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.
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 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.
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
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.
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 $0$
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.
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.
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.
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,....
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.
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.
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.
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.
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.
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 $A \cap I$ ...
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 $I$.
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
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
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
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
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
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.
@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.
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.
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.
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.
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.
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 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.
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.
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.
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 $A \cap I$ ...
