Mathematics - 2022-02-21

geocalc33

01:16

how would you accomplish a surface of revolution in $\Bbb R^2_{\gt 0}$?

say $\Bbb R_{\gt 0} \times \Bbb R_{\gt 0}=\Bbb R^2_{\gt 0}$

Hitshot

03:48

Let E be a (analytic) rank 1 elliptic curve over Q with nontrivial torsion and E' be a Q-isogenous curve to E.

If Ш(E / Q) is nontrivial, does that imply that Ш(E' / Q) is also nontrivial?
If so, does it also imply that ord(Ш(E / Q)) = ord(Ш(E' / Q))?

blah

06:03

Hello, I posted this question last friday but I haven't found an answer for it yet after scouring the internet. I was wondering if you guys could help. It's basically about finding the new (still proportional) top left (x,y) coordinates of an inner rectangle after the outer rectangle is scaled.

https://math.stackexchange.com/questions/4385268/calculate-inner-rectangle-new-top-left-position-after-outer-rectangle-is-scaled/4385383#4385383

Someone was able to send me a formula that could center the inner rectangle but what I want is to position it still proportional to its original position

mohan10216

If not it is probably impossible to determine, in my opinion.

I'm pretty sure we would need the length & width of the rectangle.

blah

I have the width and height of the rectangle, just need to calculate the new top left (x,y) of the inner rectangle

mohan10216

Then do the algebra

blah

Sorry I haven't done algebra work for years now, it just so happened I needed to solve something like this in this website I'm developing @.@ not very good with math, very sorry about that.

What formula should I use for this?

mohan10216

After that generalise it and prove that result :)

How would i know? I'm in school right now bro. Do some research.

if I knew the formula I would have just told you lol.

blah

06:23

yeah that was kinda what I was asking for...

mohan10216

But I mean you could just find it yourself with a little algebra

In all honesty it is really not that advanced, I think the generalisation part is the hard part.

let me annotate ur image a bit.

give me a sec my online whiteboard isn't loading...:(

@blah

Can you give me some numbers to work with.

blah

06:39

yeah just a sec

outer width - height: 480.0237645698818 148.93819110344342
outer x - y: 3.499999947845936 142.5
inner width - height: 85.53882724620006 71.45007922917887
inner x - y: 275.5 142.5
=====================
outer width - height: 477.0285656073882 148.00886312346634
outer x - y: 3.499999947845936 116.46973255832062
inner width - height: 85.00509157407078 71.00425296187088
inner x - y: 5.246434139941414 116.46973255832062
=====================
outer width - height: 471.7086609301619 146.35824280432695
outer x - y: 3.499999947845936 115.5733811664008

here are some, values of inner x- y are wrong though after the first one

mohan10216

06:57

Easier to generalise

wow that is really specific. I'm just gonna go with integers for now.

mohan10216

08:14

wait i'm confused, can you label them in terms of their colors in your diagram

@b

@blah

blah

outer = blue rectangle
inner = red rectangle

mohan10216

ok, give me some time.

blah

08:40

Thanks dude, I appreciate it, really. No pressure btw, you can just stop if you wanna. I'm also tryna solve it on my end

Huy

I want to write a short code using the `Manipulate[]` function in Mathematica. Namely, I want to be able to vary the degree of the Taylor polynomial `n` but with (for now) fixed function and fixed expansion point. If `n` is fixed, then it is simply:

`t[x_] = Normal[Series[f[x], {x, 0, 6}]];`

Note that using `:=` (`SetDelayed`) will cause an error here.
Now the issue is that if I simply replace the `6` with an `n` and use `t[x_,n_]` instead, the `n` would expect a `:=` (`SetDelayed`) instead of `=`. However, that is incompatible with the `x_`.

Balarka Sen

@Huy long time

how's it going

Huy

08:56

@BalarkaSen hi, you must be like grown up by now! I'm obviously still teaching. Trying to teach some basic numerical mathematics this semester, so I need update my Mathematica skills again. I used it like 10 years ago in uni and have forgotten everything.

Balarka Sen

im a first year grad student currently. cool idea to teach the kids mathematica (i don't know any of it, tbh).

Huy

where are you studying if I may ask?

Balarka Sen

still in india; tata institute of fundamental research.

Huy

nice. good to see some familiar faces in here. I missed Ted's birthday apparently.

Balarka Sen

yup. great to have you back

i lost the audio dramatization of the Durrenmatt play you sent me; do you still have it with you by any chance

Huy

09:00

I also have to teach multivariable calculus soon, so I'll probably need some help there too

Die Physiker?

Balarka Sen

cool stuff!

@Huy yup

Huy

I'd have to search at home. did I send it to you back then ?

Balarka Sen

yeah you did

i listened to it too, it was a good one

Huy

interesting. I forgot a lot of things. I'm getting old.

was it in English?

Balarka Sen

yeah

Huy

09:01

ok. I'll see if I can find it when I'm home.

Balarka Sen

thanks!

Astyx

09:58

@Huy I'm not sure I understand. Is f defined prior to this? What is the desired result?

Huy

@Astyxy: yes, it is.

f[x_] := Sin[x]/x;
t[x_] = Normal[Series[f[x], {x, 0, 6}]];
Plot[{f[x], t[x]}, {x, -2*Pi, 2*Pi}, PlotRange -> All]

something like this works. I want to use Manipulate[] and change 6 to a variable n

Astyx

10:17

f[x_] := Sin[x]/x;
t[n_, x_] := Normal[Series[f[y], {y, 0, n}]] /. y -> x;
Manipulate[
Plot[{f[x], t[n, x]}, {x, -2*Pi, 2*Pi}, PlotRange -> All], {n, 0, 6,
1}]

This isn't very clean but it works

Huy

10:45

thanks, that helps. if you come up with a cleaner solution, I'm very happy to hear about it

Astyx

The same thing but without a free variable. The issue you have is that Series does not return a function but an expression in some variable x, and you then want to specify it to x=x0 for some number x0

f[x_] := Sin[x]/x;
t[n_, x0_] :=
Module[{x}, ReplaceAll[Normal[Series[f[x], {x, 0, n}]], x -> x0]];
Manipulate[
Plot[{f[x], t[n, x]}, {x, -2*Pi, 2*Pi}, PlotRange -> All], {n, 0, 6,
1}]

A slightly different version for which t[n] returns a function in x that is the truncated series.

f[x_] := Sin[x]/x;
t[n_] := Function[{x0},
Module[{x},
ReplaceAll[Normal[Series[f[x], {x, 0, n}]], x -> x0]]];
Manipulate[
Plot[{f[x], t[n][x]}, {x, -2*Pi, 2*Pi}, PlotRange -> All], {n, 0, 6,
1}]

Jaakko Seppälä

Let $B^3=\{x\in\mathbb R^3:|x|<1\}$, $A\subset \mathbb R^3$ of the form $\bar{B}^3\cup C$ where $C$ is a Jordan curve and $\bar{B}^3\cap C$ is a singleton. How one can show that $\pi(A)\approx \mathbb Z$?

Astyx

Beyond that I'm not knowledgeable enough to bring much more insight

robjohn

11:27

@LearningCHelpMeV2 $$\frac1{n(n+1)}\overset{\substack{\text{$1/x$ is}\\\text{decreasing}\\\downarrow}}{\le}\frac1{n^2}\overset{\substack{\text{$1/x$ is}\\\text{convex}\vphantom{\text{dg}}\\\downarrow}}{\le}\frac12\left(\frac1{n(n-1)}+\frac1{n(n+1)}\right)$$

$$1+\sum_{n=2}^\infty\frac1{n(n+1)}\le1+\sum_{n=2}^\infty\frac1{n^2}\le1+\sum_{n=2}^\infty\frac12\left(\frac1{n(n-1)}+\frac1{n(n+1)}\right)$$

$$\frac32\le\sum_{n=1}^\infty\frac1{n^2}\le\frac74$$

love_sodam

12:18

If $s>1$, $\\sum_{n>0}n^{-s} = \prod_p 1/(1-p^{-s})$ where the product is over all primes $p$. Letting $s\to 1$, LHS diverges and RHS becomse $\prod_p 1/(1-p^{-1}) = \prod_p\sum_{k = 0}^\infty 1/p^k$. Does this imply $\sum_p 1/p$ diverges?

Jakobian

12:31

@JaakkoSeppälä from what you wrote, $A$ is a singleton, so its fundamental group is $0$

Astyx

@Jakobian The intersection is a singleton, but A is the union

However $\bar B^3$ is contractible, so A is homotopically like C

Jakobian

Ah. I am on phone so I read raw LaTeX

Why don't you write any contraction of B^3 onto the point in intersection and extend it to A

Contraction which fixes the point of the intersection

Muzammil ahmed

12:57

Can anyone help me out with this one?

robjohn

13:14

@Muzammilahmed Not with the information given. What are $\vec x$, $\vec y$, $\vec z$?

@Jakobian you should be able to use ChatJax on your phone. What kind of phone do you have?

Muzammil ahmed

$\vec x$, $\vec y$, $\vec z$ are not given

@robjohn any idea now?

robjohn

nope

Martin Sleziak

13:28

@Muzammilahmed Doesn't the response you got in the linear algebra room help with this?

Jakobian

13:41

@robjohn android + google chrome

user193319

What exactly is a line in projective space, say in $\Bbb{P}^4$?

Like, would the sets $\{[0,0,1,z] : z \in k\}$ and $\{[0,0,z,1] : z \in k\}$ define lines in $\Bbb{P}^4$?

robjohn

14:04

@Jakobian did the instructions for android on the installation page not help?

Leaky Nun

14:20

@user193319 no, because your sets are copies of A1

you want copies of P1

oh, you mean them combined

then yeah that would be a line

@user193319 a line in P4 is the same as a 2-dimensional subspace of k^5

user193319

COmbined meaning union?

Leaky Nun

remember the interpretation of P^n as lines in k^(n+1)

@user193319 yes

Jakobian

15:15

@robjohn i haven't used them, and they say it's a chore for android with chrome, so I'll pass

mohan10216

15:55

@blah I managed to get somewhere, except I didn't use your coordinates, made my own. I got to get some sleep now, let me try generalize any result I may have.

mohan10216

16:06

hello Ted

goodnight

Koro

16:40

a non constant continuous periodic function has a smallest fundamental period.

how do i show this?

Xander Henderson

@Koro That isn't true.

Oh...

wait... you said continuous.

I missed that.

Koro

:)

Xander Henderson

Suppose it doesn't. Then for any $\varepsilon > 0$, there exists some $T > 0$ such that the function is $T$-periodic.

Koro

there is an x such that f(x)$\ne 0$ so suppose wlog f(x)>0

Xander Henderson

For each $n$, choose $T_n < 1/n$ so that your function is $T_n$-periodic.

Then $f(0) = f(T_n)$ for all $n$.

From which it follows that there is a dense set of values such that $f(x) = f(0)$ (since $f(kT_n) = f(0)$ for all $k\in\mathbb{Z}$ and $n\in\mathbb{N}$).

But $f$ is continuous, so it must be constant.

Koro

16:52

@XanderHenderson $\ddot\smile$

thank you so much.

I don't know why I didn't think of contradiction here.

Koro

17:08

For any $x$ and any $\epsilon\gt 0$ there is a $d>0$ such that
$|t-x|<d\implies |f(t)-f(x)|<\epsilon$
Choosing n so large as $\frac 1n<d/3$. We define $S:=\{k T_n: k\in \mathbb Z \land kT_n \le x-d\}$
S is bounded above so has a supremum element in it say $k_m T_n$. It's easy to show that $(k_m+1)T_n\in (x-d,x+d)$.
$|f(x)-f(k_mT_n+T_n)|<\epsilon\implies |f(x)-f(0)|<\epsilon$
Since $\epsilon\gt 0$ is arbitray, it follows that $f(x)=f(0)$ for all $x$. This contradicts the hypothesis that f is non constant.

@XanderHenderson: I have completed the proof based on your suggestions. I hope it is correct. Thanks again :).

Too late to edit now!
$S:=\{k\in \mathbb Z: kT_n\le x-d\}$ and not the way done above.

ypercubeᵀᴹ

17:45

Question about irrational/algebraic numbers, in case anyone knows. Is this sum algebraic?

$$
\sum^{\infty}_{n=1} {2^{-\frac{n(n+1)}{2}} }
$$

Leaky Nun

@ypercubeᵀᴹ you can try to see if you can prove that your number is a Liouville number

leslie townes

18:03

seems tough to me. i don't think it converges "fast enough."

wolfram alpha gives an evaluation in terms of 2^(7/8) and the value of a theta function, or a modified theta function, at (0, 1/sqrt(2)). some stuff is known about theta functions.

geocalc33

19:13

Is there a closed form for this? $$ \sum_{n=1}^\infty \sum_{k=1}^\infty \sum_{c=1}^\infty 2^{-nkc} $$

Ted Shifrin

19:23

Have you started by doing a double sum?

leslie townes

wolfram alpha will probably spit something out. which isn't to say that it's useful.

geocalc33

with the double sum $$\sum_{k=1}^\infty \sum_{c=1}^\infty 2^{-kc}$$ Wolfram alpha says the closed form is a q-polygamma function lol

leslie townes

and if you toss in -2kc, -3kc, etc you probably get similar closed forms for those. hooray!

just add 'em all up and then close the form.

Ted Shifrin

Yup, because you're going to end up needing $\sum \frac 1{1-2^{-k}}$.

geocalc33

nice

Ted Shifrin

19:33

Well, more precisely, $\sum \frac{2^{-k}}{1-2^{-k}} = \sum \frac1{2^k-1}$.

Not sure why this is polygamma.

Houshou Rattengod

20:02

Is there a way to send someone on Math.stackexchange a private message? Like if I wanted to reach out to an individual rather than make a post?

Ted Shifrin

Not unless they list an email in their profile ...

geocalc33

20:31

what

when you take a collection of flow lines that form a surface of revolution

well not really

a flow on the surface of a sphere maybe that vanishes at the antipodes

described by the following $\vec{V}=k(-\cos \theta \sin \phi, -\sin \theta \sin \phi, \cos \phi)$

user10478

21:21

Why is the Fourier Transform of $u_x$ a multiple of the Fourier Transform of $u$, whereas the Laplace Transforms of derivatives have extra initial value terms?

Vrouvrou

hello, i need an idea to prove this " Let $u\in C^2([0,+\infty))$, such that $\lim_{t\to+\infty} u^{(n)}(t)$ exists then $\lim_{t\to+\infty} u^{(n)}(t)=(n-i)! \lim_{t\to+\infty}\frac{u^{(i)}}{1+t^{n-i}} ,\, \forall \, i\in\{0,\ldots, n-1\}$

i can prove it by recurrence ?

Ted Shifrin

@user10478 multiple is misleading … but do you know definitions of these transforms?

user10478

Yeah, I know there are multiple definitions.

Ted Shifrin

Not my point. Fourier transform is an integral from what to what? Laplace?

user10478

I know the Laplace is generally one sided and Fourier generally two sided, and the definition of Fourier I'm using has $\frac{1}{\sqrt{2\pi}}$ outside the integral of both the Fourier and Inverse Fourier Transforms, but I know different definitions can kind of distribute those coefficients differently.

Vrouvrou

21:27

please the recurrence must be on i or on n ?

monoidaltransform

If the metric tensor is independent of the t coordinate, why is $\frac{\partial}{\partial t}$ a killing vector?

Ted Shifrin

I don’t care about the constants. So do integration by parts and answer your own question.

Definition, monoidal.

user10478

@TedShifrin Regarding your prior question, the Laplace is generally from $0$ to infinity and the Fourier from $-$infinity to infinity. This double-sidedness actually trips up my integration by parts a bit, as things blow up when I try to plug the infinities.

Ted Shifrin

There are decay conditions at infinity for the transforms to make sense.

user10478

Like $u(|∞|, t) = 0, u_x(|∞|, t) = 0$?

geocalc33

21:34

Xander, do you know if there's an integral transform allowing one to obtain the analytically extended expression for certain classes of spectral zeta functions (with explicitly known spectra)? I recall you saying that the Mellin Transform can be used to relate the geometric zeta function to the spectral zeta function...

Ted Shifrin

Don’t you have more rigorous hypotheses in your book/course?

monoidaltransform

@TedShifrin sorry, I don't see where the metric being independent of the coordinate comes in

Ted Shifrin

What is the definition of a Killing field?

monoidaltransform

$\nabla_{\mu}X_{\nu}+\nabla_{\nu}X_{\mu} =0$

user10478

@monoidaltransform That actually wasn't provided with the Fourier Transform itself. There were conditions introduced with the PDE being solved via FT that I think represent what you're talking about. The exact symbols it uses are $u(x, t) -> 0, u_x(x, t) -> 0; as |x| -> ∞$.

monoidaltransform

21:38

@user10478 what I said was directed to Ted, not you... sorry! different things

user10478

I also meant to @ Ted, my bad.

Ted Shifrin

That’s not the definition I know, monoidal.

monoidaltransform

en.wikipedia.org/wiki/Killing_vector_field @TedShifrin

Ted Shifrin

So for your purposes, assume the infinity terms disappear, user10478, and don’t worry about them.

Read what you linked me, monoidal.

monoidaltransform

It sais $L_{X}g=0$ is equivalent to in local coordinates to the expression I gave above

Ted Shifrin

21:42

But maybe the actual definition is more useful? Don’t be blind.

monoidaltransform

Yes, that I agree with. But the definition in class I was given is the one I gave and the question I asked was treated as a triviality; one which I do not see

Ted Shifrin

Work out the equivalence, then.

monoidaltransform

@TedShifrin what does $g_{\nu \mu,0}$ mean?

$\nabla_{0}g_{\nu \mu}$?

Ted Shifrin

I would assume so.

monoidaltransform

Oh it actually means $\frac{\partial}{\partial x^0}g_{\nu \mu}$

user10478

21:51

@TedShifrin Okay, so if I recall, that part (the non-integral part of the integration by parts) is where the initial value terms of Laplace Transforms of derivatives come from, so the reason they don't appear in Fourier Transforms of derivatives is that the conditions for a Fourier Transform to converge are more restrictive in the first place, in such a way that the cases where initial value terms are nonzero are excluded?

Ted Shifrin

Some texts use semicolon for covariant derivative. You are responsible for the notation of your course.

monoidaltransform

this isn't notation in my course, this is wikipedia notation

Ted Shifrin

There us no “initial value” for Fourier, only for Laplace!

monoidaltransform

but my course sais that $\nabla_{\mu}g_{\rho \sigma}=\partial_{\mu}g_{\rho \sigma}$

(By definition of connection)

Ted Shifrin

That’s false unless you’re at the central point in normal coordinates.

user10478

21:55

Okie, thanks

monoidaltransform

yeah, I know. But in this course, the lecturer stated we define the connection on smooth functions to be the partial deriviative

but yeah in normal coordinates, the christoffel symbols all vanish at the point

so i'm not sure what he's thinking or what he's working with

darn physicist

this is hilarious, because i'm actually taking graduate differential geometry alongside this course you'd think the formalism developed in the DG course will help me understand the math in this physics course

but no

at times they're treating the supposed same object completely differently

Ted Shifrin

Functions, yes. Tensors, no!

monoidaltransform

isn't $g_{\mu \nu}$ a real valued function?

Ted Shifrin

No!

Components of a tensor.

Functions don’t depend on coordinate systems.

monoidaltransform

ahh, in the DG course, it is written that $g_{i,j}: U\rightarrow \mathbb{R}$ where $(U,x^1,.....,x^n)$ is a chart

Ted Shifrin

22:04

Yes, but they transform when you change chart. A function does not.

monoidaltransform

OH I SEE

THANK YOU

Ted Shifrin

Write down how your DG course tells you to take the covariant derivative of the tensor $\sum g_{ij}dx^i\otimes dx^j$ and compare with the GR course.

leslie townes

22:19

hah, ted, remember when we were discussing inverse trig functions? i love the title of this question.

5

Q: What's the Deal with Inverse Cotangent?

So, I was minding my own business and I thought I had defined inverse cotangent in the natural fashion. In particular, we define inverse tangent as the inverse of tangent restricted to $(-\pi/2, \pi/2)$. We all know this. So, I thought, $y = \cot(x)$ has vertical asymptotes at $x = n \pi$ for any...

Ted Shifrin

22:31

Yeah. I agree with $[0,\pi)$. It’s with csc and sec that we have problems with connectivity.

Xander Henderson

I make is an exercise for the students because, frankly, I don't care about cotangent. It is a silly function for silly people.

Most of my students choose $(0,\pi)$ as the "principal" domain of the cotangent function, so that it is connected.

A few choose $(-\pi/2,0)\cup(0,\pi/2)$. And every once in a while, some joker picks something like $(-3\pi/2,\cup -\pi) \cup (4\pi, 9\pi/2)$.

Clarinetist

Apologies if this is an obvious question, but summing over $n \geq 1$, how do I show divergence of $\sum \dfrac{(n-1)^4}{n^5}$?

.. hmm. I wonder if I'm going to have to just expand that numerator for the integral test

leslie townes

limit comparison with 1/n

or direct comparison, but limit comparison is more robust for this kind of thing

Clarinetist

@leslietownes Nice, this works. Thanks

geocalc33

22:51

has anyone ever seen a sum like this? $$ h(s)=\sum_{m=1}^\infty m^sK_{-s}(2m^s) $$

leslie townes

yes, about a minute ago.

geocalc33

has anyone ever seen a sum (not counting here)?

ohh

I forgot to premultiply the sum by $2$

Xander Henderson

@geocalc33 Yes.

I've seen many sums.

All over the place.

geocalc33

have you ever seen this one though

in the context of research?

"I've seen many sums. Sums all over the place."

Xander Henderson

Which one?

And what do I win if I say "yes"?

geocalc33

23:05

$h(s)=\sum_{m=1}^\infty 2m^sK_{-s}(2m^s)$

you can only say 'yes' if you've seen the sum

there's no prize money involved

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$Mathematics$ Mathematics