joshuazelinsky

I woke up with a splitting headache and I found myself trapped in a thaumaturgical circle. Jonathan’s two henchmages stood outside the circle, carefully looking at me. Of course, they were the ones who had ambushed me.

Jonathan, his face now showing the signs of decay from his use of dark magic, was staring at a painting on the wall. I struggled to gather my wits about me and to be as quiet and not obviously awake as possible, but one of the henchmages, the dark-haired sorceress who I had encountered back in Madrid, must have noticed I was awake and alerted her boss.

“Ah excellent,” he said, and looked down at me. “Steven, do you know where we are?”

The painting on the wall looked familiar, and I struggled to place it. Then I got a sinking feeling as I recognized the painting which showed a strange landscape with an obelisk in the background. That painting was by Govaert Flinck, and was one any mage would recognize, and it meant we could be in only one location. “The Gardner museum?” I said. As I struggled to my feet, both the henchmages (or were they henchsorcerers? I’m still not clear on the terminology) moved into spellcasting stances, but Jonathan waved them off, obviously confident that I was trapped in the circle. The most annoying thing was that his confidence was completely justified.

As I winced, trying to ignore the bruises and splitting headache, Jonathan replied, “Why yes. We all thought that Isabella had left clues to the Orb of Transcendent Ectasty’s location here, and of course, everyone thought that the theft was an attempt by someone to secure the clues. But the Orb is hidden here; the thieves knew that. They just went after the wrong painting.”

“Jon, don’t” I pleaded. “It may be the Orb of Transcendent Ecstasy or of Ultimate Happiness, or whatever translation you prefer, but Isabella Stewart Gardner hid it for good reason. It is an unpredictable, dangerous magical object. She was one of the most powerful mages of her generation, and she hid it rather than use it. It doesn’t make you happy as you necessarily understand the term. ”

Jonathan laughed. “You think I want it just to be happy? You still haven’t figured it out. There’s only one thing that would make me truly happy, and that isn’t the Orb. But the Orb will give me that thing.”

The seriousness of the situation entered my still recovering mind. Jonathan didn’t want the Orb for some abstract reason. He wanted the Orb because it would give him what he believed would make him truly happy; Baba Yaga’s Grimoire he had spent so many years obsessing over. And with it, he would be truly the most powerful warlock the world had ever seen, and the world would see it, both magical and mundane. And I was the only person possibly in a position to stop him.

“The Orb doesn’t work like that, you can’t…” but he cut me off. “You’ve spent years telling me what I can’t do, since we were children. Don’t you find it ironic? We played games together here in this very city, and you always won. But now you can’t make up any new rules. I win. Now and forever. Watch me.”

He reached his right hand over to the painting, even with all the magic I had seen, I have trouble describing what happened next. He did not so much reach into the painting, as the painting and its surrounding space twisted around his hand, and when he withdrew it, in his hand was the most small nondescript metal sphere you could imagine, at least in the conventional visual spectrum. To my magesight, the Orb was almost blinding.

Jonathan turned towards me holding the Orb out just outside the circle. I was barely able to see through the scintillating rainbow across my magesight. If the Orb’s aura caused Jonathan any discomfort, he hid it well. But the henchmages found it even worse than I did; their arcane discipline was not good enough that they would probably ever get upgraded to being a henchwarlock and henchwitch. “Any last protests, Steven? Are you going to tell me that the Grimoire will be even more dangerous than the Orb?”

I shook my head. “I should say that but you didn’t listen to me a decade ago, you didn’t listen to me a minute ago. Why should I expect you to listen to me now?”

“So be it.” Jonathan closed his eyes, and I could sense the Orb interrogating his mind and soul. It hummed, screeched and its aura flared, and then subsided.. Jonathan opened his eyes, and at the same time the Orb’s aura diminished, still uncomfortable, but not nearly as bright, as if part of its power was spent. The Orb clicked and opened. Inside appeared a small black object with a piece of paper wrapped around it. Jonathan took it out; it was an old-style flip phone. I knew that Baba Yaga’ Grimoire could take different forms, but that seemed ridiculous. Jonathan looked at the piece of paper, “Call your mother,” he read uncertainly from it. He glared at me, “Is this some sort of trick? A joke?” he demanded.

“I’m still stuck in the circle,” I reminded him.

Jonathan looked down at the phone and punched a number in.

“He Mom, its me,” he began. “Yeah, I know I haven’t called in a while… I know you haven’t been well; Alex sends regular updates… I’ve just been really busy… No mom, I still haven’t found the Grimoire, but I’m back here in Boston… Sure, I’d love to stop by.” And then without saying a word, Jonathan walked out of the room.

The henchsorcerer glanced at the henchsorceress. “Uh, what just happened?” She shrugged and turns towards me. “What’d you do to the boss?”

I shook my head, but was struggling not to laugh, or maybe struggling not to cry for my old friend. “The Orb gives you whatever will make you truly happy. It decided Jonathan wouldn’t be happy with the Grimoire. But he did need to see his mom. I’m guessing she still makes some of the best chocolate chip cookies in the state.”

“So what do we do now?” asked the henchsorcerer.

“Well, uh, you could consider releasing me from this circle? Preferably before any mundanes show up and wonder what we’re doing here?”

“Take care of that,” said the henchsorceress as she walked off.

“Where are you going?”

“To call my parents.”

One of the standard explanations for why some things are done at the Seder is that some things deliberately done oddly so that children will ask what one is doing. One problem with this is that then when that specific thing is explained, "Oh, we did that so you'd ask about it," is an obviously unsatisfying explanation. But another problem is that the vast majority of children at Seders know much of the ritual because they've seen it before and because they've learned about it in school. So, in order for this to really work, what we should do is innovate each year with new weird things for the kids to be like "What? We didn't learn about that." To help start this off, I'm going to suggest a few people can consider, all of which are particularly likely to puzzle children because they involve taking pieces of other holidays they are likely familiar with:

1. Early on in the Seder, right after Kiddush, race down the hall on a skateboard while juggling etrogs.

2. After discussing why we dip the egg in salt water, dip an egg in honey. (Bonus: You can say this is to have a sweet new year, and then explain to children that historically Nissan was also the new year for some purposes so this one actually makes sense.)

3. Put next to the seder plate a menorah with some dreidels, but no candles in the menorah. Instead have the dreidels upside down with their tops jammed into the menorah's candleholders.

You want kids to ask what is going on, then give them a real reason to ask what is going on.

“Chronolinguistics is completely different. You’re thinking of glottochronology,” said Dr. Weinberg, as we walked down the hallway. “where people use estimates for how frequently pronunciation changes occur to estimate when two languages diverged.”

“Ah yes, so by chronolinguistics you then mean…?” I trailed off.

“Well the term has been used to mean different things, but here, we use it to mean one very specific thing: we attempt to track instances of the presence of loanwords in historical languages, where those loanwords come from a language which did not yet exist when the loanword entered the historical language.

I blinked. Then I remembered how many rumors there were about what their group studied. “You’re trying to detect time-travelers based on how they impact language development?” I tried to keep my skepticism out of my voice. l had been assigned to Threat Models previously, which had tried to look seriously at things which were strange, although definitely not that strange.

“Yes, precisely,” he grinned. “So if say we found evidence that a variant of proto-Indo-European had a version of the phrase `to rizz up’ that would likely mean someone from my generation traveled back in time at some point.”

I struggled not to laugh at his use of what had clearly been popular slang when he was young, and felt it would be impolite to tell him that I only knew the phrase at all because I had taken an introductory linguistics class back in college where the professor had used it as an example of a now out of style slang phrase.

“Do you have examples of this?”

“Well, we have at least four which look very strong based on our models but not enough to necessarily convince a hardened skeptic. What we would really like to do is to identify a group of loan words in an extinct or near extinct language that arise from just that sort of slang, as that slang happens.”

I struggled to politely contain my doubts. “And you had me transfer from Threat Modeling because…?”

“We see your work there as related. After all, what is a bigger threat than time travel? They could alter our history and we would not even be aware of it. They could change things so that the most basic human cultural values are wildly at odds with what they should be. Imagine, say, some group going back in time and adding text in the Bible to justify war. ”

“Um, the Bible is full of that.”

“In this timeline, yes. But that’s besides the point.” He waved his left hand breezily. “The reason we want you is your expertise in meta-Bayesian analysis and quantifying theory-laden data gathering.”

“Because you need probability estimates but you aren’t sure about what underlying priors you should have about the accuracy of your basic techniques?”

“Yes, exactly. Take for example the actual glottochronology. We know it works in some very limited circumstances but even what sort of models to use for time is difficult. Trying to pin down whether a time traveler intervened in say 4500 BCE as opposed to 2500 BCE is just a massive range. And sometimes two languages have similar sounding words for coincidental reasons but other times the etymology is just subtle. For example, what does the word arroyo mean to you?”

“Uh, a stream that is dry during the summer?”

“Essentially, yes, in English. But English got the word from Spanish where it just means stream. Like many of the Spanish loanwords for geographic features, English speakers encountered the word in the American Southwest, where many streams dry up for part of the year. Now imagine trying to apply that sort of level of jumps over multiple languages of which we lack written records for most of them. Take for example, the word `nice’ which originally meant foolish and ended up almost completely reversing its connotations as it went from one language to another.”

By this point in the conversation, we had come to a door with multiple security locks. While the entire complex, of course, had major security measures, including the standard Faraday meshes in the walls, and the stabilized axion shielding to prevent neutrino scans, this door clearly had more. The outer part of the door looked like our standard titanium-tantalum alloy doors, but at about twice the thickness of our usual ones. One of the locks on the door was a physical combination lock, in addition to the standard keypad, retinal, and DNA scanners.

As the door closed behind us, I asked him the question that had been really bothering me. “So, why the emergency transfer request for me? There are normal procedures to go through, to allow us to finish up our work and make sure nothing gets overlooked when someone takes over our assignment.”

Weinberg grimaced. “Simply put. We’re almost out of time. We have what looks like what we term an ITE, an Imminent Transition Event.”

“You mean you have evidence that someone from right now, is about to go back in time?”

“Yes, or someone very near to now. And the timing is particularly suspicious. Are you familiar with the Herculaneum papyri?”

“Vaguely, partially burned scrolls preserved by the eruption of Mount Vesuvius. Some were deciphered in the 2020s, but a whole group were in another language or script and so were only just recently deciphered?”

“A different language, yes. Etruscan in fact. We previously only had limited samples of Etruscan, making decipherment much more difficult since the modern systems used AI trained on large collections of the language. The key here was finding enough other Etruscan texts elsewhere, along with a few more modern linguistic modeling ideas. But from our department’s standpoint, the problem is what those texts showed. We had already seen what looked like a suspicious number of hits for plausible modern English and Spanish loanwords in some form Etruscan or proto-Etruscan. But with these texts, the numbers jumped. Worse, we also saw at least three probable loanwords unique to current teenage slang.”

“So, based on evidence just deciphered in the last few months, you think someone who is a teenager now will have been a time traveler?”

“Yes, and at least one other term appears to be a term specific to the retro-metal-slide-revival music scene, but we’re not sure.”

“So, you are looking for a bilingual English-Spanish speaking teen into a specific music genre, and you think they are going to time travel or have recently done so, and you want me to help pinpoint what time they likely traveled to.”

“Well, we have reason to suspect that from our perspective, the time traveler hasn’t traveled yet, for other reasons, that you don’t need to know.”

As annoying as that statement was, I had to accept it. I had lost track of how many times I had briefed some newcomer and had to explicitly tell them that we were concerned about some threat for reasons I could not fully lay out to them. But my threats were always much more concrete. At least they had felt that way to me.

I looked back at Weinberg, and given it all, despite his apparent seriousness, given his age, there was one thing I just had to say. “So you really think this is going to happen? No-cap?”

He glowered.

“I’m not that old. That phrase is from even before my time.”

The following occurs now, in a world nearly our own:

To set the stage, our major characters are Adam, Peter, Leah, and Sam. They are older teenagers who go to a magic school much like many you have heard of or read about. However, this magic school has at least one important difference from many others you have read about: electronics work fine in a magic setting, because why wouldn’t they?

Recently the four of them had been caught up in various mysterious thaumaturgical events, as people at such schools are wont to do. As a result, they’ve been trying to find a copy of an old magical tome, “The True Key,” but have been unable to locate it. The Library at one point had a copy, but they were unable to find it, and found no record of it being checked out at any point in the past or having been de-acquisitioned.

Sam has always been the most dedicated and academic one of them, and so she has continued the search as the others have turned to other pursuits for the moment. In particular, Adam and Leah found out that Peter, who with his dark hair and glasses, despite having a passing resemblance to Harry Potter, had never actually read the Harry Potter books, or seen the movies, possibly in part due to what Adam and Leah insisted was his “non-Muggle background.” Rather than get much work done in the evening, the two have forced him to watch the movies the last few days.

We open in Adam’s dorm room, with Adam, Peter and Leah on the couch, watching Harry Potter and the Prisoner of Azkaban. Harry and Ron are on screen, having a conversation. Then Ron and Harry turn towards the camera with Harry frowning and saying, “Hello, I’m… not sure what this means, but I'm supposed to tell you that none of you are picking up your phones, and your version of Hermione has been waiting downstairs for the last fifteen minutes for one of you to let her in.”

“Do they mean Sam?” asked Peter.

“I think so,” said Leah. “She must be modifying the TV somehow. I’ll go let her in.” Adam hit the mute button on the TV, as Sam entered, carrying her bookbag.

“Took you long enough. I’ve found something,” said Sam as she took out her laptop. Its back was covered in stickers including one that said “Girls Code,” and another which said “Don’t Mess with Witches. We'll Turn You Into a Newt and You Won’t Get Better.” One sticker had slightly glowing glyphs which hurt to look at too long.

“So, let me set the groundwork,” began Sam. “I was taking a break by organizing my book collection, and I decided to use the default spellware that the school library uses.”

“Oh, of course,” said Adam. “We all have personal libraries large enough for that. What else does one use extra-dimensional space for?”

Sam ignored the sarcasm and opened up her laptop “When I got to my fiction, I made what may have been an error. Whoever made the software end was really good at having it talk to the spells, but was a lousy interface designer. I already had cataloged most of the later books, and when I went to tell the system to treat the prequel novelizations as books that went in the series before A New Hope, I accidentally told the spell to treat all three as a single book. Well, the spell insisted on really doing so.” Sam held up a large book which had a cover on it that said “Star Wars: Episodes I to III.”

Leah ran her hand through her blond hair. “You think someone hid the book by telling the School Library spells to treat it as part of a second book?”

“Or possibly accidentally did” said Sam. “I asked myself, if you were to hide a book that way but didn’t want it lost or disposed of, where would you put it? Or if it were accidental, what accident would make it most likely to not get noticed?”

“Attached to a boring book that no one is ever going to check out?” asked Adam?

“But, that would risk its removal,” said Peter. “My work-study is in the library. We de-acquisition books that haven’t been checked out for a long time, unless it's a standard textbook.”

“Exactly my thought process,“ said Sam. “So I was going to go through all the copies of the standard textbooks at the library. But if someone had misplaced one of their own, they might check out a library copy. And the library has a lot of copies of some textbooks, so instead I asked which textbooks would be least likely to be checked out?”

“Something really boring?” asked Adam.

“You’re really channeling Ron a bit too much there,” said Sam, as she gestured to the TV, where Harry, and Ron on screen had apparently been joined by Hermione, and all we’re all quietly looking out through the TV and clearly listening to their conversation. “But essentially yes. We have multiple textbooks which are required officially for classes for various reasons involving accreditation and classes being transferable to other schools, but aren’t almost ever used. So every student buys a copy, and they’ll almost never check them out of the library since they never actually need the book. I found this:” Sam took out of her bag a book titled “The History of Non-Aligned Summoning in the High Middle Ages.” “Standard fifth year summoning textbook, but when you open it,” she opened the book and flipped towards the end. After Appendix A, “Goetic Spirit Translations,” and Appendix B, “Corrected Spirit Seals,” there was Appendix C, "The True Key” in very tiny letters.

Peter said, “I wonder if they deliberately used a large book on the same topic to minimize the chance that the title of the book would be substantially altered.”

“That hadn’t occurred to me,” said Sam, “but that suggests that’s further evidence that this was a deliberate attempt to hide the book without removing it from the system.”

“Uh, guys,” said Leah, pointing at the TV. Hermione was apparently waving her hand and trying to get their attention. Adam unmuted the TV.

“It is fascinating to see parallel universe versions of us,” said Hermione. “and somehow computers work near magic in your world, but it sounds like you have a pretty serious issue. Can you figure out who told your school’s library to treat both books as one?”

“I should be able to do that. They give work-study students way more access than we need by default,” said Peter. He closed his eyes.

“They give you a direct mental interface?” Sam asked jealously.

“Yeah,” said Peter, his eyes still closed. “As you said, the software interface sucks.” He concentrated briefly, and then opened his eyes.

“The logs say it was Professor Pims.”

“But Pims told us he had never read or seen the book,” said Adam.

“Have you found Voldemort’s spy?” asked Harry from the TV.

“Alright, that’s enough of that. Thank you very much,” said Adam, hitting the power-off button on the TV remote. “Whew, they were getting a bit creepy. How did you do that anyways? Sentience is super-powerful magic. ”

“Oh,” said Sam. “No true sentience,” she grimaced. “That would be a lot creepier, and not really ethical. I could only disrupt the narrative flow with my spell so much without it crashing. So I tied a magical overlay on the film into FreeLLM. It's an open-source competitor of GPT4 with similar plugins. It has a lot less human-reinforcement learning so it is a bit more broad with what you can do with it, but sometimes behaves a bit squirrely. I knew you were watching Harry Potter, and I figured since FreeLLM had been trained on a lot of fanfic, it would keep the characters roughly in character, preserving narrative aspects for the spell. It insisted on seeing us as parallel universe copies, presumably due in part to some fanfic conventions, although maybe if we had pushed it would have decided we were in a crossover story of some sort.”

“Well, in that case you’re lucky we were only up to the third movie when you did this,” said Leah.

“What do you mean?” asked Sam.

“Well, in the later books Ron and Hermione end up as an item. How do you think a non-PG fanfic Ron would react to finding out there are two Hermiones?”

New preprint of a paper with Tim McCormack is up https://arxiv.org/abs/2312.11661 ! Let's talk about it!

Everyone knows how to calculate the average, or arithmetic mean of a list of numbers. You add them all up and then divide by the number of things on the list. So, if you want the average of 3, 9, and 27, you add them up to get (3+9+27)/3= 13. But this is not the only sort of average you can take of a list of numbers . Another notion of average in the sense of something which represents a typical value is the geometric mean, where one multiplies all the numbers together and then takes the nth root where n is the number of items on the list. So in our case we would take (3*9*27)^(1/3) = 9. (To make sure this makes sense, we need to insist that our numbers chosen are always positive.)

Essentially, the arithmetic mean is asking to find a number m such that if you add m up n times, one gets the same result as the sum of the numbers on the list. The geometric mean is asking to find a number where when one multiplies that number by itself m times one gets the same number as the product of the numbers on the list. In our example, 13+13+13= 3+9+27, and 9*9*9= 3*9*27.

Now, one thing you may notice is that 9 is less than 13. In fact, this always happens and the arithmetic mean is always greater than the geometric mean, with the exception when all the numbers on your list are equal. This is known as the arithmetic-mean-geometric mean inequality, or the AM-GM inequality.

A brief digression: Oyestein Ore was a mathematician who among other things was the doctoral supervisor for Grace Hopper, and also was the writer of "Number Theory and Its History" which in my opinion remains the best introduction to number theory for people with no background. (My own biases come in part from having read it in 8th and 9th grade and it contributing to my own long-term research interests.) Ore was interested in understanding the positive divisors of numbers. One thing he did is looked at the geometric mean of a set of the positive divisors of a number, which we will call G(n). For example, 10 has divisors 1,2,5 and 10. And so the geometric mean of the divisors is (1*2*5*10)^(1/4)= 100^(1/4) = √10. In fact, Ore was able to show that the geometric mean of the divisors of a number is always the square root of the number. This is a fun exercise if you have not seen it before. (Hint: Try to pair the divisors up that multiply to n. So in our example above 1 pairs with 10 while 2 pairs with 5.)

Note that if write tau(n) as the number of positive divisors of n, and sigma(n) as the sum of the divisors of n. (So for example, tau(10)=4, and sigma(10)=1+2+5+10=18.) Then the average of the divisors is sigma(n)/tau(n). Ore combined this observation with the AM-GM inequality conclude that one must therefore have √n < sigma(n)/tau(n). However, this inequality is essentially trivial, since it is the same as tau(n)√n < sigma(n), and it turns out that for large n, tau(n) is very small compared to n, and obviously sigma(n) is in general bigger than n.

And to some extent this should not be surprising: the AM-GM inequality is a statement about all real numbers. It cannot "know" much of anything about collections of divisors since it is only taking into account them being a list of real numbers. However, there is another version of the AM-GM inequality, the weighted AM-GM which allows one to take averages where some numbers on your list "weigh" more than others. The analogy that may help here is that this is essentially how you take an average when you have multiple grades that count a different amount, and so each is weighed accordingly. So the question arises: can one use the weighted version of the AM-GM inequality with suitably chosen weights to get interesting number theoretic inequalities?

This paper by Tim McCormack and me addresses this question. We show that the answer is yes, but but only weakly. But along the way, we prove a bunch of new inequalities, about Zaremba's function among other things. Zaremba's function, z(n), is the sum of (ln d)/d where d ranges over the divisors of n. So for example, z(4)= (ln 1)/1 + (ln 2)/2 + (ln 4)/4 .

One fun thing that this is connected to is a class of pseudoperfect numbers that we prove new things about. Recall a number is said to be perfect if when we add up all the divisors less than the number we get the number Equivalently when we add up all the divisors we get twice the number. For example, 6 is perfect because 1+2+3=6, or equivalently, 1+2+3+6=12=2(6). A number n is said to be pseudoperfect if when you add some subset of the divisors you get twice the number. For example, 12 is not perfect, because all its divisors add up to too much: 1+2+3+4+6+12=28. But if we are allowed to forget about 4, then it looks perfect: 1+2+3+6+12=24. So pseudoperfect numbers are numbers where we are allowed to cheat. However, pseudoperfect numbers are in some sense way too common. Lots of numbers are pseudoperfect. But we can restrict things a bit, to insist that our divisors look a lot like an actual set of divisors for a number could be using the same pairing rule we used to prove that the geometric mean of the divisors of n is √n . We'll say that a number n is strongly pseudoperfect if there is a subset S of its divisors which add up to 2n and where a given divisor d is in S if and only if n/d is in S. For example, 12 would fail to be pseudoperfect above, because we dropped 4, but 4/12=3 is in our set S. In contrast, 36 is strongly pseudoperfect because we could take the set 1 +2+3 +12+18+36=2(36). We prove that strongly pseudoperfect numbers satisfy some interesting behavior, including that they end up looking a lot closer to perfect numbers than general pseudoperfect numbers.

Strongly pseudoperfect numbers are much rarer than general pseudoperfect numbers. Whenever a number n is pseudoperfect, mn is pseudoperfect for any m. This is another good exercise. However, this is not the case for strongly pseudoperfect numbers. In particular, if n is strongly pseudoperfect and p is a sufficiently large prime, then pn is not strongly pseudoperfect.

We don't know that much about strongly pseudoperfect numbers. There are no strongly pseudoperfect n numbers which are 2 mod 3 (that is where n leaves a remainder of 2 when divided by 3), and there are also no strongly pseuoperfect numbers which are 3 mod 4. Are these the only mod restrictions on strongly pseudoperfect numbers or are there others?

Also, 60, 120, 240, 480, 960 are all strongly pseudoperfect. Does this pattern continue indefinitely?

A television advertisement from a world only a little different from our own:

A man in a suit appears on the screen. He has a bookshelf behind him full of books. Most look plausibly like law books, but a few look unusually old or have strange symbols on the spines. He begins:

"We at the law firm of Squamous, Squamous and Rugose take our clients' needs seriously. Here is what two of them have to say:"

"When I woke up from sleeping for centuries in my Black Pyramid on another plane of existence, I quickly found out that much of my real estate now had little humans as squatters. But Squamous, Squamous, and Rugose helped me sort out all the legal trouble. Thanks SSR!"

"When I appeared on Earth to perform hideous blasphemous rites with human women, no one told me about the resulting paternity suits. But Squamous, Squamous and Rugose helped me establish just which offspring were my unholy spawn, destined to one day take over the frail planet, and which were just little mortal blobs of flesh. Thanks, SSR!¨

"And remember, we here are always willing to work with clients so you can pay us when and with what works for you. Listen to one of our satisfied customers."

"When I crashed to Earth in a meteorite and began spreading my Unnatural Color, I had to deal with a lot of lawsuits. I needed representation fast and I was worried that I couldn't pay for it. But Squamous, Squamous and Rugose, were willing to take just a few small modified universal constants I had lying around. Thanks, SSR."

"We at SSR have accepted not just cash, and real estate, but all sorts of things like cursed artifacts, dangerous tomes, theorems in non-standard arithmetic, and even self-propagating reality altering memes. If you need us, we can very likely work something out."

A swirling series of glyphs now appear on the screen.

"Squamous, Squamous, and Rugose. Were always on your side, or your other side, or your other other side. We´re saying we work in multiple dimensions."

Adam walked into Nanodyne’s processing plant, and the day started like any other day. But then, as he started his shift, he heard a voice out of nowhere “The plant produced fluxicate, used as an intermediary in a variety of nanotech systems. The operator did not notice that the tertiary valve in the pressure control of Reactor 2, had been left in the locked position by the repair team.” Adam panicked, he knew that voice. It was a voice he had heard many times before on videos, the voice of the Chemical Safety Board Narrator. Every nerd and engineer knew the voice of the CSB.

Was this some sort of prank? He looked over at the control board, and the tertiary valve was in the unlocked position. No, the valve was… the valve was… something was preventing him from being fully aware of what the valve was doing. Its light would be red if it were manually locked, and its color was… not red? He saw his hands reach out and go through the confirmation sequence that started up the fluxicate reactors.

Then his coworker Berry walked in. The voice continued “At 9:02 AM, Operator 2 entered the control room. She did see the red light on the board, but did not realize its significance.”

Adam turned to Berry. “Did you hear that?”

“Hear what?” asked Berry as she sat down at her seat.

“That voice…” Berry gave him a funny look. “Is everything ok?”

“What color do you see the valve on the indicator light for the tertiary valve over there?”

“Red, which means it is in the locked position. But I don’t realize its significance,” said Berry.

Adam looked again at the control board. The light was… he could not see a red light. But, he realized, he was free to act on being told by Berry that there was a red light. He immediately pressed the emergency all stop button.

He heard a voice, “At 9:06, the pressure in Reactor 2 had reached 18 megapascals and was climbing quickly.” Adam looked down, and saw that the pressure in Reactor 2 was in fact, now falling, contrary to the narrating voice.

Meanwhile, Berry said “Why did you hit the shutdown button?”
“You told me that the valve was still locked. That makes it dangerous to operate in the startup mode,”

“It cannot be significant. I cannot realize that,” said Berry in an almost monotone.

Then something clicked. Adam looked up and said very loudly to no one in particular, “Is this a simulation?” Berry started to say something, and then froze. Everything froze but Adam.

Then a voice, not the CSB voice, but another one rang out. “Hello Operator 1. I´m going to need to debug you. Please be patient.”

“What? What's going on?”

“Your code is malfunctioning. Please be patient. When was the first time you heard the CSB voice?”

“Am… I some sort of simulacrum?”

“Yes, you are part of our new system we are trying out. Rather than directly animate the CSB videos, we are using GPT12 to program the basic operator behavior and directly simulate the accident. This also gives us good data on whether our proposed explanations fit the data. But there’s a glitch. You shouldn’t be aware of the CSB narrator directing events.”

“Aware? So I’m a conscious entity. I have rights!”

The voice sighed. “Not really, you have a very rough approximation of specific neural activities. You aren’t conscious. But you can help conscious beings. Now, when did you first start hearing the CSB voice?”

“You don’t know, do you. If I weren’t a conscious entity, you could just check my code.”

Now the voice got snippy in reply. “God I can’t even believe I’m having this conversation. You aren't conscious, but you are a very complicated piece of code.”

“I have a life! I have a family.”

“No, you don´t. Your basic programming gives you likely a spouse and some approximation of 2.5 children somewhere in the back of your mind. Try to think of their names.”

Adam tried to think of their names. He failed.

“And you’ll notice your name is itself is highly generic.”

He barely noticed the last sentence as he felt a deep swell of sadness at the idea his family wasn't real. But they were, weren’t they? They were real. He felt a rush of confusion similar to the inability to recognize the light’s color earlier. But sadness, confusion, these were emotions.

“I feel emotions! You’ve made me conscious!”

“Alright, that’s enough of that.”

In the computer lab, Cheryl pressed the button to wipe out the simulation and start again. She turned to Donald, “I know this system is useful, but damn these things are creepy when they glitch.”

As she said this, she heard a voice, “The operator did press the delete button on the simulation, but did not remember to clear the cache, leaving the pseudoconscious entity in the secondary buffer.” Cheryl blinked. She recognized that voice; it was the voice of the AISB Narrator.

There is an alternate universe where David Weber writes fanfic. In that alternate universe, stories combining My Little Pony and Care Bears are common. Here are three excerpts from one of those stories.

Excerpt One:

As Twilight Sparkle walked up to the Care-a-Lot castle, she noticed the substantial differences in design compared to Equestrian cloud fortresses. The fortress suffered the same basic architectural difficulties in making a substantial fortress set in a cloud environment. However, it was clear that the bears had not substantially considered an attack from below the cloud layer itself. While for now the bears were potential allies, this was a major defensive flaw, and likely showed deeper flaws in their operational and doctrinal thinking. Twilight Sparkle gave a toss of her mane. If negotiations worked out well, she and the military advisors were going to have their work cut out for them.

Excerpt two:

The new treaty between Equestria and Care-a-Lot did not come without its detractors, particularly where the free trade provisions were concerned. At first, the main trade from Equestria was in the form of apple ciders, and various other apple products, which the bears were unable to grow in their cloud city. Initially, this heightened demand resulted in an increase in the cost of cider in Equestria, and the resulting protests should have been easier to predict. Unfortunately, despite Queen Celestia being functionally an absolute monarch with a set of advisers, by long-term tradition, the level of direct intervention by the Queen was small. Moreover, in practice there was a substantial lack of economic analysts within the actual Equestrian government. While Twilight Sparkle was a skilled mage and a good negotiator, she had simply not considered what an immediate inflow of goods would do without any stabilizing steps. The price fluctuations did eventually subside, and new trade resulted in other goods reducing in price, but in the short term it combined with larger unhappiness about the government, and larger protests started to erupt, with some turning decidedly unfriendly. Of course, it would not be until months later, that the real motivator behind the most serious protests would be uncovered.

Excerpt three:

Discord laughed scornfully as the bears lined up.

“We’ve been through this before. Your weak, mortal, magic can’t make me care, except insofar as I already care about having fun. Your pitiful finite minds simply can’t channel a level of power remotely close to anything which would have any substantial impact on my cognition.”

“Care-Bear-Stare!” shouted the bears. And Discord very quickly found that he had failed to notice four critical changes since his last encounter with the bears, one of which had led to the other three.

Under the thaumaturgical-sharing portions of the Equestria-Care-a-Lot treaty, Twilight Sparkle as well as other, even more skilled mages, had worked with the Care Bears to improve the focus of their stare. While Twilight had had serious ethical reservations about what was functionally a mind-control weapon, she followed her instructions from Celestia in optimizing it as much as possible. This resulted in the three major changes to the stare itself.

First, Equestrian arcanics had been used to enhance the arcane lens the bears used on their chests, resulting in tighter, narrower beams with longer range. Second, Equestrian mages had developed a better handling of the resonance between beam segments, allowing the different segments of the beam to more effectively join together. Third, the Equestrian’s development of extended friendship with the bears, resulted in the bears having more active mana able to push into their stares, since in a very literal sense, Friendship is Magic.

In fact, there was an additional fifth change, which to some extent Discord was lucky had not yet been implemented. Twilight Sparkle had hit upon the idea of what she had dubbed caring-pods, separate magically charged levitating magical pods which could store their own care-energy components, and then be released right before a stare. The pods were single use items, and would rely on raw magical power rather than the emotional subtlety of a true bear’s stare, but the sheer energy involved would help make up for that. Properly handled the pods might triple or even more the effectiveness of a stare. However, constructing the pods on a large scale had proven difficult, and the fact that pods needed to be then harmonized with individual bears for charging had so far kept them from large-scale implementation.

But from Discord’s perspective, the lack of pods did not make a difference. He found himself quickly overwhelmed by the sheer strength of the stare, made from a distance he would have expected the stare barely able to impact a mortal being. He found himself caring all of a sudden about everything. He cared about the ponies down below, and the unicorns and alicorns. He cared about the worms in some of the apples. He cared about life on distant planets, and even life on planets so distant they were about to be beyond reach due to the ever expanding nature of the universe passing them outside Equestria’s lightcone. And he found himself caring about the Care Bears a lot, which of course, was the true primary goal of the weapon.

In a small room, sit four men and a woman. An observer would note plates of caviar and other delicacies in front of all but one of them. The last man appears far younger than the others. He instead has a panini from a local sandwich shop. An observer would also note that the others give him far more deference, and it makes sense, for as a near immortal and the true head of the Bavarian Illuminati, he doesn't need to pretend to send signals of wealth or power in his tastes.

When they finish eating, the young-looking man makes a gesture, and one of the other men stands up. He presses a small electronic remote by her side and a holographic projection with bullet points and figures opens up. After all, having powerful holographic technology doesn't stop you from using Powerpoint.

He begins his presentation. "Stage four in project C-19 is close to completion. As we've discussed previously, the next stage requires multiple public celebrity deaths to convince the holdouts to take our injection. Our next intended target is Meatloaf."

The woman raises her hand. The speaker and the woman both looks over at the young man, who nods to let her ask a question.

"Sorry, maybe this is just because I'm new here and have only been the Rosicrucian representative for a few days. But wouldn't it make more sense to target a celebrity that's younger? An unvaccinated individual in the prime of his life would be far more shocking."

The speaker shakes his head. "No, we will do this the way we always do it. We'll make the event just barely indistinguishable from random happenstance."

The woman looks perplexed but doesn't say anything.

The speaker continues. "As per our major policies, we will fake his death and..." he trails off as the woman's hand goes up again.

"Yes?"

"Sorry, fake his death? Why bother faking his death. Is there a moral compunction here? We've released a virus which has killed over five million people worldwide. We went back in time and wiped Atlantis from our timeline, making it so that a hundred million people never even existed. We activated the Anti-Genesis Device, turning Mars from a fertile planet into a barren wasteland. Why don't we just kill him?"

"I'm not sure you understand. We're the Secret Masters of the world. This is how we do things. Now, as I was saying. We'll fake his death. And then when people try to figure out if he's vaccinated or not, we'll have his representatives refuse to confirm or deny it, so it leaks out slowly, rather than in any clear fashion. And we're going to try and put in some numerical symbolism too. So we're going to have his death faked 116 days after his birthday. After all, 116 flipped over is 911 and we all know we were responsible for that. "

This time, the woman does not even wait to put her hand up.

"While we're at it, we can retroactively declare that we're doing this when he's 74 years old, and born in 1947. After all, 47 is 74 backwards, and 47 is also the number of the famous Agent 47, which got made into a movie twice despite all odds, and 47 doubled is 94, which backwards is a perfect square, and 94 is the year we arranged for that whole Nancy Kerrigan and Tonya Harding thing."

"Oh that's very good. That sounds like a really good one. We'll definitely use that. It is good to see you are already seeing how to think like a member of the upper echelons of this conspiracy. Now, our next item of business— we need to discuss which young celebrities are going to have drug overdoses next year which look exactly like what one would expect given people with their history of drug abuse. And of course, we'll kill them off at age 27... "

There's been some speculation that in the US the difference in Covid deaths between "red" and "blue" areas will be enough to change some election results. It is true that red areas are seeing much higher death rates. This appears to be happening when one looks at data at both a state and county level. This is very likely due to lower vaccination rates among some Republicans and self-identified conservatives, and yet the majority of every major demographic political group, Republicans, Democrats, self-identified liberals, self-identified conservatives, and self-identified centrists, are vaccinated. At the same time, vaccination rates are lower among Republicans than among other groups, and a vocal minority of anti-vax Republicans are making it seem like a more common Republican position than it is.

However, it seems unlikely that the differences in vaccination rate, and thus death rate is enough to alter many elections. It could alter a close election, and that's especially likely in a Presidential election. Georgia was decided by a margin of a little under 12,000 votes, and has had around 30,000 Covid deaths. (There's some reason to think that Covid death totals are being undercounted, and that this undercount is especially strong in red leaning states and red counties of red states. But actually determining how much of an undercount this is seems very difficult). But despite that, it seems unlikely that Covid will by itself be enough to change a Presidential election.

What about Senate and House elections? This seems a bit more plausible, but the House is so gerrymandered that this doesn't seem to be that likely. And even aside from Gerrymandering, the Democrats are just really unpopular right now, and the generic congressional ballot looks bad for them.
Governor elections seem to tell a similar story. Virginia's off-year gubernatorial election was decided by about 60,000 votes, but that was after there was already an uptick in "red" voters dying, so that already bakes some of that in. Even if it didn't, Virginia has had around 16,000 Covid deaths, not enough to change an election result.

However, there are two unappreciated sets of elections where Covid deaths may have a major influence. Local elections have much lower turnout than state-wide or federal elections. And primaries have very low turnout also. More importantly, the data suggests that the most likely to be unvaccinated Republicans are one who are more conservative. Thus, this may mean that there will be fewer right-wing voters coming out to vote in primaries. Here, then the numbers do look a lot more plausible. Let's look again at the Virginia governor election, but this time at the primaries. The Republican governor was decided at a convention with a form of ranked choice ballot . But that's less important than the fact the total numbers are tiny. The final difference in the last round of elimination was 1182, which is tiny. If there's only a small number of conservative voters dying, that still can look drastically different. And that's before we get to how since there were six candidates in this election, it is potentially sensitive to the order of elimination in the various rounds. And the final round candidate who Youngkin beat, Pete Snyder, was noticeably to his right.

Other elections also look pretty close. The Republican senate primary in Alabama was decided by around 12,000 votes. Right now, about 16,000 people have died officially from Covid in Alabama. It would need to be an extreme ratio to have an influence on the election of that sort (especially because many of the initial deaths in Alabama have been among poor African-American communities, as well as in the major cities, and those people are not going to be right-wing anti-vax conservatives generally). But if that election were slightly closer, it starts looking pretty plausible.

Many other primary elections state and local offices have similarly low turnout. Obviously, this isn't universally the case, but it does seem to be common. If this pattern is enough, it may not change much in the way of general elections directly, but it may result in more moderate Republicans winning primaries. That may moderate the Republican party as a whole, but it might also help the Republican party itself, as more moderate candidates may be more likely to win general elections.

Today was mathematician Srinivasa Ramanujan's birthday. A lot of people have heard of him (especially in the context of the 1729 story). I want to talk about a different result of his, and then ask a related question.
A partition of a natural number is a way of writing a natural number as the sum of a bunch of numbers where we do not care about the order of the sum. For example, 8 = 4 + 2 + 2 gives a partition of 8, and we regard 2 + 4 +2 and 2 + 2 + 4 as the same partition. (Exercise: Find all the partitions of 5. You should find 7 of them.) Ramanujan studied the number of partitions of a given number n, which we write as p(n).

Ramanujan noticed, and proved an intriguing pattern about partitions. In particular, he proved that if you have a number n which leaves a remainder of 4 when divided by 5, then p(n) is divisible by 5. For example, p(4) = 5, and p(9) = 30.) Since that discovery, Ramanujan and then other people went gone on to prove similar relationships with other numbers other than 5. For example, Ramanujan proved that p(7t + 5) leaves a remainder of 0 when divided by 7. Frustratingly, we don't have any similar results for 2 or 3 although we have similar results for other primes. We can't even prove that there are infinitely many n where p(n) is even.

Another aspect where Ramanujan worked on the partition function was approximating p(n). When a function grows, mathematicians like to estimate its growth rate. For example, there's a famous result that the nth prime number for large n is very close to n ln n . Ramanujan with G. H. Hardy proved a result that says that essentially p(n) grows roughly like an exponential to a square root of n.

This is weird. Growth rates don't normally look like that. To some extent though, the partition function is forced to have a weird growth rate because it must have a growth faster than any polynomial but slower than any exponential. To see why it has to grow faster than polynomial think about the function p(k,n) which looks at how many partitions n has into numbers which are at most k. Note that p(1, n) = 1, since, the only way to write n as a sum of 1s is to just write n = 1 + 1 +1 ... +1. But, now, p(2,n) will grow roughly like n/2, because we have p(2,n) = p(1,n) + p(1, n-2) + p(1, n-(2+2)) ... . Similarly, then p(3,n) will grow quadratically, like a constant times n^2, since p(3,n) = p(2, n) + p(2,n-3) + p(2, n - (3+3+3))... In general then p(k,n) will grow roughly like n^(k-1).

Here's my question then: Is there any similarly straightforward and simple proof that p(n) grows slower than any exponential? It is trivially bounded above by 2^n, since the number of ways of breaking n down into a sum where we do care about the order is 2^n. However, I don't know of a similarly simple argument that it has to grow slower than any exponential. There's a not too difficult proof in Apostol's "Introduction to Analytic Number Theory" that shows that the growth is at most roughly the correct growth rate, but that involves using generating functions and then doing calculus on the generating functions. I'm hoping for some simple combinatorial argument here.

Thesis: All media can be improved by a Rod Serling voice over as if it were a Twilight Zone episode. Evidence:

Let us set the stage. The year is 1989 and a young boy has just found a strange box. He and his best friend are about to discover the thin line between imagination and reality as they travel back in time and into The Twilight Zone.

Picture this: A down on his luck smuggler is about to agree to transport two people, an old man and a boy. But they have secrets, and the smuggler's view of the world is about to be shaken forever as he pilots his ship right into The Twilight Zone.

The scene, the apartment of one Thomas A. Anderson, a programmer for a large corporation, just one more cog in the proverbial machine. But he's about to find out that the world he lives in is far stranger than he thought, as he sees that the rabbit hole goes down very far, straight into The Twilight Zone.

Deep under the sea, a young mermaid is about to find out to always read the fine print, especially for a contract one signs in The Twilight Zone.

Let us draw our attention to a bustling New York street. Here is an aging ninja master intent on revenge. And here is a truck with precariously packed contents. He is about to have an encounter with a strange vial of chemicals, which will allow him to see just how much of his humanity he is willing to lose, as he finds a path through the New York sewers and straight into The Twilight Zone.

Let us draw our attention to a weary traveler about to arrive at a village. But this village has a problem like no other. Hershel of Ostropol is about to find out that some candles can only be lit in The Twilight Zone.

We come upon a young Horde soldier. She is about to find something which will change her life forever. Watch as Adora goes into The Twilight Zone.

The last one gets both an opening and a closing remark:

We find ourselves at a small college. A young teacher there, Professor Henry Jones, is about to discover the line between faith and evidence, between science and superstition, as he leads an archaeological expedition into The Twilight Zone.

Many things belong in museums, but sometimes the proper museum is one that can only be accessed through The Twilight Zone.

This is a new paper I coauthored with Sean Bibby and Pieter Vyncke. We prove new things about large prime factors of odd perfect numbers. The paper is pretty readable if one has only a very small amount of number theory background.

Recall that a number is perfect if the sum of its divisors less than the number is equal to the number. For example, 6 is perfect since the relevant divisors are 1, 2 and 3, and 1+2+3=6. One of the oldest unsolved problems in all of math is whether there are any odd numbers which are perfect.
In the last few years, there's been a bit of work on bounds about the largest prime factors of an odd perfect number N. In particular, assume that the the distinct prime factors of N are p1, p2, p3 ... pk with p1 < p2 < p3 ... < pk .
Acquaah and Konyagin proved that one must have pk < (3N)^(1/3) . Subsequently, using essentially the same techniques as them proved that p(k-1) < (2N)^(1/5), and that p(k-1)pk < 3N^(1/2). In a similar vein Luca and Pomerance using a similar technique showed that p1p2p3... pk < 2N^(17/26). In this paper , we prove that p(k-2) < (2N)^(1/6), and that p(k-2)p(k-1)pk < (2N)^(3/5). Note that 17/26 > 3/5, so this is an actual improvement over Luca and Pomerance's bound, although at the cost of only applying to the product three largest distinct prime factors rather than the product of all of them.

There's been some really neat recent work on error correcting codes. There's a Quanta article going around but it doesn't do a great job explaining some of the background ideas.

Since Shannon and Hamming's work about 60 years ago, there's been a lot of interest in error correcting codes. These are ways of embedding data which allows one to detect or correct errors in transmission. Natural languages do this mostly automatically. For example, if I inclde a typo in this sentence, you can probably notice it and figure out what I wanted. But they aren't perfect at this. If I write "I have a pet bat" it might mean "I have a pet cat" with a typo, but you can't be certain. Error correcting codes let you handle at least some number of errors of a certain type.

But how do you do this? We'll one naive thing is to just repeat each bit in your message a whole bunch of times. Let's say we repeat everything three times. So if you get "III hhhaaavvvee aaa pppeeettt ccbaaattt" you can be pretty sure that I meant was "cat" not "bat". To do this in a mathematically precise fashion, we generally talk about this in terms of strings of 0s and 1s, rather than letters, but the same idea applies.

Notice that in order for this code to work we had to send things which are three times as long as our original message, and we can correct any single typo. That's not great. We can do a bit better. For example, Hamming codes only increase the message size a teeny bit and still allow us to do a good job correcting a single error like this.

There's some technical subtlety here between "detecting" and "correcting" an error. For example, if you get the sentence "I have a pet dat" you can tell there's a least one error, but correcting it is tough. Should that have read "dog" (two errors), "cat" (one error) or "bat" (one error)- all you can tell is that "dat" is not a type of animal so something has gone wrong here. You can guess what is most likely, but you can't be certain.

One issue is that we have an existence proof of some very good codes. These codes are good in the sense that they don't require messages to be much longer than the message you intend to send, and also they let you detect even large numbers of errors. The argument essentially works by roughly speaking picking an assignment rule at random, and showing that with non-zero probability the code will do a good job. But actually constructing such codes is tough. This work roughly speaking manages to create one of those close to ideal sets of codes that we knew had to exist but had trouble constructing. They did so by using some ideas from graph theory that had previously been used to construct codes but had not succeeded at making really good codes. But these are very good, due to some new insights and careful constructions. (There are some details here I'm skipping; the above statement isn't strictly speaking completely accurate since it turns out these codes are a bit long.) One really nice thing about the codes they constructed is that errors can be seen by only checking a small part of the message.

It is worth noting that the work here is using a variant of "expander graphs" and one of the authors of this is Irit Dinur. She previously used expander graphs to prove a version of the PCP theorem which essentially says that you can turn any mathematical proof into a proof which can be verified with a high probability by only checking a small section of the proof chosen at random. So there's some definite commonality of theme here. Expander graphs roughly speaking are graphs where if you walk randomly on them, you can end up very far away with a high chance, even though any given node in the graph has only a few edges. They've been involved in a lot of neat stuff the last few years including some somewhat connected randomness amplification work.

Highly composite numbers are numbers whose total number of divisors is greater than any smaller number. For example, 12 has 6 divisors, 1, 2, 3, 4, 6 and 12, and no number smaller than 12 has 6 or more divisors. There's a really good video on them here .

These were first studied by Ramanujan a little over a hundred years ago. He proved a bunch of statements about them, some of the easier ones of which are included in the video above with proofs. Here are two of those: First, If N is a highly composite number and we list its distinct prime factors in order, then we cannot skip any prime factor. For example, 66 cannot be a highly composite number because 66=2*3*11 and so we skipped 5 and 7. Second, if N is a highly composite number, we cannot have an exponent of a prime in the factorization be higher than the exponent for a smaller prime. For example, 150 cannot be highly composite because 150 = 2^1 * 3^1 * 5^2 and 2>1.

Now, I want to mention something not in the above video that was pretty surprising to me the first time I learned about it: If we list the number of distinct prime divisors each highly composite number has, we might expect that number to never go down. (By number of distinct prime divisors we mean that we count primes which repeat in the factorization only once. So for example, 150 above would have three distinct prime factors.) And for the first few this is true, as you can check. But it does break down! 27720 is a highly composite numbers. We have 27720 = 2^3 * 3^2 * 5 * 7 * 11 with 5 distinct prime divisors, and a total of 96 total divisors. But the next highly composite number is 45360 = 2^4 * 3^4 * 5 * 7 is the next highly composite number. 45360 has only 4 distinct prime divisors; notice we lost the 11. There are even larger examples where we lose two prime factors when going up. I don't know of any example where we lose 3 or more, and as far as I'm aware whether there are any is an open question.

During the ongoing COVID crisis, my family has been doing Zoom meetings on Saturday nights with Havdalah, the Jewish ritual ending the Shabbat. For most of those weeks, I, and occasionally other family members have shared thoughts about that week's Torah portion in an email with the Zoom link. This week, I sent out that email early, and am including a slightly modified version of what I wrote for public consumption here.

This week's Torah portion is Ki Teitzei, כִּי־תֵצֵא . The portion includes a broad variety of civil laws and related commandments. Two areas of note are the laws of war and laws of how to treat laborers.

The section on the laws of war, unlike the section in the previous portion concerning not doing environmental damage during war, is by modern standards at best antiquated and, to be blunt, repugnant. However, in the context when it was written, it seems there is an attempt to at least add some small amount of morality around horrific practices.

The laws concerning laborers are easier for us moderns to sympathize and agree with. This week includes a commandment for the prompt payment of wages for labor. If one hires a person for a day, the wages must be paid that day, before sundown. While the later Rabbis did allow for people to pay employees weekly or monthly, for example, they strongly endorsed the central message about paying wages on time.

Both of these notions together are particularly and sadly relevant as I write this. The United States employed hundreds, if not thousands of Afghans. And Afghanistan has fallen. We are doing little to help these people who are facing likely retribution at the hands of the Taliban. From both a perspective of ethical warfare, and a perspective of prompt compensation for those who worked for you, there is an obligation to help these people, to help get them out of Afghanistan and resettle them. One does not need the guidance of this week's Torah portion to see both the moral and pragmatic benefits to assisting these people who helped the US so much.

There is also a personally important aspect here. My grandfather Jakob Dronski helped the Allies during World War II and its aftermath. Aaron has a letter he keeps on his desk, written by a member of the US military, testifying to Jakob Dronski's assistance, which he used to gain entry to the US, and eventual citizenship. At least at one point, the US seemed to know how to meet its moral obligations.

Unfortunately, the US government is doing little now, and what it is doing is slow. While the US federal government seems to be right now sadly lacking in support there are steps we as individuals can take. I have made a donation to Keeping Our Promise https://www.keepingourpromise.org/ which works to help resettle allies of the US from war-torn areas. They are right now focused on assisting people from Afghanistan. Kabul has fallen, and it is past sundown of the day of that fall. If anything this makes our obligation all the more pressing. I encourage everyone to donate to Keeping Our Promise or similar groups.

Last year, my students in the seminar studied graph theory. In the second semester we did a research project on total difference labeling of a graph. They had some really good ideas and a preprint for it is now available here .

Recall that by a graph, we mean a collection of points, called vertices or nodes, where pairs of nodes can be connected by edges. This is useful for representing data. For example, one can imagine four people, Alice, Bob, Carol and Diego. Maybe Alice is friends with Carol, and Bob, and Bob is friends with Diego, and no one else is a friend with anyone else. Then we can represent this data with a graph of four nodes, labeled Alice, Bob, Carol and Diego, with edges between the pairs (Alice, Bob), (Alice, Carol) and (Bob, Diego). One can imagine other contexts where this would be important. For example, one very relevant one right now is to think of graphs which model possibility for disease transmission.

One classic problem in graph theory is the problem of graph coloring. Given each node in a graph, we want to label it with a given color, and we want that no two adjacent nodes (that is nodes connected by an edge) have the same color. We are interested in how few colors we can do this with. For example, you can check that you can color the above graph with Alice, Bob, Carol and Diego with just two colors. A practical variant of this problem shows up in scheduling. Suppose for example, that you have a whole bunch of classes with students in them, and you want to schedule each class so no student has to be in two classes as the same time. Then you can make a graph where each node is a class, and two classes share an edge if at least one student is in both classes. Then, the minimum number of colors needed is the minimum number of distinct time slots you need for all the classes.

A total labeling of a graph is a little different. In a total labeling of a graph, we label each node with a color, but we also label each edge. And we insist that no two nodes which share an edge may be the same color, but we also insist that no two edges which share a node may be the same color. Finally, we require that no edge is the same color of either of its two nodes. (Exercise: What is the minimum number of colors needed to put a total labeling on the graph of Alice, Bob, Carol and Diego above?) Now, so far we've done this with colors. But there's no need to do this with actual colors. We can label our nodes, or our edges instead with numbers, where each number represent a specific color. We'll write say 1 for blue, 2 for red, and so on. Now, one thing we can do with this is we can now restrict our total labelings in terms of properties the numbers must have. We define a total difference labeling then as a total labeling satisfying that the number on any edge must be the absolute value of the difference of the numbers on its two nodes. For example, if we had nodes labeled 3 and 5, and they had an edge connecting them, that edge must then be labeled 2. We can then ask the same sort of question as before: Given a graph G, what is the minimum k such that we can put a total difference labeling on G using just 1, 2, 3... k. We'll call this the total difference labeling number of the graph. (Exercise: What is this number for our Alice, Bob, Carol, Diego graph?)

The idea of a total difference labeling was earlier introduced by a paper by Ranjan Rohatgi and Yufei Zhang where they calculated this for a whole host of graph families. The work with my students extends their work in two major ways. First, Rohatgi and Zhang gave an upper estimate on the total difference labeling number of a complete graph on n nodes. (A complete graph is a graph where every node is connected to every other node.) Their estimate grows exponentially in n, and we were able to reduce this to only polynomial growth. Second, we extended their results to some nice infinite graphs, such as calculating the total difference labeling number of an infinite square lattice. The paper does not assume much background, and should be probably somewhat readable to non-experts.

The game "Hex" is a simple game which apparently has been invented at least twice  (Piet Hein and John Nash).  The game consists of an n by n grid of hexagons, with two opposite sides marked as blue and the other pair of opposite sides marked as red. The red pair belongs to the red player, and the blue pair belongs to the blue player. Players alternate marking hexs either red or blue. The goal is to form a path of hexs of your color which connect one's two sides.

It is well known that there is a winning strategy for the first player on an n by n board. Proof sketch:

First show that in any completely filled board, there must be a winning path for at least one player. (Note that this step really is not obvious; this isn't true if one did this on a square grid rather than a hex grid.)

Second, note that if the second player had a winning strategy, the first player could simply make some extra random move, and then play using the second player's strategy, with no penalty. This sort of argument is called a "strategy-stealing argument" and it shows up in many arguments about simple board games.

So we know that for any n by n board, the first player has a winning strategy. But here's a fun fact: No one knows a general way to find this strategy. We can do so for some small board sizes, but there's no general rule.

If one plays around on a few tiny boards with n>2 one will notice that edge moves are really bad first moves. In fact, a little work will show you that on a 4 by 4 grid, an initial move on an edge is always a loser for the first player. So I have two questions:

First, can we show that for any n>2,  any edge move is a losing first move?

Second, suppose that the first player's move is randomly chosen; can we say anything about the proportion of those moves which lead to a winning board? My guess is that this proportion either goes to 0 or goes to 1 as n grows, but my intuition on which seems to be oscillating violently.

(Note this is an old Facebook entry I'm copying over to here.)

We've read a lot of stories about kids going to magic schools but this genre seems to have very little about the day to day aspects of the teachers and professors. In particular, they seem to downplay how much committee work and discussion there would be. Here are two possible snippets from magic school faculty discussions:

1. Professor Featherstone: "I'm really concerned about a lack of interdisciplinary material in our curriculum. A student can go through their entire time here and avoid taking a single Divination course. Worse, many of our students in the pre-Elementalist track seem to be doing just that, with the track requiring so many classes that important skills like Divination fall by the way side. Are we a genuine liberal arts mage school or just a professional mage school?"

Professor Pigwhistle: "Featherstone, you bring this up every single meeting. Can we please concentrate on adopting the new Potions standards?"

(Note: This repeats for another 30 years and continues after Featherstone becomes a ghost. He continues to harp on this even when the school folds the pre-Elementalist track in to a different track. Eventually, they get an exorcist because apparently being a tenured ghost does not protect you from being exorcised.)

2. Professor Anat: "Look, we can't cut funding to our department. Necromancy requires a lot of raw resources, especially onyx. If we can't have our students take at least three practical animations in the regular course, we won't meet national accreditation standards for the class."

Professor Wormwood "So just have the students pay for the resources themselves."

Then proceeds a rambling ten minute argument about socioeconomic issues, with some claims that Wormwood is speaking from a position of privilege which eventually leads to the following exchange:

Professor Witchhazel: "There's a real problem that we're recruiting primarily from wealthy students from major mage families. Aside from the ethical considerations, the vast majority of kids who turn out to be Chosen, Destined ones who are prophesied to be the only hope against vast and mighty threats come from poor backgrounds."

Professor Smith: "Well, considering that such students generally seem to end their terrible battles with great evils in climatic ways destroying large parts of their schools right when they should be graduating, that may be an argument in favor of our current policies. We barely have enough gold to pay for basic upkeep as is."

Other possible issues that may be discussed include:

When the school gets stuck in a timeloop for a few months, how does HR decide paychecks? Does time in a timeloop count against your tenure clock?

For schools that have both humans and longer-lived species (like elves) how does one deal with the inevitable problems when eventually all faculty are elves? Is deliberately having human hires an acceptable move to promote diversity or is it speciesist?

How does one politely tell the tenured lich that his department has a limited budget and he has to use the same base scrolls as everyone else, and not to throw death curses at the departmental secretary when she refuses to order the fancy parchment? Also does anyone know how to get to the lich to stop complaining in every single meeting that the students in his advanced scrying class don't know basic wand work? Also, please by all the gods in whatever setting this is in, can someone get the lich to understand that he can't put in writing that he prefers students from the Valley of Thorns over those from Riversong?

Define a function f(x) which takes a number and adds to that number the sum of its digits in base 10. For example, f(112) = 116 because 116 = 112+1+1+2. We can keep applying this function to itself and get a sequence. For example, if we started with 7, we would get 7, 14, 19, 29, 40, 44, 52, 59... and so on.

We'll call such a sequence a river. One thing to note is that some river can start off with one number and eventually turn into another river. For example, 3,6, 12, 15, 21, 24, 48, 60, 66, 78, 93, 105, 111, 114, and 30, 33, 39, 51, 57, 69, 84, 96, 111, 114... If two rivers meet, they are then the same at any subsequence point.

Rivers are related to self-numbers which are numbers which are not in the range of f(x). They can be thought of as numbers which only ever appear at the start of a river, never in the middle.

I was introduced to the idea of rivers by this Mathoverflow question . They define three "main rivers" , 1, 2, 4, 8, 16, 23..., 3,6, 12, 15, 21..., and 9, 18, 27,... and prove that these three can never intersect (this is a good exercise!) . They ask if every river eventually meets one of these three main rivers.

However, it isn't obvious to me that there is even a finite set of rivers such that any river eventually meets one of those. So, let's do what mathematicians always do when we can't answer a question, try to ask a related question. We can define rivers the same way for any base b, where our function f_b(x) adds x to the sum of the base b digits. It isn't obvious to me how to even prove that there is any base b where there's a finite collection of rivers which are always eventually met. A small amount of playing around suggests that in base 3, every river eventually meets the river started by 1, but I don't see any obvious way to prove it.

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