Mathematics - 2016-10-29 (page 2 of 3)

Josué

5:00 AM

@Kaumudi What is $n$?

arctic tern

@Josué an integer

Josué

@arctictern Right. I was just curious about exactly how his book is defining $n$. It's possibly arbitrary.

arctic tern

it's defining integers mod n

Josué

@arctictern Oh. Duh. You're right. Haha.

TheGreatDuck

Is there anything flawed with my question?

1

Q: How do I solve this second-order homogeneous differential equation with piecewise constant coefficients?

I am trying to solve the following differential equation (for fun, mind you): $$\lfloor x \rfloor y'' + 2\lfloor x \rfloor y' + \lfloor x \rfloor y = 0$$ Because of the piece wise coefficients, my gut instinct is to first solve the associated implied differential equation. This means that I ca...

I want to make sure I didn't write anything confusing. This subject has a tendency to confuse people in my experience. Also, sorry for the large link.

Josué

5:10 AM

Wikipedia calls a topological space $X$ contractible if its identity map is null-homotopic. Consider the identity map $i$ on $\mathbb C^*=\mathbb C\setminus\{0\}$ and define $H:X\times[0,1]$ by $(x,t)\mapsto tx$. This looks continuous despite $\mathbb C^*$ not being contractible.

TheGreatDuck

@Josué um what?

Josué

$H(x,0)\equiv0$ and $H(x,1)\equiv i$

Mike Miller

@Josue Whwre does everything go at t=0?

user228700

@BalarkaSen I don't understand the definition is what I meant. (This is part of a question, so that is why I kept asking "What is it asking of me?", sorry.

arctic tern

@Kaumudi what part don't you understand? do you know what a relation is? do you know what <=> means? do you know what n|(a-b) means?

user228700

5:15 AM

No, I don't understand what $n|(a-b)$ means.

arctic tern

then say that

it means n divides a-b

i.e. n is a divisor of a-b, or a-b is a multiple of n

user228700

@arctictern OK, sorry. Thanks!

Josué

nvm

Oh, shoot, I forgot that the codomain must be $\mathbb C^*$. Ignore my naïveté, lol.

TheGreatDuck

I will take that as a no then...

user228700

Does anybody know which set is represented by the letter $S$?

TheGreatDuck

5:26 AM

I know s^2 is spherical plane

user228700

Um, no, I don't think that's what my textbook is going for...

TheGreatDuck

context then?

cause it looks familiar

wait

user228700

Oh, nvm.

TheGreatDuck

nevermind

It looks familiar cause that was the symbol I used for complex space when I was messing around. XD

I'd hope your textbook didn't somehow use the same symbol and meaning I made up on a whim. XD

Josué

I don't think there are universal letters for sets other than $\mathbb Z$, $\mathbb R$, etc., and those aren't even letters.

TheGreatDuck

5:30 AM

well...

most people tend to think of them as Z and R

Josué

I've seen them written like that, but bolded.

TheGreatDuck

I mean that most people call them "Z" and "R'

verbally not in writing

I know the symbols are different

Josué

I get you. I was defining a letter as being an element of the set $\{A,B,\dots,Z\}$, lol.

TheGreatDuck

oh well that's silly.

the set of letters would be a set of geometric figures.

specifically, figures in the euclidean plane

built entirely from line segments

and circles

ccorn

7:19 AM

Interesting geometry question, a generalization of Pascal's theorem to three conics.

DHMO

8:14 AM

We have seen many false "proofs" that 0=1... is anyone interested in a real proof?

Balarka Sen

I am not.

user147690

8:40 AM

@BalarkaSen And existence of these corresponds to having $w_2(M)=0$ right?

Balarka Sen

Yes.

user147690

@BalarkaSen I just gave a talk(on Wednesday) on SW classes for AT, and I was asked about these afterwards and didn't know them.

Balarka Sen

SW classes are good stuff. What did you learn?

user147690

I'd been looking at Milnor&S and I'd done essentially the first 4 chapters

Balarka Sen

Hi, btw. Tobias was looking for you a few days ago.

user147690

8:44 AM

So I haven't done Steenrod & Thom isomorphism yet

Balarka Sen

Me neither. Also working through Milnor-Stasheff here.

along w/ other stuff

user147690

Oh I see, thanks. @TobiasKildetoft I'll be online more frequently again, classes have ended and finals in 2 weeks.

user147690

@BalarkaSen I want to go through Serre's Linear representation theory over christmas and do most of the exercises from Dennis & Farb's noncommutative algebra

user147690

M&S is pretty nice

user147690

Still trying to work out what I will do for my honours thesis next year.

Balarka Sen

8:47 AM

@AlexClark It is. I am complementing it with Hatcher's notes, which is also good.

user147690

His K-theory and vector bundles?

Balarka Sen

Yah

I haven't made much progress on M&S the past few days because I was looking at some diffgeom. Maybe I should read a bit

user147690

A friend did his talk on Chern classes, using that as a main reference, but he had to blackbox some things. It was only a 20 min talk, so I just gave axioms, did some consequeces, showed whitney duality, and blackboxing cohomology of $\Bbb{RP}^n$ did all the whitney classes of $\Bbb{RP}^n$

user147690

(all in M&S as you'd know)

Balarka Sen

mhm

user147690

8:51 AM

I thought I'd need to speak very fast to cover all that, but I made 17 min :P

Balarka Sen

cool

user147690

@BalarkaSen What diffgeo?

Little Alien

9:24 AM

Did you hear about this Elvis problem: "Twins are estimated to be approximately 1.9% of the world population, with monozygotic twins making up 0.2% of the total---and 8% of all twins.''

I wonder how do they get .2/1.9 = .08? I am getting something closer to .105.

DHMO

10:34 AM

> We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?" - Gian-Carlo Rota

4

s.harp

that is a good quote

Fargle

I like it and I hadn't heard it

overtly profound math quotes are what I live for

DHMO

> If I have not seen as far as others, it is because there were giants standing on my shoulders.

--Hal Abelson

s.harp

11:11 AM

what would be good way to motivate the tensor product for undergraduates?

something that scratches this very vague itch of "understanding what it is about"

I gave an exercise group on friday any they wanted to know, I felt the things I said in class were extremely unsatisfactory and told them I'd send them a write up today

Mahmoud

11:40 AM

In.

Ramanujan

@DHMO hi

After a long time

Secret

11:57 AM

Day 20+ on group theory:
The classical groups have very lengthy articles which highlight properties that are interesting in studying those groups, but not very helpful in creating a catalogue of group properties

DHMO

@Ramanujan hi

Mahmoud

12:48 PM

Hello.

Mees de Vries

@s.harp, is something near the universal property out of the question? Or, depending on what kind of tensor product you are teaching, extension of scalars?

s.harp

@Mees the basic plan of what I'll do is like this:
1. given a basis of $V$, $W$ introduce $V\otimes W$ as a vector space having basis $e_i\otimes f_j$, show how terms $v\otimes w$ can be viewed as elements of this space. This way they have a feeling of how to manipulate vectors in the tensor product
2. Give a formal definition via a quotient of the free vectorspace on $V\times W$.
3. Explain how the tensor product lifts bilinear maps to linear maps, this being the primary reason we want to look at this thing, note that this uniquely determines the tensor product as VS

syzygy

1:15 PM

explain the trace in terms of the tensor product

(real definition of the trace)

canonical isomorphism of M* tensor V with Hom(W,V) this is how you use tensors in differential geometry

where M=W lol

are they physics students?

s.harp

yea they are 3rd semester physics undergrads

syzygy

how about mention that if S,S' are quantum mechanical systemts with state spaces H, H' then the composite system has state space H tensor H'?

Mees de Vries

What is $F$? The dual space or something?

syzygy

I'm sure you can explain that basically

but as far as i know it is an axiom

physically*

not basically

Secret

Actually, we knew in general $A\otimes B \neq B \otimes A$ but why in quantum people seemed to interchange the subsystems freely as if the tensor product is commutative?

s.harp

1:19 PM

yes that is basically a consequence of $F(A)\otimes F(B) = F(A\times B)$ when $F$ is a suitable function space

Mees de Vries

Have they had any QM? If so, they might appreciate that e.g. $|00\rangle \otimes |01\rangle = |0001\rangle$.

syzygy

tensor product IS symmetric

@Secret

s.harp

@Secret because $A\otimes B \cong B\otimes A$

Secret

ah i see

syzygy

perhaps you can also explain braiding

in the end the following constructions are the fundamental ones in linear algebra

direct sum; quotient space; Hom; tensor

Mees de Vries

1:21 PM

Maybe it's easier to start with general monoidal categories :-)

syzygy

nothing more, nothing less

:D

tensor hom adjunction is also very useful to know

but seriously i think to explain it for modules would be more useful

because in differential geometry you readily encounter modules over C^infty(M), M some base manifold

s.harp

this is a third semester analysis course for physicists, they do not like abstraction and need to have something they can immediately calculate with. Introducing it via basis is I think the best thing.

Mees de Vries

I definitely agree.

syzygy

me too. in the end most we mentioned above is only useful in higher mathematics

Mahmoud

Back.

s.harp

1:26 PM

I like the definition of the trace, I need to show them that $\mathrm{Lin}(V,W)=V\otimes W^*$ for finite dimensional spaces to do that though. This is also nice, but I promised to send them my write up today, and at $8$ I have a date with alcohol

Mees de Vries

Maybe I underestimate physics students, but I don't think many would appreciate $\mathrm{Hom}(V,W) \cong V^* \otimes W$.

s.harp

Ah, I messed up where the star goes

and if I look at their next exercise sheet they will need that statement, so looks like I gotta do it :S

Secret

Currently, I lost track of things whenever I saw the words "homology", "homotopy", "morphisms" in a body of text I am reading in maths

I still yet to make them more tangible to stay in memory

s.harp

have you done complex analysis?

DHMO

@Secret do you know Green's theorem?

Secret

1:36 PM

Never had the chance formally (due to undergrad class crowding), but I do end up reading a couple of lecture notes myself from other unis, thus I knew a couple of things from cauchy integrals to residuals

Green's theorm is taught in multivariable calculus courses

DHMO

@Secret Cauchy's integral theorem?

Secret

yup

DHMO

Another way of stating it is that "line integral is homotopy-independent"

would that help you develop some intuition on what "homotopy" means?

Secret

Wikipedia gave a nice pictorial intuition on homotopy It's when they start going into homotopy groups the whole thing become quite abstract as they are buried in those $\pi_i$ maps

DHMO

alright

Secret

1:39 PM

When reading classical groups like GL, U, SO etc., this keep popping up

but they are relatively tame compared to the bazillon types of morphisms

Everytime I saw a group $G$ doing something like $\textrm{Aut}(G)$ my brain starting drawing arrow from a featureless sphere and it end up going nowhere

While I knew what Aut means, it is sometimes hard to follow all the morphisms of a group, let alone "visualise mentally" or folow by pen and paper its structure

Basically, I am bad at visualising something that is "in progress". Unlike things like groups, rings etc., which kinda "sits there" and then you investigate it, maps and morphisms act between two things thus kinda "something in progress". This is the exact same reason I am bad at real analysis

Because most things in analysis don't "sit there" Take limits of a sequence for example

Secret

Meanwhile I am reading this

Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the ancient Greek word, "harmonikos," meaning "skilled in music." In physical eigenvalue problems it began to mean waves...

which is a nice folow up after potrygian duality

Mees de Vries

2:07 PM

I am trying to explain something to someone in the comments of a question, and it is getting a little long, so I want to use the "move this discussion to chat" option

But it just gives me an error message, "Moving discussion to chat failed".

Semiclassical

2:26 PM

@MeesdeVries link?

Mees de Vries

The space of sequence is extremely elusive a concept to me. The sequence ($x_{n}$) are elements in a field of scalar. If, say, $x_{1}$=1. What exactly is $x_{n=1}^{k}$ for all natural number k? How does one even have a sequence for a point? — Mathematicing 11 mins ago

Oh, huh. It changes it to a quote automatically. That's new to me.

0celo7

@Semiclassical New QM problem set ;_;

First problem is linear algebra because everyone got the linear algebra question wrong on the midterm. Ah! The midterm

@Semiclassical I won't scan it because he's collecting it again.

But I can tell you what was on it

@Semiclassical We were given an operator on a 2D Hilbert space. We had to show that a certain construction with it is a projection operator. We had to write down the matrix rep for the operator and its adjoint. We have to compute the eigenvalues and eigenvectors of a Hamiltonian constructed with it

We had to compute a probability at time $t$ using this Hamiltonian, and had to explain why some other construction with the operator is an observable

That was problem 1

Problem 2 was an SHO problem, construct a maximum $\langle x\rangle$ state using only $|0\rangle$ and $|1\rangle$

Compute the time evolution of this state, then compute $\langle x\rangle_t$ in the S and H pictures

Now compute $\Delta x^2(t)$ using both pictures

Problem 3: box potential, what is the condition for bound states

Problem 4: solve the double delta potential S eq for a bound state

shai horowitz

2:45 PM

at what speed would you have to listen to a audiobook i.e. 1x is normal 2x is twice as fast, to remove all the redundancy from English so its as if your listening to an efficient book at 1x speed?

s.harp

2:56 PM

100000000x

Semiclassical

3:13 PM

@0celo7 Sounds tedious but straightforward

in fact, that pretty well describes all of those problems

0celo7

I think everyone got it wrong. Too much algebra

Semiclassical

perhaps, yeah

i imagine it has something to do with the fact that any diagnolizable operator can be written as a sum of projectors

0celo7

@Semiclassical Have you ever seen the matrix $U=(a_0+i\vec\sigma\cdot\vec a)(a_0-i\vec\sigma\cdot\vec a)^{-1}$?

Semiclassical

I feel like I ought to have

0celo7

I recall it from QFT, perhaps. Maybe I should check Weinberg

But it's on my QM homework. I'm thinking one needs $a_0^2+\vec a^2=1$ for it to be unitary

but that restriction is not stated

Secret

3:16 PM

The stuff in brackets reminds of quternions

Semiclassical

It should. That's basically just a quaternion written in terms of pauli matrices.

DHMO

is there any connection between polynomial convolution and convolution?

Semiclassical

I expect that you can reduce that down to the case of $(a_0+i \sigma_3 |a|)(a_0-i \sigma_3 |a|)^{-1}$ by an appropriate rotation

Secret

polynomial convolution is basically a discrete version of convolution. In fact multiplying two polynomials is a discrete convolution between the two

DHMO

@Secret thanks

Secret

3:18 PM

Expanding terms is also a thing I don't like as I always make mistakes and miss terms

writing the multiplication as a cayley table format does help somewhat in keeping track of the terms

0celo7

apparently one doesn't need that condition

back to the drawing board

Secret

Meanwhile: PHYSICS!

Special unitary group

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on Cn. It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ GL(n, C). The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and...

this is where I am now in this more than 2 weeks of group theory self study

0celo7

3:33 PM

Good god. @Semiclassical was your grad QM class this computation intensive?

Semiclassical

eh, is that really so bad? You've just got $R=e^{i \vec{\sigma}\cdot \frac{\vec{\phi}}{2}}\oplus 1$ or something like that

0celo7

No it's not so bad if you write out the exponential using the usual identity

Semiclassical

Plus, the left-hand side is pretty much begging to have the BCH identity used

s.harp

Does somebody know of an example of a function space where $\mathcal F(A)\otimes\mathcal F(B)$ has a canonical inclusion into $\mathcal F(A\times B)$ that is not also an isomorphism?

0celo7

$\exp(-i\sigma\cdot n\phi/2)=\cos(\phi/2)-i\sigma\cdot n\sin(\phi/2)$

@Semiclassical No, that's overkill

You can write down the exponentials in one shot

Semiclassical

3:39 PM

true enough

0celo7

Then just matrix multiply

Semiclassical

Seems pretty straightforward.

0celo7

Straightforward but tedious

I want more conceptual homework

s.harp

I find the font in that image to be ugly

really ugly

but its probably just computer modern at 11pt blown up to 30pt or something

Secret

I am guessing for the 2nd question, rewrite it in index and use $[\sigma_i,\sigma_j]=i\epsilon_{ijk}\sigma_k$

Mahmoud

3:47 PM

Hello

.

Hey @Semiclassical :D

Balarka Sen

@DHMO What you suggested is also a good gateway to understand cohomology too. Fix a subspace $U$ of $\Bbb R^2$; Let $C_0$ be the vector space of smooth functions on $U$, $C_1$ vector space of "1-dimensional integrands", $f dx + g dy$ (think of it as a vector field $X = (f, g)$), $C_2$ be the vector space of "2-dimensional integrands", $f dx dy$. Then define $d : C_0 \to C_1$ as $df = f_x dx + f_y dy$, and $d : C_1 \to C_2$ as $d(f dx + g dy) = df dx + dg dy = (g_x - f_y) dx dy$.

Using the identification with vector fields earlier, the former is gradient, the latter is curl. Check that $d^2 = 0$ (curl of grad is 0!). Green's theorem becomes $\int_D d\omega = \int_{\partial D} \omega$, where $\omega = f dx + g dy$ is in $C_1$. If $\omega \in C_1$, $\ker \omega = 0$ means the corresponding vector field is irrotational. $\omega \in \text{im} d$ means it's conservative.

A natural question to ask is, "which irrotational fields are conservative", which is precisely what $\ker d/\text{im} d$ (two different d's) measure. That is the 1st cohomology group $H^1(U)$ of $U$.

Semiclassical

hi @mahmoud

teadawg1337

Hey everyone, it's been a while...

DHMO

@BalarkaSen sorry i know you used a lot of effort to type that but i must say that i do not understand...

Semiclassical

speaking of cohomology, here's something neat I've been playing with (though not very elegant)

Balarka Sen

3:49 PM

Sure, I just wrote it down, didn't expect you'd understand.

DHMO

good

Semiclassical

suppose I've got a Riemann surface $y^2=x(x-1)(x-\lambda)$. That's an elliptic curve, and since it's got genus 1 the first homology group is 2D. hence, so is the first cohomology group.

as i understand it, a basis for that is $dx/y$ and $x dx/y$

What I've been grappling with is how one practically uses that fact, e.g. how to express $x^2 dx/y$ in that basis (up to an exact form)

Balarka Sen

shrug

Semiclassical

currently the only way I know how to do that is to observe that $dx/y=d(x/y)-x d(1/y)=d(x/y)-x dy/y^2$

Mike Miller

pick a diffeomorphism with R^2/Z^2, and then integrate it along the two obvious loops. if the result is (a,b), it's equivalent to adx + bdy

Balarka Sen

3:53 PM

he has weird basis though

Mike Miller

better yet, just pick any basis for homology, and integrate those two forms on it, dx/y and xdx/y

then integrate x^2dx/y along them, and figure out what linear combo you have to be

Semiclassical

Easier said than done, I think. If I do those integrals, I get elliptic integrals and it's not obvious how to break them apart

Hence why I wanted a way purely at the level of algebra

but, I wasn't quite done

Mike Miller

i see no particular reason that there's an algebraic way of finding a decomposition of a form given by rational polynomials

just have matlab do them? :)

Semiclassical

in what I wrote above, i've got $dx/y=x dy/y^2$ up to an exact form (minus sign error earlier)

but I know that $2ydy=d[x(x-1)(x-\lambda)]=p(x)\,dx$ for some quadatic polynomial $p(x)$

hence $\frac{dx}{y}=x\frac{dy}{y^2}=\frac{x p(x)}{2y^3}dx$

Mike Miller

you've convinced me mainly that i don't wanna do this

:)

Semiclassical

3:59 PM

since $xp(x)$ is cubic, I can write $xp(x)=Q(x)y^2+R(x)$ where $Q(x)$ is linear and $R(x)$ is quadratic

teadawg1337

Does anyone remember me? I gave up on mathematics for around a year, but now I'm in between jobs with nothing else to do

Semiclassical

so therefore $\frac{dx}{y}=Q(x)\frac{dx}{2y}+R(x)\frac{dx}{2y^3}$

Mahmoud

@teadawg1337 I don't, but welcome back to the Mathematics Community. :)

Semiclassical

I can do that similarly for any $x^k (dx/y)$ and so obtain linear relations between forms $x^k dx/y$ and forms $x^k dx/y^3$

and if I get enough of those, I can express everything in terms of just $dx/y$ and $x(dx/y)$ purely from algebra

teadawg1337

I haven't been on here in over a year, so I wouldn't expect many people to remember me :P

Semiclassical

4:02 PM

name sounds vaguely familiar but I don't really remember :/

for instance, I get $$x^2 \frac{dx}{y}=\frac{2}{3}(1+\lambda)x\frac{dx}{y}-\frac{\lambda}{3} \frac{dx}{y}$$

Which is simpler than one might've expected

Mahmoud

Is my virtual existence on this server annoying to anyone ? I mean all what I do is ask questions and post paradoxical math subjects.

DHMO

@Mahmoud "virtual existence" is an oxymoron

Mike Miller

sorry, @Semiclassical, it seems like you just answered your own question?

Mahmoud

Exactly ! Is this annoying ?

DHMO

@Mahmoud not at all

Semiclassical

4:07 PM

I did. I was more wondering if there was an obvious way to make it simpler

Mahmoud

So it's annoying ...

Semiclassical

i'm guessing the answer there is "i dunno"

DHMO

@Mahmoud it means it is not annoying at all

english

Mahmoud

No it means it's not at all annoying, but a little.

Mike Miller

@Semiclassical yup

DHMO

4:08 PM

@Mahmoud don't twist my words

Semiclassical

fiine

Mahmoud

You should have written No, at all. :P

Anyway, it's just chat.

Mike Miller

@Mahmoud what he said was perfectly good english to mean that you're not annoying

DHMO

@Mahmoud it's just a translation of "pas du tout"

@MikeMiller thanks

Mahmoud

O.O

French ?

Mike Miller

4:09 PM

@Semiclassical this does give you some nice solutions to elliptic integrals

Français ?

it does

@DHMO

that's the motivation, in fact

though I'm trying to move towards a case that'd be periods of a non-hyperelliptic curve rather than an elliptic one

whiiiich is pretty terrifying

DHMO

@Mahmoud I know some French

Mahmoud

4:11 PM

C'est une surprise pour moi !

Mike Miller

i realized a problem in a proof i was explaining to my advisor yesterday, and was terrified for a bit

DHMO

@Mahmoud pour

Mike Miller

and this morning i realized there's a more subtle and harder problem

(but the previous problem probably doesn't matter)

Mahmoud

@DHMO :D

DHMO

@Mahmoud alors tu parles trois langues?

Mahmoud

4:18 PM

Français, Anglais et Arabe, oui mais je n'aime pas parler en français beaucoup.

DHMO

alright then

Mahmoud

English is much more elegant and it is a whole lot easier.

Even french people have to learn English :P

DHMO

elegance and easiness are subjective

Mahmoud

Well at least in my point of view, I meant

I don't have to include that in my speech, it is automatically determined once you realize that it is ME who wrote that down.

DHMO

@Mahmoud alright

Mahmoud

4:22 PM

@DHMO Any ideas about INVENTING usernames ?

For YouTube channels ?

DHMO

@Mahmoud no idea

Mahmoud

I driving myself crazy about finding the perfect unique username.

DHMO

perfectness is subjective

Mahmoud

Well almost everything is subjective,

Science is not.

DHMO

precisely

teadawg1337

4:25 PM

You'll never come up with a single username if you're searching for "the perfect one"

Mahmoud

@teadawg1337 I just want it to be unique and has a ambiguity in it.

But that requires days and days of thinking and searching for inspiration.

teadawg1337

Everyone has a unique personality, so try to find something that expresses yours

Mahmoud

They get bulky.

Because I don't know many English words

teadawg1337

Who says it needs to be written in English?

DHMO

@Mahmoud then produce them in Arabic (and translate to English)

teadawg1337

4:29 PM

You could produce a random string of characters and call it a username

Mahmoud

@DHMO Genius ! I can even write it down without translating !

0celo7

@s.harp it's not compute modern

@Semiclassical If I write down an $\mathrm{SU}(2)$ matrix and try to identify it with a rotation $(\hat n,\phi)$, I get a sign ambiguity from some square roots. That's just because it double covers $\mathrm{SO}(3)$, right?

Semiclassical

probably? i haven't done the stuff lately

but overall signs don't matter when it comes to wavefunctions anyways

Mike Miller

that's correct

Josué

hi

JeSuis

4:45 PM

hi

DHMO

hi

0celo7

@Semiclassical I should pin a table of trig identities to my desk

Semiclassical

not a bad thought

Mahmoud

4:56 PM

Hi.

Josué

If $f:X\to\mathbb C$ is holomorphic, is it always the case that $f|_Y$ is holomorphic for every $Y\subseteq X$?

I think yes since $f$ being holomorphic implies that for every $z\in X$, the Cauchy-Riemann equations $u_x=v_y$ and $u_y=-v_x$ are satisfied, where $f(z)=f(x,y)=u(x,y)+iv(x,y)$, which also holds for every $z\in Y$.

0celo7

@Josué If $Y$ is open, I would expect so. Complex differentiability is a local condition.

If $Y$ is not open you'll have problems defining derivatives on it.

s.harp

5:11 PM

You can define complex differentiability on any subset of $\mathbb C$, namely $f$ is complex differentiable at $z$ if $\lim_{z'\to z} \frac{f(z')-f(z)}{z'-z}$ and the limit is taken so that only points the set you are working with are considered

and in this case it is obvious that $Y\subset X$ implies $f\lvert_Y$ is holomoprhic

but you now also have a little pathology: if $\bigcup_i A_i = X$ and $f$ is holomorphic on all of the $A_i$, then $f$ is not necessarily holomorphic on $X$

if the $A_i$ are open, then this pathology goes away

Josué

So it's like a necessary but not sufficient condition.

s.harp

You should be aware that one almost never calls functions defined on a non-open set holomorphic, regardless of whether or not they are complex differentiable in the above sense.

But yeah, in the above sense knowing that $f$ is complex differentiable on subsets that cover the space is not enough to know that $f$ is complex differentiable on the space. But $f$ cannot be complex differentiable if it is not so on all subsets.

Josué

I see. Thank you.

Tobias Kildetoft

@AlexClark Neat. Do you still have access to some computers that can run GAP? I have some code I would like to run for a while, and I will soon lose access to my office computer as my employment ends (and I only now finally got around to setting it up on that). This will require an old version of GAP though (however, I think Peter McNamara could be interested in the results of these).

Edward

5:46 PM

I don't visit here often, but thought that some of you might be interested that one of the most popular recent questions on CodeReview is arguably more about number theory than programming.

55

Q: Disproving Euler proposition by brute force in C

I wrote a little bit of code to brute force disprove the Euler proposition that states: $$a^4 + b^4 + c^4 = d^4$$ has no solution when \$a\$, \$b\$, \$c\$, \$d\$ are positive integers. I'm no mathematician, but reading around this, at least one solution was found by Noam Elkies in 1987...

beginner c multithreading time-limit-exceeded mathematics

@NoamD.Elkies: You might be interested in reading through some of the answers and comments to that question since it addresses your 1988 paper.

Mike Miller

@Edward That won't ping him.

Edward

I didn't know that. Anyway, it's more for his amusement than anything else. Not critical.

I read the associated paper (or attempted to!) and learned a couple of new terms, but haven't yet fully understood them.

A "pencil of conics" was new to me.

Mike Miller

There's quite a lot of number theory that effectively proceeds by starting by working in algebraic geometry and reducing to a computationally simpler problem. See also "Twists of X(7)" by Poonen-Schaefer-Stoll for a variation on a theme: there they calculate all the integer solutions to $x^2+y^3=z^7$.

Mahmoud

6:03 PM

Can anyone please tell me how to use programs on GitHub, sorry it's out of the subject.

Josué

It's usually not straightforward. Most of what you'll find at GitHub is source code, which means that you'll have to download and compile it yourself before you can run it.

Tobias Kildetoft

@Mahmoud depends on what has been uploaded. Mostly you would need to compile and run them yourself

Mahmoud

@TobiasKildetoft It's written in python.

Balarka Sen

I don't have much of a clue on what kind of tools in algebraic geometry helps figuring out rational solutions. At a glance and without reading anything, Elkies 1988 seems to be setting a birational equivalence between $X^4 + Y^4 + Z^2 = 1$ and some elliptic curve, where the story is known, I guess.

Mahmoud

I don't even know where to start. :(

Edward

6:06 PM

@Mahmoud Generally git clone xxx where xxx is the URL for the project. Then navigate into the project. There is often a README file.

Tobias Kildetoft

@Mahmoud well, then you need to figure out how to run python code

Josué

Python is interpreted. In the best case scenario, you can copy the code and paste it into an interpreter.

Mahmoud

I'm so not familiar with this.

I think there is a way to run it into GitHub's desktop program.

Edward

@Mahmoud Try this: try.github.io/levels/1/challenges/1

Tobias Kildetoft

@Edward that seems to be precisely what he is not looking for :)

Edward

6:11 PM

:)

Mahmoud

@Edward :) I want to use an existing program :)

Tobias Kildetoft

@Mahmoud If you are not familiar with how to run source code then for the most part, GitHub will be the wrong place to find stuff

Edward

@Mahmoud Everything I know about git, I learned from this book: git-scm.com/book/en/v2

Read chapters 1 and 2 and you'll probably know everything you need to know to fetch the program you want.

Mahmoud

Thank you guys that was helpful, just replying to my stupid question was already amazing :D

Tobias Kildetoft

@Edward These are not really Git questions at all. They are essentially unrelated to Git

Edward

6:15 PM

@TobiasKildetoft You may be right.

Mike Miller

@Balarka There is a complicated general story that I would know if I was a number theorist.

Mahmoud

At least how to use this one github.com/3b1b/manim

Edward

@Mahmoud If you've already read and tried the instructions there, asking more specific questions in a more suitable venue about whatever particular problems you've encountered is likely to be more fruitful.

Mahmoud

@Edward How do I run the commands ?

NaCl

yo

Can somebody give me a hint on how to find a inverse function if the original function $f$ maps from $A\times A\to A$? (It's assumable that $f$ is inversible)

It's pretty easy for functions where the "from-dimension" and "to-dimension" is equal, however, how to proceed here?

0celo7

6:39 PM

Do you have a specific function in mind?

@NaCl [citation needed]

NaCl

Take a look at this: szudzik.com/ElegantPairing.pdf (page 8)

Tobias Kildetoft

@NaCl That last is not true unless the functions are of some "nice" form

NaCl

How did Szudzik find the inverse here?

0celo7

try to invert something horrible like $\Gamma(x)+x^2$

Tobias Kildetoft

@NaCl I might have been interested in reading more than a few lines of that, except he very quickly refers to that stupid book by Wolfram

0celo7

6:42 PM

Stupid book?

Tobias Kildetoft

@0celo7 A new kind of science

0celo7

Yeah I figured out what book, but what's wrong with it?

Tobias Kildetoft

@0celo7 More or less everything from the point of view of mathematics. There are some rather scathing reviews of it by mathematicians

Edward

Short version: shell.cas.usf.edu/~wclark/ANKOS_reviews.html

0celo7

Ouch

uh

Those links are all broken

> In this note, we disprove 44 claims in [4] on minimal Boolean
formula size of one-dimensional two-state nearest neighbor cellular
automata as well as set a new upper bound.

lol

Tobias Kildetoft

6:46 PM

I am only now realizing that his manner of writing is very similar to the way in which Trump speaks.

0celo7

What's wrong with having the best words?

Tobias Kildetoft

Having the best words is obviously the best thing.

0celo7

Maybe we should build a wall around your country

NaCl

we should build a wall around trump