2:00 AM
i don't know stochastic calculus. for several semesters, larry evans taught a class on it in the room right before one of my classes. that guy had the best boardwork of all time. on the basis of that, if he wrote a book, please read it.
hendrik lenstra also had amazing boardwork. and he helped me clean up a coffee spill that neither of us had created. probably the best number theorist on that basis alone.

1 hour later…
3:15 AM
All other number theorists are filthy coffee spillers.

hahahaha
I don't know what this was about, but that's a wonderful line
thank you for the reference, @leslietownes. I plan to get through Varadhan's lecture series, first.
oh, I stopped reading half way through leslie's comment.. I get the coffee spill comment now..

@Ted 'evening

Hi, @robjohn

hi Ted
how are things

3:34 AM
@TedShifrin The weekend is over too soon.

so here's an elementary probability question i'm too lazy to work out

@Semiclassical laziness is to be tolerated, now?

yes
suppose i have an urn full of balls, labelled 1 to N. what probability distribution models the number of times i'd have to sample that urn, with replacement, to get all N outcomes?

@Semiclassical $N \log(N)+\gamma$
approximately

gross
well. i guess that's pretty nice all things considered
and doesn't scale as badly as some things

3:38 AM
Exactly, $NH_N$

oh, nice

where $H_N$ are harmonic numbers

is that the expected number of samples, to be clear?

@Semiclassical yes

makes sense, as i think of it

3:41 AM
For a die, you need $14.7$ rolls, on the average, to get all $6$ faces.

the context i'm running into this was a video game minigame, hence why i wasn't willing to put effort into it
130-sided die :)

@Semiclassical I had a screen with 1024 pixels and a program that set a random pixel at a time and I wanted to know on average how long it would take to fill the screen.

oh nice
apparently the odds in the case of interest here are not quite just "sampling with replacement"

@Semiclassical how so?

well, the source I'm looking at for the video game put the odds of getting a new result as follows:

3:46 AM
Oh, you want the distribution.

first, you can choose to bet more than one "coin" at a time; this will raise the odds (slightly). also, not all of the possible outcomes are all unlocked when you first get access to the minigame.
apparently the formula for the probability of getting a new outcome is (99+S-100*C/U)%, rounded down to the nearest whole percent. here S is how much you bet, C is how many you've found by now, and U is how many outcomes have been unlocked
which, hmm
if you just bet S=1, then that reduces to 100*(U-C)/U %. which is just the usual sampling probability, modulo the rounding
main thing is that betting more than S=1 doesn't increase your odds enough for it to be worth it
except maybe near the very end

4:16 AM
if this has anything to do with capturing pokemon, you have done an admirable job of concealing that

$\mathbb{Z}_{1}$ is integers right.

hrm, with the subscript 1? difficult to know.
some people use the subscript to indicate a quotient ring of integers mod subscript. more commonly among professionals it is used for p-adic integers, which wouldn't make sense with p being 1. or maybe it's a subtle number theory thing i don't understand
with no subscript it's definitely the integers.
our automatically generated avatars are close to mirror images of one another. i think that makes us enemies.

HAHAHHA!
we allies my dear friend.

works for me.

4:33 AM
If you mean $\Bbb Z$ modulo $1$ then you get the zero ring, not all of $\Bbb Z$
you're quotienting by the relation $a \equiv b \bmod 1$ iff $1 \mid a - b$, which is the case for any pair of integers $a, b$, so everything is congruent to everything, and in particular to $0$
(assuming you mean $\Bbb Z/1$)

yeah. only a pervert would mod out by 1, but that's definitely what you get if you do it. is there more context?

rofl

5:25 AM
Hi All.....

a nihilist surely rather than a pervert...

heh

5:41 AM
you make a good point. why don't we just mod out by everything is a nihlist perspective. we should just quotient by everything and then nothing will happen. which is already what is happening.

$\mathbb{Z}_{69}$?

even in Z, what's the point. 1, 2, 6, 10? give me a break.

69-adic integers are 3-adic x 23-adic integers

this is true

my response was not meant to be serious...

5:44 AM
nor mine

you two can back out of your statements, but i'm all in on this.

hullo

grüß Gott

yeah right, the double positive

lmao

5:46 AM
wonder how that expression evolved?

lmao?

yeah, head i can see, but ass?

i think it is just the tendency of the English language to wind up in the gutter. body functions, body parts. the head isn't as funny as the ass. so the idiom wound up there.

that's nuts

i was not going to bring nuts into this. that is your doing.

5:48 AM
No clue, it has no meaning to me anymore, I'm not really laughing my ass off, I just exhaled through my nose

sorry, should be careful of allergies.

my daughter, who is 2.5 years old, has noticed that some words get more attention than others. the other morning she snapped at me, "don't say f*ck!" i hadn't said it. nobody had said it. i'm imagining it was something she was told at day care.

that's funny

it's really weird to see a human be programmed with societal instructions from day 1. why shouldn't she have a potty mouth. i don't know.

I'm the son of a sailor, so I was never programmed in this way

5:52 AM
at home, the rule is that it's ok to use profanity as long as it is contextually appropriate. you can't drop f-bombs at random. but if you physically drop something on your foot and it hurts and you're surprised by what has happened, it's fine to let it fly.

when my daughter was around that age i would take a trip to a metro (bart) station and back with her, just because she liked it. one day i missed the return train (not a biggie, just a wait in the cold) and shouted fck. instantly my daughter started shouting fck, fck, fck,....

hahaha

i used to live right next to the rockridge bart station. sometimes i load youtube videos of the sounds the trains made.

now she has quite the potty mouth when she needs to (part of her irish heritage i suppose), but is generally careful.

What's a bart station? Just a train station?

5:54 AM
metro

bart is the system that serves the SF bay area. it isn't really like the NY or other systems, in that it mostly intersects the cities it serves in a straight line. there isn't much of a network.

if parent.say("fck"){
child.say("fck");
}

Like a subway? lol
Oh I see

i never heard my parents curse.
other than 'jesus, mary & joseph'.
but when we heard that we ran as fast as we could

lol

5:56 AM
my mother was very foul mouthed. had an enormous vocabulary of invective. it was poetic. also irish american.

My parents swore all the time but never ever used the "c-word"

we used to call my mother 'poison tongue'.

my mother used it the way people, i don't know, tell the time. it's ten o clock is something natural for you to say, the c word was something natural for her to say.

she could cut you to ribbons without a single word out of place.
but the worst was silence.
thermonuclear silence

she was a nurse and spent a lot of time joking about people dying. it was weird.
yes, silence was worse than anything spoken.

5:59 AM
funny. my mum was a dr.

i think if your job requires you to see people at their worst, it changes you in some way.

funny to reminisce. time for wine.
i agree. working in eda does that.
jk

@leslietownes it's also natural for me now, but never in front of my mum lol

less than a decade ago i let loose an obscenity at the dinner table when i was back visiting family. my youngest brother turned around and asked
are we openly cursing in front of mum now?

when my soon-to-be wife met my mother, it was something else. she (wife) suddenly understood why i was as weird as i was. i think the first night they met she (mom) was joking about corpses. it was really dark stuff. my wife understands me now.

6:02 AM
he was 40 at the time.

lol funny that you carry things like that into adulthood

irish mothers can be fearsome.

my mom lived in a mobile home park for seniors, people were dying there all the time. there were also weird transient people in the park. the police killed somebody in her driveway once. another time, bullets hit her windows. this was pure comedy for my mother.

possibly a result of a misogynistic society

damn

6:04 AM
guns were never an issue in ireland growing up (unless you were part of a sub group)
knives, serious beatings were normal, but never guns

Same in England
outside of London

sadly everybody was and is armed in california

i suspect strong cultural similarities and variances
i mean uk & ireland
thankfully the darker parts of the relationship are in the past

Hopefully it remains that way

fingers crossed. i grew up in a strong ira town.

6:07 AM
Did you witness much violence?

they more or less kept to themselves. and everyine else was fine keeping it that way

as in, Troubles related violence lol
Okay fair

not becasue they were in the town
you did not mess with them.
ira & unemployment are strongly related

I grew up in a city with a very, very large naval base
so it was mostly full of drunken sailors, both English and foreign

sounds like fun.

6:10 AM
:-). portsmouth?

most of my neighbors growing up worked in a nearby shipyard. when the government closed it, things were very awful for a while. then they mostly moved away.

I attended a party once where there were two 30something German sailors trying to hit on 16 year old girls lol
@copper.hat The big mouth, Plymouth

:-)
i sailed from cardiff to cork a few years ago in a 26' sailboat.
say hundreds of dolphins and a few really large whales!

that's awesome.

Nice :D

6:12 AM
it was fun. i was surprised my wife let me, we have two small kids at the time and i had never sailed with the 2 lads i was with.

sci-hub.se

wow :D

now we're talking. copyright infringement.

thank yoyu so much

learn something new every day...

6:19 AM
sci hub is illegal

so it driving over the speed limit.

so is $\epsilon<0$

its only illegal if you get caught.

3 hours later…
9:28 AM
0

Suppose that $A = \Bbb{P} \times \Bbb{P} \times \Bbb{P} \subset \Bbb{Z}^3 = R$ the $\Bbb{Z}$-module. Let $f : R \to \Bbb{Z}$ be the $\Bbb{Z}$-module homomorphism that sends $(x,y,z)$ to $x - y - z$. Then $\ker f = I$ is a $\Bbb{Z}$-submodule of $R$. The twin prime conjecture is that $A \cap I$ ...

Gonna need your guys's help to not get this downvoted.
I've finally done it
I told you I could do it
@JackOhara are you awake?
@jcora

9:45 AM
Okay, they found an error and I deleted. But I'm going to try and repair the error

If $I,J\subset R$ are ideals where $R$ is a commutative ring with unity, then $V(I) = V(J)$ implies $I = J$? where $V(I)$ is a set of prime ideals that contains $I$.

@love_sodam Nullstellensatz: I(V(J)) = rad(J)
ergo V(I) = V(J) <=> rad(I) = rad(J)

Ah, that's why I couldn't prove

2 hours later…
11:30 AM
What does $\wedge$ mean in $x+1\le0\wedge x-1<0$?

Thanks :-)

12:29 PM
How can I prove that if $R$ is a local ring whose maximal ideal is principal then that maximal ideal is a unique nonzero prime ideal of $R$?

12:40 PM
Never mind. I think I know how to do it

12:52 PM
No, I don't know

I think you need $\cap \mathfrak m^n = \{0\}$

1:14 PM
For a counterexample consider the ring of germs at 0 of $C^{\infty}$ functions on $\Bbb R$
The principal maximal ideal is the ideal of functions that vanish at 0, a non-zero prime ideal is the functions with derivatives of every order canceling at 0

1:59 PM
@love_sodam the statement you want is true with additional noetherian and domain hypotheses by Krulls Hauptidealsatz.
(if it isn't an integral domain then 0 isn't a prime)

2:14 PM
Hey @Leaky do you want to think about some ANT?

sure

So I'm working in $R=\Bbb Z[\sqrt{-3}]$. I have the ideal $a=(2,1+\sqrt{-3})$ and I showed that $a^2=2a$ (and that $a\neq(2)$). Supposedely I should now deduce that in $R$ I don't have unique factorization of ideals into prime ideals. But $(2)$ is not prime, so I'm confused
In fact $(2)$ doesn't factor as a product of prime ideals at all in $R$, so I can't say $a^2=a(\text{stuff})$ where stuff is a product of prime ideals to get a contradiction

@AlessandroCodenotti the correct ring is $\Bbb Z[(1+\sqrt{-3})/2]$
because $-3 \equiv 1 \pmod 4$

No I'm not working in the ring of integers of the extension

A characterisation of dedekind ring is that every fractional ideal is prime, so you'd find a = 2

2:20 PM
I'm told by the exercise to work in $R$

Which as you pointed out is absurd

But $R$ is not a Dedekind ring @Astyx

Precisely
So it doesn't have unique ideal factorization

Aren't we going in circles? I'm confused now

If you know that R is not a dedekind ring, you already know it doesn't have unique prime ideal factorization
But I'm guessing that's what you want to prove

2:23 PM
@AlessandroCodenotti so you're done right? if R has unique factorization then a^2=2a would imply a=2, contradiction
what's your question

yeah but my definition of Dedekind domain is Noetherian integrally closed domain in which prime ideals are maximal
and the only prime ideal I have here is maximal
@LeakyNun But 2 is not prime, so a^2=2a does not contradict uniqueness of factorization into primes

yeah but you can factorize 2 into primes if it had unique factorization
so 2 = q1 ... qr
then a^2 = a q1 ... qr would imply a = q1 ... qr, contradiction

Ok that makes sense
But so the issue is that in this ring some ideals don't factor as a product of primes, rather than having some ideals that factor as a product of prime in two different ways
I mean what's failing is existence of the factorization, not uniqueness like the exercise lead me to believe

oh...
idk then, maybe they assume uniqueness implicitly implies existence

@LeakyNun Hm but I'm confused again, why can I simplify an a on both sides if I have no uniqueness
Oh ok I'm assuming uniqueness to get a contradiction

2:32 PM
right

I'm convinced now, thanks!

great

2:46 PM
hmm
would this count as a boundary value problem: math.stackexchange.com/q/4053768/137524
i think the fact that the solver wants boundary conditions applied at two different times would make it such, even though they're not being applied to the same component

3:25 PM
i would consider it as such
it doesn't fit the conditions you would often find generally stated in a book, but it definitely is one
at least it is to me. often books don't treat the topic because they focus on when you can prove existence of solutions, and these things may not have them.
and that's just a depressing thing to put in a textbook or course notes

yeah
that said, the fact that it's of the form $v'(t)=Av(t)$ for diagnolizable $A$ keeps this instance from being too bad

I wrote: I admire Dr. Seuss and his work.
Base 12 is superior to base ten.
It is two messages, not one.
It is not a couplet. It does not rhyme, as you pointed out. It also does not have the same number of syllables per line.
Anyway, I was too distracted at the time to reply, but I thought I'd reply now.

"Base 10 is a mere human quirk." would do

one base two base
red base blue base

I thought it was mildly funny to presuppose that it was a couplet

3:36 PM
The way I typed it, it did look a bit like a couplet. Besides the number of syllables being wrong, the stress pattern does not sound good at all.

hmm, "Base 10 is just a human quirk" sounds better come to think of it

Base 10 is just a human quirk. If you should like it, you're a jerk.
more hostile than seuss but at least it scans.

"Base 10 is a mere human quirk." would do very well indeed.

whenever this comes up, though, i'm reminded of the ancient question: "who the hell came up with 360 degrees in a circle"

3:37 PM
One fish two fish red fish blue fish.

1 base, 10 base, red base blue base

i think one theory is that it's to do with 365 days in a year, so 360 as an ancient approximation to that

Like it!

it was the babylonians, i think. they were keen on base 60 and its variants. i think the attraction was the large number of divisors of 60.

though i do have to appreciate the fact that triangles, square, pentagons, and hexagons all have integer-degree angles.

3:39 PM
babylonians had 60 fingers

The number of degrees in a circle should be a power of two.
One is a power of two.

they weren't fingers, they were more like talons, because they were also lizard people. but they used lizard magic to appear like humans

they still do

A great unit for angle is the turn.

i do think that 60 is the smallest number with as many divisors as it has. there may also have been numerology in wanting specific divisors.

3:42 PM
it is funny how it's both utterly ubiquitous and utterly lost to history
though language is like that sometimes

60 has only three prime factors.

for instance: no one really knows where the word strawberry comes from

i find it a little odd how worked up some people get about arbitrary choices. my daughter is apparently going to learn cursive in her day care / school, and my mother in law was so happy about that. i can't imagine why. who gives a toss. i can pay in dollars or pesos, i can measure meters or feet. all of it feels the same to me.
although it's also funny, all of this weird heritage we've accumulated for no reason. like maybe some guy who thought he was a wizard needed a base divisible by both 4 and 15 and suddenly there's 360 degrees in a circle.

ideally, i feel like cursive would have the advantage of being faster to write out than print...but that requires getting good at cursive

@leslietownes I feel like I didn't thank you enough, or protested too much, when you said I was a crank of the second order, but then you added that you were also. So, thanks (a little) for that (grudging?) concession.

3:45 PM
(and thus i don't cursive)

i am definitely a crank of the second order. it is a happy brother and sisterhood.
a lot of cursive lettering is designed for the right hand. the motions are easier to perform when pulling the pencil from left to right than pushing it. so it is slower for me unless i shape the letters slightly "incorrectly."
there is an enormous amount of counterclockwise movement in cursive. it's slower to do that left handed.

cursive may have been my entry point to being a crank of the second order.

I didn't know that about cursive and handedness.
Interesting.

clearly, it's your problem for being a left-handed deviant /s

3:48 PM
if you look at all of the letter forms, most of them prioritize counterclockwise movement. my cursive o goes clockwise, which is not how they teach you to do it, because the other way is harder. it's also a letter in my name so i have to write it a lot.

"Curse you, cursive" is a slogan I've heard of.
But the reason given for hating cursive is that supposedly kids are better off just sticking with printing.

using mathrm d and putting dx before the integrand. that's two things i don't like. i didn't even notice the minus c. you, sir, are a raging psychopath.

meh
i see $\int dx f(x)$ too much in physics to be upset about it

i'm actually going to start writing minus C. that is the kind of crankery i can get behind.
and a lowercase c at that. that is a work of genius.

Try writing Russian in cursive

3:50 PM
madness

I wonder whether Arabic is also easier on right handers?
Arabic is written from right to left.

Hindi in cursive is impossible

that's an interesting question. i don't know enough about languages written right to left. it would be funny if it was still 'customized' in some way for the right hand.
my guess would be that righties always win
don't even get me started on ballpoint pens that unscrew and fall apart in your hand if you use them left handed.

there are some right-to-left languages of course

Is that true about the ballpoint pens?

3:53 PM
it seems to be true only of a lower class of pens. the kind you might get for free in a bank, or something. where there's a clicky button at top, and the body of the pen is two pieces screwed together. if the object is poorly machined, right hand writing will be fine with it, but some forms of left handed writing will unscrew the pen.
being left handed is a crash course in the significance of all forms of orientation.
which doesn't explain why i am so bad at issues of orientation, factors of -1, 2, etc. i hated orientability in differential geometry.

I think there should be a lot more photographs oriented due north.
Likewise videos where the camera faces due north.

i asked a guy in insurance if lefties were more likely to get into accidents than righties, which seems possible given the way cars are configured, and he didn't know. although the biggest thing (shifting on a manual on a US car with driver on the left) is likely not much of a thing anymore and probably doesn't contribute to accidents, and basically anybody learns where the brake pedals are.

Don't forget that the driving seat is on the other side in many countries.

i like the riddle of figuring out where the camera was. ideally also the riddle of figuring out the settings of the camera from the photo. it spoils the game to provide that information.

amusingly, i deal with camera stuff like that a lot lately

3:59 PM
it's like hearing a guitar solo and guessing the make of guitar and which pickups were being used from the sound, and being right about it. that's a lot of fun.

mostly because i coordinates intro physics labs
and a lot of them amount to "track the motion of this point on a screen, and analyze the resulting coordinates as functions of time"