Notes

Troy, (2004) dir. Wolfgang Peterson

This will be the greatest war the world has ever seen. We need the greatest warrior.

- Nestor to Agamemnon about Achilles.

Achilles: They'll be talking about this war for another thousand years.
Hector: In a thousand years, the dust from our bones will be gone.
Achilles: Yes prince. But our names will remain.

page: Are the stories true? They say your mother was an immortal goddess. They say you can't be killed.
Achilles: I wouldn't be bothering with the shield then, would I?
page: The Thessalonian you're fighting... he's the biggest man I've ever seen. I wouldn't want to fight him.
Achilles: That's why no one will remember your name.

Distilling Stockfish into a Transformer

I recently came across this paper (Grandmaster-Level Chess Without Search) by Google DeepMind and my interest was sufficiently piqued to warrant a close read. A first skim gave me the impression that the transformer learned to play chess by looking at a very large number of board positions + the corresponding best move. To do this, the researchers fed FEN strings of the boards before and after the move as input and output respectively to the feed-forward transformer network. However, the paper does not claim this at all, and instead does something else entirely, which is to approximate the position-evaluation capabilities of Stockfish 16 (a chess engine that evaluates chess positions using tree search) with a feed-forward transformer network.

It is my opinion that this is not the same as learning how to play chess, but that is a topic for another day. First of all, I'll just write briefly about what the paper actually does before adding my thoughts to this. The dataset is prepared by randomly choosing board positions from games played on LiChess and using Stockfish to evaluate board positions. Pairs of [positions, moves] are prepared and also evaluated (about a billion or so of these). Three predictors are trained using this dataset. The action-value predictor predicts the evaluation given a position and a move. The state-value predictor predicts the evaluation given a position. The Behavioral Cloning predictor predicts the best move given a position.

This whole process leads to a predictor that gets very close to approximating SF behaviour on board states, both seen and unseen. However, because the model does not use search during inference, it fails in very specific ways. One example discussed in the paper is blindness to impending three-fold repetition, this happens because the model has no record of past moves but treats each board position as an independent sample. The "without search" claim just pertains to the fact that the model doesn't explicitly search through possible solutions during inference. But a criticism that I have seen and I agree with is that using SF as an oracle (a fancy term which means they use it to annotate their dataset) means that we use the tree-search capabilities of Stockfish directly. And there are cases where the model requires hard-coding a search in order to finish games. Some details about the game are hard-coded into the data, for example, the FEN strings (Q: What would happen if PGN strings were used instead? A: Ask Nicholas Carlini, who created a LiChess bot that does next move prediction with PGN strings) contain information about whether castling is allowed.

The only thing the paper overclaims is about Grandmaster level play. The reported 2895 Elo is achieved via online blitz play and Grandmaster norms are given out for over the board classical chess games. The ELO calculation is also not consistent. The paper uses BayesElo with anchors as the LiChess ratings (which themselves, are calculated via the Glicko2 method). Technical details aside, the paper shows games where the agent misses #3 (checkmate in three moves) because it finds an alternate #5 mating strategy that ends in five moves. Human GMs don't play this way, and most beginner endgame exercises are aimed at learning and identifying quick mates. I think this follows a general trend of engines resorting to really suboptimal play in poor positions. GMs (or any chess player) don't suddenly drop their level when the position gets bad.

In conclusion, I liked this paper a lot and I think it is written very well. The paper doesn't claim that the agent learns the rules of chess, but it is definitely written in a way that suggests it does. The agent distills the evaluation capabilities of Stockfish 16, a tree-search based chess engine, into a transformer. During inference, it outputs an evaluation given a board state. With scale, the model should only match the capabilities of SF16, and I don't see why it would be better than SF in any way. Practically, I don't see it deployed anywhere for analysis purposes considering that inference would need GPUs and there wont be any significant new ideas.

One implementation detail that I did not understand initially was the use of value binning. Even though the predictors output a real number (winning percentage), the range (0-100%) is divided uniformly into $K$ bins, with $K$ being a design hyperparameter. This is done only for the action-value and state-value predictors. But I have a theory that value binning is done simply because log-loss works better while training transformer networks even though an MSE feels more intuitive. Another theory is that it is better to bin because a 97% winning percentage is not that different from a 90% winning percentage when it comes to chess. Both positions would be annotated as "completely winning" by an expert chess player.

Anyway, I have spent a lot more time on this than I originally planned to, but I think there is a lot of benefit from reading papers like these, a lot of the heavy lifting behind some of these results is just really good and inspired design choices and after a point, you need to be good at noticing these details from papers rather than just the transformer and the losses. End of discussion. Write to me if you want an annotated copy of the paper.

New+Mersenne+Largest Prime Found

A new prime has been found! This rumour was floating around for a few days now on the mersenne.org forums but all tests are complete and they have decided to announce it. It is a gigantic number. $$2^{136279841} - 1.$$ There is a collaborative project called the Great Internet Mersenne Prime Search where people all over the world contribute computational resources to find more prime numbers. The details are in the press release and will no doubt be picked up by most media outlets + pop-science media pretty soon so I wont do too much exposition and waste my time. I will, however, write why this is exciting to me, and should be to anyone that likes science.

When the last Mersenne prime was discovered (these primes are easier to search for and easier to test for using the Lucas-Lehmer test), I was really into Number Theory and my friends and I really put a lot of thought into how one would find the next Mersenne Prime. To give some context, three Mersenne Primes were found in three years making it seem, to the amateur mathematicians in us, that this was not too unachievable. As it turned out, it took almost six years for the next one to be discovered. There are a few things that have changed since then, first is that they use a probabilistic search to filter out non-primes quicker, and secondly, this is the first time GPUs were involved in this project. Usually, these initial breakthroughs in science are followed by a string of other results improving on these methods. So, we might get a few more primes in the next two or three years. I wonder how the Nvidia stock price will react to this information.

India win everything at the Olympiads!

Gold in the Open section with two rounds to spare. Gold in the Women section. Board 3 gold for Divya with a TPR of 2608. Board 4 gold for Vantika with a TPR of 2558. Board 1 gold for Gukesh with a historic TPR of 3056 (second on the all time TPR list). Board 3 gold for Arjun with a TPR of 2968. The Gaprindashvili Cup for best combined performance. Third place and fifth places respectively for the Arjun and Gukesh in the live rating charts, and as a bonus, Vishy snuck back into the top 10 :)

Links from the Monsoon of '24

Vettel's eyes, they never lie :)
Haemoglobin-based Oxygen Carriers: a substitute for blood?
The Talk, smbc comics, if you don't talk to your kids about quantum computing, someone else will.
The Times' Profile on Hannah Neeleman: yikes!
Jean-Pierre Serre is still doing math research, he's 98 now!
Mixed-input matrix multiplication performance optimizations, this is such a nice blog post from Google Research that discusses ways to perform efficient mixed-input (S8 and F16) matrix multiplication. Very reminiscent of the old Google.
Marina Abramovic anecdotes are all wonderful, what an artist: The bittersweet story of Marina Abramović's epic walk on the Great Wall of China
Not only is this paper completely unhinged, but they have a wonderful line in it, "Because this paper was written in 2024, we include an obligatory section involving generative AI and LLMs". An Abundance of Katherines: The Game Theory of Baby Naming
Sabalenka's topspin stats at the US Open 2024: super interesting reddit thread about the technical differences between men's and women's tennis.
Grammar-checking Taylor Swift
The FBI files on Paul Erdos

An Elliptic Curve with Rank >= 29 has been found!

The elliptic curve

y² + xy = x³ - 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497

has rank at least 29.

https://web.math.pmf.unizg.hr/~duje/tors/rk29.html

- robinhouston (@robinhouston) August 30, 2024

This Reddit thread from r/math is very interesting. Elliptic curves are super-important in number theory. Many unsolved problems depend on our understanding of elliptic curves. I remember reading all I could about Fermat's Last Theorem in 2021 and learning the basics of elliptic curves so I could get a rudimentary understanding of the Taniyama-Shimura conjecture. Most elliptic curves have ranks of 0 or 1, so finding curves with higher ranks is very hard. I mean, look at the coefficients!

on the sums of multiples of two numbers

I came across a strange pattern when computing the sum of multiples of 2 and 3 below 1000, below 10000 et cetera. The sum of multiples of 2 and 3 less than increasing powers of 10 are as follows, $\{32, 3317, 333167, 33331667, 3333316667, \dots\}.$ This intrigued me and I wanted to check if this behaviour held for other primes as well and found that it does for a bunch of others. Examples: $f(2,5)=\{25, 2950, 299500, 29995000, 2999950000,\dots\},$ and $f(3,5)=\{23, 2318, 233168, 23331668, 2333316668,\dots\}.$

def sumMul(p1: int, p2: int, b: int, n: int) -> int:

    A = (b ** n - 1) // p1
    B = (b ** n - 1) // p2
    C = (b ** n - 1) // (p1 * p2)

    print(A, B, C)
    
    return p1 * A * (A + 1) / 2 + p2 * B * (B + 1) / 2 - (p1 * p2) * C * (C + 1) / 2


res = []
for i in range(1, 9):
    res.append(sumMul35(3, 5, 10, i)) # change this line

print(res)

But all primes don't follow this pattern. As you may expect, the first prime to give me trouble was 7. After some more experiments, I wondered why I couldn't try it for all natural numbers and computed a few more numbers. I couldn't come up with any rules for why some pairs had these patterns and why others did not but I noticed a few interesting cases. For example, pairs like $(2,2^k)$ always gave a lot of zeros at the end of the number, and that behaviour did not replicate among powers of 3 or 5. To investigate, I looked at the actual number of terms that each summation had. Concretely, given two numbers $p_1$ and $p_2,$ we first find the number of terms in each summation $A, B, C$. $$A=\left\lfloor{\frac{10^n - 1}{p_1}}\right\rfloor, B=\left\lfloor{\frac{10^n - 1}{p_2}}\right\rfloor, C=\left\lfloor{\frac{10^n - 1}{p_1p_2}}\right\rfloor$$ $$S=\sum_{i=1}^Ap_1i+\sum_{i=1}^Bp_2i-\sum_{i=1}^Cp_1p_2i$$ $$S=p_1\cdot\frac{A(A+1)}{2}+p_2\cdot\frac{B(B+1)}{2}-p_1p_2\cdot\frac{C(C+1)}{2}$$ For most numbers, the numbers $A,B,C$ don't change apart from being multiplied by 10 and so we can see patterns in the resulting sum. So that mystery is solved. But what I don't get is what is so special about powers of 10 that this works. For example, changing the base of the exponential to 7 (and computing the sums less than powers of 7) or any other number doesn't work. Is there anything obvious that I am missing here?

$x^n/n!$ and Stirling's formula

I saw this nice problem on Mathstodon a few months ago and decided to take a look today. It turns out that the curves of $x^n/n!$ have very similar shapes but shifted. What is the size of the shift between the curves?

The easiest way to do it is by using Stirling approximations, on which I did some rudimentary speed analysis sometime before. Assume that the value we're trying to find the spacing at is $t.$ Then we need to find the corresponding value of $x$ for two successive curves. $x^n / n! = t.$ Recall Stirling's formula and substitute it into this: $$n! \approx (n/e)^n \sqrt{2\pi n}$$ $$x^n / (n/e)^n \sqrt{2\pi n} = t$$ Two things happen here when $n$ is large and $t$ is small, we can ignore the effects of $\sqrt{n}$ in the product with $n^{n}$ firstly and notice that $t^{1/n}$ equals one. So we get $$x = n/e$$ and the resulting spacing is simply $1/e.$ I thought this was very neat, and a good example of street-fighting calculations that are seemingly too complex. I would say that the hardest part in this entire thing was plotting the graphs, they needed some programming acrobatics to get the axes right.

def comp(n: int) -> list:
    
    x = []
    y = []
    for i in np.linspace(0, 10, 1000):
        if ((i ** n) / math.factorial(n)) <= 2:
            x.append(i)
            y.append((i ** n) / math.factorial(n))
    
    return x, y

for i in range(2, 20):
    plt.plot(comp(i)[0], comp(i)[1])
plt.xlabel("x")
plt.ylabel("x^n/n!")
plt.show()

An interesting fact here is that every curve intersects with the next one only at an integer. This is fairly easy to show, $$\frac{(n + 1)^{n + 1}}{(n + 1)!} = \frac{(n + 1)(n + 1)^{n}}{(n + 1)(n)!} = \frac{(n + 1)^{n}}{(n)!}$$ This means that the $n^{\text{th}}$ curve and the $(n + 1)^{\text{th}}$ curve will intersect at $x = n + 1.$ Cute.

Olympic Medals per Capita

Olympics Medals per Capita measures the number of Olympic medals won by a country relative to its population. They also have a weighted measure, and another metric relative to the GDP of the country. All of these metrics are better than raw medal tally. They have a description in their 'about' page which goes:

As you scan the Olympic medal tally, one thing stands out: larger countries tend to win more medals. There are a few notable exceptions, though, including New Zealand, which wins many more medals than you'd expect of a country its size.

However, they are very kind to India, which is definitely an exception, seeing how we regularly have nothing to show for our booming population. We need to get an award for transcending the law of averages by not even accidentally producing a good bunch of Olympic athletes over the last thirty odd years.

[wip] Thoughts on Novak Djokovic

There is a very strong chance no one will have his resume ever again. 24 slams and counting, 7 year-end championships, and now the Olympic Gold. History was made at Philippe Chatrier, an unlikely place for Novak to do it, considering that there was another man for whom this court has been a kiddie playground for almost twenty years. Novak Djokovic has always been an interesting player to me, he's definitely the greatest, but it feels like he needs to do something more than win for him to be in the debate with Federer and Rafa.

I remember reading an article about Rafael Nadal sometime ago where the writer was talking about how Rafa was perpetually in second place all his career (apart from in Roland Garros, and a couple of years here and there). First to Roger Federer and then to Novak Djokovic. The writer wondered how it must feel to always be the person that comes up short. How it must feel to always die at the end of the book.

With Novak, it is quite a different question. He has always been second favourite, first to the Roger-Rafa fans and now with the Sinner-Alcaraz fans. He is continually the anti-hero for the crowds. How must it feel when everyone is rooting for you to die at the end of the book?

Maybe that is why he fights so hard. Maybe it is when his back is against the wall, that he becomes Novak Djokovic.

Colonization of Venus

I came upon this delightfully technical article on the possibilities of a Venus colony, wait for it, not on the surface of the planet but in the atmosphere of Venus! around 50 km above the surface. The rationale is that while the surface is very hostile to human life, the zone 50 km above is the most earth-like in the solar system. I did not know this, and I wonder how they ascertained that. First principles wise, space is hard. There's the minor issue of oxygen, but we also need carbon, hydrogen, nitrogen, and basically every element that we have evolved to live in harmony with on earth. Apart from all that, winning the under-appreciation contest, is gravity. We're going to go all loopy, literally and figuratively, if we don't have gravity.

Long story short, the surface of Venus sucks. Surface temperature of 737 K (300 K on earth), surface pressure of 9.2 MPa (0.1 MPa on earth) and an atmosphere with 96.5% CO$_2$ which, get this, exists in a supercritical state at the surface because of the pressures. The worst part, there's no sun, because of all the poisonous cloud cover.^*

But 50 km above the surface is quite a different story. Apart from the acidic clouds (H$_2$SO$_4$), the conditions are very earthlike. Temperatures around 320 K, pressure of 0.1 MPa, gravity is a little less than what we get on earth but manageable, and we get some sun. We still cannot breathe though, and heat waves would take out the entire population if there was no protection. But this is as good as we have it under the sun. Literally, this is as good as we have it in our solar system.

The proposal is this. We colonize the air of Venus. We build floating cities, much like vimanas or the golden floating city of Tripura. Considerations include breathing (oxygen supply), solar radiation shielding, anti-corrosive measures in the air, pressure differentials and repairs in case of gas leaks, length of the solar day (116.8 times that of earth) and resources (both for maintenance and exploration purposes)

Space is wonderful and it is probably my fear of getting nerd-sniped into eternity that I don't spend more time on these ideas.

References:
1. Venus' atmosphere: Composition, clouds and weather on Space
2. Constraints on a potential aerial biosphere on Venus by Dartnell et al, in Icarus
3. Colonization of Venus by Geoffrey Landis

Like a river of steel it flowed, grey and ghostlike.

The entrance of the German army into Brussels has lost the human quality. It was lost as soon as the three soldiers who led the army bicycled into the Boulevard du Régent and asked the way to the Gare du Nord. When they passed, the human note passed with them. What came after them, and 24 hours later is still coming, is not men marching, but a force of nature like a tidal wave, an avalanche or a river flooding its banks. At this minute it is rolling through Brussels as the swollen waters of the Conemaugh Valley swept through Johnstown.

At the sight of the first few regiments of the enemy we were thrilled with interest. After they had passed for three hours in one unbroken steel-grey column we were bored. But when hour after hour passed and there was no halt, no breathing time, no open spaces in the ranks, the thing became uncanny, inhuman. You returned to watch it, fascinated. It held the mystery and menace of fog rolling toward you across the sea.

The grey of the uniforms worn by both officers and men helped this air of mystery. Only the sharpest eye could detect among the thousands that passed the slightest difference. All moved under a cloak of invisibility. Only after the most numerous and severe tests, with all materials and combinations of colour that give forth no colour, could this grey have been discovered. That it was selected to clothe and disguise the German when he fights is typical of the German Staff in striving for efficiency to leave nothing to chance, to neglect no detail.

After you have seen this service uniform under conditions entirely opposite you are convinced that for the German soldier it is his strongest weapon. Even the most expert marksman cannot hit the target unless he can see. It is a grey-green, not the blue-grey of our Confederates. It is the grey of the hour just before daybreak, the grey of unpolished steel, of mist among green trees...

This army has been on active service three weeks, and so far there is not apparently a chin-strap or a horseshoe missing. It came in with the smoke pouring from cookstoves on wheels, and in an hour had set up post office wagons, from which mounted messengers galloped along the line of column distributing letters and at which soldiers posted picture postcards.

The infantry came in files of five, 200 men to each company; the lancers in columns of four, with not a pennant missing. The quick-firing guns and field pieces were one hour at a time in passing, each gun with its caisson and ammunition wagon taking 20 seconds in which to pass.

The men of the infantry sang “Fatherland, My Fatherland”. Between each line of song they took three steps. At times 2,000 men were singing together in absolute rhythm and beat. When the melody gave way, the silence was broken only by the stamp of ironshod boots, and then again the song rose. When the singing ceased, the bands played marches. They were followed by the rumbles of siege guns, the creaking of wheels and of chains clanking against the cobble-stones and the sharp bell-like voices of the bugles.

For seven hours the army passed in such solid column that not once might a taxi-cab or trolley car pass through the city. Like a river of steel it flowed, grey and ghostlike. Then, as dusk came and a thousands of horses’ hoofs and thousands of iron boots continued to tramp forward, they struck tiny sparks from the stones, but the horses and men who beat out the sparks were invisible.

At midnight, pack wagons and siege guns were still passing. At seven this morning I was awakened by the tramp of men and bands playing jauntily. Whether they marched all day or not I do not know; but for 26 hours the grey army rumbled with the mystery of fog and the pertinacity of a steamroller.

- Richard Harding Davis

Links from the Summer of '24

Millikan's Oil Drop Experiment is flawed (?)
Illuminating (and sometimes funny) discussion on Russians and voting
Nice video about the XOR perceptron problem: Sisyphus Perceptron
I think Elle Cordova is just wonderfully clever. I love watching her videos!
JavaScript Bloat in 2024, if I could have math typesetting without JS, I'd get rid of JavaScript entirely
I learnt a lot about potato botany from this question on potato evolution in The Martian
A Tour of Johannes Vermeer narrated by Stephen Fry.
Generate random mathematics papers
David Hume, the Buddha, and a search for the Eastern roots of the Western Enlightenment by Alison Gopnik
No Regrets: Boris Spassky at 60: a fascinating interview from the 10th World Champion
Why do doctors still use pagers?: a good example of communicative friction being a good thing
Karpov, Kramnik, Anand vs. Karjakin, Nepo, Dubov
Gene therapy reverses hearing loss: we are on the brink of a great golden age in medical science
Being 13: Social media is the worst thing that has happened to teenage girls

Geometry Puzzle

I found a nice puzzle on Twitter that I want to add some thoughts about here. The problem is that you need to find the shortest route from A to B by touching the perimetre of the circle on the way. We can solve this analytically, by considering a parametric form of the circle and minimizing the distance from the two points but we would ideally require a ruler and compass construction of the solution. Apparently, this problem can be traced way back to Ptolemy in 150 AD.

References:
1. Alhazen's Billiard Problem on Wolfram Mathworld
2. Circular Billiard by Drexler and Gardner
3. Twitter thread

There are three special cases that I want to note here:
1. If B=A, then the shortest route is the round trip from A to the nearest point on the circle.
2. If B or A is on the perimeter, then the shortest route is the straight line from A to B.
3. If B or A is on the center of the circle, then the shortest route is along the radius of the circle.

Vielen Dank für alles, Herr deutscher Scharfschütze

Toni getting emotional after the second goal against Bayern was the clearest of the signs. In the ten years I've seen him play, the man has never let his mask down. Up by four goals or down by two, he remained the eye of the storm, a calm and unruffled presence in the center of the pitch. Turn on a Real Madrid game, and in the centre of the pitch is The German Sniper, standing over the ball, his eyes piercing the opposition defence, tracing the path that the ball will soon follow.

Never has a player of this calibre, since Zidane, retired at the peak of his career. Toni goes out at the top, retiring from a starting position at the greatest club in the world. He kept his promise to himself, to retire at Madrid. No one expected this, all eyes were on Luka and all along, it turned out to be Toni that hung up first. It's been like this at Real Madrid for a few years now. It started with Cristiano leaving, and Toni is gone now. I watched them all play football, I am the lucky one.

Like Luka Modric wrote, there won't be another Toni Kroos. Toni Kroos was the orchestrator of victories, he controlled the Madrid midfield for a decade, holding the center of the squad, usually as the last man. The greatest deep-lying playmaker of all time. I remember so much about this man, his millimeter-precision passes, the trademark Kroos feint while he receives the ball, his diagonal switches to Carva. Teams these days switch play by passing one man to another, and I often wonder why they're so slow, but they don't have a Toni Kroos to play that pin-point diagonal ping. No one presses the Real Madrid backline for this one reason, because all Real need to do is get the ball to Toni, and he's going to send it to the wing, and we score within a handful of touches. There were midfielders that were stronger than him, or faster than him, or far more skillful. But he beat all of them because he could outthink them.

In his 10 years at Real Madrid, he had pass completion percentages of 92%, 92%, 94%, 92%, 93%, 93%, 94%, 94%, 95%, 95% with an average of around 2100 passes per season. Do you know what means? Toni Kroos misses around 100 passes, over the entire league season! I will let you read that again. Remember that he plays in Real Madrid, a team that loves to score goals, and he has to make a large percentage of those passes in attacking transitions. He is unrepeatable, I hope he wins the 6th UCL, then wins the Euros at home, and gets that Ballon D'Or that he deserves so much for the career he's had.

Remember a ball skimming along the grass into the goal. Remember a pass squeezing between the gloves of the keeper and the far post. Remember a goal scored from the corner spot. Remember a vertical pass beating three lines of defenders. Remember a feint in the penalty box to shake off the press. Remember the ball fizzing in the air to the right back's feet. Remember the white shoes with the blue tongue and heel. Remember the ice-cold composure on and off the pitch. Remember the personal integrity upheld all through the fame and the success. Remember Real Madrid's greatest number eight. Remember Toni Kroos.

Sebastian Vettel Drives Senna's McLaren Mp4/8 in Imola

Seb drove Ayrton Senna's McLaren MP4/8 in Imola before the race last night. What a magnificent beast that car is, the wail of the naturally aspirated V8 strikes straight into the heart. The gunshot sounds while the car downshifts are beautiful. Seb is probably the best driver to do this stuff, not only does he have tremendous respect for the history of the sport and its legends, he genuinely loves racing in those cars. He was properly throwing the car into every corner around the circuit. But the best part was when he brought out the Brazilian and Austrian (for Roland Ratzenberger) flags. He held out the flags, and proceeded to drive two laps steering with one hand. He was just twenty seconds off the pace too!! He talks about how it felt in the cockpit and why he got out the flags in the interview just after he did donuts on the main straight and gave 2013 Indian GP flashbacks by bowing to the car.

The race itself was a slow-burner, Max looked very comfortable until the final fifteen laps when he had to hold off a very rapid Lando Norris. That McLaren really came to life in the second stint. Both Oscar and Lando were extremely quick. A couple more laps and Lando would've had two race wins in a row.

International Women in Mathematics Day!

Happy Birthday to Maryam Mirzakhani, mathematician, and Fields Medallist.

One of my greatest inspirations. Watching Secrets of the Surface reminded me why I loved mathematics so much. Her story will live forever.

The Games of Robert James Fischer

I've resumed my chess practice after months of disinterest and I'm going about it by playing classical games and analyzing former world champion games to pick up a few things here and there. I'm starting with Bobby Fischer and my references are two books: My 60 Memorable Games by the great man himself, and Russians versus Fischer, which I think I first heard about in an interview by five time world champion Viswanathan Anand and has stuck with me ever since.

Recreational Math Dump

Some fun questions I've been trying to solve on and off this month.

* The Fibonacci numbers count the number of ways a natural number can be expressed as the sum of 1s and 2s while considering the order. What is the average number of 1s and 2s needed to create a sum equal to any natural number? Ref: an average number of 1s and 2s by Joshua Bowman.

* Choose random numbers between 0 and 1, uniformly and independently, until the sum of the numbers exceeds 1. What is the expected number of terms?

* Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that for all real numbers $x$ and $y,$ $$f(x+y)-f(x)-f(y)=xy+x^2y+xy^2$$ and $f(6)=69.$ Find $f(2\sqrt{6})$.

"Do you really wanna know where I was April 29th?"

Liquefying Silica

So much of the heavy lifting in the previous note about topographical heights involved calculating the liquefying energy needed for a silica molecule. I wanted to expand on that a bit in this note so I can provide myself closure about this topic that consumed me for a few hours.

All depends on the structure of the silica molecule. In the crystalline form, the SiO$_2$ lattice has each O atom bonded to two silicons and each Si atom bonded to four oxygens. For simplicity, we can consider this configuration to have zero entropy ($S=0$). In a liquid state, we can think of some fraction $(f_N)$ of the total molecules ($N$) being "disordered" by moving into holes, and then the entropy will be $S=k_B\ln{N\choose f_N}$. Another simplification ensues after we assume the liquid to be in the highest disordered state possible and the change in entropy per molecule will be $\Delta s\sim k_B\ln{2}$.

There are three steps in the process now, first we need to detach single SiO$_2$ molecules from the lattice, but to do that, we need to break bonds, or more accurately, overcome the binding energy of the bonds. Finally, we need to ask whether it is actually advantageous to completely detach every silica molecule from the lattice.

Each oxygen atom is bounded to one silicon atom in the molecule, but in the lattice, it is bounded to four silicon atoms. So we first need to break $3/4$ bonds. But each oxygen atom is part of two molecules, and so we're double counting. Therefore, we only need to break $3/8$ bonds.

Next, from high school physics, we remember that breaking a bond at the Bohr radius of $0.5$ Å is 13.6 eV. Silicon is on the second row of the periodic table and so we can assume a linear addition of an angstrom. And because two atoms form a bond, the binding energy of a single bond is $$E_{b}\sim Ry\cdot\frac{0.5\text{ Å}}{2\times1.5\text{ Å}}\sim2\text{ eV}.$$ But do we really want the binding energy? Breaking the lattice would mean converting the crystal to a gas, but we only want to achieve liquefaction. Assume that bond length changes by some $p$ % where $p<5$. Then we get the new binding energy $$E_{b}\sim Ry\cdot\frac{0.5\text{ Å}}{2\times1.5\text{ Å}}\cdot\left(\frac{0.0p}{1.0p}\right)\sim0.1\text{ eV}.$$

Isostasy and Mountain Heights

Earth's mantle is mostly silicate rock and behaves as a viscous fluid in geological time (think large time scales). Oceanic crust and continental crust is primarily formed by partial melting of the mantle. There's a gravitational equilibrium between the crust and the mantle and different topographical heights exist on earth because of this static equilibrium. This equilibrium depends on the thickness and density of the crust.

Apparently, the Himalayas are not in isostatic equilibrium (which is a fancy way to say that they're still rising) and mountains primarily rise that high because they're low density. Think of this as them being huge, but having less mass, so they don't crumble. One important point we'll come back to is that mountains fail at their bases. When a mountain gets too high, they suffer plastic deformation at the base.

There are two major theories of isostasy. One that treats the lithosphere (the crust basically, but the more you read about it, the more you want to use the scientifically precise term) as a sheet floating on the mantle with different densities at different heights. This theory straight up applies Pascal's Law (hydrostatic pressure is the same at the same elevation for a fluid in static equilibrium) to all points in the crust. The second treats the lithosphere as an elastic plate that bends and thereby distributes loads.

If the maximum height a mountain can reach is $H$, this is the height at which the energy lost as the mountain "falls" equals the energy needed to cause plastic deformation at the base. Doing the computations for each molecule, we have $$mgh=E_p,$$ where $m=AM$, the atomic mass units in the molecule, $h=H$, the maximum height before deformation, and $E_p$ being the energy needed to cause plastic deformation. Note that Weisskopf uses $E_1$ for this. To cause plastic deformation, we need to liquefy the molecule without breaking the atomic structure. This will require a small fraction ($\epsilon$) of the binding energy $B=\gamma Ry$ where $Ry$ is the Rydberg unit. If we assume that the mountain is a block of SiO$_2$, the atomic mass will be 60 amu. And so the maximum height will be $$H=\epsilon\frac{\gamma Ry}{AMg}\sim \boxed{10\text{ km}}.$$ Later in the paper referenced below, Weisskopf shows a wonderful argument about the maximum height ($H$) of a topological structure on a planet being inversely proportional to the third root of the number of nucleons ($N_p$) in the planet. $$H\propto\frac{1}{N_p^{1/3}}.$$

Update: it turned out that this nucleon argument is neither as fancy or convuluted as I first thought it would be. Rahul Goel pointed out that for any object, $V\times\rho=m$ and mass is just the number of nucleons in an object, and volume is length cubed. And therein lies the intuition for the above equation. Other questions that were raised in the discussion were 1) How do you know that plastic deformation is what occurs when mountains fail at their bases and 2) Is it true that huge mountains usually grow isometrically?

References: Mantle (geology) | wikipedia.org, Isostasy | wikipedia.org, Of Atoms, Mountains, and Stars: A Study in Qualitative Physics | science.org by Weisskopf.

The Tortured Poets Department is out!!

The Tortured Poets Department. An anthology of new works that reflect events, opinions and sentiments from a fleeting and fatalistic moment in time - one that was both sensational and sorrowful in equal measure. This period of the author’s life is now over, the chapter closed and… pic.twitter.com/41OObGyJDW

— Taylor Swift (@taylorswift13) April 19, 2024

It’s a 2am surprise: The Tortured Poets Department is a secret DOUBLE album. ✌️ I’d written so much tortured poetry in the past 2 years and wanted to share it all with you, so here’s the second installment of TTPD: The Anthology. 15 extra songs. And now the story isn’t mine… pic.twitter.com/y8pyDK8VTd

— Taylor Swift (@taylorswift13) April 19, 2024

2023 Turing Award

Avi Wigderson has won the 2023 Turing Award "for foundational contributions to the theory of computation, including reshaping our understanding of the role of randomness in computation and mathematics, and for his decades of intellectual leadership in theoretical computer science"

Avi also won the 2021 Abel Prize! Making him the first person to hold the double (me when?). Probability is having its moment in the sun this spring.

Temperature, and some more thermodynamics

Thermometers wouldn't work if energy didn't spontaneously flow from two objects in contact with each other, and temperature is objectively, just something you measure with a thermometer. (of course, we're talking about traditional mercury thermometers here, and not stuff like infrared thermometers)

One definition for the average kinetic energy of an ideal gas (with three degrees of freedom) is $$\bar K=\frac{3}{2}k_BT$$ and we can get a bad definition of temperature by moving the terms around. Note that we got this equation from the equipartition theorem which states the average energy of a system in which individual elements have $n$ degrees of freedom is $\displaystyle\frac{1}{2}nk_BT$.

Fundamental Assumption of Statistical Mechanics: All accessible microstates are equally likely for a system in thermodynamic equilibrium.

Entropy is a statistical measure of how many possible arrangements exist for a system. Define multiplicity $\Omega$ for an Einstein Solid with $N$ quantum harmonic oscillators and $q$ units of energy as $$\Omega={q+N-1\choose q},$$ and energy units will flow in such a way as to maximize the product of individual multiplicities of the objects in contact with each other. Concretely, there are systems $A,B$ with $N_A,N_B$ oscillators and $q_A,q_B$ units of energy in contact with each other. They have individual multiplicities of $\Omega_A,\Omega_B$ and the units of energy $q_A,q_B$ will flow in a way that maximizes $\Omega=\Omega_A\cdot\Omega_B.$ The constraint obviously being that $q_A+q_B$ is always fixed. The graph of $\Omega$ versus $q_A$ looks like a Dirac Delta function with a very narrow peak near the middle, which is intuitive (if you accept that multiplicities should be maximized, because the $\displaystyle{n\choose k}$ function has it's maximum at $2k=n$)

Now, entropy is just the natural log of this multiplicity and has units [energy]$\times$[temperature]$^{-1}$.$$S=k_B\ln{\Omega}$$ The Second Law of Thermodynamics: (1) The entropy of an isolated system tends to increase. (2) A system in thermodynamic equilibrium is most likely to be found in the macrostate (configuration of $q_A$ and $q_B$ for example above) of highest entropy.

Finally, temperature is defined as $$\frac{1}{T}=\frac{\Delta S}{\Delta E}.$$ Which is why temperature is a measure of the tendency of an object to give up it's energy to another object it comes in contact with. Reference/Inspiration: Temperature as Joules per Bit

Infinitesimal Worms

I've always loved the joke where you bite into an apple and find a worm but then you have to be relieved about it because that is still better than biting into an apple and finding half a worm. Extending this further, you would prefer half a worm in your apple than a quarter of a worm. So on and so forth, until it means that the worst case scenario is when you don't find any worms in your apple.

What is worse than finding $1/n$ of a worm in your apple? Finding $1/n^2$ of a worm in your apple. Hilariously, this works for cases where $n\in\mathbb{R}^+$ except that the grammar doesn't hold up. Example: for $n=1/7,$ what is worse than finding 7 worms in your apple? Finding 49 worms in your apple.

2024 Abel Prize

Michel Talagrand has won the 2024 Abel Prize "for his groundbreaking contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and statistics."

- The Law of Large numbers and Talagrand's inequalities
- Stochastic Processes
- Concentration of measure: The sum of a large number of sufficiently independent random variables will be concentrated in an interval which is much narrower than is intuitive. (Think of the curse of dimensionality)

The citation had callbacks to applications of Talagrand's work in Large Language Models and I'm wondering how much of that was a publicity stunt by the Abel Prize committee. I'm sure there are far more exciting applications and that the inclusion of LLMs at a time when their hype is at its peak is just a coincidence.

The Ides of March are upon us!

Links from the Spring of '24

Every Author as First Author
Problems with grading students on a curve
rotation with three shears
Roger Federer's Neo Backhand in the 2017 Australian Open
52 cards win a dollar: You have 52 playing cards (26 red, 26 black). You draw cards one by one. A red card pays you a dollar. A black one fines you a dollar. You can stop any time you want. Cards are not returned to the deck after being drawn. What is the optimal stopping rule in terms of maximizing expected payoff?
Songs with parenthesis in the titles: or another analysis where Taylor Swift shows up as a statistical anomaly.
Thermodynamic Linear Algebra: Harmonic Oscillators ftw
Intelligent Machinery: Intelligence is a search problem
1012 and other such numbers
AOH1996: something to watch out for, Phase 1 Clinical Trials have begun

Richard Hamming on Claude Shannon

He wants to create a method of coding, but he doesn't know what to do so he makes a random code. Then he is stuck. And then he asks the impossible question, 'What would the average random code do?' He then proves that the average code is arbitrarily good, and that therefore there must be at least one good code.

Who but a man of infinite courage could have dared to think those thoughts?

[wip] Dimensional Analysis on Loss Functions

One of my favourite tricks to minimize errors while doing high school physics was to use dimensional analysis as a sanity check on all my equations. It is easy to see that gradient descent is not dimensionally sound, and so I have some thoughts on whether we can reinvent concepts like loss functions using dimensional analysis of gradient descent. For example, if we're trying to minimize the loss on a physical quantity like length, what dimensions would the gradients be, and what dimensions would the learning rates be. What happens to dimensions in neural networks. This is something for me to think about and update later. Watch this space!

Silman's Complete Endgame Course

A few days ago I got my hands on a copy of Silman's Complete Endgame Course, and I'm finally starting to dig into it. I think I'm going to learn a lot from it. The only endgame book I have really studied before was a slim book by Averbakh called Chess Endgames: Essential Knowledge, and I really do not think that Averbakh's book takes you through to master level the way that Silman promises to. I decided to jump into Silman's book at the class “A” level, because it looked as if I knew everything in the chapters up to class “B,” but already in the class “A” chapter there were some things I didn't know.

The class “A” chapter starts out with rook and rook-pawn versus rook positions, and already it has some great stuff. How many open files do you need to free your king if it is stuck on the eighth rank in front of the pawn? (Four.) How do you draw the game if you have the rook and your opponent has the rook and rook-pawn, and his rook is in front of the pawn? If the pawn is on the seventh rank, you want your rook behind the pawn, of course — that much is obvious. And you have to keep your king on the opposite side of the board from the pawn — something that is not obvious to a lot of players.

The thing I didn't know is what to do if the pawn is on the sixth rank. Silman shows that in that case you want your rook attacking the pawn from the side, not from behind, and he shows something called the Vancura position that I had never heard of.

This is all really cool stuff, but I do have one slight reservation. I've played loads of games, I believe, without playing a single game that went down to rook and rook-pawn versus rook. So it's unclear that knowing these endings will make a big practical difference to me (or to you). Also, the problem with learning rules like “you need four files between your king and the enemy king” is that in a game, you usually don't have control over this. The main thing to know is that you want to cut off the enemy king as far away from the passed pawn as possible. If you can get four ranks, then great. If you can only get three, then knowing that you need four is not going to be a big help to you.

Navier Stokes and smoothness

Basically, a function is differentiable iff it is continuous. Now if there exists a $k$-th derivative for a function, it means that the function is continuous at least $k-1$ times. They are using this to group functions into differentiability classes. A function of class $C^k$ has derivatives $f',f^{\prime\prime},\dots,f^{(k)}.$ Now, a smooth function is in the differentiability class $C^{\infty}$ which means that you can differentiate it as many times as you want. Which means it is always continuous.

Now that we know this, we should first try and understand that operator in Terrence Tao's paper $$\partial_t u=\Delta u+B(u,u)$$ and he says this $B$ is equivalent to the energy equation. As far as I understand, Tao doesn't assume the equation to be smooth, instead of $B(u,u)$ he assumed an averaged $\tilde{B}(u,u).$ And then he constructs a smooth solution and demonstrates finite-time blowup.

References: Navier-Stokes existence and smoothness on Wikipedia and Tao's paper: Finite time blowup for an averaged three-dimensional Navier-Stokes equation

Engine efficiencies and my attempts to relearn thermodynamics

Heat engines convert heat to energy, specifically mechanical energy, which can be used to move stuff around $(W=F\cdot\Delta x)$. It does this by using a substance to move things, while cooling the substance. A heat source initially heats the substance, the substance moves whatever it needs to move while losing heat. This substance is anything that has a heat capacity. Some heat is lost to the surroundings, and there are practical issues of friction and drag.

Note that heat pumps (refrigerators) do the exact opposite by performing work on a substance. They usually compress (do work) on a refrigerant (the working substance) and draw out heat. This is a diversion I do not wish to explore.

Carnot engines operate on the carnot cycle. The Carnot cycle has in order 1) isothermal expansion, (where it expands in volume by pulling in heat) 2) adiabatic expansion, (where it loses temperature because all the heat has to be dispersed among the expanding molecules) 3) isothermal compression (where it compresses by expelling heat) and 4) adiabatic compression where it gains temperature.

Lets see what happens if $Q_C=0$ (which means our engine doesn't lose any waste heat). Then we'll have negative entropy because $\Delta S=\displaystyle\frac{-Q_H}{T_H}<0$ which violates the second law of thermodynamics. Which means we have to consider waste heat. So the maximum work we can get is $\Delta W=Q_H-Q_C$. Now, considering both sources: $\Delta S=\displaystyle\frac{-Q_H}{T_H}+\displaystyle\frac{Q_C}{T_C}$ and we know that $\Delta S>0$. For the entropy to be positive, the minimum value of $Q_C$ will be $\displaystyle\frac{Q_H}{T_H}T_C$ and this means that $$\Delta W=Q_H\left(1-\displaystyle\frac{T_C}{T_H}\right)$$.

This means we can't have 100% efficiency because we can't exhaust the heat sink to 0 K, and we will definitely lose waste heat to the universe due to the second law of thermodynamics. Also, this is a purely theoretical concept, taking a Carnot cycle through its steps without losing any additional energy will require us to do it infinitely slowly, and it is impractical to build an engine that takes forever to do its job. Another way to think of the whole entropy argument before is to look at the S-T diagram and realize that for the cycle to complete we need to gain back all the entropy we lost.

Bonus: TIL Diesel is apparently the name of a dude called Rudolf Diesel who invented the Diesel Engine that runs on Diesel.

Automatic Content Selection

Content on the internet should be permanent, available, and addressable until the original author of the content chooses to remove it. Automatic Content Selection by Stephen Wolfram

• Even if you don't explicitly know something (say about someone), it can almost always be statistically deduced if there's enough other related data available
• Even given every detail of a program, it can be arbitrarily hard to predict what it will or won't do
• For a well-optimized computation, there's not likely to be a human-understandable narrative about how it works inside
• There's no finite set of principles that can completely define any reasonable, practical system of ethics
  
  
Stephen Wolfram suggests two options to solve this problem: 1) serve algorithmically ranked content, but allow users to choose the final ranker and 2) allow users to choose constraints on what they want to see, and then use a universal ranker.

Scaling Carbon Capture

• As solar power gets cheaper and oil becomes more scarce, at some point this decade it will be cheaper to electrically extract carbon from the air than to drill mile-deep holes in the crust on the other side of the world.
• We ideally want a Haber's process for carbon dioxide
• CO$_2$ is usually a by-product, and it can't be used for fuel anymore than any other fully oxidized stable chemical can.
• We need to put back the energy that was released when it was produced. And then some more to remove the oxygen atoms.
• Sabatier reaction: catalytically react CO$_2$ with hydrogen, producing methane (CH$_4$) and water vapour.
  • Doing this requires a very large supply of pure hydrogen, ideally generated electrolytically, which requires an enormous supply of electricity.
  • $\Delta H=-165.0$ kJ mol$^{-1}$
  • This happens at very high temperatures, so $T\Delta S$ is very high and $\Delta G$ will be positive
• Overall efficiency is at 15% to 35% at best.
• Three big challenges
  • CO$_2$ concentration
  • CO$_2$ reduction
  • Electricity Cost

MSTD sets

I was reading this interesting paper by Hegarty discussing Some explicit constructions of sets with more sums than differences and I found that the smallest set you could construct was $\{0,2,3,4,7,11,12,14\}$ and it is not possible to construct smaller sets (of course, you could create other sets from this one using affine transformations).

For example, take the set $\{2,5,6,8,11\},$ the sumset will be $\{4, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 22\},$ and the difference set will be $\{-9, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9\}.$ The sumset has 12 elements and the difference set has 15 elements, so this set is not an MSTD set.

For a set with $n$ numbers, the sumset will have an upper bound of $\displaystyle\frac{n(n+1)}{2}$ and the difference set will have an upper bound of $n(n-1)+1.$ We can understand this by drawing a matrix and seeing that for the sumset, we'll have a symmetric matrix and for the difference set, we'll have a skew-symmetric matrix. We just need to ignore the zeroes in the diagonal for the difference set. Because we lose half the numbers from the sum-set, we need to construct MSTD sets by having as many duplicate values in the difference matrix as possible. An example where both are equal is the set $\{k,k+1,k+2,\dots,k+l\}$ which has the sumset $\{2k,2k+1,2k+2,\dots,2k+2l\}$ and the difference set $\{-l,-l+1,-l+2,\dots,l\}.$ Both have a cardinality of $2l+1.$ Another example of a set where both are equal is the set $\{a, b\}$ where the sumset will have three elements, and the difference set will have three elements. The sumset will be $\{a+b, 2a, 2b\}$ and the difference set will be $\{a-b, b-a, 0\}.$

def sumdiffset(data):

    sumset = set()
    diffset = set()
    for num in data:
        for other_num in data:
            sumset.add(num + other_num)
            diffset.add(num - other_num)
    return {'sumset': sumset, 'diffset': diffset, 'set_counts': {'sumset': len(sumset), 'diffset': len(diffset)}}

data = [0, 2, 3, 4, 7, 11, 12, 14] # change this
result = sumdiffset(data)
print(f"Sumset: {result['sumset']}")
print(f"Diffset: {result['diffset']}")
print(f"Set Counts: {result['set_counts']}")

How do we lose body mass?

• People who wish to lose weight while maintaining their fat-free mass are, biochemically speaking, attempting to metabolise the triglycerides stored in their adipocytes. The three most common fatty acids stored in human adipose tissues are oleate (C₁₈H₃₄O₂), palmitate (C₁₆H₃₂O₂), and linoleate (C₁₈H₃₂O₂), which all esterify to form C₅₅H₁₀₄O₆.

• The complete oxidation of a single triglyceride molecule involves many enzymes and biochemical steps, but the entire process can be summarised as: C₅₅H₁₀₄O₆+78O₂→55CO₂+52H₂O+Energy.

• Stoichiometry shows that complete oxidation of 10 kg of human fat requires 29 kg of inhaled oxygen producing 28 kg of CO₂ and 11 kg of H₂O. This tells us the metabolic fate of fat but remains silent about the proportions of the mass stored in those 10 kg of fat that depart as carbon dioxide or water during weight loss.

• To calculate these values, every atom's pathway out of the body was traced using labelled heavy oxygen (O¹⁸).

• A triglyceride's six oxygen atoms will therefore be shared by CO₂ and H₂O in the same 2:1 ratio in which oxygen exists in each substance. In other words, four will be exhaled and two will form water. Note that the Da stands for dalton, a unit of mass.

• The proportion of oxygen that becomes CO₂ is (661 Da (C₅₅)+64 Da (O₄))/(861 Da (C₅₅H₁₀₄O₆))$\times$100=84%

• The proportion of mass that becomes water is (105 Da (H₁₀₄)+32 Da (O₂))/(861 Da (C₅₅H₁₀₄O₆))$\times$100=16%

• Calculations show that the lungs are the primary excretory organ for fat. Losing weight requires unlocking the carbon stored in fat cells, thus reinforcing that often heard refrain of “eat less, move more.” The authors recommend these concepts be included in secondary school science curriculums and university biochemistry courses to correct widespread misconceptions about weight loss.

Links from the Winter of '23

The Art of Distraction: Creativity, memory and our relationship with time
Youngest World Champion, Aditi Swami
Following the money behind Premier League betting sponsors
Algorithms for Toddlers
Luis Caffarelli has won the 2023 Abel Prize
50th anniversary of the OEIS a New York Times profile by Siobhan Roberts
Epicycle clock in vector graphics and animated time offsets by Sophie Houlden
Bad Science and Room Temperature Superconductors: what a put-down!! how not to do science.
A Mechanical Binary Adder!
Embedded Möbius strips made out of paper can only be constructed with an aspect ratio greater than $\sqrt3$
The 8000th Busy Beaver number eludes ZF set theory
Joseph Brodsky's 1987 Nobel Lecture
CAR-T therapy forces autoimmune diseases into remission
A bell that rings two notes at once
Wavy brick walls use fewer bricks than straight walls

F. Scott Fitzgerald in Tender is the Night

“See that little stream — we could walk to it in two minutes. It took the British a month to walk to it — a whole empire walking very slowly, dying in front and pushing forward behind. And another empire walked very slowly backward a few inches a day, leaving the dead like a million bloody rugs. No Europeans will ever do that again in this generation.”

“Why, they've only just quit over in Turkey,” said Abe. “And in Morocco —”

“That's different. This western-front business couldn't be done again, not for a long time. The young men think they could do it but they couldn't. They could fight the first Marne again but not this. This took religion and years of plenty and tremendous sureties and the exact relation that existed between the classes. The Russians and Italians weren't any good on this front. You had to have a whole-souled sentimental equipment going back further than you could remember. You had to remember Christmas, and postcards of the Crown Prince and his fiancée, and little cafés in Valence and beer gardens in Unter den Linden and weddings at the mairie, and going to the Derby, and your grandfather's whiskers.”

“General Grant invented this kind of battle at Petersburg in sixty- five.”

“No, he didn't — he just invented mass butchery. This kind of battle was invented by Lewis Carroll and Jules Verne and whoever wrote Undine, and country deacons bowling and marraines in Marseilles and girls seduced in the back lanes of Wurtemburg and Westphalia. Why, this was a love battle — there was a century of middle-class love spent here. This was the last love battle.”

[wip] Some ideas on improving semantic segmentation with SAM

Use some pre-existing models like DeepLab and OneFormer as baseline semantic segmentation models. Get bounding box prompts by using semantic segmentation masks from the baseline models. Use the prompts to get SAM masks, these masks are not labelled well. Now use IoU score between both masks to label the SAM masks, and we end up with labels from the baseline models and masks from SAM. Test this hypothesis on around 100 images per dataset

Division Algorithm Theorem

Theorem: For all $a,b\in\mathbb{Z},$ with $b>0,$ there exists unique $q,r\in\mathbb{Z}$ such that $a=bq+r$ with $0\le r<b$

Proof: Basically, $q$ is the quotient and $r$ is the remainder. Consider the set $S={a-bx|x\in\mathbb{Z},a-bx\ge0}$. For this set, we will first choose $a, b$ and vary $x$ to get elements of the set.

In general, to find the smallest element $r$ in $S$, we have to argue that the set $S$ is not empty.

Claim: $S\neq\phi$
Case 1: $a\ge0$, we set $x=0$, so that we get $a-b(0)=a\in S$. The set is not empty
Case 2: $a<0$, we will set $x=a$, so that $a-ba=a(1-b)\ge0$. The set is not empty

Let $r$ be the minimum value of $S$ and $q$ be the corresponding quotient for this value of $r$.

We have $r=a-bq$. Towards the contradiction, assume that $r\ge b$
$r=a-bq\ge b$
$r-b=a-b(q+1)\ge b-b\ge0$

This means that $r-b\in S$ because $r-b=a-b(q')$ where $q'=q+1$. But on the other hand, $r-b<r$ which contradicts our assumption that $r$ is the minimum value of $S$. This implies that $0\le r<b$

Now, all that is left is to prove the uniqueness of $q$ and $r$. Suppose that $q,q',r,r'$ are such that $a=bq+r=bq'+r'$. Assume $r'\ge r$, this assumption can work either ways. We have $b(q-q')=r'-r$. The LHS is a multiple of $b$ and the RHS follows the inequality $o\le r'-r<b$. The only way both can be true is if LHS $=$ RHS $=0$. This implies that $q=q'$ and $r=r'$. $\blacksquare$

1989 (Taylor's Version) is out!

✨🫶 My name is Taylor and I was born in 1989 🫶✨https://t.co/klomIqGx38 pic.twitter.com/sWofWRjpvN

— Taylor Swift (@taylorswift13) October 27, 2023

How do you know what you care about?

This was an answer I wrote to a friend that asked me how we're supposed to know what we care about before we choose to work on something? These were my thoughts...

The first heuristic that comes to mind is Feynman's. What do you find yourself treating like play? I always think that when you feel like you have to separate "work" from "free-time", you're not doing something that you care about. Because when you're working on stuff you care about, every minute is literally free time. Heuristic 1: What we like to play with is what we care about most.

There are certain things you can't do without thinking a few standard deviations differently to your peers. For example, academic or industrial research, founding startups, creating art et cetera. This implies that the barrier to entry is sufficiently high for these fields of work. Crossing this barrier to entry implies that you really care about what you're doing. Heuristic 2: An inclination to do things with an unusually high barrier-to-entry.

Specific curiosity. Everyone has general curiosity about lots of things, but the curiosity is usually superficial and limited to one or two layers of understanding. When you care about something, you will display specific curiousity. For example, when I wanted to get better at shooting the football, I asked lots of people about lots of parameters, from their weight training regimen to the kind of shinpads they wore in the hope that I get better at it. Heuristic 3: You display an intense level of curiousity when you care about something.

Corollary to heuristic 3: The feeling of being a newbie never leaves you when you care about something. You are always aware that there's more to learn. Heuristic 3.1: You care about things you're most insecure about.

Book 22, The Odyssey

You thought I would never return from Troy;
and so - you dogs - you sacked my house, you forced
my women servants to your will and wooed
my wife in secret while I was alive
You had no fear of the undying gods,
whose home is spacious Olympus, and no fear
of men's revenge, your fate in days to come
Now all of you are trapped in death's tight embrace.

The entire book leads up to these lines and when they finally arrive, they are so delicious!!

Katalin Kariko and UPenn

The University of Pennsylvania is acting proud of Katalin Karikó now that she's won a Nobel. But they kicked her out of her research assistant professor job when she insisted on doing the work that won her that prize:

"She recalls spending one Christmas and New Year’s Eve conducting experiments and writing grant applications. But many other scientists were turning away from the field, and her bosses at UPenn felt mRNA had shown itself to be impractical and she was wasting her time. They issued an ultimatum: if she wanted to continue working with mRNA she would lose her prestigious faculty position, and face a substantial pay cut.

”It was particularly horrible as that same week, I had just been diagnosed with cancer,” said Karikó. “I was facing two operations, and my husband, who had gone back to Hungary to pick up his green card, had got stranded there because of some visa issue, meaning he couldn’t come back for six months. I was really struggling, and then they told me this."

"While undergoing surgery, Karikó assessed her options. She decided to stay, accept the humiliation of being demoted, and continue to doggedly pursue the problem. This led to a chance meeting which would both change the course of her career, and that of science."

Elsewhere she recalled:

“I thought of going somewhere else, or doing something else. I also thought maybe I’m not good enough, not smart enough."

She's now an adjunct in UPenn's neurosurgery department. Will they fast-track her for tenure now that she has a Nobel, or just live with the shame?

Both quotes here come from interesting stories. The first is from here:

https://www.wired.com/story/mrna-coronavirus-vaccine-pfizer-biontech/

The second is from here:

https://billypenn.com/2020/12/29/university-pennsylvania-covid-vaccine-mrna-kariko-demoted-biontech-pfizer/

- John Carlos Baez (@johncarlosbaez) October 03, 2023

Links from the Monsoon of '23

Machiavelli: A Very Short Introduction by Quentin Skinner
Ferrari vs. Mercedes along Eau Rouge, 2017 to 2019
Manchester United vs. Barcelona, UEFA Champions League Final, Wembley 2011. Tiki-taka masterclass by Pep Guardiola.
Mechanical Watch by Bartosz Ciechanowski
The Physics of Racing by Andre Marziali
What Does Any of This Have To Do with Physics? by Bob Henderson
How to Be a Mathematician (or Theoretical Computer Scientist) by Ryan O'Donnell
Math Has a Fatal Flaw by Veritasium
The Silver War, 2014 and 2016
The Real Problem at Yale Is Not Free Speech by Natasha Dashan
Joe Gebbia with Tim Ferriss
Last of the Czars: Part 1: Nicky and Alix, Part 2: The Shadow of Rasputin, Part 3: The Death of the Dynasty
36 all-out to Mission Melbourne, The Great Sidney Robbery and The Gabbatoir Breach. The greatest Test series win, relived by Ravichandran Ashwin and R. Sridhar.
Tribute to Christopher Hitchens by The 2012 Global Atheist Convention the future linus torvalds

Max-bounds on information travel speeds

You know how the light we see from stars was emitted ages ago and we're only seeing the past because of the distance. How do you find out whether a star exists at this moment in time? Because information can't travel faster than the speed of light, we are fundamentally constrained.

Let's talk about the sun because it's closer to us, and more intuitive to think about. The sun is 499 light-seconds away from us (what an annoying number) and we want to find out if the sun actually exists now or if something happened to it in the 499 seconds since it last emitted light. One thing we could do is to send someone up to the sun and report back with real time updates (ignore the fact that the sun would burn up this person, or if you can't ignore it, imagine a heat-proof sensor or something). But this fails because the fastest way the person would send us the information would still be slower than the speed of light.

After more failed thought-experiments, I thought of one that I can't seem to refute. Tie someone to a rope that is exactly the distance between the sun and the earth and fly them out to the sun. When they reach the sun, the rope will be taut. Now, when the sun disappears, all this person has to do is to pull on the rope which will instantly alert us, 499 seconds before things actually go dark suddenly. Obviously, I'm ignoring something, because nothing actually can travel faster than the speed of light.

What am I missing? The only thing I can think of is that pulling on the rope doesn't transmit forces instantaneously as I hope, but why would it not be instantaneous? It's a rope, why will the length or anything else matter? I'm being incredibly myopic about something, but can't figure out what.

Update: It takes time for an impulse to travel along a physical object. This can be seen with high-speed differential cameras. I feel like I've won the Nobel Prize for Stupid.

rant: linkedin

In the everlasting spirit of keeping things petty here, let's talk about LinkedIn. How many times have I had people I've barely known ask if I could "connect" (what a repulsive word) on that abomination of a site and I felt every shred of respect leave my body. LinkedIn needs to be mocked mercilessly for its assumptions that our worth as humans is solely tied to the work we've done. It needs to be ridiculed for how it functions as a platform for people to continually debase themselves in public. It is a humiliation ritual, that people willingly take part in. Actually, no, scratch that, I don't think there has been a single person that woke up and went, "let me create a linkedin account today" So what is it? What is it that makes the most normal person you know get "beyond excited" to tell the world about their leadership skills or be "thrilled" (seriously, do people even know what this word means?) to explain how they "create value". The most honest answer given to me when I asked a friend was that having an account makes it easier to stalk other people on LinkedIn. At least that's a noble cause.

Few things are more icky than people using social media to write motivational quotes and reposting company posts in order to show how much they love their new job. It just doesn't matter, so close your account and find platforms where people get to know you as something more than the number of connections you have (if that's what is important on LinkedIn). Don't become a career.

2024 Breakthrough Prize

Simon Brendle, Columbia University. For transformative contributions to differential geometry, including sharp geometric inequalities, many results on Ricci flow and mean curvature flow and the Lawson conjecture on minimal tori in the 3-sphere.

A Matter of Concern...

Scientists publish studies on how too much sitting will kill you, and how standing desks aren’t much better … but how about a study of people who plant their feet against the nearest surface and tip back and forth on their chair, the way their teachers always told them not to?

- Greg Egan (@gregeganSF) September 08, 2023

rant: short form video content

Notice how short-form video clips taken from longer videos/interviews used to be clips, then they became clips with subtitles, then clips with coloured/highlighted subtitles, and now they have background music or special effects added. This will only get worse.

Short form content is so repulsive to me, it is an incremental commoditization of our attention. A relentless attack on our senses, until the baseline is far removed from the norm. I'm going to write an essay about this if I can sufficiently marshall my attention.

The Making of the Atomic Bomb

Finished reading The Making of the Atomic Bomb and it's a solid 5/5. Richard Rhodes gets the balance -between writing science, and the human stories- perfect. All the brilliant scientists I'd grown up reading about come together and their science fits in like a perfect mosaic to split the atom.

The chapter on the bombing and effects on the Japanese cities is traumatic to read, a reminder about man's inhumanity towards man. Reminded me of Voltaire's, "It is forbidden to kill; therefore all murderers are punished unless they kill in large numbers and to the sound of trumpets."

When you read Feynman's accounts, you have this impression that life at Los Alamos revolved around him. So it was surprising to me that he's only mentioned thrice in the entire book, and only one of those times, he's described as doing anything (setting up the radio before the Trinity test)

In chronological order, the architects of our understanding of the atom. Democritus, Aristotle -> Newton, Dalton, Avogadro -> Thomson -> Rutherford (nucleus) -> Plank, Einstein, Bohr -> Rutherford (proton), Chadwick -> de Broglie, Schrödinger, Heisenberg.

And for the splitting the atom. Becquerel, Curie, Rutherford -> Fermi -> Hahn, Meitner, Frisch -> Szilárd -> Bohr, Anderson -> Teller, Wigner, Einstein-Szilárd -> Roosevelt, Oppenheimer, Groves, Manhattan Project.

The Shoulders of Giants.

Für Therese

Wow! I never knew that Beethoven's "Für Elise" was probably written for a woman named Therese - and the letters these two names have in common:

E S E

may have inspired the alternating sequence of notes that starts this piece:

E E♭ E

Yes: in German E flat is called "Es" and pronounced like the letter S.

I also didn't know that Beethoven never published "Für Elise" in his life: he gave the score to Therese Malfatti, and it was transcribed and published by someone else after his death, and hers. Beethoven almost published a second more complicated version of the tune in 1822... but at the last minute decided not to.

I learned all this stuff from this video, which also includes a performance of the second version of "Für Elise". If you like the famous version, which is rather simple, this one may seem ludicrously fancy. But it's interesting.

In 1810, Beethoven wrote a letter which ended like this:

"Now fare you well, respected Therese. I wish you all the good and beautiful things of this life. Bear me in memory—no one can wish you a brighter, happier life than I—even should it be that you care not at all for your devoted servant and friend, Beethoven."

https://www.youtube.com/watch?v=jblFQ1whX5s

- John Carlos Baez (@johncarlosbaez) August 17, 2023

Intermediate activations in Llama 2.7B

By layer 24, the model is quite certain about the correct answer, and the remaining computations are mostly redundant, mainly re-weighting alternative less obvious completion paths such as 'The capital of Germany is {a, the, one, home, located...}'. Interestingly, the model becomes less certain about 'Berlin' from layers 24-31 as it figures out more alternative options. The attention output of layer 24 of the llama 2 transformer consistently represents relevant information related to countries, even when neither the prompt nor the higher probability completions are related to countries

the inexorable march of time...

can someone tell me where the phrase, "the inexorable march of time" originated? I absolutely love it, but can't seem to find who came up with it. G. K. Chesterton has "relentless march of time" and Walt Whitman has "long march of time".

at my wits' end...

Update: I asked this on Mathstodon, and Prof. Michael Kinyon from the University of Denver found an 1876 reference by a R. W. Shufeldt of the U. S. Navy.

Notes on Nausea

Prodromal phase of emesis: Salivation increases, sometimes a lot, pupils dilate, heart races, blood vessels constrict, cold sweat. Attention and focus go down, and fatigue starts to set in. Stomach has electrical activity, and relaxes to prevent emptying. Oesophagus contracts to pull the stomach into the chest through the diaphragm (like a funnel). Retrograde giant contraction: small intestine pushes contents back to the stomach. In the lower small intestine, contents are pushed to the colon

Vomiting act: Retching involves muscles of the abdomen, oesophagus and respiratory system, but nothing comes out. During expulsion, intense pressure is generated in the stomach after the diaphragm and the abdomen contract

Nausea: From the Greek word meaning ship. Intensely aversive, used in behavioural change therapy (rather controversially). Aside: potatoes during pregnancy is a bad idea, unless you want your child to have a neural disease. Pregnancy nausea as a evolutionary defence mechanism, because the embryo is sensitive to toxins in food. Women with moderate to severe emesis during pregnancy have lesser miscarriages than women with mild to no emesis. But we don't know why nausea happens with motion sickness.

Motion sickness happens when the motion that we experience is not the motion that we are expecting. No convincing explanation for why anxiety, or the sight of blood, or the sight of vomit causes nausea. Injury from sever vomiting can occur: rupture of the oesophagus, lung collapse, tearing of the spleen. Some doctors think pregnancy sickness is due to unconscious rejection of pregnancy (originally touted by Freud). Vomiting and nausea are not connected always, you can have one without the other, considering how common this is, we still don't have a good antiemetic drug.

John Bardeen

To understand what LK99 was all about, I was looking at how superconductors work (revising the basics) and I found out that John Bardeen came up with the Theory of Superconductivity. The John Bardeen! The very same man that invented the Transistor with William Shockley!! He's the only person to win the Physics Nobel twice.

Wow, superconductivity AND the transistor. Where does he stand in the most influential person of all time rankings?

Symmetry and Newton's Laws

A little classical mechanics problem you can solve without doing any calculation:
Consider the hyper-simplified problem of a bell-shaped hill, and a point rock that can slide without friction up and down the hill. If you start with the rock at the bottom, and give it exactly the kinetic energy needed to arrive to the top and stop there without sliding on the other side, how long will it take to arrive there?

- j_bertolotti (@j_bertolotti) August 03, 2023