Mathematics - 2022-04-14

« first day (4271 days earlier) ← previous day next day → last day (754 days later) »

12:01 AM

Jesus, I was struggling to tie that back to what you were originally telling me, but now I see that for any point on the circle you get a line, so I see it now. I guess it's just a trivial fiber bundle. But for unoriented lines it would be $[0, \pi) \times \mathbb R$, so then it doesn't seem to correspond to a circle anymore.

I guess I have to do something in the unoriented case to make a circle appear somehow.

You identify $\pi$ and $0$, so that gives a circle.

You mean all antipodal points, or just $\pi$ and $0$?

Either way.

So this equivalence relation doesn't meaningfully change the (trivial) fiber bundle?

I wonder if this has to do with the quotient topology... I need to go read about that.

12:18 AM

Well, when you identify $0$ and $\pi$ think about how we are dealing with the lines at “distance” $r$.

Well, before you made the identification you'd have two distinct lines (corresponding to their orientations), but after the identification you treat them as the same.

So it's still a product space, except you're saying the line tacked on is not unique at every point on the circle.

No, that won’t do.

The line at distance $r$ at $\theta = 0$ is the line at distance $-r$ at $\theta = \pi$?

monoidaltransform

Removing an open ball from a torus gives us a contractible space?

No.

that doesn't seem right.

12:26 AM

Very much no.

it oughta have fundamental group same as a wedge of circles or something.

@TedShifrin So this would be like lines $y = 1$ and $y = -1$, i.e. they're in different places in the plane. So definitely not the same line. But I need to figure out how this relates to what I wrote.

@leslietownes or not something.

@Novice OK. I won’t say that this sounds like comparing cylinder and Möbius.

yeah, ok. i was taking a hole punch to the middle of a square with the opposite sides identified and retracting that thing onto the boundary of the square.

or whatever the word is.

Yes, so delete “or something.”

monoidaltransform

12:31 AM

Wait, a ball is contractible, so $T\B$ is homotopy equivalent to $T\point$, no?

Oy.

Do not pass go.

Is a point the empty set?

monoidaltransform

I meant $T\point$, not $T$

Well so?

@TedShifrin Oh, so the case involving unoriented lines leads to a non-trivial fiber bundle. I guess I need to see why that's true.

I gave you the hint, but this is a good concrete example to think through.

12:42 AM

I'll have to think about it more. Thanks.

12:55 AM

Hi, my apologies for this crude description:

Let there exist a point that grows two line segments, at the end point of each of which two more line segments sprout and this pattern continues unimpeded ad infinitum.
My questions are these:
1) If I give you a row on this tree, can you give me a count of how many line segments (branches) there are (ie, the population in each generation)
2) For fun, the total count of branches in the tree overall.

you might google 'binary tree' or add modifiers to that for many variations on this problem

What areas of math might I enjoy if I liked complex analysis?

derivative: more complex analysis?

just kidding. it might depend upon why you liked it.

i much prefer curves to lines, what was i thinking

@copper.hat yeah!

1:03 AM

ted: my wife was taken aside at pickup today because my daughter apparently introduced the phrase 'what the heck' to the class and everybody was saying it. she also told the teacher that my wife was expecting (she isn't, as far as we know).

trying to upload a picture of lake anza for Leslie

LOL

… Munchkin is Patty Hearst in training.

@leslietownes just to skip into the meat, I'd like to estimate populations where the number of children from each node are between, say, 0 to 15. What should I be looking for?

the teacher asked how far along my wife was, which is a fairly advanced question for a toddler. my daughter responded by saying "it takes 40 weeks." which is approximately true but presumably is something else she learned at school.

@leslietownes excuse my thumb, imgur.com/a/BmAoG1J

1:05 AM

wow, looks great.

last week the entire lake was completely covered in the green vegetation

the high winds cleared it up a little. there is a mallard working its way through the stuff

Ugly thumb!

my right one is so much prettier

nick: think powers of 2. at the nth level above the point you'll have 2^n nodes from which 2^(n+1) segments sprout. for "branches in the tree" you'll need to clarify whether you mean the total number of line segments up to some level in this diagram, or total number of paths from the root to the tip (up to some level) in this diagram, or something else.

ktvu.com/news/…

1:08 AM

1 + 2 + 2^2 + ... + 2^k = 2^{k+1} - 1 might help with some of that.

Wow. Deep.

copper: kinda cool that i guess it's ecologically fine and native. i could imagine a lot of not-so-fine stuff being dumped in that lake

@leslietownes what do you mean?

also what does more complex analysis look like?

derivative: i mean, different people teach complex analysis very differently, and like it for different reasons

when i was an undergrad it was notoriously taught by many visiting faculty as a calculus class with contour integrals instead of calc 2 integrals. if someone liked that, they'd want to, i don't know, teach calculus for a living. if someone liked the geometry or the analysis part, they would want to study geometry or analysis

@leslietownes it was a bit of a shock to see it completely convered, my first reaction was how terrible, covered in algae, so was relieved to discover otherwise.

1:17 AM

@copper No climate change here.

@leslietownes eh I liked the analysis parts, and also the applications to number theory. I wanted to know more concretely (as in book titles, etc) what studying this type of thing entails

Analytic number theory seems like an obvious answer.

you might like analytic number theory. i don't know of a book.

hah, ted.

The analysis leads to several complex variables, but that is a huge step up.

a lot of first course in complex analysis textbooks have sections that the instructor doesn't cover. those might be a glimpse of what more 'just complex analysis' would look like. conway's functions of one complex variable II is an example of such a thing.

1:21 AM

the thing about number theory is that if I study more number theory I'll graduate feeling that I don't know any math except number theory

nevanlinna had a book or maybe a two-volume set on single variable complex analysis that went deeper than the average intro.

Well, take resl analysis, topology, diff geometry, probability, differential equations.

Not really, leslie. I took a course from that.

Stein/Stakarchi has more number theory than average.

really? i'm thinking of the one that has a lot of his value distribution theory.

Misspelled

i haven't seen that in any other books.

1:24 AM

That’s a totally different book,

ok, i'm just confused then.

Oh, mine was Nevanlinna & Paatero.

finns sure like double letters.

I may have misspelled.

1:46 AM

Whenever I see Nevanlinna I think Pick

apparently pick came up with it first, although i usually see nevanlinna's name first. poor pick.

same pick as pick's theorem for the area of lattice polygons. died in theresienstadt at age 82, which is awful.

no more wikipedia for me today.

2:40 AM

Suppose that $(x_n)$ is a sequence of real numbers, and $\lim_n (x_{2n}+x_{2n-1})=2003, \lim_n (x_{2n}+x_{2n+1})=315$, can it be concluded from this information that $x_{2n+1}\to 0$ or $\infty$?

I ask this because $\lim_n \frac {x_{2n}}{x_{2n+1}}$ is to be found by using Cesaro Stolz.

The solution uses Cesari Stolz on $a_n= x_{2n}, b_n= x_{2n+1}$ so that $\lim_n \frac {a_{n+1}-a_n}{b_{n+1}-b_n}$ exists and equals $\lim_n \frac{a_n}{b_n}$

$\lim\limits_{n\to\infty}((x_{2n}+x_{2n+1})-(x_{2n}+x_{2n-1}))=\lim\limits_{n\to\infty}(x_{2n+1}-x_{2n-1})$

$\lim\limits_{n\to\infty}((x_{2n}+x_{2n+1})-(x_{2n+2}+x_{2n+1}))=\lim\limits_{n\to\infty}(x_{2n}-x_{2n+2})$

I think that the book has a typo in the solution.

2:55 AM

So, I've been thinking about this on and off for the last couple hours. If we skip a lot of the fiber bundle formalism, then we can think of the cylinder as the product $[0, 2\pi) \times (-1, 1)$. For the Möbius strip, I think you have to do something like taking the unit square and identifying $(x, 0)$ with $(1 - x, 1)$.

For the "oriented lines + circle", we decided that we can roughly think of this as $[0, 2\pi) \times \mathbb R$, so it's like an infinitely long cylinder. So naturally, one wonders about an infinitely long Möbius strip, but I can't see that in my mind and I don't know if …

3:13 AM

@Koro there must be a typo?

@Novice i would switch coordinates. We want the circle “first.” And I would write $S^1\times\Bbb R$ …

Oh, OK, I missed the $\pm$ difference.

Prithu biswas leftmse

3:50 AM

Can someone tell me what is the statement of the chain rule (for single variable)?

Look in any calculus book or on line?

Prithu biswas leftmse

@TedShifrin Well, wikipedia doesn't seem to be clear on what kind of function f and g should be.

Spivak also doesn't seem to have that info.

i somehow doubt that, but if you're considering f(g(x)) at the point c, then g is usually assumed differentiable at c and f assumed differentiable at g(c)

Nonsense.

(Not to leslie, for a change)

Prithu biswas leftmse

@TedShifrin why?

4:00 AM

so domain wise you probably have g defined in an open interval around a and f defined in an open interval around g(a)

maybe with some exceptions and bookkeeping for one-sided intervals

ted: i even extended spivak the rare courtesy of 'somehow doubt[ing]' that he forgot to include hypotheses for the chain rule.

Prithu biswas leftmse

Spivak: If g is differentiable at a, and f is differentiable at g(a) , then f ∘ g is differentiable at a and (f ∘ g)'(a) = f'(g(a)).g'(a).

@TedShifrin the book uses $a_n= x_{2n}+x_{2n+1}, b_n= x_{2n+1}+ x_{2n+2}$ in Cesaro Stolz but has written that as $a_n= x_{2n}$.

Nonetheless, Cesaro Stolz requires b_n to be decreasing to 0 (or increasing to infty) but I don’t know how they applied Cesaro Stolz without even knowing b_n is increasing to infty or decreasing to 0.

4:15 AM

@leslietownes He did not.

@Prithubiswasleftmse Precise and correct. How can you say he omits hypotheses? That’s annoying as hell.

Prithu biswas leftmse

@TedShifrin what is f and g? are they ℝ→ℝ ?

@Koro I never have used C-S and do not know it.

You need to check the definition of “diff at $a$“ … the domain is irrelevant as long as that holds.

4:35 AM

anyone know differential eq here? i need to know if im on the right track for doing simple equation

i have dy/dx = y-2y^2, do i just multiply dx and divide by y-2y^2

and then integrate both sides?

something like $\int \frac{1}{y-2y^2}dy=\int dx$ ?

@AidenChow exactly

@LeakyNun ok thanks!

5:42 AM

Is it correct to abbreviate Central Limit Theorem as If $(X_n)$ is iid then $\bar{X_n}$ converges in law to $N(\mu,\sigma^2/n)$?

5:57 AM

@Prithubiswasleftmse you can consider f to be defined on an open interval containing g(a).

1 hour later…

7:08 AM

for $\lim_\limits{h\to0}\left(\frac1h\int_1^{1+h}e^{-t^2}dt\right)$, why can u directly plug in $t=0$ into $e^{-t^2}$ to get $1/e$ as the answer?

aiden: if F(x) is a function satisfying F'(x) = e^(-x^2), then that definite integral is F(1+h) - F(1) by the FTC. and that limit is then the limit definition of F'(1)

6 hours later…

1:16 PM

Can you tell me what the formula for the derivative of a quaternion with respect to the angle of rotation will look like, please?

2 hours later…

3:18 PM

@Ted: Morning.

3:58 PM

Morning, @robjohn!

It's cold here this morning. How are things there?

Not cold, but under 60.

4:15 PM

It was 43° when my puppy and I went walking at 6:30.

It is currently 59°; I consider that chilly.

1 hour later…

5:36 PM

user image

5:52 PM

I think we're taking $[0, 2\pi) \times \mathbb R$ (or $S^1 \times \mathbb R$) and identifying $(\theta, r)$ with $(\theta + \pi, r)$ for $\theta \in [0, \pi)$.

So if I'm on the right track, it doesn't quite seem to be an infinitely long Möbius strip, but I do sense a twist in the construction.

Maybe it would be a good exercise to now go back to the $(E, B, \pi, F)$ definition of a fiber bundle and try to make these examples fit that definition.

How does this differ topologically from the usual polar $(r,\theta)$ making up $\mathbb{R}^2$?

well, $\mathbb{R}$ would need to be changed to $\mathbb{R}^+$

6:11 PM

I'm not sure if you're trying to point out an error in what I've done, but in polar coordinates, my recollection is that you allow $\theta$ to go over the whole circle and restrict $r$ to positive values. Here, I'm restricting $\theta$ to go over half the circle, and allowing $r$ to take any value.

I've never tried to connect polar coordinates and point-set topology, though.

6:45 PM

I'm looking at the definition of a fiber bundle on Wikipedia. It looks like the homeomorphism $\phi \colon \pi^{-1}(U) \to U \times F$ is supposed to represent the un-twisting.

So for the Möbius strip, we take a point $x$ on the circle, then consider an "open interval" $U$ around $x$, and then $\pi^{-1}(U)$ is the section (slice) of the strip containing $x$. In my mind I imagine $\phi$ un-twisting that section of the strip.

I'm still mulling over the commutative diagram, though. I've seen them before but I don't have much experience with them.

5 hours later…

11:37 PM

What would you call a shape in $n-$dimensions formed by taking an $n-$dimensional square/cube/tesseract/whatever, and, for each edge/face/cell/whatever, adding a single square/cube/tesseract/whatever which shares that edge/face/cell/whatever?

If $n=2$, then you get a plus-sign-looking-shape

Xander Henderson

@Rithaniel Samantha?

Interesting name

"An $n-$dimensional Samantha"

Xander Henderson

Seems as good as any other name.

0

Q: Name of a cube with cube attached to each face?

Is there a name for the shape that results from gluing a cube to each face a cube? I can't come with a natural description for it, and google doesn't seem to give any results (the closest being a depiction of a 7-point finite difference stencil). See figure:

I'm not entirely sure why you would need a name for such a figure, but an "$n$-cross" would, I think, be reasonable.

Or an "$n$-dimensional cubic cross" (since, I suppose, you could construct a cross out of other polytopes).

Well, having terminology is useful when you're searching for stuff.

Xander Henderson

I don't think that it is an interesting enough object to have been named.

Or, rather, it is not interesting enough to enough people to have received a widely disseminated name.

11:46 PM

My overall goal is to see if anyone knows whether you can tile different spaces using concave polytopes like this. I know you can do it with $2D$ crosses, but not sure if you can in $3D$ space or hyperbolic space

Also, I figured it wouldn't be widely disseminated, if a name did exist

It's just, this is where I trust people's opinions on math stuff the most, out of my online haunts, so this is where I bring math questions :]

11:59 PM

Tiling questions are very challenging. (Some) Old stuff may be found in Hilbert Cohn-Vossen.

« first day (4271 days earlier) ← previous day next day → last day (754 days later) »

Transcript for

Apr '2214

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

join this room
about this room

00:00

06:00

12:00

18:00

all times are UTC

$Mathematics$

site design / logo © 2024 Stack Exchange Inc; legal

mobile