.
So in AoC you have to assume that the drawers are not empty because you can't pick a sock from an empty drawer.

2025-07-02 12:10:13 UTC

Axiom of Choice in HoTT. For any family of sets Y parameterized by elements of a set X, there exists a function below.
Naively, this says that if all sets Y(x) are inhabited then all sets Y(x) are inhabited. Duh!
The tricky part is the vertical bars. They signify 'propositional truncation', meaning: we know it's true, but we forgot why. So we know that every Y(x) is not empty, but we don't have the witness of this fact.
The axiom of choice says that the witnesses exist: if you know that the sets are non-empty, it's possible to choose one element from each set.
The propositional truncation on the right means that, trust me!, the witnesses exist, but you can't get at them.
https://media.mathstodon.xyz/media_attachments/files/114/783/493/006/659/908/original/cc4314a7f9b195e2.png

2025-06-29 14:47:14 UTC

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I'd rather buy drinking credits from teetotalers

2025-06-29 09:24:02 UTC

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So you still have 10 years of drinking to look forward to? 🥂

2025-06-29 08:17:44 UTC

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Don't forget about an occasional leap light-second.
https://en.wikipedia.org/wiki/Leap_second

2025-06-28 08:54:42 UTC

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If only everybody took a 2-hour course on fraud awareness, this could have been avoided 😉

2025-06-27 14:46:58 UTC

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They are trying to exorcise the "evil" from category theory. https://ncatlab.org/nlab/show/evil

2025-06-27 14:32:04 UTC

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They are arguing that "equals" was a loose badly-defined concept, so they are not so much "redefining" as "defining" it. And the push came from attempts to codify mathematics so it would become acceptable to computers--which are much pickier than human mathematicians.

2025-06-27 13:10:09 UTC

I just realized that my whole adult life I've been studying precategories. In an "actual" category, if \(a \cong b\) then \(a = b \).

2025-06-24 19:25:25 UTC

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wrote "But there must be people who don't feel intelligent enough - much as I don't feel athletic enough, or healthy enough. Right? Or does *everyone* think they're intelligent enough? "
People don't think they lack intelligence. They might say "I'm bad at math, or physics." But they think they have other powers, and they have access to privileged information (favorite conspiracy theory). They have a "gut feeling" about some people (politicians, billionaires, celebrities) that overrides common sense. When confronted with logical arguments, they often dismissively say "You and your logic..."

2025-06-24 19:10:43 UTC

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I was going to say that if you have intelligence, and you are interested in money, you can easily double it. But it's not that simple.
Intelligence means you're good at solving puzzles. Life is not a puzzle. You have to be able to work with incomplete information and manage risks. I know a lot of people smarter than me who are very bad at managing their affairs.

2025-06-22 11:55:08 UTC

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Encourage the invasion of Taiwan by showing indifference towards Russian invasion of Ukraine.

2025-06-06 14:36:45 UTC

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For comparison, the nearest star is 270,000 AU from the Sun.

2025-06-05 11:03:55 UTC

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All these models are incomplete without accounting for stupidity. Stupid people act against their own utility. Also:
Cipolla's law of stupidity: A stupid person causes harm to others without any gain to themselves.

2025-05-20 12:22:36 UTC

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Yes, I'm also used to up/down summation-- I had to force myself to lower them all ;-).
A quick check for this formula is to realize that both sides must be anti-symmetric in free indices, and that \(\epsilon_{1 2 3}\}\) is equal to one.

2025-05-20 09:30:06 UTC

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In my time, I had to do a lot of vector calculus so I learned a few tricks. I would rewrite the equation in components, and express vector product using the Levi-Civita symbol. There is just one basic identity to remember:
\[ \epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{jl}\delta_{im}\]
With this and:
\[\partial_i(v_mv_m) = 2v_m \partial_i v_m \]
you can derive the identity you were using.

2025-05-12 20:53:14 UTC

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"More electrons will land on one side than the other, by chance."
I think they land on the side that faces the negative pole of the current source. It's not random. They get stuck there for a while. They are repelled by the negative pole, so the drop will start turning.
"When they reach the bottom of the droplet, some will fall off and continue moving through the oil."
Actually, the droplet is also a conductor, albeit slower than the oil. So the current inside the droplet will slowly neutralize these surface charges--as the droplet turns.
The key is that there is a difference in conductivity and permeability between the two mediums.

2025-05-11 12:04:42 UTC

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Consider the extremes:
(1) the charges are moving fast inside the droplet and not at all in the liquid. The droplet would quickly become a dipole in the "right" direction. That's a stable situation.
(2) the charges are moving fast in the liquid, but not in the droplet. It's like having a hole inside the dielectric. The surface of the hole becomes polarized in the "wrong" direction. Or, conversely, the liquid on both sides of the droplet gets polarized in the "right" direction.
The charges from the liquid tend to stick to the droplet, thus giving it the "wrong" polarization.

2025-05-11 11:27:53 UTC

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Exactly! In the hunter-gatherer society, light was the regulator, and there was boredom after sunset. Artificial light and screen devices changed all that. It's like giving children unlimited access to meth.

2025-05-11 11:20:05 UTC

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It seems like the charges move slower in the droplet than they do in the liquid. This leads to charges accumulating on the surface, creating a dipole moment. But this dipole is oriented in the "wrong" direction, leading to instability. The dipole tips over in a random direction. The viscosity means that the droplet starts slowly rotating in that direction.That triggers the dipolization process again, trying to recreate the "wrong" dipole by moving charges. The bulk of the droplet rotates in one direction, while the charges are moving in the opposite direction.

2025-05-04 09:03:33 UTC

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I used to watch his YouTube videos, but I got gradually disillusioned by his ideology. He's actually motivating his philosophy of mathematics by his religious beliefs. So when he says a number "exists" he means it was created by God.

2025-05-01 10:05:50 UTC

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Maybe the problem is that we teach about energy first, get used to the idea of kinetic energy and potential energy, conservation of energy, etc. Then we spring up the Lagrangian in terms of energy and circle back to the Hamiltonian.
Maybe it would make more sense to think of the Lagrangian and the action as more fundamental and introduce energy (and momentum) as a result of Noether's theorem. After all, conservation of energy is not a fundamental law, as seen in General Relativity.

2025-04-30 19:45:54 UTC

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Roughly speaking, potential energy is the thing that flips the sign when you go between the lagrangian and the hamiltonian. So it has something to do with the Legendre transformation. But what's the meaning of the Legendre transform?

2025-04-30 19:33:34 UTC

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I was thinking more in terms of field theory, where the quadratic part is the kinetic energy and the higher order terms are interactions.

2025-04-30 17:45:19 UTC

In all my years of doing physics, I never figured out the intuitive meaning of the lagrangian (or the action). Why should the difference between kinetic and potential energy play such a prominent role? What separates kinetic from potential energy?
I was always afraid to ask this question, not to sound stupid. 😕

2025-04-09 13:39:25 UTC

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She hands me a cigarette lighter. "Here, take this. Its flintstone contains cerium and lanthanum, rare earth elements."

2025-04-06 19:26:06 UTC

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I found this help:
On Windows:
Go to Settings > Devices > Bluetooth & other devices.
Find the adapter, click it, then choose "More Bluetooth options".
In the Devices and Printers section (Control Panel), right-click the adapter > Properties, and rename it under the Bluetooth tab.

2025-04-06 18:06:27 UTC

I'm so fed up with Bluetooth headphones!
They take a few seconds to connect. When I'm streaming YouTube and I disconnect, it sometimes stops the playback, other times it doesn't. I'm using B&O headphones. Is there a better alternative?

2025-03-21 08:36:51 UTC

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I have 34k followers and I'm almost exclusively posting about politics. I have a feeling that Musk (if he paid any attention) would rather have me quit. It makes no difference that I'm doing it from France. Living in the US was just too depressing.

2025-03-20 18:57:07 UTC

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The reason I think engagement is so important is that people will trust personal exchanges more than they trust mass media. A tweet from John Baez is more trustworthy than an article in NYT because you don't have an economic incentive (maybe even the opposite). You do it because you are genuinely worried about the state of the country. And your voice also encourages other people to speak their mind.
You can always block the idiots who are disrespectful.

2025-03-20 18:28:42 UTC

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The same goes for emigration--it makes sense only in extreme conditions, when you're actually facing persecution (that was my reason for leaving Poland). Of course, moving to Europe for other reasons is a personal choice (which I'm glad I made). Not to mention that Europe needs all the help it can get, now that it can't count on the US.

2025-03-20 18:17:57 UTC

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I'm worried about people reacting to Trump excesses by withdrawing to safe environments--ceding territory to the enemy. It makes sense to boycott Tesla, but it's counterproductive to withdraw from Twitter, which is a huge public forum. As long as Musk is not blocking critical voices, we can engage and be heard by lots of people. Our voices have to be louder, not quieter. Every negative action by Trump, Vance, or Musk must be met with strong reaction.

2025-03-20 10:44:31 UTC

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Speaking as a survivor or a totalitarian system:
The things that work are protests and solidarity.
They will try to go after individuals, so solidarity is very important. An attack on one is an attack on all. Don't be a bystander.
The more public the protests are, the more effective they are. Protesting inside gated communities and ivory towers doesn't work.
This is why I see no sense in boycotting Twitter.
You might feel weak as an individual against a mob, but as a community we are very strong. A society would collapse without scientists.

2025-03-17 10:10:58 UTC

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I think physics has accumulated much more technical debt than mathematics. It made a lot of sense as long as we could fall back on experimental verification. But the ice is getting thinner.

2025-03-16 21:14:23 UTC

There is a common misconception that physicists create mathematical models of reality. In actuality they use a system of heuristics that imitates mathematics. It can be used to calculate things while skilfully ignoring internal inconsistencies. Physicists hallucinate mathematics.

2025-03-16 20:27:05 UTC

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Kenny G and David Sanborn created the genre of music that I call jazzak (a portmaneau of jazz and muzak)

2025-03-11 18:54:51 UTC

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He's like this great Roman Emperor. What was his name? Nero.

2025-03-07 09:19:25 UTC

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Yes, I've heard about the Steele's dossier. It was also investigated in the Mueller report. I don't think it's a very reliable source.
Also, I'm not sure if Trump can be compromised. Even if the golden shower tapes were released tomorrow, he would say, "That's just what guys do in Russian hotels" and the MAGA people would heartily applaude.
I'm more inclined to speculate that Moscow offered him power and money. And they are delivering on it.

2025-03-02 17:10:22 UTC

A pretty harsh takedown of Sabina that to a large extent supports my own growing disenchantment with her increasingly click-baity and anti-scientific posts.
https://www.youtube.com/watch?v=nJjPH3TQif0

2025-03-02 10:44:39 UTC

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The literally create negative vacuum

2025-02-28 21:22:30 UTC

I keep telling myself that it's ridiculous to think that the US president could be a Russian agent. I'm not going to fall for a stupid conspiracy theory. A much simpler explanation is that I'm just going insane.

2025-02-24 09:22:27 UTC

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What Europe should, and hopefully will do soon, is to confiscate frozen Russian assets worth 210 billion euro and give it to Ukraine. https://www.euronews.com/my-europe/2025/02/20/tusk-calls-on-eu-to-confiscate-russias-frozen-assets-to-provide-ukraine-with-financial-ass