Mathematics - 2024-07-15

copper.hat

05:34

can you use the fact that $\exp$ is increasing?

psie

06:17

> Definition 8.20 Let $X$ be a random variable with values in $\mathbb Z_+$. The generating function of $X$ is the function $g_X:[0,1]\to[0,1]$ defined by $$g_X(r):=\mathbb E[r^X]=\sum_{n=0}^\infty \mathbb P(X=n)r^n.$$

Why is the domain of $g_X(r)$ only $[0,1]$? If it has radius of convergence at least $1$, shouldn't the function be defined for at least $[-1,1]$?

Soumik Mukherjee

08:27

Reddit confession led to Indian student’s deportation from US: ‘I planned something very wrong'

This is hilarious

Jakobian

10:41

@psie to be non-negative

No one says you can't plug in values other than that, you can

psie

ok, thanks, I guess it's what you say, for convenience to be nonnegative

Jakobian

I actually don't know why authors restrict themselves like this. One thing that comes to mind is that it might be because $g_X\circ \exp = M_X$ where $M_X$ is the moment generating function of $X$

So to further make clear the duality between the two someone could have done that, since $\exp$ is a positive-valued function.

Of course this equality is valid on $(-\infty, 0]$

psie

11:00

ok 👍

Jakobian

11:21

Another issue might be that they only know Abel's theorem for non-negative coefficients of the power series

So to prove $g_X$ is continuous they make such assumption

But, oddly enough answered by me on this site, there is an alternative approach using Weierstrass $M$-test which shows its continuous in the ball $\{z\in \mathbb{C}: |z|\leq 1\}$

Jakobian

12:22

-1

Q: What are the criteria for a subject to be under the domain of mathematics

It is to my understanding that mathematics is in some way the domain of all logical systems. However unconventional, as long as certain criterias are met, they could be considered as part of mathematics. For example, subjects like ring theory or knot theory are quite far fetched from the "normal"...

Binky McSquigglebottom

13:28

$\int \int_D \frac{xe^{2y}}{y+2}dxdy$ D is the flat domain included between the unit circle with the origin as its center and the ellipse of semi-axes a = 2 and b = 1 and contained in the first quadrant.

How can i find the domain ?

Sine of the Time

14:13

@BinkyMcSquigglebottom did you draw the region and the equations of the circle and ellipse?

Binky McSquigglebottom

15:44

@SineoftheTime the First Is $x^2 + y^2 = 1$ , the second is $\frac{x^2}{4}+\frac{y^2}{1}=1$

However, I only have to consider the first quadrant

It should be the area between these 3 points

Sine of the Time

can you write the domain using the equations?

Binky McSquigglebottom

16:03

$D=\{ x^2 + y^2 ≥1,\frac{x^2}{4}+\frac{y^2}{1}≤1,x≥0, y≥0\}$

CowperKettle

He is the top-rated user by reputation score on MathOverflow - MindScape interview

Binky McSquigglebottom

@SineoftheTime

Sine of the Time

In the first quadrant you can write the circle and the ellipse as functions

Xander Henderson

@SineoftheTime Though, arguably, it might make sense to work in polar coordinates. Otherwise, you have to consider $[0,1]$ and $[1,2]$ separately (which might be fine, too).

Sine of the Time

16:20

@XanderHenderson Looking at the integrand, I don't know if using polar coordinates it's the simples way. Instead we can write the region as $\{ 1\le x \le 2, g(x)\le y \le h(x)\}$

but we have to consider also $x\in [0,1]$ as you've said

Xander Henderson

@SineoftheTime Oh, I don't even know what integral we are worrying about. I'm too lazy to read that far back up the thread.

Sine of the Time

maybe it's best if we fix $y$

Xander Henderson

@SineoftheTime This doesn't seem to be correct, if we are only interested in quadrant 1.

You need to consider $y$ between the two functions on $[0,1]$, but $y$ between the $x$-axis and the ellipse on $[1,2]$.

Sine of the Time

$\{0\le y\le 1, a(y)\le x\le b(y) \}$

Xander Henderson

@SineoftheTime Oh, derp. That's the thing to do.

Sine of the Time

16:23

By the way, does this type of domain have a name?

Like normal domains?

Xander Henderson

@SineoftheTime What do you mean by "type"?

And what do you mean by "normal domain" in this setting?

Sine of the Time

For example $\{a\le x \le b, f(x)\le y \le g(x)\}$

Italian wiki page: it.wikipedia.org/wiki/Dominio_semplice

In italian are called "normal domains"

But I've never heard something similar in English

Xander Henderson

"Normal domain" is a bad translation, as "normal domain" has a more standard meaning in algebra. Google translate suggests a "simple domain".

In any event, I've never heard a terms for such domains. It seems like it might be pedagogically useful, but there is nothing particularly interesting about them mathematically which would necessitate a term (in my opinion).

Sine of the Time

@BinkyMcSquigglebottom for example, we call this region "normal domain with respect to the $y$-axis" because $y$ is bounded by two numbers and $x$ is bounded by two functions of $y$

Binky McSquigglebottom

Yes

Its clear

Xander Henderson

16:38

@SineoftheTime I would suggest not calling it such in English. You are likely to confuse people.

Sine of the Time

16:49

I don't use this term, I was just curious if there's a similar terminology in English

Xander Henderson

22 mins ago, by Xander Henderson

In any event, I've never heard a terms for such domains. It seems like it might be pedagogically useful, but there is nothing particularly interesting about them mathematically which would necessitate a term (in my opinion).

Jakobian

@SineoftheTime sometimes we call it like that too

but not everyone uses this type of nomenclature

i suspect the same thing exists in English, just Xander doesn't know about it

Xander Henderson

@Jakobian I mean, that is basically the content of the comment I just quoted...

Jakobian

@XanderHenderson where in your comment does it contain my opinion that it probably exists in English?

psie

Consider the Laplace transform of (the law of) $X$, i.e. $$L_X(\lambda):=\mathbb E[e^{-\lambda X}]=\int_{\mathbb R_+} e^{-\lambda x}\mathbb P_X(\mathrm{d}x).$$ It is claimed $L_X$ is convex. Is it clear to someone why that is?

Xander Henderson

16:58

That being said, I've taught a lot of calculus, multivariable calculus, and analysis classes out of a variety of books and at several institutions. If such a term were widely used in the US, I imagine that I would be aware of it.

@Jakobian I am not going to engage with you any longer. You are clearly trying to pick yet another fight, and I don't have the energy for it.

Jakobian

@psie Should follow from Jensen's inequality

psie

ah good idea, I'll look it up

Jakobian

@psie actually no, not Jensen's inequality

psie

yeah, Jensen's has as assumption convexity

I think the statement that $L_X$ is convex is a deeper result in convex analysis

nickbros123

@psie what book are u using to study probability / stats (if I can assume thats what youre studying)

psie

17:03

Currently, and this is where my question comes from, I'm using this book.

Jakobian

@psie this should just follow from $\lambda\mapsto e^{-\lambda x}$ being convex for any fixed $x$

psie

I think you're right, but is it because integration preserves convexity? After all, we are integrating $x\mapsto e^{-\lambda x}$.

Jakobian

its linear and preserves inequalities, so yeah

we are integrating $\omega\mapsto e^{-\lambda X(\omega)}$

psie

ok 👍

Jakobian

@XanderHenderson I also don't have energy for it, as I am sick and coughing.

psie

17:14

ok, I guess I should have said, we are integrating $x\mapsto e^{-\lambda x}f(x)$, where $f$ is the density of $X$. Then it would be correct :)

Jakobian

$X$ is has a density, yes

nickbros123

17:49

i was wondering about non zero characteristic fields cuz of this problem in Sheldon Axler LADR: the union of 3 vector subspaces is a subspace if and only if one of them contain the other two. testing it for a vector space containing $a_1,a_2,a_1+a_2,0$ over the field ${0,1}$ the theorem didnt hold, which is interesting

Jakobian

18:04

@nickbros123 yeah, iirc this doesn't hold for finite fields

the proof that union of two proper vector subspaces not containing each other is not a vector subspace requires that your field is infinite

PNDas

Hi! I was reading a paper. They show $(-\Delta-|x|^{\alpha})u\leq0$ and $(-\Delta-|x|^{\alpha})v>0$ in $\mathbb{R}\setminus B_r(0)$. Then they are using maximum principle on $\mathbb{R}\setminus B_r(0)$ to get $u(x)\leq \frac{u(r)}{v(r)}v(x).$ Does anyone know which maximum principle they are are using?

I think they have rescaled to get 0 on boundary.

Soumik Mukherjee

18:36

@Jakobian wish you get well soon!

Binky McSquigglebottom

@SineoftheTime so $\{0\le y\le 1, \sqrt{1-y^2}\le x\le 2\sqrt{1-y^2} \}$ ?

Obliv

What does it mean for "every statement becomes a theorem if a contradiction is true"?

this question comes from this page on dialetheism

Sine of the Time

@BinkyMcSquigglebottom yes

nickbros123

@Obliv in this context i think it means if a statement X and its negation not(X) are true, then every conceivable statement becomes true

Jakobian

@SoumikMukherjee thank you, I will eventually

Obliv

18:45

@nickbros123 yeah i don't understand why

I don't get why in hungerford, he says "A paradox occurs in an axiom system when both a statement and its negation are deducible from the axioms. This in turn implies (by an exercise in elementary logic) that every statement in the system is true"

How come $A\land \neg A$ being true means we can imply $A \land \neg A \implies X$ for all statements $X$ in the formal system?

Don't theorems have to be stated, or can be derived, from axioms? (Or I guess they can be independent statements)

So if a contradiction exists within the system, why does this now "explode" the entire system? Wouldn't it only be limited to the statements connected to the contradiction?

Binky McSquigglebottom

Im stuck here

Pls help , i dont know what to do

CroCo

Dear all, I need to prove $f(x)=e^x$ is a convex function. I could use the derivative test, but I would like to use the actual definition for practice. The definition says let $\lambda\in[0,1], x_1,x_2\in\mathbb{R}$, a function is convex if
$$
f(\lambda x_1 + (1-\lambda) x_2 \leq \lambda f(x_1) + (1-\lambda)f(x_2).
$$
In essence, I need to show
$$
e^{\lambda x_1 + (1-\lambda)x_2} \leq \lambda f(x_1) + (1-\lambda)f(x_2).
$$
To prove the above inequality, let $a\in[1,\infty)$, I start with this
$$

Obliv

idk but I see $(1-y^2)$ and I think of $y = \sin\theta$ maybe

nickbros123

@Obliv "with P {\displaystyle P} standing for "all lemons are yellow" and Q {\displaystyle Q} standing for "Unicorns exist". We start out by assuming that (1) all lemons are yellow and that (2) not all lemons are yellow. From the proposition that all lemons are yellow, we infer that (3) either all lemons are yellow or unicorns exist. But then from this and the fact that not all lemons are yellow, we infer that (4) unicorns exist by disjunctive syllogism. "

Obliv

19:06

@nickbros123 Do contradictions come from axioms? Also how can we infer that statement $P\lor Q$ though? In math, let's say we have axioms $A_1$ through $A_9$. If we have $A_1$ true and $\neg A_1$ is also true, then I can just say $A\lor B$ is true no matter what $B$ is?

Jakobian

every statement comes from some string of implications that follow from axioms. But not necessarily directly. In fact, usually this string of implications is large

nickbros123

@Obliv contradictions come from what is regarded as "wrong" assumptions, or wrong axioms, so yes, they come from axioms

Obliv

@Jakobian what is meant by not directly? As in, there are many ways to reach the same theorem/statement or something?

Jakobian

When we obtain a contradiction then it only means that we found a statement $A$ such that both $A$ and $\lnot A$ are true. This $A$ doesn't have to be an axiom

nickbros123

@Jakobian generally when we start a proof by contradiction by "assuming" a particular thing, arent we setting it as an axiom for the time being?

Obliv

19:11

Ah okay. So there is like a path from the axioms to reach $A$ and $\neg A$. In the shortest path, $A$ and $\neg A$ are axioms.

Jakobian

yeah, like a "path" or formal deduction

@nickbros123 no

Obliv

Are there multiple paths to the same statement? Or do they just look like different paths/proofs but they're actually the same

Jakobian

of course, there can be multiple

an example is set theory

there is such thing that Kunen (iirc) calls Basic Set Theory or BST

CroCo

Sorry for interrupting, is it difficult to show
$$
\lambda e^{(1-\lambda)(x_1-x_2)} + (1-\lambda) e^{\lambda(x_2-x_1)} \geq 1.
$$

Jakobian

The reason is that there's multiple things that can be proved either from power set axiom, or from axiom schema of replacement

for example, you can prove the existence of Cartesian product $A\times B$ just from BST

that is, by assuming either power set or replacement, we obtain two different proofs that it exists

so there can be different "paths" to prove a theorem

Obliv

19:20

Hmm okay I will look into Kunen, thanks.

Sine of the Time

@BinkyMcSquigglebottom what is the source of the original integral?

PNDas

19:36

I haven't seen TedShriffin's messages for so many days.

Is he okay?

Binky McSquigglebottom

@SineoftheTime $\int \int_D \frac{xe^{2y}}{y+2}dxdy$ D is the flat domain included between the unit circle with the origin as its center and the ellipse of semi-axes a = 2 and b = 1 and contained in the first quadrant.

Sine of the Time

source of the problem?

Binky McSquigglebottom

from an exam trace

I think this integral cannot be done

Sine of the Time

it seems strange, since the result is related to the exponential integral

Pizza

I put it on wolfram, I get this notation

How should it be calculated? I didn't understand this notation

Sine of the Time

19:53

Ei is the exponential integral

Pizza

$\text{Ei}(x) = -\int_{-x}^{\infty} \frac{e^{-t}}{t} \, dt$ for $x > 0$

CroCo

I'm not sure if this replay for my question. if so, please take a look at the question in here
https://math.stackexchange.com/questions/4946393/prove-ex-is-a-convex-function-using-the-convexity-definition

Pizza

@SineoftheTime Indeed

Sine of the Time

I tried to switch the order of integration but I get the same result, so it's highly improbable we've made a mistake

Binky McSquigglebottom

20:29

so what should I do ? :⁠-⁠(

Semiclassical

21:11

@BinkyMcSquigglebottom to clarify, what does "flat domain" mean here?

also, where is said ellipse centered?

my guess would be that the domain is this:

this also recovers the same result found by Pizza (using mathematica)

Sine of the Time

@Semiclassical this should be the region

Semiclassical

yeah

i guess flat might just mean 2D here

but that's redundant here

Sine of the Time

yes, but it's not necessary

Semiclassical

i guess. strange question

Binky McSquigglebottom

21:40

┻⁠┻⁠︵⁠ヽ⁠(⁠`⁠Д⁠´⁠)⁠ﾉ⁠︵⁠┻⁠┻

So it can't be done?

Sine of the Time

$\ddot{\smile}$

@BinkyMcSquigglebottom It can be done, but the result involves special functions

Binky McSquigglebottom

@SineoftheTime ─=≡Σʕっ•ᴥ•ʔっ

@SineoftheTime But how do I understand where to use the special function?

૮ ⚆ﻌ⚆ა

Sine of the Time

21:59

@BinkyMcSquigglebottom you usally don't need this knowledge

when solving double and triple integrals

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