Mathematics - 2019-04-07

(I couldn't easily find it)

@Mikhail: I don't understand your reasoning. You're restricting the linear map to an $s$-dimensional subspace. Every matrix is a collection of columns (or of rows). Now you're looking just at the first $s$ columns.

12:16 AM

Basically, I'm confused because the matrix $A_s$ is a has less columns than elements in the vector $y$, how do you even matrix multiply them together :-)

No, no. They said explicitly to multiply by $s$-dimensional vectors $y$.

Yeah. Anyways, I'm still struggling to understand the meaning of the bound... Why is the left most term not the same as there right most term...

Huh? It is an approximation to the identity, but it might shrink vectors or stretch them (not to mention rotations).

I should say that $A_s^*A_s$ is close to the identity, so it can either shrink or stretch.

So, $d_s$ is a number like 0.3, the 2 norm of y is a also a number so you have number <= something <= same_number

Huh?

12:21 AM

The RIP bound

$(1-\delta_s)\|y\|_{2}^2 \le \|A_s y\|_{2}^2 \le (1+\delta_s)\|y\|_{2}^2$

No, it's the same thing I said. There's $1-\delta$ on the left (allowing shrinking), $1+\delta$ on the right (allowing stretching).

Oh, darn. Somehow my mind didn't parse the +/- sign. Deeply embarrassing :-)

Funny, cuz you typed it fine. :P

I just copy and pasted from wikipedia :-)

Ah, there's a moral to that story.

12:25 AM

I accidentally took math class I was completely unprepared for, we just talk about papers instead of learn anything. Also first class the guy ever taught. So the homework takes like 20+ hours, and everybody dropped the course :-/

You couldn't figure out the first few days that you weren't prepared?

Of course, reading papers is learning things ... in grad school, that's the way a lot gets learned.

In grad school we have this agreement, they don't teach us and they don't ask us anything.

I know I was one of the few faculty to assign homework exercises in advanced grad courses. Most faculty are too lazy to do that.

I'm pissed off that they still ain't teaching us, but now he's asking something! Everybody dropped but I was at a conference, and was anyways required to take a course for a fellowship. Course started with 40 people, down to 8

Ugg. Well time to learn all of compressive sensing in 2 days.

One of my former students is doing his PhD in that.

Good luck.

12:29 AM

Its all made obsolete by machine learning. Like I can (and already did) just train an neural network to do everything.

I don't know enough to debate this.

From what I can tell, the new thing is to work out the math behind somebodies neural network. In a kind of unsolicited fashion, where you walk up to them a year later and explain to them why their cure of cancer works.

Lucas Henrique

Hi chat.

Hi Lucas

12:48 AM

So, when are gershgorin discs typically covered/mentioned/encountered in the math curriculum? Maybe 3rd year undergrad?

Applied linear algebra course, usually.

They don't seem very applied. More like a way to make a somewhat loose bound?

Sure.

I never saw it as an undergrad, but it's in more advanced/applied linear algebra books.

Érico Melo Silva

ive never heard of those

heya Eric

Érico Melo Silva

12:52 AM

ive also never taken an applied class

hi Ted

The proof is very easy, Eric. I even stuck this as an exercise in my linear algebra book, in the complex matrices/eigenvalues section.

They keep coming up in sparsity stuff where you need to determine the conditions on the spark. Basically, you build a gramiam matrix and use them as a bound on when the gramiam matrix is strictly positive.

The eigenvalues are contained in disks centered at the diagonal elements of radius the sum of the absolute values of the non-diagonal terms in the row.

Page 58 on this guy princeton.edu/~yc5/ele538b_sparsity/lectures/…

Érico Melo Silva

cool

1:01 AM

I found half of my homework problems online in lecture notes or papers, then spend hours staring at them trying to figure out what the cracker jack the authors did...

1:12 AM

Okay guys, I need some help understanding a variable in this answer, what is $\sigma_{\max}$ and $\lambda_{\max}$ are these some kind of threshold functions? math.stackexchange.com/a/2137408/12631

$\sigma$ is singular value, $\lambda$ is eigenvalue.

1:38 AM

given $\phi(n)$ as the euler totient function, how many perfect squares are values of $\phi(n)$? I think that there are a lot of perfect squares and probably infinitely many

Karl Kronenfeld

2:26 AM

@Ultradark All odd powers of $2$. Can you find a way to generalize this?

2:38 AM

Anybody know how they find the dual problem (from fact 8.7 to 8.8) in princeton.edu/~yc5/ele538b_sparsity/lectures/…

2:58 AM

Hey to the chat.

I will try to find a way to generalize this

Silent

I read on wikipedia that every holomorphic function is equal, locally, to its own Taylor series.

I don't get why we need `locally' in above statement.

@Silent try $f(z) = 1/(1-z)$ around $0$

the Taylor series is $1+z+z^2+z^3+\cdots$

the radius of convergence is $1$

Silent

3:20 AM

So, for $|z|\ge1$ , the give taylor series does not work, is that the point? that's why locally?

Semiclassical

3:56 AM

@TedShifrin I made myself feel silly by doing a calculus problem wrong earlier. In my defense, it was a long day

Basically: I wanted to find the range of values for $f(x,y,z)=\cos x+\cos y+\cos z$ for $x,y,z\in[0,\pi]^3$ subject to the condition $x+y+z=2\pi$

@Silent yes

Semiclassical

So I solved for $z=2\pi - x-y$, then imposed $\partial f/\partial x=\partial f/\partial y=0$

which you can check to be satisfied when $x=y=z=2\pi/3$. That much is nice and easy

Karl Kronenfeld

@Silent It boils down to the fact that power series naturally work in disks, while holomorphic functions can have singularities.

Semiclassical

where I got myself into trouble: In terms of $\alpha,\beta$, the allowed domain is the region $\alpha\leq \pi,\beta\leq \pi,\alpha+\beta\geq \pi$

And $f(x,y,z)=-1$ on the entire boundary of that domain

But, of course, taking derivatives isn't going to tell you about that boundary...

So I got myself good and confused about something as silly as "a local min/max can occur at a critical point, but it can also occur at an endpoint" :S

soooo that's a bit embarrassing

Akiva Weinberger

4:20 AM

Fun fact: if you misinterpret a message as aggression, and respond to it as if it were an attack, the other person will become aggressive

War, in a nutshell.

@AkivaWeinberger what do you mean the other person will become aggressive??? how can you assume such things??? ridiculous!!

Martin Sleziak

5:44 AM

@Isa I left a few comments in calculus chatroom: chat.stackexchange.com/transcript/14150/2019/4/7

0xbadf00d

6:10 AM

Could anybody take a look at my question about a generalized central limit theorem: math.stackexchange.com/q/3175554/47771?

7:21 AM

So, for a ring $R$, if $S$ is a multiplicative closed subset and $I$ is an ideal with the property than any ideal which properly contains it must intersect $S$ nontrivially, then let $x,y\in R\setminus I$ and we know that $I+(x)$ and $I+(y) must intersect $S$. I want to use this to show that $I$ must be prime, but I am having trouble seeing the thread that would lead to that conclusion.

8:03 AM

Darn, I should have double checked that before clicking away. Now it's been too long to edit.

9:17 AM

I am being dim but why does $1+\ln(2)/(k/2) = (k+1)/(k+1-\ln(4))$ ?

@Anush where do you see that claim?

oh they are no identical

oops

they are just very close when k is large

so maybe I should have asked why the ratio tends to 1

because they both tend to 1

@LeakyNun good point.. hmm.. thanks

@LeakyNun they do seem very close to each other thought when k is large. Is there a sense there they could be asymptotic to each other not only because they both tend to 1?

they both tend to 1, period.

9:25 AM

@LeakyNun e.g. when k = 500 they are 1.0027781450122644, 1.0027802973432982)

@LeakyNun ok thanks

ok

if you subtract one from both sides

then they become $\dfrac{\ln 4}{x}$ vs $\dfrac{\ln4}{x+(1-\ln 4)}$

so they're just slight horizontal translations of each other

is that a more helpful answer?

@vzn New paper on why some theories of quantum gravity fail

oh cool that's very helpful

thank you

vzn

2:32 PM

in The h Bar, Apr 1 at 8:45, by bolbteppa

↑ still some fractal fans around? authors seem to dis/ ridicule fractals... those are fighting words! (which reminds me of a recent experience, lol!) o_O :P

3:30 PM

anyone familiar with the shortest vector problem?

3:40 PM

I'm just trying to understand the basics of the problem. It states: Suppose we are given a long basis for some lattice L. Then the shortest vector problem asks us to find a grid point in L as close as possible to the origin point.

Karl Kronenfeld

@Rithaniel What did you mean to edit?

Lattice problem

In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability of such problems is central to construction of secure lattice-based cryptosystems. For applications in such cryptosystems, lattices over vector spaces (often Q n {\displaystyle \mathbb {Q} ^{n}} ) or free modules (often Z n ...

It's NP

4:31 PM

are "a fortiori" and "therefore" the same?

4:42 PM

@LeakyNun I don't think so. From my understanding, "a fortiori" roughly means "from the stronger principle"

Or maybe just, "from the stronger."

Quick short ramble about infinity before go to sleep as I had 3 past days of crap and things only just get solved:

So one key feature for an object to be a potential infinity is that it is a procedure P and and some set S such that given any $s \in S$, $P(s)$ is injective and monotonic (meaning it will never retrace elements that it has already visited, meaning no cycles).

@Secret What about a cyclic universe and Poincare's recurrence theorem? I would consider such a universe an object that is potentially infinite, yet it doesn't seem to satisfy your injectivity requirement (of course, it is difficult to even understand what you mean).

4:58 PM

Hmm... if we let $s$ be a specific point in each universe of the cyclic universe, and $P$ to map from one cyclic universe to another, then there should be no situation where PP(s)=P(s), because each mapping of P will send you to the next universe. But I guess the issue is that we will need to index each universe, as otherwise all those s will have to be identical based on Poincare's recurrence theorem

right, it is not going to be easy, I need to think about this...

I'm not talking about multiple universes, though; I am talking about one cyclic universe which returns to identical states after a certain period.

ok so its evolution is periodic but its size can be potentially infinite, or are you talking about a cyclic universe that takes an infinite ordinal amount of time before it repeats itself?

If P indices its evolution, and 0 is the state of the universe at time 0, then there will exists some ordinal $\alpha$ such that $P^{\alpha}(0)=0$

I'm just talking about a universe which "goes on" forever and the states of the universe repeat, in accordance with Poincare' theorem.

Perhaps. Honestly, I would strongly recommend against not thinking about this stuff. Rather than reading wikipedia articles with crackpot ideas, read legitimate works of philosophers, physicists, etc. (but more importantly, read the philosophers).

rudyrucker.com/infinityandthemind/#calibre_link-294

Will this be a good book (I am currently reading this)

Yeah, I'm skeptical when it comes to mathematicians talking about philosophy (they tend to think they know more than they really do).

5:07 PM

I see

Of course, that's just rule of thumb.

Anyway, I don't know if I will call a process that produce a sequence like 1,2,3,1,2,3,1,2,3,... as potentially infinite. Because you can produce the same sequence by cycling through the elements of the set {1,2,3} indefinitely, or you actually have an ordered tuple (1,2,3,1,2,3,1,2,3,...) and you move to the next element for each step. The latter case will be your suggestion of the Poncaire' theorem scenario. Otherwise, going to spent some time to read that book before thinking about this

Okay, but the universe exists for a potentially infinite amount of time--that was my point; and those states will be temporally ordered by an 'early than' or 'later than' relation, so if one were to "plot" the states with respect to time, non-injectivity would be observed.

Chebotarev Density

hello @BalarkaSen

hello @LeakyNun

wait somone is still shitposting about infinites

5:23 PM

Not me! I was just responding.

Chebotarev Density

Of course not you lol. I was referring to you-know-who-shitposter-about-infinties

user76284

6:03 PM

@Secret I believe that if $f : A \rightarrow A$ is injective but not surjective, then $A$ is infinite.

If that's relevant to what you're discussing.

An injective but not surjective endofunction has infinite (co-)domain.

6:28 PM

Could someone please guide me trying to differentiat this formula using the product rule, im not sure if im correct

chandx

How many irrational numbers $x \in [0,2]$ can be expressed as $a + b\sqrt{2}$, where $a,b\in \mathbb{Q}$? Countably many or more?

ÍgjøgnumMeg

@chandx I guess countably many

Sha Vuklia

hey guys, I'm trying to show that $\neg(\forall x\phi)\vdash \exists x(\neg \phi)$ through an ND-derivation

I guess I dont understand the simplification steps

Sha Vuklia

6:39 PM

but I'm kind of stuck, because I don't see how I can find some $t$ such that we have $\neg\phi\frac{t}{x}$

I was trying to derive in some subderivation that we have $\forall x\phi$, and then that would yield $\perp$, which could maybe contradict the assumption $\phi\frac{t}{x}$, which would then yield to $\neg\phi\frac{t}{x}$

but that's rather the idea, and I can't really formalise it in an ND-derivation

Alessandro Codenotti

6:51 PM

ND?

@Sam what don’t you get?

I didn't realise A(a+b+c)+A(d+e+f) = A(a+b+c+d+e+f)

I was getting confused why there wasnt 2e^x

You need to use brackets

Your answer is too hard to follow otherwise

also is that $3-\sin x$ or multiplied?

Because what you’ve initially written is $\exp (x) x^2 + 7x ...$

which is very different to every term having their own exponential

yeh my bad. It's ok now though i realised my mistake.

im Kind of baffled to what you have done tbh

Its hard to get a hand of to begin with as with everything. You’ll get better though

6:59 PM

Yup, I'm practising.

Wanna try again and I’ll check it for you?

Sure, give me a sec I'll write something out properly.

Thanks.

@chandx Well, the set of all numbers of the form $a + b \sqrt{2}$ is at most countable, right? So, any subset must also be at most countable.

chandx

oh, right

MrAP

For $\theta_1,\theta_2,\dots,\theta_{51}\in R$, let $A(\theta_1,\theta_2,\dots,\theta_{51})$ be the average of the complex numbers $e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_{51}}$, where $i=\sqrt{-1}$. Find the minimum and maximum value of $|A|$.

On applying the triangle inequality for complex numbers, I got $|A|\leq1$. Therefore maximum value of $|A|$ is 1. To find the minimum value, I put all the angles equal to $\pi$ for which each $e^{i\theta_n}$ equals -1. On further calculation, I got, $|A|$=1. Therefore the minimum value is also one.

Alessandro Codenotti

7:10 PM

Algebraic extensions of an infinite field don't raise its cardinality

7:24 PM

@JakeRose could you talk me through this one

8:02 PM

@Sam I think you should get the previous one correct first

Stop cancelling things randomly

You’re just crossing out random things it makes no sense. If you want to cancel stuff write out the fully multiplied equation

Then simplify

Do the original question first there’s already big enough issues with that

@KarlKronenfeld Just the formatting. I dislike missing a $. It turns a sentence into a string of variables and I've found that it can be difficult to read.

I think i struggle with simplification

Do the first question

OK will do it again now

8:20 PM

Looks good to me

a lot better

although it’s worth noting that if you take the exponential out as the first step the maths may be easier

always factorise first if you can

Could you help me with the second one. Struggling to simplify it.

What is the name for a network in which nodes in the network only have two different degrees. For example the vast majority of nodes in the network have degrees 4 and some have degree 10

but there are no other nodes of any other degree

plotting the degree distribution, you get two points, and obviously there's an indirect linear relationship, not a normal distribution or power law distribution

Attempt it again

and imwill

8:35 PM

OK two sec

@JakeRose i know its totally wrong

Stop writing so many words

instead of writing just write it mathemtically

writing words that is

I don’t get what you’ve done in the second or third lines

you need to multiply them out first

Which lines?

as in defining f and g

No where you do actual maths

you need to expand everything out before you start adding and subtracting stuff

8:50 PM

how would i expand (2x-2)*(cos(x)+sin(x))

2 x sin(x) - 2 sin(x) + 2 x cos(x) - 2 cos(x)

Yes

just simple multiplication

ok

You can’t simplify it more than that

you on an ipad too? :P

Yup

9:09 PM

So essentially i need to review factorisation, thanks