Mathematics - 2019-09-17 (page 2 of 2)

s.harp

19:07

@user193319 the wiki article says that $SU(2)= \left\{\begin{pmatrix} z_1 & z_2 \\ -\overline{z_2} & \overline{z_1}\end{pmatrix} \middle| z_1, z_2\in\Bbb C, |z_1|^2+|z_2|^2=1\right\}$

user193319

@s.harp Yes, and I don't see how the alleged isomorphism is in fact an isomorphism from SU(2) to S^3

s.harp

the tuple $(z_1,z_2)$ is in $\Bbb C^2 = \Bbb R^4$

and has unit norm in $\Bbb C^2$, but the norm in $\Bbb C^2$ is the same as in $\Bbb R^4$

user193319

Yes.

s.harp

hence the map (matrix of that form) $\mapsto (z_1,z_2)$ is a map into $S^3$

user193319

But is that a group isomorphism? I'm viewing $S^3$ in terms of the quaternions.

s.harp

19:11

it clearly is bijective, and you must check that its a group homomorphism by hand

( the tuple $(z_1, z_2)$ corresponds to the quaternion $R(z_1) + i\, I(z_1) + j\, R(z_3) + k\, I(z_3)$ )

user193319

I'm sure this is right, but I feel that my professor doesn't want us to do it this real. Because complex numbers don't come into play with respect to $S^3$ (of course, they do with $SU(2)$).

s.harp

the write out $z_1 = a+ ib$ and $z_2=c+id$ in all equations..

user193319

$|z_1|^2 + |z_2|^2 = 1$ implies $a^2 + b^2 + c^2 + d^2 = 1$., e.g.

Leaky Nun

they're both the quaternions with norm 1!

Ultradark

19:26

given an $n \times n$ integer lattice in the euclidean plane can you define a directed graph from this

such that the flow takes larger ordered pairs to lower ordered pairs, for example the point (8,8) would flow into two parts, (8,7) and (7,8)

user193319

So, the group structure on $S^3$ thought of as consisting of ordered pairs $(z_1,z_2)$ is what exactly?

Ultradark

It's a commutative diagram

Ted Shifrin

19:43

@user193319 The quaternion is given by $z_1 + j z_2$, with $z_1,z_2\in\Bbb C$.

Leaky Nun

I like $z_1 + z_2 j$ more

user193319

So $(z_1,z_2)$ is thought of as $z_1 + jz_2$ and we use "regular" multiplication of numbers?

Sayan Chattopadhyay

@TedShifrin I managed to convince my universities library to get three copies of your multivariable calculus book. I have been trying that for a long time and now it has finally happened.

Sir Cumference

20:10

@AlessandroCodenotti Oh neat, so we don't need choice?

Ted Shifrin

@LeakyNun I agree. I vote with you.

@Sayan: Wow. Three might be overkill, but I hope you aren't too disappointed.

@user193319 Well, we're going to have to use $ij=k=-ji$, $j^2=-1$, etc., to get the rule worked out.

ÍgjøgnumMeg

Evening all

Rithaniel

Evening, Meg

ÍgjøgnumMeg

Heya @Rithaniel

Rithaniel

How's it going in Germany?

ÍgjøgnumMeg

20:29

Haven't moved yet, one week to go!

ish

Rithaniel

Ah, I am eternally off when it comes to predicting when things have happened.

Next week I'll start saying "ah, they probably haven't moved yet."

Semiclassical

20:55

@TedShifrin the weird bit of this is that I think you do see orthogonality used with some frequency

There’s a whole Wikipedia page on the “orthogonality principle” in stats and signal processing

Shadow

Will you have a look at this "https://math.stackexchange.com/questions/3360228/some-extended-question-on-bounds-of-rational-map"?

Ted Shifrin

@Semiclassic: That's step 1 of being geometric. There are more steps. But, of course.

hi @Rithaniel and @ÍgjøgnumMeg.

amanuel2

@TedShifrin Sorry didnt refer to this

but I got the solution. Thanks!!!

Project was due 2 hour ago lol

Ted Shifrin

LOL, good. :)

Ultradark

21:14

best part of my day was (doing math)

math never dissapoints

Semiclassical

@TedShifrin yeah. It’s just odd to me that it goes there but no farther

That said, I get plenty of workout from orthogonality in physics in QM and electromagnetism

And while there’s vector space ideas built into stuff like Fourier analysis, I’m not sure how much intuition I actually get out of that

So me complaining about stats not being sufficiently geometric is probably a bit hypocritical

@ted on a related note, I finally figured out a simple way to do one of my geometric computations

At which point I realized that what I’d done was equivalent to scalar triple product = volume of parallelpiped

And felt silly for missing something so obvious for so long

johnny09

22:02

hi everybody

could anyone provide some help/input in the question below?

1

Q: Optimization problem with inequality constraints

Suppose we have $\theta=(\theta_1,\ldots,\theta_n)$, with $v_i:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ being a continuous, differentiable, concave function. Now I want to solve the following maximization problem: $$ \max_{\theta} \sum_{i=1}^{n} v_i(\theta_i,a_i) $$ subject to $$ \theta_i\geq 0,\...

basically I am trying to solve a convex optimization problem where some constraints are relative to the first variable of the objective function and then there is a constraint on the second variable.

Semiclassical

22:40

There's an abuse of notation I know I use: If I take the quotient Z/3Z, then strictly speaking the elements are not 0,1,2,3 but rather the congruence classes 0+3Z, 1+3Z, 2+3Z, 3+3Z

with 0,1,2,3 as convenient representatives of these classes

Is there an easy way to say "When I talk about elements of Z/3Z, I'll only talk about the representatives"? It's a common enough abuse of notation that (in my context) I want to acknowledge it and move on

Ted Shifrin

@Semiclassical Just the determinant ...

Semiclassical

yep

Hence why I felt quite silly upon realizing how simple it was.

Ted Shifrin

I use $\bar 0$, etc., for the elements of the quotient. I marked my students down badly for leaving off the bar. Without the bar, you've chosen a representative and you're no longer working in the quotient.

Pedagogically it's very bad to confuse the two.

Students really mess up and don't understand what they're doing.

Semiclassical

Yeah, I get that. But in the context of what I'm writing up, there's no way I'm going to introduce notation to distinguish them. (it's not the integers mod n in my case, to be clear. it's "random variables are equivalent if they're equal with probability 1.")

Ted Shifrin

Well, it's standard in Lebesgue theory to work with functions defined a.e.

If you'd done measure theory, you wouldn't be pondering this.

In other words, if $f=g$ a.e., they're the same function.

Semiclassical

22:52

Isn't that the same abuse of notation, though?

I don't mind that abuse of notation, I'm just trying to find a quick way to acknowledge it and move on.

Ted Shifrin

Yes, but by the time you're a grad student doing measure theory, you're well beyond someone just learning cosets. And in your setting you really don't prove theorems about equivalence classes; you prove theorems about functions.

Semiclassical

Sure. I get why that's a smart thing to do. I'm just trying to figure out how to say it (in a footnote) without belaboring the point

Ted Shifrin

I think the probability abbreviation is a.s. as opposed to a.e.

Semiclassical

yeah, almost surely

Ted Shifrin

That's what you're talking about.

So you can say that everything you're talking about has an understood a.s. unless otherwise specified.

Semiclassical

22:56

for the context of this, I'm looking at an old paper of de Finetti (1937) where he deals with this stuff

I guess the simplest procedure is just to cite this statement of his:

"Since we can consider linear combinations of random variables, we can interpret them as vectors in an “abstract space". Considering $\sigma(X)$ as the modulus of the vector $X$, and, correspondingly, $\sigma(Y – X)$ as the distance $d(X, Y)$ between the vectors $X$ and $Y$, we define a distance space, or space “$D$” in the sense of Fréchet, under the hypothesis that all the random variables whose difference is given by the same variable are represented by the same vector." (emphasis added)

Ted Shifrin

This is old-fashioned language. I actually don't understand the bold part.

Semiclassical

Yeah, old-fashioned and translated from Italian

Ted Shifrin

What is 'given by the same variable'???

To me that's garbage.

Semiclassical

for clarity, $\sigma(X):=\sqrt{E((X-EX)^2)}$

Ted Shifrin

I know what standard deviation is.

Semiclassical

23:03

so, for instance, $d(X,X+c)=0$ since $(X+c)-X$ has variance zero for any constant random variable $c$

Ted Shifrin

You're not addressing my question.

Semiclassical

hmm. I know what I -thought- he should be saying there

But as I stare at it I'm a bit perplexed myself

(Something lost in translation, possibly)

Ted Shifrin

I think you should say what I said ages ago :P

Semiclassical

lol

Ted Shifrin

Mathematics has changed over 80+ years.

Reading classic stuff is often a challenge. Some classic differential geometry books are quite challenging to read.

Semiclassical

23:07

"Considerando $\sigma(X)$ come modulo del vettore $X$, e quindi $\sigma(Y – X)$ come distanza $d(X, Y)$ dei due vettori $X$ e $Y$, veniamo a definire uno spazio distanziale, o spazio "$D$" nel senso di Fréchet, purchè si considerino rappresentati da un medesimo vettore tutti i numeri aleatori la cui differenza coincida con un numero fisso."

now if only I could read italian

(I mean, I can guess that the "numeri aleatori" means "random number")

@ted yeah, the time barrier is probably as significant as the language barrier

Ted Shifrin

You can ping @Alessandro ...But I suspect it's old-fashioned, regardless.

Semiclassical

But I was already a bit dubious of this particular translation. (in the course of fixing a math typo at one point, they lost a semicolon and so created another typo)

No disagreement there.

One point I've gotten curious about, due to that paper: This paper dates to 1937. That's after Kolmogorov published his axiomatic treatment of probability theory in terms of measure theory

Ted Shifrin

Do people acknowledge it?

Semiclassical

Which paper, the de Finetti one?

Ted Shifrin

Right.

Semiclassical

23:12

It hasn't gotten a lot of attention, no. There are some citations of it, but largely by way of the translation I'm referencing

Ted Shifrin

No, I'm asking if your paper acknowledges Kolmogorov.

Semiclassical

ohhhhh

Nope.

Ted Shifrin

That's what I figured.

Semiclassical

that paper doesn't have a lot of citations period

Ted Shifrin

Not clear how fast the rest of the world got to know Kolmogorov's work ... Russia and all.

Semiclassical

23:14

Exactly, yeah.

At a conceptual level, I'm curious how far the notion of an inner product space of random variables goes back.

Though that's a bit hard to nail down. by 1915 Fisher had a notion of "correlation coefficient = cosine of angle between vectors", but his vectors correspond to n samples of a statistical quantity rather than random variables themselves

As far as inner products between random variables go, that paper of de Finetti is the earliest one I've seen.

(old-fashioned language and all)

on the other end, Kolmogorov published a paper in 1941 on "stationary sequences in Hilbert space", where the context is that of a random process being a sequence of random variables

So by 1941 Kolmogorov definitely has that POV, and is applying it to a rather sophisticated context.

But how did that come about? That, I haven't managed to figure out.

Leaky Nun

@TedShifrin finally :P

how many people actually do the :P emoticon and stick their tongue out

Semiclassical

Kolmogorov's statement about equivalent r.v.'s, btw, seems a lot clearer: he regards "equivalent random variables (that is, random variables that differ from each other with probability zero) as identical."

so score one for the Russians over the Italians? :P

Ted Shifrin

Some Russians can be sloppy about details, too. But I think Kolmogorov's thrust was to make everything rigorous in the context of measure theory, so ...

Semiclassical

yeah

I think what de Finetti's sentence ought to say is : all the random variables whose difference is given by a constant variable are represented by the same vector

Leaky Nun

23:30

@Ted what is maths?

Ted Shifrin

@Semiclassic: But that's still not accounting for measure 0 phenomena.

Huh? @Leaky

Leaky Nun

@TedShifrin what is the common features of the categories in mathematics?

Ted Shifrin

Probably 0.

I'm not in a philosophical mood, so I'm gonna depart.

Semiclassical

later

Leaky Nun

> math.AG - Algebraic Geometry (new, recent, current month)
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
math.AT - Algebraic Topology (new, recent, current month)
Homotopy theory, homological algebra, algebraic treatments of manifolds
math.AP - Analysis of PDEs (new, recent, current month)
Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE's, conservation laws, qualitative dynamics

Semiclassical

23:33

"purchè si considerino rappresentati da un medesimo vettore tutti i numeri aleatori la cui differenza coincida con un numero fisso"

it's easier to read math in other languages when you don't have to worry about the words :P

@ted hmm, Google translates "numero fisso" as "fixed number"

which seems pretty close to what I'm saying about it being difference by the same constant rather than by the same variable

Semiclassical

23:46

@ted as for the measure zero bit, I think the point is that de Finetti actually defines the phrase "X coincides with Y" earlier in the paper as "$|X-Y|>\epsilon$ with probability zero for any $\epsilon$"

so that's just a synonym for equal a.s.

oh. and the translation doesn't use "coincide" in that context despite its clear presence in the original. so, strike two for them

Shaun

in Constructive Feedback, 37 secs ago, by Shaun

Please may I have some constructive feedback on the question above?

Semiclassical

so, my corrected translation: "under the hypothesis that all the random variables whose difference coincides with a fixed number are represented by the same vector."

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