Mathematics - 2017-01-20 (page 3 of 3)

Semiclassical

20:00

A node like that could split into a pair of branch points under a perturbation e.g. $x^2-y^2=a^2$. I presume that to be the generic case, and the node surviving said perturbation to be nongeneric.

But I don't know what the terminology for the node to 'survive' a given perturbation is.

Astyx

Because this doesn't work in any field, does it ? @Tobias

Tobias Kildetoft

@Astyx Sure it does, you just need the right topology

Ted Shifrin

Right, @Semiclassic. I'm just saying that to me saying "the singular point $(0,0)$ is stable under the perturbation" means that the point itself stays singular, not that it moves.

Semiclassical

Yeah, I understand.

Question is whether there's a proper terminology for that.

Ted Shifrin

I would say something like "perturbation among curves with nodes" ...

Consider a family of curves with ordinary nodes ...

Mike Miller

20:01

You would say that.

Ted Shifrin

@MikeM: Do you disagree (seriously)?

If you say a singular point is stable, what does that mean?

Mike Miller

Nah.

Astyx

@Tobias What topology would that be in, say, $\Bbb Z/p\Bbb Z$ ($p$ prime) ? I'm intrigued

Ted Shifrin

The usual differential topology language is that a property is stable/generic.

Mike Miller

I'm just agreeing with you.

Tobias Kildetoft

20:02

@Astyx The Zariski topology

Ted Shifrin

Oh, ok ...

Astyx

@Ted I don't get what you mean

Ted Shifrin

Forget about it, Astyx.

Astyx

Let me look that up :) @Tobias

Ok

Mike Miller

Trying to get to my hotel via london transit before my phone dies.

Ted Shifrin

20:03

Good luck!

Stop chatting.

Mike Miller

indeed

Semiclassical

The specific case I've got consists of a family of curves with parameters $g,\gamma,E,x$.

Ted Shifrin

$x$ is a parameter? seriously?

Semiclassical

ack

Balarka Sen

by physics standards that's perfectly fine

Semiclassical

20:04

Hrm, yes it is in this context.

Ted Shifrin

LOL

Semiclassical

Which is rather silly

Ted Shifrin

Not when I'm writing equations of plane curves it's not.

Mike Miller

What do you expect me to do for the next hour?

Semiclassical

In the defense of the preprint I'm reading (It's their notation)

The variables on the curve would be $(t,\epsilon)$

Ted Shifrin

20:05

Oh good grief

Balarka Sen

i wouldn't be surprised if $\slashed{\phi}_{ijk}^{lm}$ was a parameter instead

Semiclassical

And they're not doing any plane curves, I'm bringing that in myself.

Ted Shifrin

@Balarka: Go finish Gauss-Bonnet.

Semiclassical

But yeah, for the purposes of this conversation it's rather silly.

Balarka Sen

meh, I won't fix this

Semiclassical

20:06

Hence, I'll instead say I've got parameters $a,b,c,d$. :)

(They also have $e$ as a parameter. Even I can't abide that.)

Ted Shifrin

It seems like you have too many parameters for a hyperelliptic cubic.

Astyx

I'm going, good day to everyone

Ted Shifrin

Bubye, @Astyx

Semiclassical

Eh, it's more symmetric than you'd think.

Ted Shifrin

Anyhow, I'm leaving in a few minutes for lunch. So get to the punchline.

Balarka Sen

20:08

@TedShifrin Alright, thinking about it.

Semiclassical

It's the characteristic polynomial in $x$ of the matrix $$\begin{pmatrix} a-y & 0 & b c & d \\ 0 & a+y & cd & b \\ bc & cd & 1 & 0 \\ d & b & 0 &-1 \end{pmatrix}$$

Akiva Weinberger

Hielo

Ted Shifrin

@Semiclassic: Did you mean for it to be almost symmetric but not?

Hi, DogAteMy.

Semiclassical

Fixed.

Ted Shifrin

OK.

Semiclassical

20:12

When $c^2=\frac{1-a}{1+a}$, the resulting curve $f(x,y)=0$ has a singular point.

Ted Shifrin

Now what?

So you want to perturb along that hypersurface in the parameter space?

Semiclassical

Yeah.

Ted Shifrin

OK, no problem.

You could explicitly substitute $a=a(c)$ and get rid of one parameter.

Semiclassical

And I want to know how one refers to the fact that the curve remains singular along that line.

...which, maybe one just says that.

Ted Shifrin

Not line. It's a hypersurface.

Yup, you just say that.

Semiclassical

20:13

Right, I've got b,d still.

Ted Shifrin

This is analogous to the discriminant subvariety which parametrizes multiple roots.

Semiclassical

Right.

Ted Shifrin

So we're done for now? :)

Semiclassical

I'm interested, then, in perturbations which preserve singularities of the curve.

Ted Shifrin

No, that suggests that the singular points don't move.

Which preserve the singular nature of the curve.

Semiclassical

20:15

Hmm, okay.

Ted Shifrin

Language is subtle.

I'm out of here for now. Back later.

Semiclassical

I'd say "perturbations which preserve the singular-ity of the curve" except that one already has "singularity" as a noun not as an adverb.

later, @ted

Socrates

does induction work for statements about matrices?

Krijn

@Socrates You could do induction on the dimension $n$ of a matrix, yeah

Semiclassical

(not adverb. I should've said: "singularity refers the points on a curve, not to the curve itself.")

Alessandro Codenotti

20:24

@Krijn for a practical example that's how you usually prove that symmetric real matrices are diagonalizable

Daminark

Hello chat

Krijn

@AlessandroCodenotti Linear algebra seems ages ago, it was so much fun tho

Semiclassical

I've said it before, but I always find it a bit funny how 'linear algebra' means different things depending on what level you're talking about.

Balarka Sen

everything is linear algebra

2

(which explains why i am bad at everything)

Semiclassical

To a high-school student, linear algebra would be "simultaneous systems of equations." To a college student taking an intro course (not a full one), it'd probably mean "matrices." And to someone taking a serious course in linear algebra, it'd be stuff like vector spaces.

Balarka Sen

20:28

on a more serious note, i really need to get some of my linear algebra straight: i don't really use a lot of linear algebra.

Semiclassical

I use certain parts of linear algebra a lot.

Eigenvalues-eigenvectors, diagonalization, etc.

Krijn

I don't really use it because every time it comes up the prof will wave his hands and say "this is just linear algebra"

s.harp

with linear algebra do you mean finite dimensional linear algebra?

Semiclassical

And there's stuff which I don't say a lot but which I know is there, e.g. null space. (An eigenvector, for instance, is nothing but an element in the null space of a particular matrix.)

Balarka Sen

@Semiclassical yeah, that's the kind of thing i don't use

Socrates

20:32

@KajHansen hi, how is it?

Semiclassical

Can't do quantum mechanics without eigenvalues/eigenvectors.

Mike Miller

does anyone actively use that?

like that theorem

s.harp

btw if I'm looking a space $C$ of maps $X\to Y$, whats the topology called that makes all the evaluation maps $ev_x: C\to Y$ $f\mapsto f(x)$ continuous?

Semiclassical

Which theorem?

Mike Miller

or rather fact... doesn't really deserve "theorem"

@s.harp pointwise convergence?

s.harp

20:34

@MikeMiller huh, right that makes sense as a name

Balarka Sen

the new neighbors are making hella racket.

Daminark

Functional programmers, huh?

Balarka Sen

i dunno what they are. having a party it seems

Mike Miller

back to advance wars til i get there i guess

Daminark

It was a joke if they're making "Hella Racket"

Mike Miller

20:36

can't much work

Balarka Sen

i wish they'd just conk out after drinking to their fullest

Semiclassical

I finally broke down and downloaded VBA-M so that I could play advance wars.

Balarka Sen

cool

Semiclassical

I then got a bit annoyed with it and decided to download Castlevania: Aria of Sorrow instead :)

Balarka Sen

@Daminark ah i see

@Semiclassical traitor

Semiclassical

20:39

lol

Mike Miller

I'm going through the war room in aw2

Liad

Hi, i got a question im not sure of:
Let F be a field. i need to prove that in the ring $F[x,y] $ $f \in (x,y) $ iff $f(0,0) = 0 $

Assuming $f \in (x,y) $ then $f(0,0) = a_{0,0} $ why does $a_{0,0} $ must be zero?

Sie

20:54

So in the context of sets what is the appropriate way to say inbetween? The way I've been doing it is B = { x|x ϵ 1-100 }. Is that acceptable or is there a proper way to do it?

Balarka Sen

@MikeMiller Have you ever saved both the T- and B-copter in the Green Earth mission? I can't seem to be able to do it

(was replaying that one)

Socrates

@Sie $B=\{x\in\mathbb{R}|1\leq x\leq 100\}$ or with N, if you want integers

Balarka Sen

nvm, did it

Semiclassical

lol

JamalS

In the context of blowing up singularities, does anyone know what is meant by $\mathcal{O}(-2)$?

Balarka Sen

21:06

It's the cotangent bundle over $\Bbb P^1$

Equivalently tensor product of the tautological line bundle with itself

JamalS

@BalarkaSen Thank you :) Can I ask why the notation is as it is?

Oh I see

Balarka Sen

If you take a generic section of it as a real 2-plane bundle, it intersects the zero section at two points, with opposite orientation each

aka the zero section of the underlying 2-plane bundle has self intersection number -2

or it's a bundle with Euler class -2, whatever you want to call it

JamalS

I'll go with the Euler class then, I'm more differential geometry minded :)

Balarka Sen

Me too :)

JamalS

@BalarkaSen On an unrelated note: Would you by any chance know of any resources you could point me to that discuss how to compute the period matrix of an algebraic variety $X$ defined as the zero set of a polynomial, which is a smooth manifold?

Balarka Sen

21:12

I don't really know what a period matrix is. But you can wait for @TedShifrin, who's the resident complex algebraic geometer/differential geometer here.

Also I misread your previous "differential geometry" as "differential topology". I don't really know much about the former :P

Semiclassical

@JamalS You might look at Carlson's 'Period mappings and period domains' book. The first chapter, at least, is pretty readable.

JamalS

@Semiclassical Thanks, I'll have a look.

Semiclassical

@BalarkaSen The period matrix of a smooth genus-g curve has matrix elements $\int_{\gamma_i}\omega_j$ where $\{\gamma_i\},\{\omega_j\}$ are bases of cycles and co-cycles respectively.

JamalS

@Semiclassical Yes, I'm aware of the textbook definition so to speak, it's just that evaluating it by that means is not always possible, or at least not the easiest way.

Semiclassical

Don't I know it.

JamalS

21:18

Hehe

Balarka Sen

@Semiclassical Ah

Semiclassical

The main tool I know is to find the Picard-Fuchs equation for the given elliptic curve and solve the resulting ODE.

That only works in cases where you've got a family of Riemann surfaces parametrized by a parameter, though, and even there it can be pretty hard to actually find the Picard-Fuchs equation.

Joe Slater

Hi

Can someone help me understand how to go from the first equation to the second - imgur.com/a/tH7Gh ?

JamalS

@JoeSlater You can work backwards, using the formula for the sine of a sum.

The phase shift is physically meaningful, and perhaps it's better if you understand this in the context of RLC circuits, as in your problem; Young and Freedman have a good section it.

Semiclassical

It may also be useful to compute $e^{i\phi}=\cos \phi+i\sin \phi$ given that definition of $\phi$.

Joe Slater

21:30

Ok thanks. I will try to read more about it. But I am not able to see how using the eulers formula or formula for the sine of sum is useful in this case.

Simple Art

Say, should I delete this? :

-12

Q: Interesting patterns to the algebraic solutions of polynomials

In yet another attempt to find the solution to the quintic polynomial, I started looking backwards at the solutions to the quartic, cubic, quadratic, and linear polynomials to see if I could pick up any patterns. $\left\{ \begin{aligned} 0&=ax+b \\ 0&=ax^2+bx+c \\ 0&=ax^3+bx^2+cx+d \\ 0&=ax^4+...

Semiclassical

Keep in mind that $e^{i (\omega t+\phi)}=e^{i \omega t}e^{i \phi}$.

Simple Art

:D Euler's formula

Semiclassical

So you can write $\sin(\omega t+\phi)$ using Euler's formula, and then compute $e^{i\phi}$ from its definition.

Socrates

@SimpleArt could you even, if you wanted?

Simple Art

21:39

@Socrates Yes, I definitely could, but I do not want to do so rashly

Socrates

@BalarkaSen one question was titeld "doubt in linear algebra" for what it's worth lol

Kasmir Khaan

hi guys

JamalS

@TedShifrin When you have time on here, could I ask a question about period matrices?

Simple Art

@KasmirKhaan Hi!

Kasmir Khaan

in a tripple integral when we have x+y+z = 1 to define a region

and when we have it as function

can i set it to equal 1 and integrate?

tripple integral of 1/ (1+ (x+y+z)^3 , over the region x=0 , y=0 ,z=0 and x+y+z=1

my question is if we can replace x+y+z with 1 and get 1/2 as integrand

Semiclassical

21:46

@JamalS What kind of curve do you have, out of curiousity?

JamalS

@Semiclassical I actually don't have a particular one in mind; the question is motivated by reading I'm doing into the Kawazumi-Zhang invariant of a Riemann surface, which turns out for certain genera to be written as a Fourier series in the period matrix.

Semiclassical

Ah. Yikes.

JamalS

In these cases, I believe it can also be related to the Faltings invariant.

Semiclassical

The only cases I know how to calculate turn out to be (generalized) hypergeometric functions

Kasmir Khaan

anyone can help me plz?

JamalS

21:49

@Semiclassical Do you have a resource to point me to on that? Sounds interesting too.

Simple Art

@KasmirKhaan While I may be able to do random integrals with complex analysis, that's just a side thing, and I can't do triple integrals (at least not very well)

Semiclassical

It's a survey, but you might appreciate this: maths.ed.ac.uk/~aar/papers/kontzagi.pdf

See 'Chapter 2' in particular.

JamalS

@KasmirKhaan No, because the volume that you are integrating over is bounded by several surfaces, one of which is the plane $x+y+z=1$. On this plane, the equation holds, but your integration includes points in the interior (after all it is a volume integral) that do not satisfy, $x+y+z=1$. Does that make sense to you?

Kasmir Khaan

yes we get like level surfaces

or planes

@JamalS thank you ! :)

Liad

Hi, in $F[x,y] $ is it correct that $(x,y) = \{ f_1 *x + f_2 *y : f_i \in F[x,y] \}$ . (im not sure if that is the definition)

Kasmir Khaan

21:56

how to integral 1 / (1+u^3)

Simple Art

@KasmirKhaan Oh, wait, I got you!

Simple Art, America

21.6k 17 73

calculus sequences-and-series algebra-precalculus trigonometry limits real-analysis summation

The answer to your question is in my top questions list

18

Q: Evaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$

I was wondering on how to evaluate the following indefinite integral for all $n\in\mathbb R$. $$\int\frac1{1+x^n}dx$$ It seems to be peculiar in that we have $$\begin{align} \int\frac1{1+x^{-1}}dx&=x-\ln(x+1)+c\\ \int\frac1{1+x^0}dx&=\frac12x+c\\ \int\frac1{1+x^{1/2}}dx&=2\sqrt x-2\ln(1+\sqrt ...

real-analysis integration sequences-and-series indefinite-integrals

:D There you go @KasmirKhaan

Kasmir Khaan

@SimpleArt thank you sir! :)

Simple Art

@KasmirKhaan If you want to go through the algebra, you should use PFD with $u^3+1=(u+1)(u^2-u+1)$ with an arctangent integral.

JamalS

@KasmirKhaan In general, $\int \frac{dx}{1+x^n}$ is $x \, \, {}_2 F_1 (1, n^{-1}, 1+n^{-1},-x^n)$.

Simple Art

lol

Kasmir Khaan

22:01

thank you again but that is way to hard for me at this point =p

Simple Art

@JamalS Could also just take geometric series if one is headed that way

JamalS

Hehe

Actually, the fun way to do this integral is Risch's algorithm ;)

Kasmir Khaan

are these algorithms easy to learn ?

JamalS

Noooooooo

Kasmir Khaan

because we are doing many integrals ><

JamalS

22:02

The Risch algorithm requires some mathematical preliminaries other than calculus and takes 100 pages to cover a summary of it.

Simple Art

LMAO!!!!!!!

@KasmirKhaan I feel you though :P

@JamalS That is why we have WolframAlpha and such

Kasmir Khaan

haha okay ill keep using what I know

JamalS

It's definitely worth knowing the details though; I always like to know what my tools are doing. Furthermore, it can be used to prove in certain cases that no closed-form exists, which Mathematica won't do for you.

Simple Art

Lol

If it looks solvable, but Mathematica doesn't give an answer, I conclude it doesn't exist @JamalS

JamalS

Yeah, but we know that's not always the case.

Semiclassical

22:04

Yeah, that won't fly in a grad-level physics class :)

Lots of hard integrals which are best handled by citing a table of integrals.

JamalS

I always feel like I've cheated if I've used a table, though if I'm doing an immense problem where the integral is the least of my worries, then not so much :)

Simple Art

@JamalS Wolfram and mathematica are my table of values

JamalS

@SimpleArt Yeah I use Mathematica a lot

It's fun to just play around with it as well

Simple Art

;) yup

Socrates

@TheGreatDuck that's why i deleted

you know, it has a reason why I deleted :P

TheGreatDuck

22:13

@Socrates which I posted before you deleted. Otherwise, I couldn't tag you.

Socrates

ah

makes sense ;)

I can not really say what's your question?

TheGreatDuck

huh?

how does one prove proof by contradiction doesn't work?

Socrates

do you seek a proof that fails only because we look at the equivalent contrapositive?

Simple Art

huh? is the question @Socrates

TheGreatDuck

@Socrates i don't understand. We try proof by contradiction and it fails either because the original statement is false or because the statement is not provable. How does one prove it's the latter and not just lack of ability of the prover to find the contradiction?

Simple Art

22:17

@TheGreatDuck Reminds me of proof by exhaustion

Kasmir Khaan

can someone explain to me how even and odd functions work when we integrate

Socrates

I think those are called conjectures

and it would be really cool to have a way for your question, but obviously this way doesn't exist. Or why are there still conjectures?

Simple Art

@KasmirKhaan $\int_{-a}^aeven(x)\ dx=2\int_0^aeven(x)\ dx$ and $\int_{-a}^aodd(x)\ dx=0$.

Kasmir Khaan

i meant more like

is the product of even and odd = ?

and such things

does it work like normal even and odd with numbers?

Simple Art

$even\times odd=even$

lol, just like numbers XD

Kasmir Khaan

22:19

okay thank you =p

Simple Art

Oh wait, even times odd is odd

Oops, it's backwards, my bad

Kasmir Khaan

we did not prove this

omg dont comfuse me

Simple Art

You just check this:

Kasmir Khaan

am allready comfused

check what ?

Simple Art

$$\text{even}(-x)\times\text{odd}(-x)=-\text{even}(+x)\times\text{odd}(+x)$$

So it is odd.

TheGreatDuck

22:20

@Socrates what...? We can prove that contradiction shows independence in the case of the parallel postulate of geometry. Or do you claim otherwise? Surely there was a method used in that?

Socrates

@TheGreatDuck but there is no(known) method for all conjectures, if there was, we wouldn't have conjectures.

Suppose the following: i try to prove Riemann Zeta by disproving the contradiction. Now I fail, but does it mean that Riemann Zeta is true(or indeed false)?

The fact that anyone tries and fails says absolutly nothing

If there where a method to show that it's only my lack of ability, then Riemann would be proven

Simple Art

@Socrates Yes it does

It says that the first one to do it will probably get rich

Socrates

fair enough

Russians tho, tend to disagree.

Simple Art

Lol

Fair enough

Socrates

my conjecture: if all 7 billion humans on the planet would sit down to solve math, we would have went from the mathematical stone age to feudal age

Simple Art

22:28

@Socrates We'd probably all die though

Socrates

I survived till now, because someone else gave his life, not even knowing me.

Simple Art

:D

TheGreatDuck

@Socrates who, cause if I read that right said person did know you.

Mike Miller

@Balarka Level 6 or whatever? Doubt it.

Socrates

If some soldier dies to save the lifes of his country, he saved my life, but he (probably) didn't knew me.

TheGreatDuck

22:41

ah

i thought you were making a religious reference

nvmd

Socrates

ah, no thanks Jesus^^

TheGreatDuck

@Socrates regarding your point, we don't have algorithms to find all possible solutions to differential equations with closed forms. We don't know how to reduce any given number into it's factors without brute force. I'd argue that we know how to solve some of them and reduce many of them by memory. Surely a method exists. Whether it applies to everything is obviously expected to not be the case.

However, I'd imagine that one was able to prove the parallel postulate was independent therefore, some kind of method exists for at least some statements and situations.

Simple Art

@TheGreatDuck We have differential galois theory though

TheGreatDuck

huh?

nullgeppetto

Hi all, I'm looking for a confirmation on an attempt to answer a question of mine. It concerns the estimation of mean value of a function of random variables. @Batman provided me with precious info, but it would be nice if anyone else could also take a look. Could you check Second attempt? Thank you very much.

TheGreatDuck

22:46

@SimpleArt I don't follow?

fateme jl

Hi, I just looking for a solution manual for Folland real analysis, Is there a free link??

Simple Art

You said that there is no algorithm to find all possible solutions to differential equations, so I said differential galois theory @TheGreatDuck

fateme jl

this is my question, math.stackexchange.com/questions/2105270/…

thanks for your help...

TheGreatDuck

@SimpleArt fine then. What is the solution to y'' + 2floor(x)y' + floor(x)^2y = 0?

or better

y' = e^{x^2}

thought the latter has none

that we can physically but on paper

Simply Beautiful Art

@TheGreatDuck I didn't say it can solve all differentials, but I do can tell you the latter can be shown to have none

with DGE

TheGreatDuck

22:51

no

i meant FINDING the solutions

Simply Beautiful Art

Yes :D

Socrates

the latter has one, if I remember, or it was $e^{-x^2}$

TheGreatDuck

not determining if they exist

Simply Beautiful Art

@TheGreatDuck :| But if they don't exist, what am I supposed to do?

TheGreatDuck

@Socrates perhaps the latter. the first is the textbook example.

@SimplyBeautifulArt that's why i said ONLY the one's that exist. but you still have to find it using the algorithm. I'm merely saying nonexistent things give any result. Don't worry about the case of no closed form.

Socrates

22:53

well

depends on what you mean with solution

TheGreatDuck

it's just an analogy

things work sometimes and sometimes not

Simply Beautiful Art

@TheGreatDuck Idk, perhaps you could try the Risch algorithm

TheGreatDuck

and since someone has done it in the past

Socrates

in my physics class we weight paper to measure the integral

TheGreatDuck

there must be a way to prove a statement is unprovable

Socrates

22:54

was accurate

TheGreatDuck

(and uncontradictable)

i.e. neither true nor false

Socrates

no, some statements are true, but unprovable

some statements are just nonsense

think about it this way: can you prove to 100% that the universe didn't start 5 seconds ago?

Simply Beautiful Art

@Socrates Yes, assuming the universe started some time ago

Socrates

@SimplyBeautifulArt I don't think this is a valid proof

Simply Beautiful Art

Then just warp the thing you call a "second" into what I call a "second", such that 5 of my seconds equals the beginning of the universe

Socrates

23:03

If we do that. Why does "think about it this way: can you prove to 100% that the universe didn't start 5 seconds ago?" not mean "Good day fellow, how are you"?

TheGreatDuck

like i said

we're talking about math for starters

and also, the idea is "provable from what you are given"

Maks

Hi ! I want to know something

TheGreatDuck

i think it's a bit bridging on absurd to bring in philosophical questions.

after all, we can then rule out everything as given

and then nothing is provable

Maks

Do you guys know from where does all formulas you know come from ?? Or you just know they are like that and dont know their origins ?

Socrates

there is mathematical truth, because we can try axioms

that doesn't mean those axioms are good, it just means we can prove something

like the 5(?) axioms of euclid

he derived alot of that

Sophie

23:18

if $a^2+b^2=c^2$ and $(a,b,c)=1$ then one of a,b is odd and the other is even, and c is odd. Let $a\mapsto 2a$ then $a^2=\frac{c+b}{2}\frac{c-b}{2}$ and if a product of two coprime integers is a square then both are squares. Let $\frac{c+b}{2}=p^2$, $\frac{c-b}{2}=q^2$ then $c=p^2+q^2$, $b=p^2-q^2$, $2a=2pq$ which means all pythagorean triples are parametrized by Euclid's equation

I came up with that proof after this one didn't satisfy me en.wikipedia.org/wiki/…

Socrates

what means (a,b,c)=1?

Sophie

(x,y) is short for gcd(x,y)

Socrates

ok

why can't all be odd?

Sophie

odd^2+odd^2=even

Socrates

ah -.-

yeah

Kasmir Khaan

23:38

hi @TedShifrin how does the even and odd functions work in doing integrals

like if we multiplay odd * even what do we get?

does it work like normal numbers?

even *odd = even ?

Ted Shifrin

Well, check it out. If $f$ is even and $g$ is odd, what is $(fg)(-x)$?

Kasmir Khaan

even

Ted Shifrin

Huh?

Write it out.

Kasmir Khaan

g(-x) = -g(x)

Ted Shifrin

And $f(-x)=$?

Kasmir Khaan

23:41

f(-g(x)) = f(g(x))

Ted Shifrin

We're not composing. We're multiplying.

Kasmir Khaan

oh okay

Ted Shifrin

At least, you said you were multiplying.

Kasmir Khaan

yes i did

-f(x) g(x)

Ted Shifrin

So even times odd is ?

Kasmir Khaan

23:43

odd

Ted Shifrin

And what is odd times odd?

Kasmir Khaan

even

hmm thinking about x and x^3

we x^4

so it is not like numbers =p

Ted Shifrin

So the right analogy with integers is with addition, not multiplication. Can you think of a reasonable explanation?

Kasmir Khaan

oh that is good let me think

no i cant think of why

Ted Shifrin

Look at the example you just did a moment ago.

Kasmir Khaan

23:46

x and x^3?

well in power functions we add exponents

but how does it work with sin and cos

Ted Shifrin

Well, note that the Taylor series for an even function has only even powers, and the Taylor series for an odd function has only odd powers.

Kasmir Khaan

oh yeah :D

so we can reduce it all to powers

JamalS

@TedShifrin Can you point me to any resources on computing the period matrix of algebraic manifolds (specified as the zero set of a known polynomial)?

Ted Shifrin

As a mathematician, you should try to figure out more of these puzzles.

Kasmir Khaan

yes i should =p but having hard time learning about the other new topics

and i got another question about periods

Ted Shifrin

23:50

No, @JamalS, I've never even thought about this. But what do you mean by a period matrix for a general algebraic variety? This is normally used for abelian varieties (so these are algebraic varieties that happen to be tori).

Kasmir Khaan

is it the same to go from 0 to 2pi and from -pi to pi ?

i know we go around once but is it a legal step ? :)

Ted Shifrin

@Kasmir: If the period is $2\pi$, you can integrate over any period ... $[a,a+2\pi]$.

Kasmir Khaan

because that way i could use odd functions add up to 0 on symmetric intervall

Ted Shifrin

Right.

Kasmir Khaan

Thanks Ted ! that would be handy :)

JamalS

23:52

@TedShifrin Well, an example would be the Burnside curve, $y^2 = x^6-1$ with automorphism group $S_2 \times \mathbb Z_2$; I'd like to compute, for example $\Omega$ for that curve.

Ted Shifrin

Oh, you're talking about curves, so then it's the Jacobian variety of the curve which is the abelian variety.

You can probably do that bare-hands. What's the genus?

If you haven't looked at it before, you might try Phillip Griffiths's beautiful little book An Introduction to Algebraic Curves. He does a lot of concrete stuff in that book.

JamalS

I think it's genus 2.

Ted Shifrin

I think it's bigger than that. $g=(d-1)(d-2)/2$, no?

JamalS

Ah, yes, sorry.

Ted Shifrin

But look at Griffiths's book. He works out a lot of examples in there.

JamalS

23:58

I'll download it now, thanks.

Ted Shifrin

Good luck. I've actually never done this sort of computation.

Transcript for

$Mathematics$ Mathematics