Mathematics - 2017-02-27 (page 1 of 5)

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12:02 AM

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does this system have a solution?

on the nonnegative integers?

Write down the number of equations and the number of variables. If there are more variables than equations, then there are infinitely many real solutions and I would be surprised if you can't find an integer solution.

I don't know for certain, though, because I have no interest in trying to examine that more closely.

Ok.

is there a closed form for $\sum_{n=1}^\infty \frac{x^n}{2^n-1}$? Or if you replace 2 by some other number

Thank you, @ThomasWard.

Mathematica doesn't return one.

That doesn't mean there absolutely isn't one, but it gives me little reason to hope for one.

That said, Mathematica does return a closed-form solution for $x=1.$

just go a Sigma deeper

and accept it as "simple" and "closed form"

approach works on basically any hard to crack problem

12:14 AM

To be precise, it yields 1-QPolyGamma[0,1,1/2]/Log[2] for $x=1$.

does it give something for $\sum \frac{1}{x^n-1}$?

Oh, that kindof gives me hope for a hypergeometric solution

Starting at n=1 or n=0?

n=0 makes it not exist

Huh, you're right. Oddly, Mathematica gave a sum for n=0 (and for n=1).

I think it's just being silly, though.

12:18 AM

this one? wolframalpha.com/input/?i=sum+n%3D1+to+infty+1%2F%28x^n-1%29

Yeah.

12:34 AM

i'm home

12:47 AM

hey everyone

@AliCaglayan do you want to discuss something small in commutative algebra

sure

hi @Adeek

also my wrist is hurting more and more :(

hi @ZachHauk

@AliCaglayan I am just checking tensor product properties for modules. Just to verify my understanding I checked many of them there is something I am stuck on.

ok shoot

@AliCaglayan Suppose M,N, and P are A-modules.

12:50 AM

@ZachHauk haha you can interpret this sentence in whatever way you want to

...

Consider the map from $(M \times N) \otimes P \rightarrow (M \otimes P) \times (N \otimes P)$ given by $(x,y) \otimes p \mapsto ((x \otimes p), (y \otimes p)$. This map is well defined since the map $((x,y),p) \mapsto ((x \otimes p), (y \otimes p))$ is bilinear.

playing too much games is the cause

@ZachHauk do math instead of games :P

in a sec

when i finish this game

12:52 AM

so we get a well defined map from $(M \times N) \otimes P \rightarrow (M \otimes P) \times (N \otimes P)$

But then I am having troubles constructing the two sided inverse

where should we send $(m \otimes p_1,n \otimes p_2)$ ?

I don't think it has a two sided inverse

well how else is that an isomorphism ?

Well the element you gave is definitely not in the image of the map

unless p1 = p2

yeah I know but I think there is some

trick with tensors that get that element to be inside of the image somehow I don't know

user image

@AliCaglayan see this map should be an isomorphism somehow I don't know why though

@Adeek I found this arbourj.wordpress.com/2012/04/17/…

12:59 AM

oh @AliCaglayan thanks

Hi, everybody.

There are unique homomorphisms from M (X) P to (M x N) (X) P

and for N (X) P

yeah I see

I'm a physicist. I'm reasonably good at math. I would like to understand stochastic processes better than I do now. In particular, I'd like to be able to see the links between diffusion-type equations, stochastic differential equations, and Langevin type equations.

Thus you can take their sum (product) to get homomorphisms blah blah

1:02 AM

yeah

@Adeek As far as I can remember, the notation is the most difficult part when studying this

I'd really like to find a resource that explains these things in enough depth that I believe the results, and such that I can apply them to real problems. At the same time, I don't really need to see all the epsilon-delta proofs.

Anyone know of a reference like that?

yeah @AliCaglayan also there is no intuition in regards to tensor products

I mean weird things happen.

@DanielSank any physics book on the aforementioned topics?

@AliCaglayan Such as?

I'm making this request so I don't have to go on book safari from scratch ;)

1:04 AM

@DanielSank just google some, something will come up

Yes, of course, but there are more crap books than good books in the world.

higher level physics books usually don't mind getting into the maths

When people ask for resource recommendations, they're looking for more than pick-a-random-google-book, usually to save time.

I'm sure he's aware. See "I'm a physicist".

@DanielSank maybe ask on main site what would be a good book recommendation about this.

1:04 AM

For example, if someone asked about statistical physics, I'd have opinions about what to read.

@Adeek Sure. Thought I'd try here first.

@MikeMiller Yes, thank you.

@Adeek the tensor product is quite intuitive in linear algebra

I would not recommend someone to read any random stat. mech. book they find on Google.

...because most of them are crap.

its really just transfering that over

yeah but over modules it is not

hi @Daniel

1:06 AM

hello

Consider the problem $U_{n+1} = U_n + h \Lambda U_n$. $\Lambda$ is diagonizable and is eigen-decomposed to $\Lambda = VDV^{-1}$, where D and V $\in R^{d\times d}$, D is diag. How to proceed the induction to show that the Forward Euler method applied to this problem gives $U_n = V(I + hD)^n V^{-1} y_o$?

@AliCaglayan for example if you consider $(x \otimes y)$ as element in $\mathbb{Z} \times \mathbb{Z}_2$ then it is zero.

while

as element of $2\mathbb{Z} \times \mathbb{Z}_2$ it is not zero

I thought that was kinda of weird but yeah the logic follows

@DanielSank Anyway, you'll probably get an answer from either @Semiclassical or nobody here.

@ZachHauk I'm surprised you've even heard of algebraic geometry in middle school. Hah!

Amazing.

@MikeMiller Ok well thanks for pinging him/her to see the question.

@DanielSank I think you would be even more surprised at the talent of @AkivaWeinberger and @BalarkaSen

they're a bit older, but much more knowledgable

1:08 AM

I think I may have heard of algebraic geometry by high school, but only because Derbyshire's popular book on the Riemann Hypothesis mentioned it.

right now all i'm studying is non-euclidean geometries

@ZachHauk Perhaps. But comparison is unnecessary as regards my observation that it's surprising that you've heard of algebraic geometry by middle school.

then i'll be studying algebra

from Artin's

@Semiclassical Supposedly you know things about stochastic processes...

Can't say I know anything about stochastic processes, alas.

1:09 AM

@Semiclassical dang

he's the closest I could think of

Back to Physics SE.

Thanks, everybody.

@MikeMiller since i can't go, you should volunteer for mathcamp next year

and you'll also see Akiva

Depending on your background, though, you might look at the last chapter or two from Altland and Simon's book on Condensed Matter Field Theory.

"Volunteer" isn't quite the right word ;) But if I get a chance I'll try to apply to work there next year.

1:10 AM

is it paid?

brb everyone.

sorry i thought it was like something volunteer-like

Yeah, it's paid.

At least the teaching positions are.

@Adeek have you done tensor-hom adjunction yet?

By next year i'll probably have the money for the camp

1:11 AM

no not yet @AliCaglayan but I will

@AliCaglayan btw I solved first chapter of Michael atiyah

some problems were very challenging

Nice

anyway I better get going I want to prepare for my midterm for commutative algebra

Atiyah-Macdonald?

gonna ace this ***

or mac-whatever

1:12 AM

I am working through chapter 2 problems on Hartshorne

yeah @ZachHauk

hey @Akivaaaaa

Each problem takes a good few pages

@AliCaglayan I heard that was a very dry book

but probably because my writing gets bigger

1:13 AM

@AliCaglayan cool. I will be there through the summer. Also, I will working through allufi and dedicating more time to it.

@ZachHauk I kept mine dry because its inside

almost as dry as my humour

Akiva Weinberger

Yo

or as dry as my soul

@ZachHauk you are too young to know these things exist

I think most higher level math books are dry. But, I figured that to go through them with pen and papper and working every little detail makes it fun.

1:14 AM

do your trignometry homework :P

or don't exist @AliCaglayan haha

@AliCaglayan no i don't care about 30-60-90 triangles

@AkivaWeinberger what math class are you actually taking this year?

Graduate texts tend to be "dry" but thats because they are no nonsense

Trig is useful but in general not interesting.

I keep meaning to link this here:

37

Q: Does the drunk man fall off the cliff? (a random walk problem)

A drunk man stands with a cliff one step to his left. He takes steps randomly left and right. Each step has probability $p$ of going left and probability $q=1-p$ of going right. Each step is the same size. If allowed to randomly step indefinitely, what is the probability that the drunk falls off...

mathematics probability

1:15 AM

You have to really be motivated to read it

I think y'all might find the variety of solutions rather amusing.

Hi can somebody try answering math.stackexchange.com/questions/2161205/…

please ...

@Semiclassical yeah

i agree

yeah

@A---B Hmmmm, looks wrong. The gradient transforms as a covector.

1:16 AM

anyway brb @AliCaglayan I will let you know if I encounter really interesting problem in MT

@DanielSank What is a co vector ?

ok @Adeek take care!

@A---B Ah, wait a minute. This problem specifically asks about rotations.

Hi there.

Ok forget what I said for now.

1:17 AM

Any idea on how to compute $$\sum_{i=1}^{n} \frac{2n}{(-3n+2i)^2}$$?

@DanielSank I got this question from electrodynamics book.

Within the context you're working, there's not a distinction to be drawn.

(If you try a coordinate transformation which involves stretching instead of just rotating, you'll see that the transformation is not quite what you expect)

@Semiclassical Right, I shouldn't have mentioned covectors. Sorry.

Without using integrals.

Hey so I'm new around here. How do "check my work" questions work on this site?

On Physics, we're pretty harsh on them, i.e. we don't support straight-up check my work.

1:19 AM

Which isn't surprising: A rotation doesn't change the density of charge in a system, whereas a stretch would.

@Topologicalife Just curious, why no integrals?

@Semiclassical Right. Exactly.

@DanielSank I asked around and nobody I know who would know knows.

@DanielSank because that sum comes from trying to evaluate an integral using riemann sums.

@MikeMiller What are you referring to with that message?

@DanielSank I don't know but my check my work always get answered

1:19 AM

Ref request.

Akiva Weinberger

@ZachHauk Real analysis with Rudin's textbook

i mean like in school

@Topologicalife Oh, lol.

:)

Akiva Weinberger

@ZachHauk Yes.

1:20 AM

Not sure you saw what I said after that, but you might check out the second-to-last chapter in Altland & Simon's text on condensed matter field theory. @DanielSank

you have a real analysis course?

I don't know if there exists an analytical way.

Akiva Weinberger

I'm in a special class with just me and the teacher

I found another way to compute the riemann sum, but I'm curious about that sum.

aww lucky you

1:20 AM

@Semiclassical @DanielSank Atleast can you show me how to derive the equation in the hint ?

@Semiclassical I happen to have that on my shelf!

Neat.

I should say, I have the second edition in mind.

@AkivaWeinberger how did that happen?

I don't know how much nonequillibrium stuff is in the first.

@Semiclassical Will check. Thanks.

1:22 AM

(Second-to-last chapter in the edition I know is classical nonequillibrium theory, which is what you'd want. Last chapter is quantum nonequillibrium theory, stuff like Keldysh.

@AkivaWeinberger that is, how did you get the school to let you do that?

It seems there is no way to compute analytically this sum

$${\partial f\over \partial \overline y} = {\partial f\over \partial y} { \partial y\over\partial \overline y} + {\partial f\over \partial z} { \partial z\over\partial \overline y} $$ I was talking about this equation.

@A---B For what it's worth, if you go to the physics site's chat room and ask 0celo7, he will give you a good answer. In fact, I will ping him now with a link to your question.

to clarify, those overlines are for the rotated coordinates?

1:24 AM

@DanielSank what is the question?

I'm physicist.

@Topologicalife scroll up. It's about transforming a gradient under a rotation.

@Topologicalife math.stackexchange.com/questions/2161205/…

@A---B what is $\overline{y}$?

@Topologicalife y coordinate after rotation.

rotation around the x-axis, it would seem

1:27 AM

@Topologicalife It's just the new coordinates.

@Semiclassical I'm looking at Atland and Simons. The second to last chapter is on renormalization.

@Topologicalife $(x,y, z)$ would become $(x ,\overline y, \overline z)$

Drat. Must te the older edition, then.

@Semiclassical I know how to fix this. I can order a new version for my lab.

Is the cover pink and black? (I have the newer edition, so I'm having to remember.)

Akiva Weinberger

@ZachHauk It involved talking to the administration, I don't remember exactly. The fact that I took the AP Calculus BC test right before high school and got a 5 (out of 5) probably helped.

1:28 AM

It's justified since I want to know about this for work ;-)

Lol

@Semiclassical Cover is purplish and black.

So you are trying to prove the gradient has the property of rotational invariance?

hmm

@AkivaWeinberger how do i do that? sorry for being pushy, i don't want to be stuck with boring classes for the rest of high school :(

1:29 AM

Yeah, the newer edition has a green-blue front

@Topologicalife Nah, he/she is trying to prove that it's coordinates transform like a vector under rotations.

@Semiclassical To the Amazon!

Huh, there's a pdf copy of it just sitting out there on google...

that is, take the AP test

(seriously, it's the first link if I just search "altland and simons")

@Semiclassical I strongly object to that. Writing a book is very hard and very time consuming.

I know because I have an 80% done one.

1:30 AM

Yeah.

It's been 80% done for like fifteen years.

Ah.

I think that now I understand your question.

@Semiclassical book ordered. Thanks.

It's a good one, yeah.

Though I've never quite gotten as much out of it as I'd like.

I noticed just now that in a chapter I never read before, there's a discussion of dissipative quantum tunneling. I am rather interested in reading that as it's directly relevant to my day job!

1:33 AM

Oh, nice.

Okay, why don't you use the matrix transformation of a rotation? i.e $$\left( {\begin{array}{cc}
\cos\phi & \sin\phi \\
-\sin\phi & \cos\phi
\end{array} } \right)$$

Yeah.

The section on spin coherent states was directly relevant to one of my projects in the last few years.

(Though we ended up using the original source more than Altland and Simons)

@Semiclassical Neat. I actually could stand to learn about boson coherent state path integration. I saw it in a course and I understood why it was nice, but I can't reconstruct it now.

@Topologicalife Won't it give the same result ?

Yeah.

1:34 AM

My lab works on quantum computer hardware. In one particular case, dissipative tunneling is really important.

Yeah.

Akiva Weinberger

@ZachHauk I don't remember. Talk with your school about maybe taking it. But definitely study and take tons of practice tests

Sure.

I always wished I could follow the theorist's discussions more closely.

I'm sort of a wannabe theorist turned experimentalist ;-)

A lot of the stuff I've done over the last few years has been in the realm of semiclassical methods in simple quantum systems (hence the name).

1:35 AM

Ah, makes sense.

And tunneling is part and parcel of that.

My name is Daniel Sank, hence the name.

It should give the same.

I figured :)

heheh

1:36 AM

You can comprobe it doing in the way I mentioned. I didn't check your work.

@AkivaWeinberger the only thing i don't really remember is $\epsilon-\delta$ def. of limit

Oh by the way, anyone who was looking at that random walk problem, I'm particularly interested in the meaning of $z>1$ in the answer written by me.

otherwise im pretty set on calculus lol

@DanielSank actually, that riemann sum comes from consider the integral $$\int_{-3}^{-1} \dfrac{1}{x^2} dx$$ Easy one with hardest sum I've never seen :P

You'd probably recognize the kind of problem I'm working on, then.

1:37 AM

@Topologicalife No need to check my work, just show me how did we get that equation in the hint.

Lately I've been interested in the Landau-Zener problem, or more precisely its generalization to multiple levels.

@A---B that is just the chain rule.

@Semiclassical Aha! Rather important, I would say.

Quite!

By the way, if $\vec{v} \cdot \vec{w} = 0$ and $\vec{v} \cdot \vec{x} = 0$, then $\vec{v} \cdot (\vec{w} + \vec{x}) = 0$, right?

1:38 AM

@Topologicalife Interesting.

Or at least, the Landau-Zener problem is. Not sure what I'm looking at really is :P

To be specific, I'm interested if there's a meaning to $z>1$ in this answer.

@Topologicalife I don't get that part. Chain rule on what ?

Namely, with the standard Landau-Zener problem you find an exact (if difficult) answer to what the scattering amplitude is.

Now, with a generic Landau-Zener problem (multiple levels, various couoplings, etc.)

because that would show that the set of all orthogonal vectors to some $\vec{v}$ is a subspace spanned by the most possible linearly independent vectors whose dot product with $\vec{v}$ is 0

1:40 AM

Things won't be so simple, namely you can't compute the scattering amplitudes in a simple way.

@Semiclassical Just as an aside, in quantum computing, it turns out to be very important to understand the two level Landau-Zener problem with arbitrary trajectories, i.e. not just the linear one that people usually talk about.

In fact, the linear trajectory is kind of stupid.

Sure. For one, the linear trajectory has the energy separation becoming arbitrarily large.

In particular, if you want to sweep through the avoided crossing with minimal state transfer, you have to find an optimized function.

@Semiclassical yeah

But, anyways, there are models where the scattering amplitudes can be done exactly, and quite recently it's been discovered that this collection of models is actually larger than people realized.

And I'm trying to understand that.

arxiv.org/pdf/1402.5467.pdf

@Semiclassical Cool!

1:43 AM

Here's the abstract for what the above is referencing: arxiv.org/abs/1701.01870

@Semiclassical So why is everyone so interested in linear time?

Is there a real world application?

@ZachHauk yep

alright good

shrug

The dot product is a linear function.

@A---B $$\begin{pmatrix}{\overline{y}}\\{\overline{z}}\end{pmatrix} = \begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta & \cos \theta \end{pmatrix}\begin{pmatrix}{y}\\{z}\end{pmatrix} $$

1:45 AM

In other words, if you define $f_v(x) \equiv v \cdot x$, then $f_v(a+b) = f_v(a) + f_v(b)$.

@Topologicalife Yes I know.

I mean, for the two-state version one can at least claim that it matters as a linearization of real problems.

But with regards to these multi-state models...ehhhhh

All I claim is that it's a neat mathematical-physics puzzle :)

so if $f = f(y,z)$ and $ y = g(\overline{y},\overline{z})$, $ z = h(\overline{y},\overline{z})$, then...

(There are probably some better defenses of it, but yeah.)

Apply the chain rule to $f$.

You will get $${\partial f\over \partial \overline y} = {\partial f\over \partial y} { \partial y\over\partial \overline y} + {\partial f\over \partial z} { \partial z\over\partial \overline y} \tag{+}$$

That's with abuse of notation. It should be $${\partial f\over \partial \overline y} = {\partial f\over \partial g} { \partial g\over\partial \overline y} + {\partial f\over \partial h} { \partial h\over\partial \overline y} \tag{+}$$

1:50 AM

@Topologicalife I get this. Wow thanks. it was easy I guess. Can you write an answer ?

I don't have time, sorry.

But if you have any question feel free to ask it here, I'm checking randomly this chat.

2

@Semiclassical Fair enough.

@Topologicalife No problem anyway thanks for clearing it up.

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