Notes



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 $\Delta s\sim k_B\ln{2}$.

silica structure

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. $$H\propto\frac{1}{N_p^{1/3}}.$$

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!!

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!

taoc1 taoc2 taoc3 taoc4

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?

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.

P-V diagram S-T diagram

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.”

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!

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

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

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...

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

the future linus torvalds

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

Allen Mandelbaum's translation of The Odyssey

Muse, tell me of the man of many wiles,
the man who wandered many paths of exile
after he sacked Troy's sacred citadel.
He saw the cities—mapped the minds—of many;
and on the sea, his spirit suffered every
adversity—to keep his life intact,
to bring his comrades back. In that last task,
he will was firm and fast, and yet he failed:
he could not save his comrades. Fools, they foiled
themselves: they ate the oxen of the Sun,
the herd of Hélios Hypérion;
the lord of light requited their transgression—
he took away the day of their return.

Muse, tell us of these matters. Daughter of Zeus,
my starting point is any point you choose.

All other Greeks who had been spared the steep
descent to death had reached their homes—released
from war and waves. One man alone was left,
still longing for his home, his wife, his rest.
For the commanding nymph, the brightest goddess,
Calypso, held him in her hollow grottoes:
she wanted him as husband. Even when
the wheel of years drew near his destined time—
the time the gods designed for his return
to Ithaca—he still could not depend
upon fair fortune or unfailing friends.
While other gods took pity on him, one—
Poseidon—still pursued: he preyed upon
divine Odysseus until the end,
until the exile found his own dear land.

Conway's Game of Life is Omniperiodic!

IMO 1988 Day 2, Question 6

Let $a, b\in\mathbb{Z}^{+}:ab+1 | a^2+b^2.$ Show that $\displaystyle\frac{a^2+b^2}{ab+1}=r^2$ where $r\in\mathbb{Z}$.

I figured out that the solutions were $a=n,b=n^3,$ and then concluded falsely that that was it. Turns out there are infinitely more and the only reason that $(n,n^3)$ shows up as the first solution is a coincidence of the complete proof.

Zvezdelina Stankova explains on Numberphile one solution that uses induction with infectious enthusiasm. This problem consumed me, and its insides were wondrous.

John Conway's Doomsday Algorithm

Pi day (March 14) and the last day of February is Doomsday.
Apart from February, the $n$th day of the $n$th month is Doomsday for all even months.
Switch dates for months 9, 5 and 7, 11. We already know March. January has Doomsday on the 3rd on non-leap years and on the 4th on a leap year.
Doomsday advances by one day each year because 365 divided by 7 leaves 1 remainder. Doomsday advances two days each leap year.

Speak Now (Taylor's Version) is out!

John B. Goodenough

John Goodenough and his work on Li-ion batteries and computer random-access-memory. Immortal.

"a product that today touches nearly everyone's life"

on endings and beginnings

friendly reminder that the ending does not change the context of the beginning

Links from the Summer of '23

2020's biggest breakthroughs in Math and Computer Science, Physics, and Biology
Taylor Swift: folklore, evermore and Songwriting
Who goes Nazi? by Dorothy Thompson (1941, Harpers)
Dan Carlin on the Lex Fridman podcast.
Essay: How do you describe TikTok?, "The best bodily position in which to watch TikTok is supine, muscles slack, phone above your face like it's an endless tunnel into the air."
Trench verses by Belash Yuri Semyonovich
The Insect Brain
A Mathematician's Lament by Paul Lockhart
Against Dog Ownership
Documentary on Amal Kumar Raychaudhuri
The Offensive Lineman - with John Urschel on the Numberphile podcast.
A survey of some famous problems in Mathematics over the years, compiled by Alvaro Rozano-Robledo.
A perfect Suzuka qualifying lap by Sebastian Vettel in the AMR22.
some cool problems to think about... in matrix completion from LessWrong
Gilbert Strang's Final Lecture, streamed live by MITOCW. Brilliant as usual.

Harvard Careerism

(From The Crimson)“Students that don't come from familial wealth may be “disinclined to smell the flowers along the way and to take some chances,” said former Dean of Freshmen Thomas A. Dingman '67.”

“The clearest example of Harvard's careerist turn is its extracurriculars… student organizations… increasingly function as pre-professional outlets.”

Key Changes

Apparently, there are lesser key changes in the billboard top 100 nowadays due to 1) the rise of hiphop (which doesn't really care about keys as much as it cares about rhythm and lyrics) and 2) digital recording (which doesn't constrain artists to use instruments like guitar and keyboard only). One exception is Sicko Mode by Travis Scott and Drake. Also discussed was Michael Jackson's Man in The Mirror and the key change near the start of the second chorus.

From Chris Dalla Riva on Tedium.

Nicholas Mahut on Rafael Nadal's dominance at Roland Garros

When you play Roland Garros 14 times you tell yourself you had a good career. When you win 14 matches there, that's not too bad at all. When you get to the second week 14 times you are one of the great players. And when you win the title 14 times, there is no way to comprehend that. There are no words.

Informed Consent

Are patients even capable of providing "informed consent" before a medical procedure? The keywords being informed, and consent. Considering how less the layman knows about the human body compared to a trained medical doctor. What does it mean if they can't? Both legally and ethically.

Ethics of AI cloning

Should it be illegal to make software that allows the AI cloning or deep faking of the voice, image, or video likeness of real people without their consent?

— Katherine Brodsky (@mysteriouskat) May 18, 2023

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.

Verstappen Rising

There's no stopping Max this season, he's just breezing past everyone. I guess F1 has converged to a point where aero is so overpowered that there wont be any competition if you get it right. This isn't even recent phenomena. It started with Sebastian and then Lewis and now Max. One car, and usually one driver (that's in tune with that car) dominating the field.

The Cursed Child

Everyone I know that has bought The Cursed Child has the hardback edition (as opposed to the paperback that is released much later). All of us really hoped great things from that book eh?

overrated is overrated

I just don't understand the use of over-rated as critique. Isn't it enough that we say we don't like it? Why is it a problem if many others like it?

Get Some Free Speech While You Can

Freedom of expression isn't the tool of the powerful; it's the tool of the powerless. From Caitlin Flanagan's Atlantic article.

Malcolm Gladwell pointed out the absurdity of her position by tweeting, “I signed the Harpers letter because there were lots of people who also signed the Harpers letter whose views I disagreed with. I thought that was the point of the Harpers letter.” Whenever a society collapses in on itself, free speech is the first thing to go. That's how you know we're in the process of closing up shop. Our legal protections remain in place—that's why so many of us were able to smack the Trump piñata to such effect—but the culture of free speech is eroding every day. Ask an Oberlin student—fresh outta Shaker Heights, coming in hot, with a heart as big as all outdoors and a 3 in AP Bio—to tell you what speech is acceptable, and she'll tell you that it's speech that doesn't hurt the feelings of anyone belonging to a protected class.

Field Trip to Citi Neuro Hospital!

Today, we went to Citi Neuro Hospital with everyone taking the Medical Image Analysis course. There were ten of us and we booked an SUV to the place. The air was super hot outside the campus, and the first thing I noticed about Hyderabad was how undulating the terrain was. Apart from small areas of the city, most of it is super uneven. The place we went to, Banjara Hills, is literally that, hilly. Some parts of the roads were like roller-coasters, steep and winding.

Banjara Hills as a residential area is pretty posh, and I definitely might have seen some A-list celebrity homes while I passed them. We went to the hospital and I took out my mask (haha). Everyone else had to buy masks. I wish I looked more closely at the patients (the hospital was CROWDED) because this wasn't like a usual hospital, it was for neuro disorders, so I'm sure the patients would be remarkedly different. In general hospitals, people usually look pained, and in suffering. Anyways.

Dr. D. Ravi Varma who's part of the board (and he probably owns the hospital) came out to receive us. The staff and the patients gave us a wide berth when they saw who we were with. He took us first to the X-ray room. It was kinda fancy, he showed us how the walls were super thick so that radiation wouldn't leak outside, and then I saw that some walls had water seeping through them and I looked very suspiciously at that for a while. Because I was trying to think whether water would be affected by X-rays in any way. The X-ray machine was super sophisticated, it could move in two directions (say X-axis and Y-axis) and could rotate along one plane.

The room had provisions for a full body X-ray (where the patient has to lie down) and for a chest X-ray (where the patient has to stand, much like Roentgen's wife in the original X-ray experiments). I didn't really understand why the two are so different, even though he said something about magnification. The operator stands behind a lead frame? (the right word evades me at the moment) and the glass he sees through also has lead in it. There was a heavy (I tried it on, it is heavy) lead apron for pregnant women so that the foetus is not harmed.

Then we went to CT, the machine costs 1.2 crores, and they're planning to replace it with a machine that costs 3 crores. CT was pretty uneventful, except that I saw a warning on the CT machine that warned technicians to not put their fingers in the machine. I mean, some idiot must have done that for the machine to come with that warning. Also, for angiography, they use iodine (which is super dense) in order to mark out blood flow. But the body only wants limited quantities of iodine, and injecting that much iodine is super bad. So they tie up iodine in a molecule called iodohex? me forgot :(

Then we went underground, which really puzzled me, because the hospital was like 5 stories high or something. Guess why? These MRI machines are so heavy, so heavy, that the roof would fall on your head if you placed it on standard slabs. Being a civil engineer, you could see how interested I was in all this. 6 tonnes, is how heavy the machine is. And the machine was a thing of beauty. Physics and engineering in perfect tango. We had to remove all metallic objects and we went inside. There were pictures of the beach inside in all directions to help claustrophobic patients. I asked how many claustrophobic patients he gets, and he said, VERY SURPRISINGLY TO ME, that almost all the patients he gets are claustrophobic.

We spent so much time with the MRI machine, Ingenia 3.0 Tesla by Philips. It has some 3 million miles of wire about 2 micron thick running with some huge current to generate the magnetic field. The current is sustained by ensuring superconductivity of the material, liquid Helium is sustained at 4K (-269 C) by using a heavy Carnot engine, that pumps so loudly that the room vibrates constantly (thud, chain click, thud, chain click). The patient is given a pair of pneumatic headphones (which are super awesome, they convert digital sound to pneumatic sound, and the inside of the wires have loads of cilia, which propagate the sound) so that they're not affected by the magnet. I wish I asked if the patients could choose their own music.

I was fascinated by the warnings, and the backups, and the backups to the backups they had. One warning near the MRI machine showed a person being sucked into the machine because they got too close with their wheelchair, the drawing was hilarious. There were 3 separate buttons to turn off the magnet in case of such a situation. Apparently, turning of the magnet and turning it back on will cost, wait for it, drum roll, 2 crores. Self-explanatory no? THAT IS MORE THAN THE CT MACHINE.

Phew, once, they lost power supply to the machine because of some circuit breaker. An MCB, it didn't break all the way and got stuck in the middle, and because of that, they didn't get power from either the electricity board or their generator, and the machine turned off. Since then, they have installed a protocol where they have sensors that always tell them whether the board is giving them power or not. And if that protocol fails, the chief doctor (Ravi Varma in this case) has a direct line to the EB (electricity board) in charge with priority. All rather important stuff.

When I said underground, you should have thought of what I said earlier, that we were in Banjara Hills. The hospital has been built into a hill. Dug into rock, I wonder how stable the building really is. It has been 7 years, and I definitely did see some cracks in some corners. He then showed us how fast his computing machines were (they have state of the art GPUs to get instantaneous reconstructions in 3D, and in super high resolution), their data center, which gets filled by 1TB a year (that is unreal) because of all the high-quality images they store. That is that, I don't think I've forgotten anything else. One funny thing that happened was one student asked him how long the generator runs, and he replied rather drily, "as long as we pour diesel into it".

I found it cool, how well-versed he was with the physics, the engineering, the computation, the algorithms, and the logistics of his hospital. He's a doctor, and he's the first one I've seen that I would never guess was a doctor unless he told me.

Exponential improvement for the bound for diagonal Ramsey numbers

Links from the Spring of '23

Read Something Great
Save matplotlib figures as TikZ/PGFplots for smooth integration into LaTeX
Queen and Rook Endgames: A Study on LiChess
Forensics of the Subclavian Vein
The Confessions of Marcus Hutchins on Wired
Steering into the Iceberg on Common Sense with Dan Carlin

$\LaTeX$ pipeline

On Unix, install texlive-full for compiling TeX, vi as code editor, and zathura as document viewer. Running latexmk -pdf -pvc file.tex will open a .pdf file everytime file.tex is saved. EB Garamond is unmatched for typesetting mathematics.

Thoughts on Amnesia

Patients with amnesia or agnosia usually answer questions by cues rather than memory, a sort of inference. For example, identifying the month by looking out the window and identifying platonic solids by feeling them. Is the loss in these pattern making abilities greater nowadays than before? What can we recognize nowadays by cues? shapes, yes, I imagine. Seasons, maybe not? More auditory cues than haptic cues?

The Human Placenta

Placental protein 13 creates a diversion, it goes to some other part of the uterus that the placenta isn't trying to invade and attracts the attention of the immune system. Human placental lactogen can cause pre-eclampsia, dangerously high blood pressure leading to organ failure. By the third trimester, 25% of the blood flow is going into the placenta.

Links from the Winter of '22

Tom Lehrer songs archive
Life begins at 40: the biological and cultural roots of the midlife crisis | The Royal Society
Best of Nerdwriter1: Harry Potter & The Prisoner of Azkaban: Why It's The Best - YouTube, Mr. Bean Is A Master Of Physical Comedy - YouTube, The Death of Socrates: How To Read A Painting - YouTube
Theory Thursday sessions held weekly at IIITH
Reversals in psychology

Lionel Messi is a World Champion!

Some fun math puzzles

Figure out the rule that governs the following sequence and the next term in the sequence: 6, 3, 3, 4, 4, 7, 3, 9, 8, 3, 3, 4, 4.

Imagine a rectangular table made with "4920 $\times$ 2352" boxes, how many boxes does the line joining the opposite corners pass through?

Midnights is out!!

Midnights is a collage of intensity, highs and lows and ebbs and flows. Life can be dark, starry, cloudy, terrifying, electrifying, hot, cold, romantic or lonely. Just like Midnights. Which is out now

— Taylor Swift (@taylorswift13) October 21, 2022

Solutions to IYMC 2022 Qualification

Problem A: What are the roots of the function $f(x)=(\log(3^{x})-2\log(3))\cdot(x^2-1)$ with $x\in\mathbb{R}?$ $$f(x)=(x-2)(\log(3))\cdot(x^{2}-1)$$$$x=-1,1,2$$ Problem B: Find the values of the following infinite sum $$1+\frac{3}{\pi}+\frac{3}{\pi^2}+\frac{3}{\pi^3}+\frac{3}{\pi^4}+\frac{3}{\pi^5}+\cdots$$ We can rewrite the sum as $$1+\frac{3}{\pi}\bigg(1+\frac{1}{\pi}+\frac{1}{\pi^2}+\frac{1}{\pi^3}+\frac{1}{\pi^4}+\cdots\bigg)$$ Using the formula for the sum of geometric progressions and noting that $\displaystyle\frac{1}{\pi}<1,$ we can write $$1+\frac{3}{\pi}\bigg(\frac{1}{1-1/\pi}\bigg)=\frac{\pi+2}{\pi-1}$$

Multilingual Speakers

When a multilingual person wants to speak, the languages they know can be active at the same time, even if only one gets used. These languages can interfere with each other, for example intruding into speech just when you don't expect them. And interference can manifest itself not just in vocabulary slip-ups, but even on the level of grammar or accent.

Separation of languages happens by a suppression of the non-relevant languages. A kind of attention mechanism. The problem is called language-control in multilinguals. Errors are a great way to understand this. Sometimes, it seems like they inhibit the dominant language so much that they actually are slower to speak in certain contexts. A reversal of language dominance, where you end up making a mistake in your primary language

Wow, Spanish and English errors have "pero" and "but" as something common. I personally have shifted to the understanding part of language with that particular word.

Through the use of eye-tracking technology, Gollan and her team found that these mistakes were made even when participants were looking directly at the target word. She explains that when mixing languages, multilinguals are navigating a sort of balancing act, inhibiting the stronger language to even things out ​​- and sometimes, they go too far in the wrong direction

Sometimes bilinguals will produce the right word, but with the wrong accent, which is a really interesting dissociation that tells you language control is being applied at different levels of processing

As it turns out, the Italian migrants were more likely to reject correct Italian sentences as ungrammatical if these did not match correct English grammar. And the higher their English proficiency, the longer they had lived in Canada, and the less they used their Italian, the more likely they were to have found the correct Italian sentences ungrammatical

Act III Scene II, Julius Caesar

Friends, Romans, countrymen, lend me your ears;
I come to bury Caesar, not to praise him.
The evil that men do lives after them;
The good is oft interred with their bones;
So let it be with Caesar. The noble Brutus
Hath told you Caesar was ambitious:
If it were so, it was a grievous fault,
And grievously hath Caesar answer'd it.
Here, under leave of Brutus and the rest-
For Brutus is an honourable man;
So are they all, all honourable men-
Come I to speak in Caesar's funeral.
He was my friend, faithful and just to me:
But Brutus says he was ambitious;
And Brutus is an honourable man.
He hath brought many captives home to Rome
Whose ransoms did the general coffers fill:
Did this in Caesar seem ambitious?
When that the poor have cried, Caesar hath wept:
Ambition should be made of sterner stuff:
Yet Brutus says he was ambitious;
And Brutus is an honourable man.
You all did see that on the Lupercal
I thrice presented him a kingly crown,
Which he did thrice refuse: was this ambition?
Yet Brutus says he was ambitious;
And, sure, he is an honourable man.
I speak not to disprove what Brutus spoke,
But here I am to speak what I do know.
You all did love him once, not without cause:
What cause withholds you then, to mourn for him?
O judgment! thou art fled to brutish beasts,
And men have lost their reason. Bear with me;
My heart is in the coffin there with Caesar,
And I must pause till it come back to me.

Sebastian Vettel's Greatest Hits

38 laps on the soft, in Bahrain 2018
Ferrari screw ups in Australia 2016, Canada, Germany 2016, Germany 2018, Italy 2018, China 2019, Russia 2019, Mexico 2019, Silverstone 2020 costing valuable points
Monza 2008 win
Brazil 2012 win
Most number of pole positions in a season (15) in 2011
Most number of wins in a season (13) in 2013
Most number of consecutive wins (9) in 2013
Youngest pole sitter in 2008
Youngest world champion in 2010

The Genius Of The Crowd, by Charles Bukowski

but there is genius in their hatred
there is enough genius in their hatred to kill you
to kill anybody
not wanting solitude
not understanding solitude
they will attempt to destroy anything
that differs from their own
not being able to create art
they will not understand art
they will consider their failure as creators
only as a failure of the world
not being able to love fully
they will believe your love incomplete
and then they will hate you
and their hatred will be perfect

like a shining diamond
like a knife
like a mountain
like a tiger
like hemlock

their finest art

Endgame puzzle

White can win this endgame with perfect play! It seems obvious now, that I need to move the white king all over to f6, while guarding the c pawn with the bishop. Then black's bishop will be overloaded trying to defend the f pawn and also prevent the h pawn from moving up. But I drew this position while actually playing it by giving away my h pawn after which the position is equal, even though white has a pass pawn.

Sebastian Vettel F1 retirement

Just a really sad day :(

Cristiano Ronaldo in The Player's Tribune

But when you're a kid, you don't care about money. You care about a certain feeling. And on that day, this feeling, it was very strong. I felt protected and loved.
When you lose, it's like you're starving. When you win, it's still like you're starving, but you ate a little crumb.
When I was 15, I turned to some of my teammates during training. I remember it so clearly. I said to them, “I'll be the best in the world one day.”
They were kind of laughing about it. I wasn't even on Sporting's first team yet, but I had that belief. I really meant it.
It is like a final reminder … a final motivation. It says, “El sueño del niño.” The dream of the child.

German names

Today I learnt that Michael Schumacher is pronounced as mika-el schoomahhhuh and Sebastian Vettel is pronounced zebaastyan fetl. Nice. The rabbit hole that led me there was Juergen Schmidhuber's blog on which he has a page on Schumi where he focuses on details about Schumi's money and fame rather than his race stats and achievements.

Maryam Mirzakhani anecdote

After completing an undergraduate degree in mathematics at Sharif University in Tehran in 1999, Mirzakhani went to graduate school at Harvard University, where she started attending McMullen's seminar. At first, she didn't understand much of what he was talking about but was captivated by the beauty of the subject, hyperbolic geometry. She started going to McMullen's office and peppering him with questions, scribbling down notes in Farsi.

“She had a sort of daring imagination,” recalled McMullen, a 1998 Fields medalist. “She would formulate in her mind an imaginary picture of what must be going on, then come to my office and describe it. At the end, she would turn to me and say, 'Is it right?'' I was always very flattered that she thought I would know.”

2022 Fields Medals!!

James Maynard for distances between primes, existence of infinite decimal representations of primes with no digit 7 (or any other digit).
Maryna Viazowska for sphere packing in eight dimensions.
June Huh for proof that absolute values of coefficients in chromatic polynomials are unimodal and log-concave. (chromatic polynomial for a complete graph is $x(x-1)(x-2)\cdots(x-n+1),$ for a graph with no edges is simply $x^n,$ and for a tree on $n$ vertices is $x(x-1)^{n-1}$)

Links from the Summer of '22

Some cool math puzzles from Quanta Magazine
People dislike their political opponents for views that most don't actually hold. [N=4993] on Twitter
The Math Doctors
Wittgenstein at War from The New Statesman
Rapoport's Rules for debate from Daniel Dennett's "Intuition Pumps and Other Tools for Thinking"
Botticelli's The Birth of Venus was inspired by Phyrne of Thespiae
Your Local Epidemiologist by Dr. Katelyn Jetelina. Paxlovid (88% reduction, 89% efficacy) and Molnupiravir (30% reduction)
Complex-Valued Autoencoders for Object Discovery published in Transactions on Machine Learning Research (TMLR)
The Art of the Command Line

Nicer ceil function in C++

The inbuilt ceil function was being very unwieldy. I prefer defining my own inline function now. 

int ceil(int numerator, int denominator)
{
    return (numerator + denominator - 1) / denominator;
    // in case of suspected overflow
    return numerator % denominator == 0 ? numerator / denominator : numerator / denominator + 1
}

The Hellespont

The Hellespont is the ancient name for the Dardenelles Straits, they separated Europe from Asia. Alexander crossed the Hellespont from Greece and threw his spear into Persian soil, thereby declaring war on the Achaemenid Persian Empire. Personally, I think this act was what defined Alexander. Alexander could have stayed home in Greece, raised a family, been a great king, and died a celebrated man. But this was not him. By crossing the Hellespont and daring to invade Persia, he transcended his mortality. It is a reminder to me that I have a Hellespont to cross.

Rafael Nadal wins the French Open!

Real Madrid win La Decimocuarta!

Hala Madrid!

Productivity Culture

The fatal flaw in the productivity culture: Its mandate is never, “you figured out how to do my tasks more efficiently, so you get to spend less time working.” It is always, “you figured out how to do your tasks more efficiency, so you must now do more tasks.”

Links from the Spring of '22

The Best of Dan's Narration from r/dancarlin
Brief Intro to Game Theory on SSRN
Date Formatting in Javascript on StackOverFlow
The Fading Battlefields of World War I from The Atlantic
Where do Birds Go? from xkcd
The Machine Learning Community has a toxicity problem from r/MachineLearning
Colour photographs from The Great War, these are nothing like I expected them to be. Granted, the soldiers were posing but these photos give off an idyllic vibe. Very unsettling.
Elon Musk and Sal Khan talk about Entrepreneurship on YouTube
Career Panel with Yoshua Bengio and Richard Sutton on YouTube
Eagle's Nest, Hitler's HQ and all of Mark Felton's videos
GitHub Stars are overvalued
Typesetting before $\LaTeX$
Save matplotlib figures as TikZ/PGFplots for smooth integration into LaTeX.

Nietzsche's "Morality is Cowardice"

If you don't have the courage to commit a crime, it doesn't mean you're moral for not committing a crime. It just means you're afraid of doing it. We can see this in mob violence, people riot because they don't think they'll get caught.

Thoughts on Privacy

People assume that if they're not doing anything unlawful, there is no threat to their privacy. If there is unlawful activity, it is all the more reason for the activity to not be private. I read some version of this thought on an SSRN paper, I can't recall which.

Rafael Nadal wins the Australian Open!

Some Kimi Räikkönen stats

Most number of fastest laps in a season (10) and third on the all-time list of fastest laps (46)
Most number of podiums after not starting from the front row (72)
Only driver to win for McLaren, Ferrari, and Lotus
Only driver to win races in the three different engine regulations eras (3.0 litre V10, 2.4 litre V8 and 1.6 litre V6 turbo)

If, by Rudyard Kipling

If you can make one heap of all your winnings
And risk it on one turn of pitch-and-toss,
And lose, and start again at your beginnings
And never breathe a word about your loss;

Modular Arithmetic

(a + b) % m = (a % m + b % m) % m is fairly obvious to derive.
(a * b) % m = ((a % m) * (b % m)) % m can be derived by writing out the multiplication as repeated addition.

Fermat's Little Theorem states that if $p$ is a prime, then for any $a\in\mathbb{Z},$ $$a^p\equiv a\mod p.$$ (a / b) % m = (a % m * b^-1 % m) % m can be derived from Fermat's Little Theorem. The multiplicative inverse of $b$ modulo $m$ is $b^{-1}$ such that $b\cdot b^{-1}\equiv 1\mod m.$

How Many Recursive Calls Does a Recursive Function Make?

The paper looks a the number of function calls made to itself by a function calculating the Fibonacci numbers recursively. There is a linear relationship between the Fibonacci numbers and the number of recursive calls. The number of function calls for a number $n$, $G(n)$ is related to the Fibonacci number $F(n)$ by the relation $G(n)=2F(n)-1.$ Achieved by adding a global counter to the recursive function and incrementing it each time the function is called.

So, you think you're bad at math...