Mathematics - 2016-04-14 (page 2 of 2)

This challenge is to write fast code that can perform a computationally difficult infinite sum. Input An n by n matrix P with integer entries that are smaller than 100 in absolute value. When testing I am happy to provide input to your code in any sensible format your code wants. The default wi...

Quill

@JasonBourne I kinda styled the long parts out of it, it's still around about the same length. interesting that you remember it ;p

Anush

the mathematica code (siegeltheta) is very slow sadly

user174558

@Quill I remember everyone in this room. =)

Anush

slow and somewhat buggy

Semiclassical

that's just as well, my mathematica is tied up with iterative pde solving right now

Anush

1:28 PM

3

Q: SiegelTheta throws errors from calling Range with complex arguments

This may or may not be related to the bug reported in this question. I was trying to verify the results of this challenge over on Code Golf with the following code: p = {{5., 2., 0., 0.}, {2., 5., 2., -2.}, {0., 2., 5., 0.}, {0., -2., 0., 5.}} SiegelTheta[I*p/Pi, ConstantArray[0, Length@p]] W...

bugs error special-functions

that's the buggyness

Mats Granvik

@Anush What is that about?

Anush

but the codegolf code is frankly amazing :)

Semiclassical

i really need to find a way to run it remotely on the school pc's rather than on my laptop, though.

Anush

@MatsGranvik we are discussing math.stackexchange.com/questions/1741157/…

any ideas gratefully received

user174558

@Anush I like amazingly frank code.

Anush

1:30 PM

:)

Semiclassical

btw, is there any reason for your matrix to have $k$ be an integer?

Anush

not really

Semiclassical

it seems unnecessary

mmkay

Anush

I didn't want to worry about teeny tiny eigenvalues

Semiclassical

ahh

well, you've got a matrix of the form $I+(k-1)uu^T$ with $u_k=1$. so that's pretty nice

one should be able to compute the inverse and determinant of that matrix explicitly, btw

if for no other reason than the eigenvectors being so simple: $u$ itself is an eigenvector, as are the column vectors with entries $e_{k+1}-e_{k}$ for $k=1,2,\cdots,n-1$

Mike Miller

1:35 PM

@Krijn I think thwy're just not the same thing

Like they're defined with different sites or whatever.

Krijn

@MikeMiller Yeah but Wikipedia and Hartshorne are both pretty unclear about this

Something something uses étale to construct $\ell$-adic something something

Well, We'll see

Anush

@Semiclassical sounds good

Mike Miller

Neither of those are textbooks on the subject, are they? ;P

@Balarka: Yes, it is true that homotopic diffeomorphisms of closed surfaces are isotopic. But you can do it for the torus with your teeth.

I guess you can do it for everything with your teeth, but the torus is a nice test case for the visualization.

Mambo

2:02 PM

Hey ya

moonshine

hi people, I'm an ultranewbie

I wanted to ask you something

If I knew the answer to my question

should I answer it myself?

Or just let it go?

or even deleting it?

Mambo

answer it

moonshine

@Mambo I see

well thanks

J. M.

2:25 PM

@Anush, you have looked into the paper by Deconinck and others, I presume?

that's essentially the state-of-the-art. Maple also uses that method.

Anush

3:02 PM

@J.M. yes I believe the codegolf answer is much faster

@J.M. although I don't have access to maple. If you do, could you try it out?

skill patrol

@moonshine answering it will help others who have the same question :-)

robjohn

@Anush Do you have any numerical evidence for a large difference? We might be able to use Poisson Summation to convert to a sum which might more easily be approximated by the obvious integral.

J. M.

3:20 PM

@Anush Those authors were prioritizing stability and reliability, tho. No use returning crap quickly.

Mike Miller

3:44 PM

there was a gauge theory question on main

i don't get that a lot!

Huy

happy birthday Mike

Mike Miller

thanks

Jessy Cat

4:16 PM

0

Q: Prove that the $c_{n}$ in $\frac{1}{z-z-z^{2}}=\sum_{n=0}^{\infty}c_{n}z^{z}$ satisfy a Fibonacci-like recurrence relation

I need to prove that the coefficients $c_{n}$ of the expansion $\frac{1}{1-z-z^{2}}\sum_{n=0}^{\infty}c_{n}z^{n}$ satisfy the recurrence relation $c_{0}=c_{1}=1$, $c_{n}=c_{n-1}+c_{n-2}$, by multiplying $(1-z-z^{2})\sum_{n=0}^{\infty}c_{n}z^{n}$. I'll admit that the coefficients look rather Fibo...

complex-analysis complex-numbers recurrence-relations taylor-expansion fibonacci-numbers

Happy birthday, dude.

Mambo

@MikeMiller Happy birthday man

Mike Miller

It's not actually my birthday.

Huy

:(

user174558

4:37 PM

Smile.

user174558

You're on candid camera.

user147690

AG is very terminology dense

Mike Miller

What's not?

user147690

I mean particularly so

user174558

AG can also stand for arithmetic geometry.

user147690

4:46 PM

I'd say first two courses in functional analysis wouldn't be anywhere near as terminology dense as AG

user174558

@AlexClark Interestingly, Algebraic geometry = AG = Andreas Gathmann.

user174558

@AlexClark I would say that depending on the courses, a second course in functional analysis could be as dense if not denser than a first course in AG.

user147690

Acronym Guy ^

user147690

¯_(ツ)_/¯

skill patrol

Aussie Guy^

:P

user147690

4:48 PM

I mean it's only really particularly terminology dense (learning-wise) when you hit sheaves

user174558

I like Athletic Guy.

user147690

And affine schemes are fairly nice, and with sheaves already done, schemes shouldn't be too bad

user174558

Sheaf, scheme, germ, stalk, variety

user174558

Math is interesting right?

user147690

Variety, sheaf(stalk, germ), scheme :P

user147690

4:51 PM

Definitely very fun

user174558

Are you staying in a proper place now?

user147690

Definitely. Most proper since I was 7ish years old

user174558

Good. I worry about you all the time. =)

user147690

Well you need not now :P. The government still pays me, and I am a TA for calc1, so I have money :P

user174558

I think I should worry about myself more. =)

user174558

4:54 PM

Who knows what will happen to me in a year, or five, or ten?

user174558

Too late for me to win the Fields medal, I will try the Abel prize instead.

user174558

Why are you not sleeping?

user147690

Wow 600k pounds is the monetary aspect of the Abel prize, nice.

user147690

I am just doing an all nighter tonight, just so I can get some serious work done for next tues+wednes

user174558

I see. Then out of the chat room and into the books. Keep in touch. Your birthday next month right?

user147690

5:02 PM

@JasonBourne You can guess that from my email address :P. Indeed, I am only periodically looking here

user174558

@AlexClark Yes, that is what I guessed. I am a genius, remember?

user147690

I do not forgive, I do not forget :P

user147690

Alright, talk later, I should stay focused :P

user174558

Yes, goodnight.

skill patrol

cya later

Mike Miller

5:17 PM

Hi @AndrewThompson.

Andrew Thompson

Hellu @MikeMiller

How's life?

Mike Miller

It's ok.

Yourself?

Andrew Thompson

Quite alright. Just skimming through Hartshorne. Its more readable than I thought.

Mike Miller

I think people mostly dislike it because it puts too many important theorems in the exercises, not that it's hard to read.

Andrew Thompson

Yesh, the lecturer has written additional notes because of that.

Mike Miller

5:45 PM

fair enough!

is that what you're spending most of your time on, @AndrewThompson?

Andrew Thompson

Nope, the course (sheaves and schemes, considered half a course) started a week or two ago. My time is mostly divided between my undergraduate project, Lie theory and K-theory, although I admittedly spend too little time on all of them.

Mike Miller

gotcha

Andrew Thompson

How about you? Getting anywhere with your repvarieties?

Mary Star

Hello!!

Let $R$ be a U.F.D. and $0\neq d\in R$.
I want to show that there are finitely many different principal ideals that contain the ideal $(d)$.

A principal ideal is generated by a single element, say $i$, and so that it contains the ideal $(d)$, $i$ must divide $d$, right?
We have that $d=a_1^{k_1}\cdots a_r^{k_r}$ with $a_i$ irreducible.
Since $R$ is a U.F.D. the irreducible elements are prime. Does it follow from that that the divisors of $d$ are of the form $a_1^{j_1}\cdots a_r^{j_r}$ with $0\leq j_i\keq k_i$ ?

Mike Miller

No, my advisor told me to restrict to a simpler case for a while so I wouldn't get stuck in such details. :)

Andrew Thompson

5:49 PM

I see. Did it help?

Mike Miller

Quite a bit, yeah, but at some point I'll have to go back and muddle through some of those.

A conversation with a colleague made me realize some of my expectations about what were true were overly ambitious.

Overly hopeful? I dunno, you know what I mean.

Andrew Thompson

If it's "one hopes things would behave nicely but of course they don't", then yes.

Mike Miller

Yeah, I thought a theorem from a simpler case would generalize to mine, but nah. Kunneth-type theorem. If it still works (probably does) you'd need some gross statement like "If $I_*(P \# Q)$ exists, then..."

Andrew Thompson

Had a similar experience recently, thought I was being brilliant and was to use an old theorem of Serre, but it turned out to be senseless.

Mike Miller

Always fun huh

Semiclassical

5:58 PM

had a moment of that on the site a few days ago with a quantum question

it asked for a construction of something, i thought i had one, and then realized after it went up that while it was structurally alright it didn't actually do the thing it set out to do

or more directly: i managed to factor a certain matrix as $UDV^T$ but $D$ wasn't actually diagonal :/

Mambo

Singular Value Decomposition ?

@Semiclassical

Semiclassical

yeah

Mambo

I have question on functional analysis

TheGreatDuck

Hi

skill patrol

6:14 PM

quack :P

Mary Star

Hello @TobiasKildetoft !! Did you see my question: chat.stackexchange.com/transcript/message/29014846#29014846 ?

Tobias Kildetoft

@MaryStar No

Too tired for ring theory right now

Mary Star

Ok... no problem...

user147690

@AndrewThompson How many weeks are there for half a course?

Andrew Thompson

Lemme check.

Six.

(For this particular one, at least.)

TheGreatDuck

6:22 PM

Lol

Good on

Mambo

@TheGreatDuck Do you know fourier transform

TheGreatDuck

Maybe. Maybe not.

i don't know the term

but chances are I've inadvertently done it at some point

without knowing it

@Mambo did you are a question?

*have a question

LifeOfPai

Hello

Mambo

I actually have a question regarding fourier algebras

TheGreatDuck

Hi @LifeOfPai

@Mambo ask away.

LifeOfPai

6:36 PM

HOW I find the osculating circle if y=y(x)?

TheGreatDuck

Please don't yell

LifeOfPai

I not sure if it`s Radius of curvature or osculating circle Radius

TheGreatDuck

We don't even know what you're talking about. -_-

LifeOfPai

Osculating circle

In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. Its center lies on the inner normal line, and its curvature is the same as that of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for "kissing circle") by Leibniz. The center and radius of the osculating...

Mambo

@TheGreatDuck If you know what is left regular representation of a group, then corresponding fourier algebra is coefficients of left regular representation

TheGreatDuck

6:40 PM

Uuh

yeah. Turns out I don't know.

havent done abstract algebra at this time

Mambo

6:54 PM

@TheGreatDuck you there

Balarka Sen

@AlexClark Huh, OK.

Next chapter on Shafarevich is divisors and whatnots, but I am not going to learn this. I am having trouble with the basics.

Not enough examples and such. I'll work on the foundations for the next few days/weeks/months.

@AlexClark re "terminology dense": I found I didn't really understand much of the terminology until I had concrete examples.

Balarka Sen

7:10 PM

@MikeMiller Hmm. I'll think about it and get back to you. Do you have a hint for me, or do you think I should be able to do it by myself after sufficient thinking?

user147690

7:36 PM

@BalarkaSen Oh dangit, generally I find that you are sufficiently ahead that you give great guiding comments :D

Balarka Sen

I don't know anything about algebraic geometry to give guiding comments.

I'd probably mention as a passing comments that learning sheaves and schemes so fast without getting hands dirty with varieties is probably going to be a hard way through and somewhat unmotivated, but you should take this at face value because (1) I don't know much about sheaves and schemes (2) there are plenty of people out there who are actually doing this. Hartshorne has this approach, in fact.

Especially since you seem to know some commutative algebra already, maybe the sheaves and schemes first approach won't be so hard as it would be for me, as most of commutative algebra is dry and unmotivated for me. I motivate commutative algebra with the algebraic geometry of varieties :)

rorty

I am feeling frustrated because I don't think my problem solving skills are getting better despite many months of self study. I keep getting stuck on seemingly trivial problems, and my impulse after 30 min or so is to get hints. I want to get through a certain body of math within a time limit so I can't take forever on a single problem, but at the same time I want to improve my problem solving skills. Any advice?

Krijn

8:03 PM

@

@BalarkaSen Hi!

I should thank you

I got a great score for my Algebraic Geometry hand in

Balarka Sen

Hi Krijin.

@Krijn Nice. I didn't really help too much though: you did most of the work by yourself. Good job.

Krijn

It's a really common mistake for non-Dutch people to type Krijin for some reason

Although I can't figure out why

Balarka Sen

Oops; apologies for that.

Krijn

Also, I'll have to write a final paper on étale cohomology, quite exciting

Balarka Sen

You should tell me what it is when you learn it. I attended a student seminar on Weil conjectures a couple days ago so at least I know why one cares about it.

Krijn

8:10 PM

It's one of those things that sound supermystical in mathematics and I'm excited to learn it just from the name itself

I had the same feeling with De Rham-cohomology

terrace

hi

Balarka Sen

Good :) I am not sure if I am too excited about it right now though. But it sounds interesting... everything nice does....

terrace

I just made a few sarcastic posts and I think some moderator is angry at me and im sorry but I don't know who to ask for forgiveness lol

I lost like 100 rep all at once and it doesn't saywhy but im pretty sure it's punishment for making troll comments

Balarka Sen

I don't see any 100 reputation decrease from your reputation statistics: math.stackexchange.com/users/181915/terrace?tab=reputation

terrace

Balarka well it definitely said it

I had like 700 and it was -104 suddenly

doesn't really matter about reputation, I just want to apologize for trolling

but don't know who to say sorry to

Mike Miller

8:15 PM

@BalarkaSen did you think about it for the torus?

ramsay

hello,
i got confused!
is $|-2.313|$ defined, where $|.|$ is modulus func.

Mike Miller

@Krijn How do you pronounce your name?

Is the j pronounced like an english y?

Krijn

No not really

Balarka Sen

@Krijn So what are you studying?

Krijn

The "ij" is one sound in Dutch

Balarka Sen

8:17 PM

@MikeMiller No, I didn't. I'll try to do that.

Krijn

It's more like "train" but with a "k"

@BalarkaSen Not too much at the moment, working in Hartshorne, some commutative algebra, some Riemann Surfaces

And some symplectic geometry but I don't like that too much

Mike Miller

Sure. The point is that most native English speakers will try to read that as "Kreej-in"; they might try to make the i sound not last very long, but they'll naturally say it (since they don't know how to pronounce the ij!).

So they will naturally rewrite your name phonetically as Krijin.

Krijn

I mostly get "Kridgn" sort of, when they try to pronounce it

Ah well, it's just a name

Balarka Sen

@Krijn You're onto chapter 2 in Hartshorne, yeah?

Krijn

@BalarkaSen I did sheaves and schemes

Balarka Sen

8:21 PM

Ah, OK.

I am occasionally working on Hartshorne chapter 1, with no intention to get into further chapters for now. I was reading a bit from chapter 1 section 6. I don't really understand the purpose of the abstraction he does with algebraic curves up there.

Krijn

I really didnt like section 6 and 7

Balarka Sen

Ah? I was rather excited about getting into section 7, because apparently he does some intersection theory there.

Krijn

He does, yes

Balarka Sen

I think there is more to the notion of abstract algebraic curves than meets the eye though. I admittedly didn't read all of the section yet, but the geometric content of the definition is not clear to me.

Krijn

I had some intersection theory in my course on elliptic curves so that made it a bit easier

Do you know why Hartshorne is considerd such a classic?

Balarka Sen

8:26 PM

Nope.

Mike Miller

Before Hartshorne you had EGA.

That's a good reason.

Semiclassical

EGA?

Balarka Sen

lol

Semiclassical

(extremely god-awful?)

Krijn

@MikeMiller But surely better books have been written by now?

Mike Miller

8:29 PM

Éléments de géométrie algébrique

@Krijn Not everyone agrees with your taste! But you didn't ask why it's still so widely used; rather you asked why it's considered a classic./

Krijn

That is true.

Balarka Sen

@Krijn Hartshorne's exposition on blowups, albeit short, is nice and concrete. I preferred that over Shafarevich's explanation, which talks a lot but doesn't provide enough geometric intuition (at least for me) on that particular construct.

Krijn

Yes, the part on blowups was nice, I have to admit that

Anush

@robjohn this answer does the Poisson summation for us math.stackexchange.com/questions/1699078/…

@robjohn and it also explicitly shows how much large the sum is than the integral

I will try to produce some numerical results too to show the difference

@J.M. I believe the codegolf code has a guaranteed accuracy which is set by a parameter

Mike Miller

In any case you can certainly find people talking about their preferred algebraic geometry books anywhere on the internet.

Balarka Sen

8:36 PM

I do think both the books should have mentioned the geometric meaning of the equation $x_i y_j = x_j y_i$. This gets clearer when one goes to local blowups instead (choose a system of local coordinates $t_1, \cdots, t_n$ for your tangent space at a smooth pt $p \in X$ and define the subvariety of $X \times \Bbb P^{n-1}$ by $t_i(p)y_j = t_j(p)y_i$ - same thing but it's intrinsic to the variety now).

I guess it's trivial to figure out what's going on, but it wasn't apparent to me at first.

ramsay

28 mins ago, by ramsay

hello,
i got confused!
is $|-2.313|$ defined, where $|.|$ is modulus func.

Akiva Weinberger

@ramsay Yeah, $|-2.313|=2.313$.

ramsay

8:52 PM

thanks Akiva

Akiva Weinberger

@Krijn So like the word "crane"

Krijn

@AkivaWeinberger Almost, yeah

British pronunciation of train, that is

Akiva Weinberger

It's easier to write in IPA

/εi/ as opposed to the English-language /ei/

(Where /ε/ is the sound that "e" makes in "bed" in most dialects, /i/ is the "ee" sound, and /ei/ is the sound found in the word "crane".)

Krijn

Or just use translate.google.nl/#nl/en/krijn and press the sound button on the left side "Nederlands"

Mike Miller

or just call him Joe

in the spirit of

How to Pronounce Chipotle

►

Krijn

9:02 PM

That's cool as well, actually

Even in Dutch, it's not a common name and many people do not get it in loud areas, such as bars etc.

So most nights I just call myself some other random names

Akiva Weinberger

Eh, /kɾεin/ isn't hard to pronounce. (/ɾ/ is a flapped "r", or the flapped "t" in the North American pronunciation of "butter". Do you roll or flap your "r"s in Danish?)

*Dutch

Sorry

Mike Miller