Mathematics - 2017-05-03 (page 2 of 6)

yeah

i think this issue has a lot of stuff going on with it

you know rotation matrices right?

yeap

here's something to think about

a rotation matrix that maps an object sitting on a flat plane to the tangent plane to a parametric surface at any point on it

(either side is allowed)

you can assume whatever you wish

been racking my brain trying to figure it out. :p

a single matrix that works for any object? o.O

nonono

given a particular paremetric surface with its regular functions given to define it

use calculus to determine a function of x and y (the variables entered into the parametric surface equation) to output the sort of matrix i described

i.e. the general equation

probably just a simple differential geometry thingy

5:10 AM

oh, I think I understand the problem

been trying to do it for smooth surfaces

Hey everyone!

You're trying to push an image in a plane onto the tangent plane of the surface, right?

hello damin :)

precisely

the starting plane is z = 0

for simplicity

sure

5:11 AM

How's it going?

so basically, a matrix to let a 3d model walk on a parametric surface

been stuck on trying to do it

anyway

just a fun problem,

goodnight

yeah :)

g'night

See you @TheGreatDuck!

Hey guys :)

decently Damin

5:11 AM

How come latex isn't working?

eyy rayny

you might have to turn in on each session?

Or have you tried SteamyRoot's suggestion?

As in, you can't see it rendered? Do you have the start chatjax thing there? If so, click it

today was an awkward day of teaching, but it's over and that's what matters :D

otherwise pretty sweet

Lol, what happened?

there were senior thesis presentations today which was fun :D

ehh it's kind of a long story

the course I'm in is called 'linear algebra and differential equations' and it is precisely one of those things

so I tried, thoughout the semester, to talk about honest linear algebra

just 5 minutes per week

cause obviously it's not what I'm there for

5:14 AM

Wait so the course is mostly differential equations?

but basically it came down to a choice between stopping in some random place in the story that doesn't make any sense, or spending 15 minutes today on it

('today'='day they're submitting evaluations', by the way :/ )

Oh lawd

but I just gotta get over it because

that's what grad school is for

screwing up teaching without much consequence

But yeah I dunno, I'm of the opinion that people should learn linear algebra before multivariable calculus, and probably wouldn't be opposed to doing it before calculus period

making mistakes while the stakes are low :P

yaaaasssss

5:17 AM

I basically saw linear algebra twice, the first time being in the REU where my professor definitely taught it with the mindset of someone in discrete math

@Daminark I actually do have the start chatjax thing

let me try this: $x^TAx$

Rayny: did you try Steamy's suggestion?

Second time was in analysis, where we were just given a billion problems from Hoffman and Kunze to do weekly

Also hey @Alessandro!

Replace "2.7.0" by "2.7.1" inside the url in the MathJax bookmark. It should work then
- 20h ago by SteamyRoot

Steamy's selection? :o

Oh gotcha. Sorry, still half asleep, didn't comprehend it til now haha

5:19 AM

haha Hoffman and Kunze?

let me look that up

I'm rather fond of that linear algebra book

oh lol that's a hell of a book

It's somewhat old-fashioned, so chapter 1 is on matrices/linear equations

And it doesn't do stuff like first isomorphism theorem

looks good but not an intro course :P

But I dunno, maybe for an intro that's better

Oh lel

Eh, I find that baptism by fire can do a lot of good

:P

5:20 AM

:P

my intro-to-proofs course was anaylsis with Rudin, and I turned out okay, so I can't disagree too much :P

Oh Rudin...

but yeah I'm always on the lookout for good intro lin al textbooks

you mean this part? : "https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js

So I first started by Spivak

Because it already says 2.7.1. How problematic

5:22 AM

:(

I don't actually use it myself, so that's all I got :/

Restarting browser. Sec!

cyasoon

I've actually never looked at Spivak

Last year we went through most of Spivak, up to uniform convergence, and then we did a bit of multi. Most of the sections used Calc on Manifolds, my professor didn't feel like there was any point in dealing with open sets since we'd do it next year, and also that we needed computation practice since we were getting pretty shitty at it

I feel like I've been trained to say 'spivak is a great book to learn from', but honestly I have no idea >.<

So we used Stewart more to start with, discussing stuff like quadric surfaces and parametrized curves

5:23 AM

lel

Ah. the good ol' restart works

Awesome :)

that's simultaneously awful and excellent XD

We used a sort of mix later for things like derivatives, and then went to Spivak for integration

not you Rayny, that's just good news jeje

(Which happened for ~ 2 days)

5:24 AM

yeah! thanks :)

Woohoo @OneRayny!

yay :D

But yeah, I had already seen some calculus from IB (whose curriculum is nonsense) so I had sorta gotten bored after $\delta-\epsilon$ started making sense

oooooh let's bitch about IB pl0x

So I tried just reading Rudin directly... Yeah... I think you can imagine how that went...

5:26 AM

Yeah...

Halfway through chapter 2 I was just like "I'm not even sure what's happening anymore" so I just started sitting in on analysis.

Oh yeah IB at the time felt great but they have no idea what the hell they're doing in math/science. They have "higher level" physics which makes you think you're getting deep (so I pushed to go into honors physics when I got here), but there's no problem solving practice (so I got rekt)

And IB Math teaches you integration as anti-differentiation so FTC feels like a statement of definition when you first see it. Then only if you do the calc option will you actually see Riemann sums

I guess that makes sense. I did HL phys and didn't really think much of it. Probably that was because the course was sort of tangent to the exam after the requisite magnetism stuff

I had to miss school on request of the principal to help with a play that was being performed so I missed that part, meaning for a whole year I thought FTC had literally no content.

>.<

So second quarter calc was actually interesting, I realized that there was something in that statement so I was like "Whoa"

Other classes were alright, though. Still, if IB had a "Legit higher level physics" I'd be much happier, and if they taught calculus so that it made sense...

(Though in all honesty I'm suspicious of how we insist on throwing calculus so early)

5:33 AM

^

I mean I know why, because math education is built around engineering

idk I feel like I have no idea what was actually required for the social sciences part of IB

like I did a thing, and it was fun, but it seemed really non-standardized compared to everything else, even A1

Hi @Dami

also was ToK complete bullshit at your school too?

I really want to believe that it can be done well but...

So as for social sciences, I did HL econ

5:35 AM

oh lol

that was a lot more serious than me >.<

I did SL anthro

I did an online econ course during high school. I forget why.

And ToK was odd at my school, we bounced around between various teachers first year, and it was pretty discussion based, though the teachers had various conflicting styles

Probably I needed to take it for reqs but it didn't fit nicely in my schedule.

How's it going @Alessandro? And hey @Semi!

heyo

funny thing is that now I could probably do about any econ math problem someone would throw at me, given enough prep time to translate it.

5:37 AM

I could have taken an intro to economics course this year, but I chose functional programming instead

But yeah, we did quarters (called trimesters) there as well, so second and third quarters were normal, discussion-based ToK where we bounced teachers. I remember one who was the type to go around the room and ask everyone (including those who very clearly didn't care) what they thought, another who would just let a few of us talk for 90% of the time

First quarter 12th grade, we had one teacher and while there was some discussion, the focus was on the essay and presentation. That teacher was a philosopher and rather enjoyed ToK, and was good at it too (he was my economics teacher in 11th grade, and taught business as well)

Semic: Math had their senior thesis presentations today. The actuarial ones were fun but oh goodness the vocab was a little intense :P

Huh, damin, that sounds pretty reasonable tbh :)

Though the one who basically said "If you care, participate, if not your grade will suffer but I won't get up in your face about it as long as you aren't disruptive", so that only 4-6 of us ever talked, was probably the best. Not for that reason (though it was helpful), just that I thin she had the most experience so kinda knew how to make it work well

@Alessandro Functional programming is great

Hey @Ted!

evening @ted

Ted Shifrin

Hi Demonark, @Alessandro, @Semiclassic, @EricS. Except for Alessandro, lots of late night oil being burned ...

5:41 AM

I've done Racket, meaning to get into Haskell

yeah Ted... just got back from family outing and needed some unwinding

@Ted Lol I basically never sleep before 2AM, often 3AM :P

So Haskell is an advanced Racket, eh.

Ted Shifrin

family outing, ah

@EricS What other presentations came to mind?

5:42 AM

hehe

best presentation hands down was the one about differential equations

it was some nonlocal ODE that was vaguely Laplacian-like

I've never been so turned on by ODE in my life tbh

Nice

Least comprehensible was "factor homology and TQFTs"

I mean, also fun, but in a masochistic kind of way

Lolol, nice

Actually today in math club one of my friends gave a talk on the fundamental group and covering spaces

nayc :)

Reminds me of how I feel at some of the IMA workshops when they happen.

5:44 AM

haha

He had to go fast because we had housing lottery later, so many intermediate results were sorta handwaved, but it was pretty fun

Mostly they're outside my subject realm, but sometimes I know just enough to recognize something.

^ same tbh

I really should go to more of them. I feel like I just haven't had enough exposure to that kind of math and I really could be comfortable with it if I made the effort to go to more talks

What types of topics do they do in IMA?

their two big seminars are "Industrial Problems Sem." and "Data Science Sem."

5:47 AM

Institute of Math and its Applications.

this year they have a few programs running simultaneously: ima.umn.edu/programs/annual

Each year they've got a theme around which (most of) the workshops are organized.

Ah, I see, that sounds fun

So for instance this year's was this: ima.umn.edu/2016-2017

Kind of outside my current "line of work" if we'll be so generous as to call it that

5:49 AM

mine too, really.

Hey guys, for proximal operator $prox_{\lambda r}$, I'm trying to find a closed form solution for where $r(x) = ||x||_2$ (not squared).

the funny thing was that the ones I appreciated the best were usually the experimental ones :)

Does anyone know an easy way to separate out each element of x so I can solve it implicitly case by case like $||x||_1$$?

Or if that's not the way to go, how would one go about solving this?

Which is probably because they were just experimental enough to be grounded but still had enough theory/math to be at the workshop

I see where you're coming from for sure

5:51 AM

Plus this talk was just plain cool: ima.umn.edu/2016-2017/W3.13-17.17/25876

Though I've grown not to really like labs/experiments much

I wouldn't want to work in a lab, to be fair.

But watching a talk to mathematicians about a tabletop experiment is good fun at times :)

Which is what pushed me away from science, coupled with how I've got 0 physical intuition

That is true

It helps that some of the stuff they talked about was optical versions of stuff I've heard a lot about in condensed matter physics involving electron behavior.

@Daminark agreed

5:55 AM

e.g. surface states, topological protection...

Fun. Is condensed matter physics your thing? It seems like quite a lot of people work in that (especially solid state)

Hi @Ted, I'm not too awake either though, still haven't had my morning coffee

yeah.

Ted Shifrin

why not, @Alessandro?!

though I'm not a very applied guy, so

5:57 AM

@TedShifrin Because I'm about to :P

@Alessandro Already reached the stage where coffee becomes necessary

@Semi Neither am I :P

Though... I mean...

there's degrees, of course.

I am so far intrigued by stuff like complexity theory

My school's got a number of people doing stuff along the lines of theoretical computer science/math

That's a fairly specific definition of 'applied', of course :)

Laci is into combinatorics and computational group theory mostly, and he's god-tier. There are also people who do theoretical compsci using things like algebraic geometry and representation theory

I think all of it seems rather fun, though I'd probably be better at the latter if only because combinatorics is creativity is very hard

6:02 AM

'theoretical compsci with algeo' is currently confuzzling me

lol

a fairly specific definition of 'applied math', as I said. :)

Check out something called "Geometric complexity theory"

Interesting. My school also has a lot of combinatorial optimizations people apparently and they do theoretical CS as well

It's Ketan Mulmuley's thing

Also we have someone named Razborov who uses logic and algebraic geometry in his work

Anyway, I don't know if I'm gonna go down a more traditional math path or something like algebro-geometric complexity theory, I'll give it more time and decide, but yeah, beyond that (as applied as you may call it :P) I'm more into theory

6:17 AM

g'morning Danu

Hey @Danu!

Danu

@TedShifrin Hey, I know the Gysin sequence :P

Hey @PVAL!

And @Liad!

@TedShifrin hi

hiya

6:24 AM

@Daminark hi there!

@Dami and whoever else is here

How's it going for you guys?

i need to calculate the volume of the intersection of $\{(x,y,z) : x \ ^ 2 +y \ ^ 2 \le 1 \}$ and $\{(x,y,z) : x \ ^ 2 +z \ ^ 2 \le 1 \}$ , someone here can help? i want to do it in cylindrical coordinates

Sorry, I've got minimal experience at this stuff.

that's ok :) How is your semester is going ? @Daminark

6:32 AM

Going pretty well, over halfway done now!

(Sounds late, but we do quarters :P)

im halfway done too :P

what is derivative of z*z' with respect to z ?

I mean $z\bar{z}$ wrt $z$

My immediate thought is that this doesn't satisfy Cauchy-Riemann

If $z = x + iy$, then $z\overline{z} = x^2 + y^2$

So you're only differentiable at $0$

And that should be easy to compute outright

$\frac{f(h) - f(0)}{h} = \overline{h}$, so that the derivative at $0$ is just $0$

Elsewhere, Cauchy-Riemann fails so this is not differentiable

thanks for the answer

No problem! Are you sure you see why this makes sense?

I didn't actually outright do the Cauchy-Riemann equations, try that

6:44 AM

Can anyone help with solving this least-squaresy problem? $min_x (\lambda W + \sqrt{x^TWx}I)x - \sqrt{x^TWx}y$?

yes they dont satisfy CR conditions

Oops, I guess my notation was terrible, let me rephrase:

Separate x in the equation $(\lambda W + \sqrt{x^TWx}I)x = \sqrt{x^TWx}y$.

Well, show me this. I was a bit too fast since it's 1:45 and I don't have my wits about me, normally I wouldn't have just said all that, I'd have walked you through more carefully so that you would've come up with it

@Rajesh

So let's just make sure you at least understand one part, do find the relevant partial derivatives and show where they satisfy C-R

7:07 AM

@Daminark I am taking partial derivatives and proceeding with my gradient descent algoritm. Not really interested in complex differentiation for y problem. I thought it would have easy notation if it were differentiable

Q: Growth of the zeroes of a family of functions.

Well, just make sure you understand each step then

Anyway, I should go to bed before I become dysfunctional even by my standards

Bye chat!

user8469759

9:15 AM

0

(I'm going to be informal...and I'm not sure of what I've done so far) Let $$ f_n(x) = 3^{x+1}+2(x-n)-1 $$ I define for $n \geq 1$ the value $x_n$ as the zero of $f_n$ (easy to prove it exists and it is also unique). I want to prove that $x_n$ grows as a log function, as $n$ diverges. My atte...

10:29 AM

@TedShifrin Which intuition? $H_1$ doesn't have torsion, right?

Also, can I do a Serre spectral sequence argument on $T_1 \Sigma$? I think so, the circle bundle is orientable so trivial over the generators of $\pi_1(\Sigma)$. That should mean $\pi_1$ of the base acts trivially on $H_*$ of the fibers.

Hi @PVAL

Even in the case the base is a sphere

you get L(p,1)

for euler class -p

Can someone help me with this Question:

which has H_1 Z/pz

If $\Sigma$ has genus > 1 $H_1$ should never have torsion, by abelianizing the homotopy exact sequence, @PVAL

For genus = 0 you get nontrivial $\pi_2$ that's all

by abelianizing the homotopy exact sequence

What does that mean?

10:35 AM

A pile has 34 coins. Two friends, Mina and Tina are playing a game in which each of them draws 2 or 3 coins from the pile of 34 coins. The person to draw the last coin is the loser. How many coins should Meena draw to ensure her win, if she is the first one to draw?

$\pi_1$ of a surface bundle is a $\Bbb Z$ extension of a surface group

How to do this one?

You get $\pi_2(\Sigma) \to \pi_1(S^1) \to \pi_1(T_1\Sigma) \to \pi_1(\Sigma) \to 0$. The last term is zero for genus > 0.

If you abelianize this don't you get $0 \to H_1(S^1) \to H_1(T_1 \Sigma) \to H_1(\Sigma) \to 0$, which splits, so you get $H_1 \cong \Bbb Z^{2g+1}$?

I guess you can't just abelianize short exact sequences.

@balarka whats the homology of the Klein bottle?

oops.

10:38 AM

The Klein bottle group is the first nontrivial $\Bbb Z$ by $\Bbb Z$ extension you learn about.

well that explains a lot

A pile has 34 coins. Two friends, Mina and Tina are playing a game in which each of them draws 2 or 3 coins from the pile of 34 coins. The person to draw the last coin is the loser. How many coins should Meena draw to ensure her win, if she is the first one to draw?'

Does no one know the answer?

@ABCD please don't spam

Sorry :/

@ABCD after the first move, Mina can always take a number of coins s.t. the total amount of coins taken by her and Tina in the last two draws is 5

10:43 AM

@PVAL Thanks, that's very interesting actually. So I guess the fiber is not torsion in fundamental group, similar to Klein bottle, but torsion in homology. Funny.

that is, if Tina picks 3, Mina picks 2, and if Tina picks 2, Mina picks 3

@Astyx that is not given in the question

What do you mean ?

@Astyx How did you come to this conclusion?

LegionMammal978

@robjohn As I figured out, it was the shutdown of cdn.mathjax.org that did it, so I just changed the URL in the bookmark

10:46 AM

If Tina picks 3 coins, in the next move Mina has the choice between 2 or 3 coins

The options are : A)2 B)3 C) A or B D) Mina can't win

@Astyx yes

I haven't solved it completely, I've just given you a hint

I am not getting it. I am stuck at it since half an hour. :/

we're not going to just do the whole problem for you

I know that.

But at least give a better clue.

10:48 AM

Okay let's get technical then. Let $m_0$ be the first move, then $t_1, m_1, t_2,\dots$ etc

Yes.

You want $m_0 + t_1 +m_1+ \dots + t_n +m_n = 33$

34 right?

I'm telling you you can always chose $m_k$ such that $t_k+m_k = 5$ whatever $t_k$ is

No, think about it @Abcd

What if both of them pick 3 coins?

then sum is 6 not 5

10:50 AM

Re read what I said

@Astyx What is "s.t."

such that

ohkay

I'm not saying the sum will be 5 whatever Mina choses, I'm saying she can always chose it so that the sum is 5

LegionMammal978

11:13 AM

Question: Is there any way to simplify $\sqrt{\dfrac{\sqrt2+\sqrt6}2}$ (i.e., de-nest the radical?) I got it from denesting $\sqrt{1+\sqrt{2+\sqrt3}}$, but I'm unsure if it's possible to go any further

There seems to be two @Eric here. Oops.

Martin Sleziak

11:36 AM

@LegionMammal978 We could rewrite $\sqrt2+\sqrt6=\sqrt2(1+\sqrt3)$, but this does not help much, since we cannot find real a,b such that $a+b\sqrt3=\sqrt{1+\sqrt3}$. (Well, unless I made a mistake.)

yeah sqrt(a + sqrt(b)) can be written as sqrt(c) + sqrt(d) iff a^2 - b is a rational

Martin Sleziak

We would need $a^2+3b^2=1$ and $2ab=1$, which would mean $1\ge 2\sqrt 3$ from $a^2+3b^2\ge 2\sqrt3ab$.

There are a few posts about denesting radicals in general on the main site. For example, Strategies to denest nested radicals..

Wikipedia might also be worth having a look: Denesting radicals.

LegionMammal978

@MartinSleziak Yeah, the quadratic algebraic integer case was used for the first denesting, but the second one is quartic

And the general algorithm involves loads of Galois theory that I tried (and failed) to understand

Wait, no, I'd have to denest $\sqrt{\dfrac{2+\sqrt2+\sqrt6}2}$

Ima go see if Landau's algorithm would work here

> input: f ( x ), an irreducible polynomial of degree n over k, and n, the lcm of the indices of the reducible radical expressions for α.

This will be fun

12:05 PM

@MeowMix Hi, how are you?

Here's a really quick puzzle: A polynomial that's not the product of two other polynomials (other than $1$ and itself) is called irreducible.

A polynomial whose first coefficient is $1$ is called monic.

Clearly, there are infinitely many monic irreducible polynomials; take $x+c$ for any $c$.

What if, instead of the coefficients being in $\Bbb Q$ or $\Bbb R$, we took the coefficients from $\Bbb Z/p\Bbb Z$? Are there still infinitely many?

Derp, autocorrect

Hehe

Although I kinda want "mimic polynomial" to be a thing now

A mimic polynomial is a monic polynomial with coefficients in $\mathbb{Z}_p$ ?

(Note, by the way, in $\Bbb Z/2\Bbb Z$, the polynomials $x$ and $x^2$ are considered to be unequal, despite having the same value no matter which element of $\Bbb Z/2\Bbb Z$ you plug into $x$.)

Cool question

12:11 PM

That's surely true over $\Bbb Z_2$

Yeah, very nice question indeed

Hrmf. We have a bachelor student who made his thesis about roots of polynomials over $\mathbb{Z}_n$, but I can't remember anything from his presentation anymore...

Why so @Alessandro ?

Well, except that he couldn't explain why $p^3 - p$ is always divisible by $3$, but that's not important

Oh, wait, I'm quite sure that's true for all $\Bbb Z_p$, can't you just take the irreducible polynomials and multiply for the inverse of the leading coefficent?

Why do you have an infinite number of irreducible polynomials ?

12:15 PM

Because the algebraic closure is an infinite degree extension

It is a two-second proof

^^Not that proof

Oh is it the same as proving there are infinitely many primes ?

Yeah.

Suppose there are a finite amount, multiply them all together and add 1

Multiply them all and add one, like Euclid did

Yep

12:16 PM

Cool

Hi Sha

Hi @Astyx

can you believe it? I'm an aspiring physicist

yesterday I started doing physics again, and it's making me really excited. i finally feel like all the math-work i've been doing is paying off:P

anyhow, how are you?

also, does anyone know how to show that the magnitude of the cross product in $\mathbb R^3$ is the area of the parallelogram of the two vectors? so I basically want to show that $\Vert a\times b\Vert=\Vert a\Vert\Vert b\Vert \cos\theta$ holds, using the definition

i tried doing this by consider two vectors in the $xy$ plane and calculating their cross product

however, I don't see immediate how $a_xb_y-b_xa_y$ is the area of the parallelogram

Haha what kind of physics did you do ? I'm good for my part

@ShaVuklia Should be sine for cross product

(Cosine is dot product)

You make your life easier by a change of base

12:31 PM

Nice @Akiva

Pick $a = (a_1,0,0)$ and $b = (b_1,b_2,0)$

Noice

haha well, I was supposed to start reading an introduction on quantum physics, but then I realised my classical mechanics was very rusty, so I've been reviewing everything, and it's all so simple now, because I've finally ditched the horrible non-mathematical books and now I'm doing physics on a slightly more mathematical level:P

@Akiva whoa man very cool

I think I like the second one better

Credit goes to Solomon Golomb (1932 – 2016, RIP).

12:34 PM

Intrinsically, isn't that by definition though ?

Amazing thanks @Akiva

Q: Square integrable continuous function and the square summability of a sequence formed from its values on a countable dense subset.

0

Consider a square integrable continuous function of the form $f: (0,1) \to\mathbb{R}$. Consider a countable dense subset $D$ of $(0,1)$ and let $k:\mathbb{N} \to D$ be its enumeration. Now consider the infinite sequence $\{x_n\}$ where $x_n = f(k(n))$, $n = 1,2,3,...$. Can we say that $\lim\limi...

real-analysis sequences-and-series

Martin Sleziak

@AkivaWeinberger From this answer? Why determinant of a 2 by 2 matrix is the area of a parallelogram?

@Akiva you prefer the second one?

but that one isn't even clear :P

the first one is great

@MartinSleziak

12:45 PM

"$\|(a_1,0,0) \times (b_1,b_2,0)\| = \|(0,0,a_1b_2) \| = |a_1||b_2| = \text{ base } \cdot \text{ height } = \text{ surface }$"

@ShaVuklia You have to think of it as taking the rectangle, cutting off a triangle each off the bottom and left sides (forming the bottom and left sides of the parallelogram) and shifting them up and to the right each (thus creating a parallelogram plus a small rectangle's worth of extra mass).

@Steamy yea, but if you're already going to make life simple, then you might as well choose $(a_1,0,0)$ and $(0,b_2,0)$ :P

You can't

oh ofc

If you had a piece of paper and scissors you'd see it very clearly.

12:46 PM

wait

i can

@Akiva yea I prefer considering the entire rectangle and then cuffing of the appropriate pieces, so without shifting pieces. but to each its own i guess:P

Not without changing the shape of the parallelogram :P

Yeah but for that one you have to do algebra

Expand out the binomial product and stuff

What's wrong with algebra :(