Mathematics - 2016-04-22 (page 1 of 2)

@Mambo Fourier algebras sound terrifying.

@Adeek yeah

1:02 AM

@Adeek: I would say you should have asked the professor. This isn't a seminar talk, right? It's for your tiny little class? So it's the polite thing to do to ask the professor in charge. But probably it's fine. Of course, that probably makes you even more nervous if they're there ...

Hi @Ted.

tylerl-uxai

Hey everyone!

I'm trying to understand Euler's growth rate. We use e to some power to show growth or something... Why do we use 2.7?

1:29 AM

Hi @TedShifrin I just got 100% for my Linear Algebra test

So thank you & the other kind people on here

1:40 AM

Congrats, @BenjaminR. I don't think we deserve any credit. Well done.

Good night, @MikeM.

You do, because just being able to come on here and ask random questions is such a big help

I have zero math confidence. I failed HS math.

Well, be careful, because soon I'll start asking you questions :)

What are you majoring in?

Now completing my Math minor for my Science degree

CompSci

Failing high school math has more to do with boredom and not being challenged than anything else. You're a smart guy.

Screw my p.o.s. teachers in HS

I think that's true for most ppl that fail math – boredom from crap teaching

1:42 AM

I'm working with high school students now that I've retired, and motivation/engagement is sorely lacking for so many. It makes me sad.

Math is amazing.

It literally gives me a high

Very proud of you. :)

natural algebraic endorphins

LOL

;-)

1:43 AM

Of course, I'd prefer geometric endorphins to algebraic ones, but mastering linear algebra is a wonderful first step :P

Joebot

Hey does anyone know what a "blob" is in the context of combinatorics? E.g., oeis.org/A003168

I prefer graph or group endorphins ;-)

Thanks again! cya

Bubye, @BenjaminR.

Joebot

The example given for n=2 seems to suggest the combinatorial object in question has 6 edges, yet according to the definition it should have 2(2)+1=5 edges

The Man Who Knew Infinity Official Trailer #1 (2016) - Dev Patel, Jeremy Irons Movie HD

I have no clue, sorry, @Joebot.

Joebot

2:04 AM

It's very odd. The sequence has been in the lists since before the OEIS was online, yet I can't find a reference for it anywhere else.

Thanks though, @TedShifrin

Rajesh Dachiraju

2:22 AM

►

0:40 What does the S stand for? Hope

lold

Adeek

LOL

HOPE @anon

2:55 AM

Hey guys! I have a question regarding linear algebra - it'd be great if anyone could take a look! math.stackexchange.com/questions/1753657/…

Have a great day and thanks :)

@OneRaynyDay pick four small numbers, put them in a matrix, see if it satisfies the identity

@OneRaynyDay trivially, the 0 matrix

@OneRaynyDay If you want to show that something is not always true, it suffices to give an example of when it is not true

Wait, but the proof says

it's not possible for every single matrix A

Not it's always true

I'm not sure whether I'm tripping

2 mins ago, by datalava

@OneRaynyDay If you want to show that something is not always true, it suffices to give an example of when it is not true

Read before typing please.

or whether I didn't convey the idea correctly

@anon I did read it, and I'm saying that I'm not trying to prove

that it's not always true. In fact, I'm trying to prove that it's never true

3:07 AM

Do you mean, for all A, it is not the case that A^2-A=I?

yes

so you mean always not true, not not always true

logic.exe

@anon Yes, my bad - I was bad at explaining it

equivalently, you could say "there is no A such that A^2-A=I"

@anon Isn't this a good enough way to say that? " A2A2 - AA = I2I2 is not possible for every single possible A."

I said that in my question, and I thought it was reflective of that idea

3:09 AM

no

that's ambiguous

it could mean "it is not possible that (A^2-A=I for every single possible A)"

also, do you want to specify integer entries?

@anon Alright, my bad then, I'll edit it

No, I think anything is alright - the question didn't specify

then solve x^2-x=1 for x and then consider xI.

I don't think the statement is true if you consider a matrix over the reals...

(you get the answer in one of the comments)

@anon I see... but isn't x a matrix?

3:12 AM

@OneRaynyDay no

I'm not very clear on how matrices translate over to that equation

xI is a matrix

@OneRaynyDay you solve x^2-x=1 for the number x, then consider the matrix xI

The True/False question is true. You're trying to prove it's false

A difficult task, indeed.

@anon I'm sorry, I don't understand what you mean by "consider the matrix xI"

@OneRaynyDay I is the identity matrix, xI is x times I.

3:15 AM

Yeah, I understand that part - but what do we do in considering it?

Do we plug it into the previous scalar equation some way?

for instance, say you wanted to find a matrix A such that A^2=9I. well, you can solve x^2=9 with numbers, just use x=3. then consider the matrix A=3I. that's a solution to A^2=9I.

If so, I'm not seeing where we can do that - do we just replace all x's with xI's?

do you know how to multiply a matrix by a number?

Ah I see, gotcha

Yeah I do

Ahh okay, I understand it - that makes sense... but the test question says that it's false for any matrix A

If that's the case then we can just do what the anonymous user said in the post, but then the question would be wrong...?

Maybe it's the fact that our teacher only taught us integer matrices so far, but I'm not completely sure

Well then... In that case should I delete the question?

I'm not sure if it'd be misleading to anyone

user19405892

4:27 AM

hi

Q: Confirmation on a monotonicity formula?

@user19405892 lo

TheGreatDuck

6:18 AM

anyone think that they can answer this?

0

After a long series of difficult problems (which are completely irrelevant) I found myself experimenting with a way to convert a graphed function to a purely monotonic form (I hope I didn't butcher that terminology). I was thinking that if I took a function's derivative and then it's absolute va...

7:02 AM

@Axoren Have you heard of it earlier ?

Akiva Weinberger

7:40 AM

Given a cocycle, it's so hard for me to see if it's a coboundary

It's so much easier in homology, 'cause it's much more geometric there.

8:46 AM

That was a really good meeting with my mentor yesterday. Barring some computations, we managed to extend one of our results from holding for primes to all odd numbers.

8:56 AM

that's a huge step, no?

user147690

@TobiasKildetoft How does one obtain a mentor?

@AlexClark It is essentially the same as an advisor but for postdocs (so he does not really have any formal role, but we meet about once a week)

user147690

@TobiasKildetoft Ah, that's very cool

@AlexClark The first time here I constantly felt like I was playing catch up, trying to get a better understanding of what we were working on. But now I at least feel like I can sortof keep up (though my mentor is still wildly productive)

user147690

@TobiasKildetoft How many years of math education did you do before becoming a post doc? 4 years undergrad, 5 years phd grad?

9:02 AM

@AlexClark 5 year master's then 3 years PhD (well, about 4 due to maternity leave (or should that be paternity leave?))

user147690

It's paternity leave yep

lol

user147690

And undergrad?

@AlexClark Yeah, it is easier in Danish as the term is not gender specific

@AlexClark the master's degree was 3 years bachelor and then 2 years master, but nobody does just a bachelors degree here (and you need a master's to do a PhD)

user147690

Oh alright, interesting

user147690

9:04 AM

Whereas the people I know who stay in Australia do 3 years Bsc, 1 year Honours, 3 years PhD

user147690

But I don't feel that it's a great idea to do PhD without a coursework masters

user147690

3yr Bsc-> 1yr Hon -> 2yr Msc -> 3yr PhD

Do you have grade 13 in Australia @AlexClark?

user147690

@skillpatrol Australia not Austria :P, and no

user147690

And depending on state, we graduate highschool at different ages

user147690

9:07 AM

So we start uni in my state turning 18 in first year

@AlexClark Even in Europe, people do PhD in three years

@Mambo Except in Sweden where they do a 5-year PhD after a 5-year master's

Wow

Oh I see

(they call it a 4-year PhD but that is because you need to teach 20% of the time and they the year-count is the research part)

9:09 AM

Where are you doing your postdoc @TobiasKildetoft

@Mambo Uppsala

and originally from ?

Denmark

Uppsala looks really beautiful

Yeah, it is

Jester Tran

9:34 AM

Does anyone know modern probability? Specifically, random variables and conditional distributions?

9:45 AM

why is that modern

what's old-fashioned probability?

user147690

10:04 AM

Abstract -> Introduction -> Prelims -> Content -> Optional appendices -> References

user147690

What goes in Prelims?

user147690

And does Introduction sort of just run you really quickly through the content without explaining on it. I.e. after reading the paper you would only then be able to really 'read' the introduction

@AlexClark prelims is where you put all the setup of what all the symbols mean and so on

user147690

Any basic theorems?

you can read the intro to have a rough idea what the content will be about

user147690

10:06 AM

Could one throw PBW in there?

so like "throughout the paper, $X$ will be a widget and $Y$ will be a distinguished subwidget"

@AlexClark yeah, those would go there too if you want to state them in full rather than just refer to them in proofs

The introduction can take one of two forms: One type is pretty much only understandable to someone already familiar with the field because it uses a bunch of standard terminology without explaining it, in order to more easily explain what the paper will be doing. The other spends some time also explaining the terms to make it understandable by more people, but becomes either longer or less clear in doing so

Any sobolev space people here?

I don't live in a Sobolev space

lol Well I don't either

user147690

Thanks @TobiasKildetoft

10:15 AM

@AlexClark though this is all based on the paper being a research paper aimed at other researchers

user147690

@TobiasKildetoft That's alright. My research course is meant to have me emulate this

user147690

(although I have no novel results [ofc])

@AlexClark Then get working on those :)

is GAP still computing btw?

user147690

Ahahaha, apparently noone has shown there is a unique basis of canonical type for sl(5), so I could do that :P

user147690

@TobiasKildetoft Last I saw many hours ago it was. But I can't verify until I can get a hold of someone at uni near the computer

10:18 AM

@AlexClark That's fine, just let it run and we will see at some point

no rush

And yeah, get cracking on sl(5) then

user147690

:P

user147690

I am not sure anyone would care if I did that though, it seems so obscure

@AlexClark My first paper improved an upper bound on something from $2n$ to $2n-3$ (though there was also a bound of $n + 24log_2(n) + 364$ known so my result was only an improvement for "small" $n$).

And the bound is conjectured to be just $n$ or maybe even smaller.

Nothing is too obscure for early career researchers to do.

user147690

I'm only 3rd year undergrad(in math atleast), and I have heard that it may be bad to try to publish anything as an undergrad

@AlexClark Not so much bad as probably very difficult

user147690

10:24 AM

Alright, well that's actually very encouraging then

the result I mentioned there is from my master's thesis (and the paper you are doing calculations for me for is also based on that)

though I have extended the results form my master's considerably for that new paper

user147690

Do most researchers, atleast from your observation, continue down one specialization after their first publication?

user147690

One of my friends is doing his honours year in Lie theory, and a few weeks into algebraic geometry says he wants to do masters in AG

@AlexClark not that much no. I did complex characters of finite groups to start with, then moved to algebraic groups in positive characteristic and finite groups of Lie type, then to $2$-representation theory and related stuff

with some complex Lie algebras in there as well

Though I am not really sure how common it is to switch this much (probably it is more a matter of spreading out a little bit)

10:54 AM

Hi guys!

Could any of you help me with a physics question?

I have a problem where an object whose mass varies over time is falling under gravity

$F = \dot{p}$

Does the change of mass affect its velocity in any way?

@Huy hmm, let me see where I get with that...

plug in and see what you get

But isn't acceleration independent of mass? Isn't that what the bowling ball and feather falling experiments proved?

did the bowling ball and feather change their mass during fall?

if you want, you can do the bowling ball and feather first, also use $F = \dot{p}$ to show that the acceleration in those cases is independent of mass

11:01 AM

@Huy hmm, alright, thank you!

no problem, you've given me an idea for an exam question for my students. :P

So you already know that acceleration under gravity is independent of mass. Now ask yourself: "what if it was the color that was changing instead?"

@Huy haha, I'm not sure whether to feel happy or sad for the students :P

@TobiasKildetoft ?

@Huy you are changing a characteristic of the object which does not impact the acceleration.

11:06 AM

@TobiasKildetoft I'm not sure if I understand you correctly but if you're talking about the changing mass situation, that's wrong

@Huy It is? You mean that changing the mass will change the acceleration?

@TobiasKildetoft: $F = \dot{p}$

what is p?

momentum

p = mv

11:09 AM

@Huy Ahh, I think I see where my intuition fails. It is not really that acceleration is independent of mass, it is just the the way it is dependent cancels out for constant mass

yes.

and everybody knows a pound of feathers weighs more than a pound of lead :P

@skillpatrol No, but it does weigh more than a pound of gold

iirc you actually will have to look at a constant-mass system to apply the 2nd law properly and can't just apply the product rule

but the masses will certainly not cancel in the varying case

what is the question you're going to ask your students?

11:12 AM

@skillpatrol Is there not something about the masses being equal, but the pound of feathers weighs less because the surface is bigger and therefore something something Archimedes' principle

@skillpatrol: just some nice differential equation with varying mass

I mean the starting rocket is the "classical" example

I was joking @Krijn :P

@Huy Am I correct in assuming that a=dp/(m*dt) then?

but I'm sure I can come up with something more interesting

"Nice differential equation", that's like "funny murder" or "cheap yacht"

11:15 AM

@Huy Be gentle on them and stick with the classics

@Mircea a = dv/dt

@skillpatrol i know that but I wanted to express it in terms of mass

@Krijn do you not like ODEs? ._.

@Huy I do not, but that's mostly the fault of the prof that thaught me a class on them

how so?

11:30 AM

@Huy hey. I'm still a little stuck... How can I derive an equation for the velocity in terms of the mass, time and g?

@Mircea: check in your notes, I'm sure something relevant must be in there

He was not a pleasant man and also made the course extremely dry and boring

I actually checked and you indeed can't directly apply Newton's 2nd law to a varying-mass system. see this wikipedia page

@Krijn: I hope that's not something my students will say about me

@Huy my notes are quite laconic...

@Mircea: then make more notes and ask questions where necessary :P

11:35 AM

@Huy i mean the notes that the professor was following. He was going over everything very quickly... And I can't really ask now since he's probably on vacation

@Huy Do you enjoy teaching them?

@Krijn: yeah, a lot actually. they're graduating in 2 months, so I will have to say goodbye soon.

hopefully, my new students will be just as nice as my current ones :D

Then they won't say that about you!

Still, it's a shame that almost nobody tells you the compliments and almost everybody tells you the complaints

@Huy heh, I'd also like to become a professor. Sounds so great to be able to help and inspire younger students. What are you teaching? :)

@Mircea: I'm not a professor, just a high school teacher :)

teaching maths of course, right now differential equations

probably gonna finish the chapter with systems of differential equations

11:40 AM

@Huy ah fair enough. I don't know anyone that did DEs in high school, what country are you from?

Switzerland

I'm teaching at a high school with focus on (applied) mathematics and natural sciences, so differential equations are very standard in the last year

@Huy ah, I see. How are the students handling them? Do they find them easy?

not easy, no, because everything comes together - calculus, complex numbers, and also linear algebra (for systems of ODEs)

I'm trying my best to show them many examples in medicine and biology though, because my students focus on biology and chemistry

@Huy do you also do the proofs for why the solutions are unque or follow a general pattern and that kind of stuff?

@Mircea: some I do, some I don't. unfortunately, there's not enough time to do everything with all proofs

alright, I have to teach soon. see you around and good luck with your physics problem!

11:46 AM

@Huy i see, makes sense

@Huy thanks, have fun.

AlpArslan

Hello everybody.

@AlpArslan hi

AlpArslan

I just have a small question regarding the theory of indefinite binary quadratic forms.

@AlpArslan huh, I know some of these words

AlpArslan

Basically, the definition I've got to use is that an indefinite form is reduced if the real centred circle passing through the roots of that form intersects the region {|z| >= 1, |Re(z)| <= 1/2}

And, given a discriminant, I need to find bounds for the coefficients of such reduced indefinite forms.

And then basically, write a program to enumerate all of them.

GPhys

2:29 PM

does this just come down to making a partition of the rectangles such that every subrectangle either fully contained or fully not contained in $R'$? i.imgur.com/4JKuUnb.png

which I can always do?

here a "rectangle" is defined on $\mathbb{R}^n$

Jeff

3:04 PM

Hi all. Does anyone know how long chat history is kept and where I can find it?

I want to find something that was explained to me possibly 1 or 2 years ago.

you probably want to use the search feature on the right. the chat history is immortal

or at least as long-lived as SE is

Jeff

Found what I wanted to @MikeMiller... TY

4:25 PM

Any help on what I've asked here would be very helpful: reddit.com/r/math/comments/4fyrt1/… . Ping me if you have a response

(The above is not a math-problem question, by the way... it is a presentation question)

@Mambo Nope, just from you asking about it. To me, it sounds like an algebra over Fourier transforms. That just sounds scary.

@Axoren lol. It is actually the set of all Fourier transforms in a sense

@Mambo It would be a shame if those didn't form an algebra. The name would be wasted.

With respect to pointwise product, they do form an algebra

@Axoren What do you do?

I'm a computer scientist in the field of machine learning. I tackle mathematics from the outside in to make practical use of cool math.

Expect a lack of rigor like no other.

4:38 PM

Good

@Axoren Ah, you're into machine learning???

@Clarinetist Yeah.

I've been trying to learn about ML. I have no idea how to start. I'm in a M.S. statistics program

In my opinion, it's easier to start with classification than regression or clustering.

I've tried reading the Elements of Statistical Learning and the intro version of that, and I absolutely loathe them @Axoren

I know a lot of the regression stuff; have already covered a good majority of it (minus the fancy penalized stuff like LASSO) in my masters courses

4:43 PM

@Clarinetist coursera.org/learn/machine-learning/home/welcome

tatan

I have $7x \equiv 1 \pmod 8$.How can I prove that it is equal to $x\equiv 7\pmod 8$?

Everyone I know tells me about Andrew Ng and how good his course on ML is

@Axoren Tried that one too. Didn't really have the patience at the time for using.. what was that language, Octave?

Yeah, Octave/Matlab

and his math symbols... don't even get me started

I know Python and R well

4:44 PM

It would be a lovely language if it just had better scoping and syntax.

Python, I recommend working with Theano once you have a better understanding of ML

If I could find a good way to start from knowing some advanced statistics, Python, and R, I would be sooooo happy

With R, it's a whole animal I know no part of.

I would just like a really good, dumbed-down explanation of what machine learning is, what the algorithms are, and how they would be programmed

That seems to be too much to ask for

@tatan What does $7x \equiv 1 \pmod 8$ mean?

and potential pitfalls of each algorithm

Elements of Statistical Learning is basically scholarly barf, in my opinion

4:47 PM

@Clarinetist For the pitfalls, you'll want to learn Learning Theory.

Which is way deep in math you'll probably never use in practice again

ML starts to become very empirical out in the real world.

And it's unfortunate when you know the math and you look at a method that "just works" and it's mathematically equivalent to a single matrix multiplication.

@Axoren I'm not at all surprised. But yes, those are my thoughts on ML. I'm just not sure how to start. I've read a lot of the popular books but none of them seem to be right for my level

But people are constructing entire multi-layer perceptrons with linear activation and learning those massive deep structures and making claims about them which are technically right.

@Axoren It seems that these days, there's a plethora of resources out there for people who don't know how to program and don't know any math

I'd like to do a little more than that

I might just look at Udacity and try Kaggle sometime

Classification: There is a predicate $P_{\mathcal{Y}}(x) = x \in Y$ that we want to learn. We instead estimate a predicate $\hat P_{\mathcal{Y}}(\hat x)$ where $\hat x$ is some description of $x$. The goal is to pick $\hat P_{\mathcal{Y}}$ such that $(\hat P_{\mathcal{Y}} - P_{\mathcal{Y}})$ is minimized over all $x$.

Obviously, the domains of those predicates are different, but there's a morphism in there somewhere I didn't notate.

That's pretty much the whole goal of that branch of ML

We want to be able to answer a Yes or No question. Obvious extensions of this are being able to answer "Maybe" and "$x \not \in Y$, but $x \in Z$", etc.

Hey

4:56 PM

$|(\hat P_{\mathcal{Y}} - P_{\mathcal{Y}})|$ *

@iwriteonbananas Hello, Mr. bananas.

Suppity sup

@MikeMiller, have you taken a course on spectral sequences at some point?

I know something (not much) about spectral sequences but I've never taken a course on them or anything like that.

Ok, too bad...I'm looking for the best way to structure my study of them. A curriculum and problem sets would be useful.

There are books you could read I guess, but do you know why you care about spectral sequences?

Not everything needs to be learned in the form of a course... why not just study other stuff, where they're applied, and learn the details as necessary?

McCleary looks pretty good, got that book from the library today. The author outlines which chapters form his book a student who wants to study SS's should learn, taking the least amount of time and incurring the least amount of pain

@MikeMiller Yeah, ok I'll do that

5:03 PM

That's more or less how I approach them, though I think I should think a little bit more about them at some point

I think that $H^*(BSO(3);R)$ problem I asked about is probably doable with some clever spectral sequences so I should think about them more

Right, I see

It probably wouldn't hurt to look a bit through McCleary and get the idea of a lot of things

I know less than I should, and I don't do algebraic topology. I think you're really into algebraic topology?

@MikeMiller I need them for my bachelor thesis, and I wanna start writing that thing...so I need to learn a bunch of stuff as soon as possible

Yeah, I like it...I wanna compute homotopy groups of spheres

You should try to prove that the J-homomorphism is not surjective onto $\pi_n^{st}$ for all sufficiently large $n$.

Sure, ok, then in that case my advice is probably off-base. One of the best computational papers I've ever seen was just like ten spectral sequences in a row, all used to calculate one differential of a particularly hard spectral sequence

You should talk to someone at your uni?

@MikeMiller Thanks, I'll write that down and as soon as I know what the J-homomorphism is and when I know more about htpy groups I'll try to prove it

5:08 PM

I mean, that should be your career goal.

haha. oh, it's one of those.

That's the exotic spheres problem.

I just asked you to prove that there's an exotic sphere in every sufficiently large dimension, more or less.

@MikeMiller I'll talk to my professor next week

Also I should have restricted to n = 3 mod 4 I think. I forget the details.

5:22 PM

My internet just died

Well, I only couldn't connect to this chat to be precise lol

Did that happen to anyone else?

no

ok, weird

i mean i guess i cant speak for everyone

poorly posed question i guess

lol

if you don't do algebraic topology, @MikeMiller, how would you describe what you're doing?

5:38 PM

Differential geometry? Low-dimensional topology? (I assume I'm not allowed to say gauge theory.)

hm, okay

I guess I gave the wrong impression this whole time :)

my world view is shattered. all this time i thought you were a full-blown algebraic topologist.

I won't be too hurt if you never speak to me again

i will cry tonight.

5:45 PM

yikes

@iwriteonbananas I emailed you.

I just registered for my first ever research hours as a grad student!

:)

6:03 PM

I finally have something I'm looking forward to...

what are you going to be working on?

combinatorial matrix theory

in particular sign patterns

what's that mean?

which part? sign pattern matrices are matrices with entries {+, -, 0}

sure, so what's exciting about em?

what do you want to find out?

6:07 PM

most of the questions are about the minimum ranks of associated matrices over the reals with the same sign pattern

well, I hope you enjoy it :)

I have a question on giving presentations. No math problem involved. reddit.com/r/math/comments/4fyrt1/…

@MikeMiller I emailed you too

congrats, @datalava

No way, @TedShifrin is here!

@TedShifrin!!!!!

lol

@TedShifrin I told my students yesterday that a solid torus deformation retracts onto any knot inside it that winds once around the torus. They were upset.