Mathematics - 2021-03-18 (page 2 of 2)

Astyx

13:00

@EdwardEvans By Lang?

Edward Evans

Washington

Astyx

I might be tempted at some point

Edward Evans

looks cool

p-adic L-functions and Iwasawa theory

Thorgott

If $Y\rightarrow X$ is a continuous map and $E\rightarrow X$ is an epi (surjective if you want) s.t. the pullback of the resulting is isomorphic to disjoint copies of $E$ as space over $E$, is $Y\rightarrow X$ a covering map?

anakhro

@Thorgott just take a wild guess: 50% chance that you were right, and that's pretty good.

Balarka Sen

13:06

That condition is almost never true right

Thorgott

I mean, it's true for covering maps

assume $X$ nice

Balarka Sen

Why is it true for covering maps?

Thorgott

pull back to universal cover?

Balarka Sen

I don't see it

Thorgott

pullback of covering map is covering map

Astyx

13:08

What is "pullback of the resulting" ?

Balarka Sen

@Thorgott I understand, but why would $E$ trivialize when you pullback along a covering map

Thorgott

so pulling back a cover to the universal cover gives a cover of the universal cover, which is just a coproduct of the universal covers projecting down via identity map since universal cover is simply connected

@Astyx resulting diagram

Astyx

ah ok

Balarka Sen

Oh I thought you meant for all $E$ it is true

Thorgott

oh nonono

Balarka Sen

13:10

That is why I said that condition is almost never true

Some quantifers are missing

Thorgott

just there exists an $E$

sorry

Balarka Sen

Yes, I agree then. This is just definition of etale.

Thorgott

it's apparently the way one defines a covering map in an arbitrary topos

I don't see the equivalence, though

Balarka Sen

I think it is true, you have to show for some $E$ this happens implies it happens for inclusions of open sets

Certainly it is true if you demand it happens for $E = \widetilde{X}$

Because you can restrict to an open set.

If you demand it happens for some covering map $E \to X$, it is true.

I am not sure about an arbitrary epimorphism.

But of course you'd like to stay in the etale site, so I don't see why you would demand an arbitrary epi.

Thorgott

but of course epis should constitute a cover in the natural coverage on the slice category Top/X, no?

user2103480

13:24

@Jakobian min(1,d_n(x,A^c))*f should be sufficient

Astyx

I don't see it for covering maps

user2103480

since A is open

Balarka Sen

@Thorgott Yeah, that's what I thought

Astyx

Are you assuming that the projection of the pullback on E is the obvious one? Is that always the case?

user2103480

@EdwardEvans normie

Edward Evans

13:27

it's cool tho

ya prick

Astyx

something ard

Edward Evans

lmao

user2103480

@EdwardEvans duh

Thorgott

the morally right thing when we have a site is to demand existence of a trivializing cover, which means that pullbacks over maps in this cover decompose as coproducts with identity projections, agree?

user2103480

normie

@EdwardEvans youtube.com/watch?v=nwomZwSHzF0

Balarka Sen

13:29

Pick any open set U of X, look at the inclusion map U -> X. Then there's a pullback diagram because E -> X is a covering map
U x Fiber ----> E
...|..................|
...V..................V
..U------------->X
Now there's a pullback diagram
E x Fiber ----> E
...|..................|
...V..................V
..Y-------------->X

Thorgott

man adamantly refuses to use tex to draw a commutative diagram

Edward Evans

@user2103480 I feel like this would be better if I knew who the people are

Balarka Sen

If you paste these diagrams side by side I think you get Y -> X is evenly covering U

Edward Evans

loool nice numerology

Thorgott

@BalarkaSen I think the point is you get local sections

anakhro

13:32

Trying to grok something: there is no way to "patch up" an essential singularity into a holomorphic function, right? On the other hand, a pole can always be patched up.

user2103480

@EdwardEvans uncultured

Balarka Sen

@Thorgott I only think in terms of diagrams

Edward Evans

just spitting the one word insults today

Thorgott

@user2103480 Edgar based

Balarka Sen

@EdwardEvans lool

user2103480

13:33

@Jakobian sorry meant min(1,n*d(x,A^c))

Thorgott

@BalarkaSen local sections are a diagram

Astyx

Like so ?
U x Fiber ----> E <---- rebiF x E
...|..................| ..................|...
...V..................V ................V...
..U------------->X <-----------Y..

Balarka Sen

rebiF lmao

Thorgott

rebiF lmao

Edward Evans

rebiF lmao

Balarka Sen

13:34

dying

swallowed my fucking chewing gum

Thorgott

dead

user2103480

@Thorgott always

Edward Evans

hahaha

user2103480

7/11 was an inside job

Donald Trump flubs terror attacks on '7/11'

►

Jakobian

@user2103480 for phi_k ?

Edward Evans

13:36

lol what a dunce

Jakobian

that should give me integral of f, but only when m = 1

Balarka Sen

@Astyx yeah, now the pullback of the top of the diagram is U x Fiber x rebiF -> rebiF x E

this must be pullback of the bottom of the diagram

so p^{-1}(U) = U x Fiber x rebiF, something like this

you get my point

user2103480

@Jakobian yeah. should converge using DCT since the absolute vaue is bounded by f restricted to A which is integrable

Astyx

Right I'm going to try and convince myself

user2103480

@Jakobian isn't the integral in m dimensions just single integrals per coordinate

Jakobian

13:39

f restricted to A isn't necessarily integrable

user2103480

ugh A isn't bounded right

Jakobian

it's arbitrary open

user2103480

but then why should this make sense

f is only locally integrable

How do you know the integral is defined

Jakobian

yeah but integral of |f| is always defined

Astyx

and sorry about your chewing gum

:)

user2103480

13:41

It is but what's the problem with proving the integral diverges then?

Balarka Sen

@user2103480 ää

user2103480

So in any case if f is integrable over A this argument should work

@BalarkaSen ääää

Jakobian

I don't see it

Balarka Sen

there will be no more ääääääääää

but you may feel a little sick

user2103480

@BalarkaSen oh god make the ääääää stop

when will it be no more

Jakobian

13:43

even if this would work, I need |f| and not f

caring for sign is my biggest problem here

user2103480

@Jakobian if A isn't bounded you may need to diagonalize with an increasing sequence that approximates A

but I don't see why sign is a problem if we're dealing with lebesgue integrals over a non-signed measure

Jakobian

if A is unbounded I can approximate it by open bounded sets, yeah

user2103480

Hm but I see your point

I think it should still work out if done carefully. A_n get larger and so the distance to the boundary of A_n get's larger and min(1,k*d(x,(A_n)^c) gets larger both when one increases n and increases k

Hasselhoff

14:03

Would someone be willing to look at my solution to this problem. I've been going back and forth with another member for about a week, but I feel we are on the wrong track.

1

Q: What is the autocovariance of $y_t = \exp{(x_t)}$ where $x_t$ is a stationary normal process.

Let $x_t$ be a stationary normal process with mean $\mu_x$ and autocovariance function $\gamma(h)$. Define the nonlinear time series $y_t=\exp{(x_t)}$. (a) Express the mean function $E(y_t)$ in terms of $\mu_x$ and $\gamma(0)$. The moment generating function of a normal random variable $x$ with ...

user2103480

@Hasselhoff what is a normal process

robjohn

@BalarkaSen I never heard the umlauts in the song. I'll have to listen harder.

Hasselhoff

in a stochastic process x_t, all x_t are normally distributed, ie. for each t, x_t ~ N(m, s^2)

user2103480

Sum of normally distributed random variables

In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships. This is not to be confused with the sum of normal distributions which forms a mixture distribution. == Independent random variables == Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if X ∼...

Hasselhoff

A@user2103480 that fact is somewhat useful here, but a normal process is just a time series which for all t, the rv are all normal as well as the entire series being jointly normal

user2103480

14:12

the expectation of Y_hY_0 is the expectation of exp(\lambda(x_h + x_0)) minus expectation of exp(\lambda x_0) squared and this first expectation you should be able to obtain using the autocorrelation of x

since x_h + x_0 is normal with variance given the sum of the variances + some correlation

Hasselhoff

yes, i agree, I actually posted a solution using what you've said and using the MGF of the normal, but I'm getting the feed back from another user that the solution is wrong. would you be willing to look at my solution to tell me if you agree with him?

user2103480

do you understand his citicism?

He is right, these random variables aren't independent and you can't just pull the expected values apart

Hasselhoff

I know that aren't independent. My solution does not suggest that they are. My solution shows that the of y_t is the variance of the log normal distribution

user2103480

look your variance in edit 3 is wrong

the variance of this sum is not just double the variance

again, these are not independent

Hasselhoff

to make that argument deductively (which is not what I've done, but is easier here) is that the autocovariance of y_t is not dependent on a lag, so that for any h >=0, the autocovariance is just the variance of 2 log normal rv.

user2103480

14:21

whatever just look at the wiki article

"Correlated random variables"

Hasselhoff

yes, ok, that is true, which is why I did Edit 4! Please stick with me, I feel we are almost figuring this out.

user2103480

edit 4 still has the same mistake

Hasselhoff

ok, thanks

user586228

15:04

Can anyone help me out with the thinking tecchnique for the problem..

Even writing about the tecnique in ac ouple of words should help..

leslie townes

there may be aspects of business law i don't understand here, but first i would work out percentage ownership, compute profit shares from that, and then do a subtraction

user586228

@leslietownes Thnaks

Why final subtraction?

leslie townes

they ask how much shyam's share exceeds that of ghanshyam.

seems like an artificial question, they just want you to do one more step would be my guess

user586228

ok I will figure out the rest myself thanks for helping

Mark Giraffe

$shenanigans$ $+$ $chat$ $=$ $fun$

user2103480

15:22

if people continue to write text bits in TeX I'mma bring back the almighty floating torus and that will benefit nobody

Hasselhoff

@user2103480 hey fixed my solution, mind taking a look. i finally saw what yall were saying I think.

Alessandro Codenotti

lmao

user2103480

@Hasselhoff the variance is right now, only weird to write $\rho$ for the correlation when you have the autocorrelation at hand

ah sorry

mean isn't zero

Hasselhoff

did i use a zero mean for something? definitely didn't mean to.

Thorgott

decent pun

user2103480

15:27

nono it's all good

using rho here is fully okay, this only reduces to the autocorrelation of the mean is zero

I thought about just plugging in the autocorrelation

I didn't check the calculations afterwards though. But should be straightforward

Hasselhoff

yea. thank you very much!

@Thorgott totally accidental.

user2103480

no problem!

user586228

15:48

@leslietownes Can you assist me with the last step..???

The ratio is 4:9:6

This is the profit ratio by how much is Shyam's profit share exceeding Ghanshyam's?

robjohn

16:01

@user586228 I get R 3600.

Astyx

@user2103480 dark times

user586228

@robjohn Correct

Please share the method

leslie townes

from the ratio i would compute profit shares. e.g. 4/(4+9+6) * [profits] for one, 9/(4+9+6)*[profits] for the other, etc. then some subtraction.

i wonder if this is going to be used in a legal dispute, to illustrate some kind of unfairness. maybe ghanshyam thought they had 'sweat equity' in the business.

robjohn

I count the time each had how much invested: $\text{R:}16000\times3+11000\times9=147000\\\text{S:}12000\times3+17000\times9=189000\\\text{G:}21000\times6=126000$

Giving a total weight of $462000$ so the share proportions at the end are $\text{R:}\frac{147}{462}=\frac7{22}\\\text{S:}\frac{189}{462}=\frac9{22}\\\text{G:}\frac{126}{462}=\frac6{22}$

this is all assuming they agreed on a simple interest at the end arrangement

leslie townes

16:16

oh good point

Adam L

17:26

THERE IS A SWEAR WORD IN THE STARRED COMMENTS MIKE MILLER SAID A BAD WORD, I DEMAND CENSORSHIP

Ted Shifrin

17:41

You demand? Go to a different website.

anakhro

I think he might be joking.

leslie townes

possibly. people who demand censorship tend not to refer to it as such

i have my suspicions, anyway.

Ted Shifrin

I don't like rude yelling, regardless.

anakhro

I only like polite yelling.

Hasselhoff

NICE TO MEET YOU!

anakhro

17:51

NICE TO MEET YOU, TOO!

Ted Shifrin

censors

Mark Giraffe

Shchiekcemn.

leslie townes

i may be a little desensitized to yelled demands. my daughter told me that she wanted a cookie yesterday. i said, is that so? she said "YES! mom said, IF YOU NEED A COOKIE, EAT A COOKIE!" upon investigation i learned that her mother said no such thing.

Astyx

Typical example of proof by intimidation

Ted Shifrin

Indeed, @Astyx. @Leslie is apparently just learning about the games children play — pitting one parent against the other.

leslie townes

18:06

also an appeal to authority. she's clever for her age, at least about trying to get cookies.

anakhro

What kind of cookie was it?

(this matters a lot)

leslie townes

six months ago she would have just blurted out "COOKIE!" now it's getting psychological

Astyx

"Heraclite once said, we never eat the same cookie twice"

Ted Shifrin

“It's ok with me if it’s ok with your mom.” “It's ok with me if it's ok with your dad.” Ad infinitum.

leslie townes

cookies come in two kinds, ones from the freezer and ones not from the freezer. freezer cookies are like, pumpkin or something. my wife baked them. non-freezer cookies are packaged store bought vanilla/lemon cookies with a trace of icing. she wanted a store bought cookie.

Ted Shifrin

18:08

I pass on both.

leslie townes

me too. the freezer ones are from thanksgiving, i think. store bought were from her grandmother. thanks, grandma.

anyway she goes from advanced game theory on the subject of cookies to crying when the cat doesn't want to drink from her water bottle. it's a strange time.

Hasselhoff

its good to play games back, like putting stuff I cook into empty store boxes and then privately gloat when they openly gloat that store bought is better than my cooking.

Ted Shifrin

LOL

Anyone who says store-bought is better than my cooking is never invited back!

Mark Giraffe

I believe the imperial system will soon be cursed.

Hasselhoff

i'm waiting to 18 to kick them out

anakhro

18:12

Can someone quickly confirm something for me? let $\ell$ be the map which takes a matrix to its lower-triangular part (the strictly upper triangular part being set to all zeros). This paper I am reading says that if $U$ is in $gl(n)$ then if $0 = tr(\ell(U)^\top\ell(V))$ for all $V\in so(n)$, then $U$ must be strictly upper triangular. I think this is an error and that it doesn't need to be strictly, but just upper-triangular (since the $V$ are always zero on the diagonal).

Hasselhoff

@Astyx Hericlite must never have had a Samoa.

anakhro

My reasoning is that $0 = tr(\ell(U)^\top\ell(V)) = \sum_{i,j}(\ell(U)\circ\ell(V))_{ij}$ for the Hadamard component-wise product $\circ$, then it's clear from inspection.

leslie townes

Heraclite said, we never eat the same row of cookies twice

Ted Shifrin

21:12

@robjohn Do you have a way to search for a deleted question? I had actually answered it and it's disappeared. I just accused the person I believe to be the perpetrator on his current "self-learning" post. The previous question was likewise a spin-off of a DoCarmo question from this chapter.

This makes me even more suspicious that answering any question is doing someone's exam question.

dc3rd

New account, question even though I don't know anything about geodesics, seems like it has one direct way of going about solving for a solution with no vagueness. Sort of thing that could be asked on an exam......

seems like the sort of question that if you know what you're doing will take 15 - 30 mins.....perfect exam material.

Ted Shifrin

I actually raised this question commenting on a different person's related question a week ago. This is actually a question in my diff geo text. There are several ways to do it, some easier than others.

I am 98% sure this is the person I answered before.

I have searched my answers/comments for that interaction a few days ago, and it's vanished.

That's why I'm asking robjohn with his super powers.

dc3rd

21:39

I take it you chose not to take on the super powers when they were offered to you?

robjohn

@TedShifrin I did not undelete the question with your comment because that was the only comment/answer there.

Govind75

Which value of $\epsilon$ gives a bifurcation in the linear system defined by the matrix $\begin{bmatrix}1 & -2\\3 & -7\end{bmatrix}$

sorry this matrix\begin{bmatrix}1 & -2\\3 & {-3+\epsilon}\end{bmatrix}

Ted Shifrin

But the answer was a valid answer. This person needs to be taught etiquette.

@dc3rd No, I never tried to become a moderator.

Govind75

I was thinking $\epsilon$ = -3 worked

robjohn

@TedShifrin yes. I have left comments on the 3 other questions which did have answers/comments.

Ted Shifrin

21:47

Oh, this person deleted 2 other questions?

robjohn

they deleted 3 of their 4 questions.

Ted Shifrin

Oh, so now it did reappear. So you did undelete?

I truly think this person has an exam and is hiding his tracks.

I missed those early ones a few months ago.

@Govind75 What makes there be a bifurcation?

robjohn

@TedShifrin I am positive of that. If you have an answer to the question about the geodesics that you commented about deletion on, I can undelete that too

Govind75

When there is a change in stability of the fixed points of the system @TedShifrin

Ted Shifrin

Oh, did he delete that one already after I commented?

I'm not going to answer, obviously.

robjohn

21:51

@TedShifrin yes, he has deleted all of the questions he has asked.

Ted Shifrin

As**ole.

robjohn

I undeleted the 3 with answers/comments.

Ted Shifrin

@Govind75 So how do we go about finding those?

robjohn

If you want to answer the other one, I will undelete it and it will be there for the instructor to possibly see.

Ted Shifrin

Most instructors don't come searching here, @robjohn, unless things have changed drastically.

robjohn

21:53

@TedShifrin I know, but the cheaters don't know that ;-)

leslie townes

guys, i just need to pass this one lame differential geometry class, give me a break. i'm not going to do math for a living, i'm going to be an instagram influencer.

another student might see it and snitch. or copy, i guess.

but if two copy that might tip the prof off.

Govind75

@TedShifrin The method I've been taught is only for 1D systems where I sketch the graph defined by the differential equation and then looking at the graph to see whether it is positive or negative near the fixed point to determine the flow of the solutions. However I do not know if this will work for 2D systems

robjohn

I love to slap those cheaters who get answers and delete the question.

Govind75

@TedShifrin I was thinking I need the real part of the eigenvalues to be of opposite sign, so do I need to find a value of epsilon which achieves this?

Ted Shifrin

Yes, but I'm not going to reward this ass by answering the question. Besides, it's not a standard question ... but is in my book. :)

robjohn

21:55

@TedShifrin okay. Just offering.

Ted Shifrin

What eigenvalues are we looking for, @Govind?

robjohn

. o O ( and getting the holey paddle ready )

leslie townes

i just want to read good differential geometry problems. thankfully i have good books.

Ted Shifrin

At the undergraduate level, @leslie, my text has a number of problems that appear nowhere else (and I think they're good).

Govind75

@TedShifrin the eigenvalues of the matrix? I also thought about calculating the trace and determinant of the matrix to determine which type of phase portrait the system would give.

leslie townes

21:57

that's cool. i forget which book i used in undergrad, but all of its problems were also in most of the other books.

not a lot of variation.

Ted Shifrin

DoCarmo is the standard ... but he wastes too much time on surfaces as 2-manifolds for my taste for a one-semester standard course.

@Govind: I'm asking how you find these fixed points. What eigenvalues are we looking for?

Govind75

@TedShifrin I don't know, I've never done this for a 2D system before. I guess I would want eigenvalues that equate the system to 0?

Ted Shifrin

Well, where is the problem coming from if you have never seen it before?

Govind75

@TedShifrin Its on a problem sheet. My only tool that I have been taught to use for 2D systems is the trace-determinant plane.

To characterise the phase portrait of the system

Maybe I just dont fully understand the what a bifurcation is

Ted Shifrin

Maybe you should ask your professor to point you in the right direction.

Govind75

22:07

Is my answer $\epsilon$ = -3 correct?

Ted Shifrin

That gives you $0$ as an eigenvalue. That would seem to be an unstable situation, yes. But I'm asking what definition you guys are supposed to be using, so you need to know that.

Govind75

So my understanding is its the point of the tuning parameter where either side of the tuning parameter the stability of the system changes

Oh would $\epsilon$ = 2 work

wait no

yh surely $\epsilon$ = 2 works because we change from an unstable spiral to a stable spiral

dc3rd

22:23

So the president of Tanzania passed away today apparently from Covid complications.....he was staunchly anti-covid and all the protocols in place to attempt to stem the spread. He only believed in "herbal" remedies......But the interesting thing is that he had a degree in chemistry and math.................I know there is huge philosophical debate all the time about religion and the sciences even among scientists.....

I'm curious from some of you folks that have been around for a few rotations on the planet, how could a person that was educated in one form of thinking (scientific) deviate so far from it?....I know it is a deep thought, but I've been trying to reconcile the dissonance...particularly since I am an aspiring mathematician (except for Q10, Sec 1.5 of Ted's Mulitvariable book)

Thorgott

the human mind is plastic

leslie townes

people's brains are like gears with large numbers of teeth. even a 99% fine gear can be missing a few teeth and will slip on some things. [academic subject of one's choosing] is not going to replace missing gear teeth. it might make you more effective at using your other gear teeth.

when i worked in a math department we would get unsolicited manuscripts of so called 'crank' stuff. some of it was what i will somewhat crudely refer to as crazy, but a surprising lot of it came from people who clearly had some, and in some cases a lot, of mathematical, scientific, engineering, etc. experience.

just a few teeth is all it takes.

that's my theory anyway

PM 2Ring

And people are good at putting things into different mental compartments. You only get cognitive dissonance if you allow the incompatible categories to interact. Still, I wonder how, eg, Young Earth Creationists with geology degrees manage that. ;)

leslie townes

it's also true that in many places, and i would include the US here, having one or more degrees in something is not highly probative of knowledge or keen judgment, even within that something.

BigSocks

we are not burdened by the need for internal consistency. we believe what is most useful. Perhaps given the sociopolitical/economic situation in Tanzania he believed a steady stream of misinformation was the most effective way to govern. looking at his remarks suggests this, but this is only speculation. Furthermore it could all be posturing because of the aforementioned. who is to say what he "really" believed

leslie townes

22:33

and yes with politics, all is possible. there are people in the US who take political positions that would seem to be at odds with their training because it works for them politically. there's often the question of whether they 'really' believe it. i think it can go either way.

dc3rd

interesting thoughts, all of them.

I'm not sure why misinformation would be a good way to govern @BigSocks, but then I am assuming that he was working with the intention of serving the people, when in truth he could've just been in it for himself.

leslie townes

i don't have to look as far as tanzania to see other potential examples of that.

some people, who tend to be more humanities driven than math/science driven, might even blame math/science education a little bit. i say this because you can get quite good at math, for example, without ever engaging at any depth with other human beings, and complexity that is not mathematical complexity.

crude example, but there were some very good mathematicians who also happened to be nazis. you'd think, how is it possible? but then you think, well, if you're just proving theorems all day, how are you going to become less stupid in ways that don't involve theorems. which is not to say i think that reading good literature or social theory is an inoculation against poisonous belief. i prefer my gear theory. gears just slip.

dc3rd

I've encountered those situations in my life as well @leslietownes, just feels this whole pandemic environment as amplified it to another power.

I was commenting on the first part by the way....not the nazis.....haven't encountered any of those....they probably wouldn't like me anyways.....

leslie townes

i've been disappointed to learn that mathematicians whose work i liked a lot, or who were friendly to me, held beliefs that struck me as repugnant, or treated people in ways that seemed at odds with a proper understanding of logic, reasoning, etc. that we supposedly cultivate in math.

dc3rd

I like the gear analogy

leslie townes

22:44

at least the tanzania guy got to be president of something, i can see the possibility for material benefit there. he probably owned a nice car. many people go way down crazy paths without such compensation.

i'm a bit down now, thanks for that.

Semiclassical

23:39

@leslietownes the main example is always Teichmuller

Brilliant mathematician, and ardent Nazi

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$Mathematics$ Mathematics