Mathematics - 2024-07-14

user 85795

3:08 AM

Ask not when politics is a stage show. Ask when is politics not a stage show.

nickbros123

4:43 AM

@copper.hat yeah definitely seems like a stage show. The unseriousness of it all is a bit hilarious

nickbros123

6:06 AM

Id like to redact my above statement

copper.hat

yeah, it was pretty bad. i am curious how an audience member was killed and another injured. one report said that the bullet that clipped Trump hit someone behind.

I did not realize that people had been killed when I wrote my comment above.

user 85795

6:34 AM

thankfully it didn't turn into a mass shooting as in Las Vegas.

user 85795

6:47 AM

He shot into a crowd of 50,000 people.

robjohn

10:17 AM

@copper.hat My wife and I went through the same thought processes.

psie

10:38 AM

In the probability generating function, when someone writes $E(t^X)$, where $X$ is a random variable, what is $t^X$? It can't be a function of a random variable, since $t$ isn't fixed.

Jakobian

11:22 AM

@psie the random variable $t^X$ for some fixed $t$

psie

yeah, it wasn't harder than that, thanks :)

Jakobian

the probability generating function is the function $t\mapsto E(t^X)$

perhaps that's whats confusing to you

psie

yeah, now that you write it it's much clearer

Jakobian

11:57 AM

0

A: Compact sets of Moore plane

I will denote the Moore plane by $M$ and the $x$-axis (that is, $\Bbb R\times\{0\}$) by $X$. If $p\in\Bbb R^2$ and $r>0$, then $D_r(p)$ is the open disk centered at $p$ with radius $r$. Let $K$ be a compact subset of $M$. Then $K$ is also compact with respect to the usual topology on $\Bbb R\time...

this answer that characterizes the compact subsets of the Moore plane got downvoted for no reason

Xander Henderson

12:29 PM

@copper.hat There were a lot of shots fired.

What really gets me about the whole thing is that a bunch of secret services officers were risking their lives to protect Trump, but rather than just GTFOing, he kept popping his head out of the scrum and raising a fist, putting those people at further risk.

Xander Henderson

12:58 PM

NY Times suggests that an AR-15 was used: nytimes.com/live/2024/07/13/us/biden-trump-election/… .

psie

1:12 PM

Silly question, but Abel's theorem says that the limit from the left at $x=1$ of $\sum_{n=0}^\infty c_n x^n$ exists if $\sum_{n=0}^\infty c_n$ converges and $|x|<1$. Now, I am looking at probability generating functions, specifically derivatives of such, where one has $$\sum_{n=k}^\infty n(n-1)\cdots (n-k+1)t^{n-k}P(X=n).$$I'm confused as how to apply the theorem here, since it doesn't appear to be of the form $\sum_{n=0}^\infty c_nx^n$.

Jakobian

1:50 PM

@psie first you can make a substitution $m = n-k$ so that $\sum_{m=0}^\infty (m+k)...(m+1)t^m P(X = m+k)$ is your series

then you see that this is $c_m = (m+k)...(m+1)P(X = m+k)$ where $t$ and $x$ are just dummy variables

psie

ah ok, thanks, this helped.

copper.hat

5:00 PM

@XanderHenderson it is rather puzzling.

nickbros123

where can I learn about Vector spaces over finite fields, Ive checked sheldon axler, hoffman, and they dont seem to talk about them in much detail

im looking for introductory level stuff, get my feet into the water

Thorgott

5:23 PM

what about them?

Jakobian

Finite geometry

A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre...

are you looking for a book on something like this

nickbros123

5:39 PM

no no

I just wanted to know what fundamentally changes (if any) when studying LA over finite fields (or general fields not R or C)

kinda found the answer here: math.stackexchange.com/q/2865696/1118236

Jakobian

5:58 PM

@nickbros123 theory of quadratic forms is interesting even for $\mathbb{R}$

e.g. Minkowski's space

Thorgott

pretty much all of basic linear algebra does not care about the ground field

quadratic forms are a good example of something that does care

Jakobian

about Moore plane though, I think its interesting to replace $\{(x, 0)\}\cup B((x, r), r)$ with $\{(x, 0)\}\cup U_{x, r}$ for some open sets $U_{x, r}\subseteq \mathbb{R}\times (0, \infty)$ such that $(x, 0)\in \overline{U_x}$ and $U_{x, r}\subseteq U_{x, s}$ for $r < s$.

It shouldn't matter for most properties of the Moore plane, but should change how things converge to points $(x, 0)$, in particular it would change what the compact sets are

Jakobian

6:16 PM

hmm... it does need that $\bigcap_{r > 0} U_{x, r} = \emptyset$

and not sure if its enough to assert complete regularity

In the argument it matters that $U_{x, r}$ are convex, but also that the balls "converge" to $(x, 0)$ in a particular way

Sonu Gupta

Hello everyone!! can anyone please explain how the probabilities can be calculated here for picking any random ball ??

Jakobian

In the argument you take some set $U_{x, r}$ and define $f(x, 0) = 0$, $f(p) = 1$ for $p\in \partial U_{x, r}\setminus \{(x, 0)\}$ (boundary in $\mathbb{R}^2$), and extend this linearly to $U_{x, r}$ and let it be $1$ otherwise. Then you need to assert this is continuous.

Xander Henderson

@SonuGupta Isn't that what you are supposed to be figuring out?

Sonu Gupta

I am just confused about the many possible permutations of the picking of ball due to special balls

Jakobian

so for $\varepsilon > 0$ there has to exist $0 < s < r$ such that $f[U_{x, s}]\subseteq B((x, 0), \varepsilon)$

Sonu Gupta

6:28 PM

@XanderHenderson yes and I really tried hard but couldnt get any answer so far. Even I tried to understand the editorials but couldnt understand them. Thats why I'm asking here

Jakobian

I wonder what would be some easy conditions for $U_{x, r}$ to have so that this property is satisfied

@copper.hat could you help? This is, at heart, about convex sets

Xander Henderson

@SonuGupta What you have posted appears (to me) to be a competition problem. I am not really interested in working a competition problem for someone else.

Also, it appears to me that it is a coding problem, not really a math problem. It seems that you are supposed to simulate it.

Sonu Gupta

@XanderHenderson No, its over. Believe me. I am just curious to understand the concept behind it which is probability.

@XanderHenderson The main issue I'm facing here is that due to the special balls in n set of balls we can have many possible combination of the balls which would lead to different expected values for each of them ?? Then how can I uniquely determine it as it is asked in the question.

Obliv

would it make sense to say polar coordinates are specified in $\mathbb{R}\times\mathbb{R}/2\pi$ or $\mathbb{R}/360$ for degrees

i guess more precisely $[0,\infty)\times[0,\infty)/2\pi$

psie

6:55 PM

I'm studying PGFs and it is claimed that if two random variables have the same PGF, then they have the same distribution. The author of my book refers to the literature for this result and remarks this follows from the "uniqueness theorem" for power series. I have been trying to look up this theorem and found two theorems in regards to this, but to a great extent different proofs. This makes me think these theorems are not the same, but I can't spot what the difference between them is.

The first theorem is from baby Rudin. It reads as follows:

> 8.5 Theorem Suppose the series $\sum a_n x^n$ and $\sum b_n x^n$ converge in the segment $S=(-R,R)$. Let $E$ be the set of all $x\in S$ at which $$\sum_{n=0}^\infty a_nx^n=\sum_{n=0}^\infty b_n x^n.$$ If $E$ has a limit point in $S$, then $a_n=b_n$ for $n=0,1,2,\ldots$. Hence the above equation holds for all $x\in S$.

The second theorem is from these lecture notes.

> Corollary 10.23. If two power series $$\sum_{n=0}^{\infty} a_n(x-c)^n, \quad \sum_{n=0}^{\infty} b_n(x-c)^n$$ have nonzero-radius of convergence and are equal in some neighborhood of $c$, then $a_n=b_n$ for every $n=0,1,2, \ldots$.

The corollary is much easier to prove it seems, so it makes me think it is a different statement from the theorem in Rudin, but I can't see what the difference is.

copper.hat

@Jakobian Sure, if I can, I will. Where does the question start?

Xander Henderson

@psie In one theorem, both series on defined on a set which has a limit point. In the other, they are defined on a neighborhood (i.e. a set containing an open set).

psie

@XanderHenderson ah ok, subtle difference, but I think I see what you are saying. $E$ in the corollary is a neighborhood around $c$ (i.e. a set containing an open set), whereas $E$ in Rudin's theorem is simply a set which has a limit point. Grazie!

Jakobian

8:54 PM

@copper.hat So consider $X = \mathbb{R}\times [0, \infty)$. For each $x\in\mathbb{R}$ you have a convex set $U_{x, r}\subseteq \mathbb{R}\times (0, \infty)$ such that $(x, 0)\in \overline{U_{x, r}}$ and for $0 < s < r$ you have $U_{x, s}\subseteq U_{x, r}$. Define $f:\overline{U_{x, r}}\to [0, 1]$ to be $1$ on $\partial U_{x, r}\setminus\{(x, 0)\}$ and $f(x, 0) = 0$, and extend linearly. What would be some simple conditions on $U_{x, r}$ to make it so that for each $\varepsilon > 0$ there exists

$0 < s < r$ such that $f[U_{x, s}]\subseteq B((x, 0), \varepsilon)$?

Example of such family is $U_{x, r} = B((x, r), r)$

By extend linearly, I mean that on the segment from $(x, 0)$ to a point $p$ on the boundary you are defining it linearly. Also assume $U_{x, r}$ are non-empty, open and bounded

so $f((1-t)(x, 0)+tp) = t$ for $t\in [0, 1]$

CroCo

11:16 PM

Hi everyone, I need to prove $f(x)=e^x$ convex without using the derivative test.

I got stuck at

$\lambda e^{(x_1-x_2)}e^{\lambda(x_2-x_1)}+(1-\lambda) e^{\lambda(x_2-x_1)} \geq 1$?

I need to prove the above inequality. Is there any hope with this?

$\lambda \in [0,1]$

Interestingly, numerical attempts show that this inequality always holds

Given $x_2\geq x_1$ and $x_1,x_2\in \mathbb{R}$.

Sine of the Time

4

Q: How to prove the convexity of the exponential function?

How to prove that the convexity of exponential function? It is not allowed to use second derivative of $e^x$.

CroCo

11:33 PM

I saw this post but I didn’t fully understand the answer.

Sine of the Time

the first one?

CroCo

Yes

Sine of the Time

I think OP meant that proving convexity is equivalent to proving the last inequality

CroCo

I feel the answer is missing details. Mine is more lucid if I could prove the above inequality.

I can’t type my full answer because I’m typing from my phone.

Sine of the Time

what are you allowed to use?

CroCo

11:41 PM

I don’t understand your question.

Sine of the Time

what tools can you use?

CroCo

Tools what?

you mean typing?

Sine of the Time

no

you said you can't use derivatives

CroCo

I need to use the definition which is the one stated in the answer you’ve linked

I feel the answer is not complete.

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