Mathematics

Leaky Nun

3:00 PM

@Silent define integrable.

Semiclassical

@Silent that’s not my issue. In what sense is infinity an integral?

GFauxPas

usaully I see the phrase "uniformly convergent" as the limit of a sequence Leyla

Leaky Nun

the dominated convergence theorem is so cool

GFauxPas

so perhaps they mean something like a sequence of

$\left \langle { \displaystyle \int_{1/n}^n t^{2k} e^{-xt^2} dt } \right \rangle$

and the claim is that this converges unformly to an integral on $[0..+\infty]$

actually that's a sequence of Riemann integrals, you can go straight to

$\left \langle { \displaystyle \int_{1/n}^{+\infty} t^{2k} e^{-xt^2} d\lambda(t) } \right \rangle$ for Lebesgue integral

then by the DCT you want a sequence of measurable functions $f_n$ converging pointwise to $f(x,t)$ and $|f(x,t)|\le g(x)$ etcetera

Silent

3:16 PM

oh, i see. if i go by Rudin PMA's definition, then integral is lower and upper partial sums. But rudin defines it for closed interval only.@LeakyNun, @Semiclassical

Leaky Nun

@Silent exactly. what you're asking is an improper integral, which is usually defined as a limit

Silent

ok,, only that kind of counterexample possible?

GFauxPas

oh I'm sorry @LeylaAlkan I think it's a sequence in $k$

Leaky Nun

@Silent maybe I can answer your questions more effectively if you make them more precise

(or others can, I don't know everything)

GFauxPas

Leaky knows everything up to a set of $\lambda$-measure $0$ though

2

Silent

3:24 PM

@LeakyNun No, thank you very much, I am convinced.

Jacksoja

{{-1, -8.4, -8.4}, {-1, -4.4, -5.4}, {1.2, 6, 7}}

does this matrix converge?

A^n ?

Leaky Nun

@Jacksoja what is its JNF?

Leyla Alkan

Unfortunately I am not familiar with Lebesgue integral yet @GFauxPas

GFauxPas

okay Leyla then let's use limits of Riemann integrals, and we're saying the following

what book is this from? you probably can do it with Riemann integrals and limits but the notation $\int < \infty$ is usually in context with Lebesgue integrals

but anyway

Jacksoja

@LeakyNun why do we need that?

Leaky Nun

3:30 PM

@Jacksoja well, converge iff all eigenvalues have magnitude smaller than one

so you don't actually need JNF, you just need its eigenvalues

GFauxPas

you want a sequence of functions $f_n$ converging to $f$ pointwise

Jacksoja

Okay let me check that

GFauxPas

okay they actually are using Lebesgue integration in a sneaky way @Leyla , implicitly

but let's pretend we don't know that and we're using riemann integrals with limits

to use theorem 7.38 you need to find a sequence of functions

$\langle f_n \rangle$

such that for every $n$, $|f_n| < g$ and you can integrate $g$ and get a finite number

Leyla Alkan

It's from Folland's Advanced Calculus

GFauxPas

lol I'm using Folland's Real Analysis to learn this stuff, high five

he's limiting the case where you can do everything with limits of finite integrals; with lebesgue integration you don't need things to have finite limits always and you have more freedom, so he's only dealing with a case where you can write everything as improper riemann integrals. make sense?

that's funny that we're using the same author

so your sequence $\langle f_n \rangle$ to use theorem 7.38 you need to bound EACH $f_n$ with a function integrable in the sense of an improper riemann integral. let me know if you're folllowing

Leyla Alkan

3:38 PM

I guess yes

GFauxPas

well let's do better than "I guess" because I have lots of free time now to make my explanation simpler or more detailed :)

know what a sequence of functions is?

Leyla Alkan

hahaha thanks then, yes

GFauxPas

okay, so

know what uniform convergence is

Leyla Alkan

yes

Jacksoja

@LeakyNun does any entries of that matrix converge?

GFauxPas

3:41 PM

so we know that if every member of the sequence $f_n$ is integrable on a closed finite interval, $f_n \stackrel{\text{unif}}{\to} f$ implies $f$ is integrable on that interval

right?

Leaky Nun

@Jacksoja no idea

Jacksoja

besides entries 1,1 and 2,1

Can you use some program?

I need to install these things

Leyla Alkan

okay @GFauxPas

GFauxPas

okay the problem here is that $[0,\infty)$ is neither closed nor bounded so that's why we need extra theorems than just "uniform limit of riemann integrable functions is riemann intergrable", because these are limits of riemann integrals, they're improper integrals, so the old theorems are not enough

okay?

geocalc33

Given a set of functions, can you take the limit of the whole set?

GFauxPas

3:49 PM

well if they're indexed by a countable set you essentially have a sequence of functions geocalc

but the concept your looking for is called "limit point"\

a set of functions can have a function as a limit point. not always, and it's not always unique

let me know when to continue Leyla

Leyla Alkan

@GFauxPas okay I'm following

Secret

@geocalc33 It will be similar to treating a sequence of functions, but you will need some notion of topology on the set before limits can be talked about

GFauxPas

so to show the convergence of the improper riemann integral in the example we use theorem 7.38

geocalc33

@Secret what sort of notion of topology would you need?

GFauxPas

your sequence is $\left \langle { t^{2k} e^{-xt^2} } \right \rangle_{k = 0}^{+\infty} $

Secret

3:55 PM

At minimum you will need some directed sets, so that some notion of convergence can be talked about. But assuming your set of function is countable, you need converging sequences.

e.g. $x^2, 0.5x^2, 0.25x^2,0.125x^2,...$ converges very slowly to $0$

GFauxPas

and the interval you're checking is $[\delta,+\infty)$ for $\delta > 0$

with me?

Leyla Alkan

yes

Secret

So, for a set of function to have a limit, it has to be a directed set which eventually converges towards some function

GFauxPas

we want a function $g$ such that $|f_n(x,t)| < g(x)$ for every $n$ and such that $g$ is integrable as an improper integral over $[\delta,+\infty)$. that's what theorem 7.38 requires, except the theorem uses $c$ instead of $\delta$

geocalc33

@Secret basically they are all the same function with different parameter values, and as the parameter tends towards infinity the area under the curve approaches zero

GFauxPas

4:01 PM

good?

Secret

right, so your family of functions is arranged under some linear order based on the parameter

Leyla Alkan

yeap

GFauxPas

all the sets are non-negative so we'll ignore the absolute value part

we want an integrable function $g$ such that

geocalc33

@Secret yeah

GFauxPas

$t^{2k} e^{-xt^2} \le g(t)$ for any $k$

the RHS isn't a function of $x$ so you need to find a bound that works for all $x \in [\delta,+\infty)$

any ideas?

does $t^{2k}e^{-xt^2}$ have a maximum on $x \in [\delta,+\infty)$ that might help?

$\dfrac {t^{2k}}{e^{xt^2}}$ has a maximum where the denomenator has a minimum in this case

Leyla Alkan

4:07 PM

so that's why we take $\delta$?

Since we can make it as smaller as possible

GFauxPas

because $e^{xt^2}$ has a maximum at $x = \delta$ so $t^{2k}e^{-xt^2} \le t^{2k}e^{-\delta t^2}$

Leyla Alkan

Yeap makes sense

GFauxPas

the RHS is integrable because $t^{2k}$ increases much much slower than $e^{-\delta t^2}$ decreases

Secret

So that means, letting $\lambda$ be the parameter, you have $\{\int f_{\lambda}\}$. Thus for $\lim_{\lambda \to \infty} \int f_{\lambda}$ to exists, this set has to have a limit point under the order topology of some subset of $\Bbb{R}$

GFauxPas

you can prove it formally by inequalities or taylor expansions or whatever but it's integrable

on $[\delta,+\infty)$

Secret

4:11 PM

I don't know how to articulate that better cause I don't know my topology well enough

GFauxPas

Then Folland says that because the convergence is uniform on $[\delta,+\infty)$, you can integrate and use limits to let $\delta \to 0^+$

and then you win the prize

the prize here, I'm guessing, is to find the value of $\displaystyle \int_{-\infty}^{+\infty} e^{-t^2}\, \mathrm dt$

or half of it,the integrand is even

Leyla Alkan

which is $\sqrt \pi$

GFauxPas

:)

I'd have to see the previous pages but he's probably justifying a step in a larger proof of that integral

Leyla Alkan

So will you study this book of Folland too :)?

GFauxPas

I'm studying a different once of his

Leyla Alkan

4:18 PM

The book that I use

Apart from the real analysis one?

Mancala

Show that a $k$-form $\omega$ is smooth iff it is smooth as a section $\omega : M\rightarrow \Lambda ^k(M)$ of $\pi$.

GFauxPas

oh, for that I ordered adult rudin, but it didnt come in the mail yet

loch

What is your definition for $\omega$ to be smooth?

Leyla Alkan

Oh , but I think Folland's way of explaining things is so cool and easy to follow.

geocalc33

Secret

4:34 PM

the L shape thing is not really a function?

geocalc33

the black curves are all functions with different values for the common parameter and the purple line is y=x

Secret

yup

and it seems they converge to some L shaped thing

geocalc33

and as the parameter approaches infinity all the curves go to (0,0)

along y=x that is

Secret

yup

geocalc33

and as the parameter approaches negative infinity all the curves go to (1,1)

Secret

4:40 PM

I don't know how to set up an order topology of $f(x,\lambda)-x=0$

because you are basically computing $\lim_{\lambda \to \infty} f(x,\lambda) - x$

o wait, the limit will be defined on the usual topology of $\Bbb{R}$ because of how the function is rearranged

geocalc33

I don't know how to turn on the math symbols in this chat

Secret

tinyurl.com/cfqcvpc

geocalc33

i clicked on that but it won't let me right click the link to add as a bookmark

Secret

what browser you are using?

geocalc33

chrome

Secret

4:51 PM

so you don't have a right click box popping up as you click on the link?

geocalc33

yeah i can right click

but there's no option to add as a bookmark

Secret

O, for chrome, you can just drag the link to the bookmark bar at the top

geocalc33

okay got it to work thanks

philmcole

5:15 PM

Hi, how do you get the bounds of an iteraded integral from the area? For example let $f: \Bbb R^3 \to \Bbb R$ and $B=\{(x,y,z) \in \Bbb R^3 \mid 0 \le x \le y \le z \le 1\}$. How do I get the bounds for $\int_B f \, \text{dvol} = \int_a^b \int_c^d \int_e^f f \, dxdydz$?

Leaky Nun

x from 0 to 1

y from x to 1

z from y to 1

oops you have them in the reverse order

z from 0 to 1, y from 0 to z, x from 0 to y

More Anonymous

Guys I'd appreciate help on:

0

Q: Regularization of Exponentially exponential series?

Question What are the convergence properties of the last equation: $$ J^{-1}(x) + \delta = \int \frac{dx}{1+ e^{x} + x + \ln{x} + \ln\ln(x) + \dots + e^{J^{-1}(x)}} $$ Is it possible to ignore $e^{J^{-1}(x)}$ under some suitable range? How can one infer the value of $\delta$? Background...

philmcole

5:30 PM

@LeakyNun thanks, but can you explain how you get them?

tatan

5:41 PM

@Lozansky I get f(x)f(0)=f(x)..

geocalc33

6:03 PM

$f(x)$

Secret

Vulta - Robotica Satanica (ft. Hollywood Burns)

►

Leaky Nun

@AlessandroCodenotti hi

Alessandro Codenotti

Hi

Silent

What does $1+2+\dots+n$ represent here?

Oh, its counted from, top right(1) upto diagonal(n). Got it

@LeakyNun, Let $M_3(\Bbb R)$ be set of $3\times 3$ matrices with real entries, and $V$ be subspace of symmetric matrices with trace zero. So, dimension of quotient space $M_3(\Bbb R)/V$ is $3^2-(\frac{3^2+3}{2}-1)$ , right? Also, if $M_3(\Bbb C)$ is considered, then also, dimension of quotient space remains unchanged, right?

Leaky Nun

6:23 PM

@Silent could you clarify the part with $\Bbb C$?

Partha Sarker

Hello, guys. My ODE textbook has a problem of finding an example of autonomous ODE i.e. $\frac{dy}{dx}=f(y)$ which all of its fixed points isolated.

Leaky Nun

@Silent the first claim is right

Partha Sarker

Now, a trivial solution to this would $\frac{dy}{dx}=0$

But can $\frac{dy}{dx}= \lim{a\to \infty}sin(ay)$ be a non-trivial solution?

I mean as the frequency of the sine function get larger and larger the zeros of the sine function also get dense. So, if the limiting frequency is taken to be infinity then the zeros would be very close.

O. Sorry. I meant the fixed/stationary points would be non-isolated i.e. there can be no open interval which contains only one fixed point.

I think my messages are becoming unclear. Let me re-organize.

My ODE textbook has a problem of finding an example of autonomous ODE i.e. dydx=f(y) which all of its fixed points non-isolated i.e. there can be no open interval which contains only one fixed point..

Silent

@LeakyNun I am sorry but internet connection was lost. Do you need to rewrite me the question, or make clear what I thought before inferring 2nd claim from first?

Leaky Nun

@Silent rewrite the second claim

Silent

6:36 PM

ok

Partha Sarker

Now, a trivial solution to this would $\frac{dy}{dx}=0$. But can $\frac{dy}{dx}=\lim_{a \to \infty} sin(ay)$ be a non-trivial solution?

Silent

Let $M_3(\Bbb C)$ be set of $3\times 3$ matrices with complex entries, and $V$ be subspace of symmetric matrices with trace zero. So, dimension of quotient space $M_3(\Bbb C)/V$ is $3^2-(\frac{3^2+3}{2}-1)$ , right?

Partha Sarker

@LeakyNun Can you help me with this problem?

Leaky Nun

@ParthaSarker $\displaystyle \lim_{a \to \infty} \sin(ay)$ does not exist

Kasmir Khaan

yo leaky

what is up ? :D

Partha Sarker

6:40 PM

@LeakyNun I know that but what I mean is as the frequency of the sine function get larger and larger the zeros of the sine function also get dense. So, if the frequency is taken to be very large then the zeros would be very close.

Leaky Nun

but they are still discrete

@KasmirKhaan hi

@ParthaSarker you would have to make your question precise in a way that it can be interpreted unambiguously

Partha Sarker

So, we can't take the limiting frequency to be infinity?

@LeakyNun Sorry about that!

Kasmir Khaan

In finding maximum of functions of 2 var

i take partial with respect to the two var and set them to 0 right?

Partha Sarker

Got disorganised while writing it

Kasmir Khaan

@LeakyNun I forgot my multi var stuff :D

Partha Sarker

6:44 PM

@LeakyNun Wouldn't Dirac delta function be a non-trivial solution to the problem?

Lozansky

@tatan If $f(0) \neq 0$ and $f(x) = f(0) f(x)$, then $f(x)$ must be...?

Leaky Nun

@ParthaSarker it isn't a function

GFauxPas

it just plays one on TV

geocalc33

@GFauxPas Hey

Partha Sarker

@LeakyNun Usually it is said to be a function.

GFauxPas

6:54 PM

yo

$\delta$ is a functional or a measure

Leaky Nun

@ParthaSarker that doesn't make it a function

Partha Sarker

And it is a generalized function I think @LeakyNun

GFauxPas

or a distribution/generalized function

Lozansky

Generalized function

GFauxPas

but it's not a function

it's defined by $\langle \delta, \phi \rangle = \phi(0)$ where $\phi$ is a function

that's the easiest way to think of it without using measure theory

Partha Sarker

6:57 PM

Okay. So is there any non-trivial function $f(y)$ for which the autonomous ODE $\frac{dy}{dx}=f(y)$ has all its zeros non-isolated?

*fixed points

Hey, guys?

geocalc33

why are closed homotopies more interesting than open ones

Lewis

7:14 PM

Ok, can anyone just check I'm not being stupid. If I have a filter with poles at: 0.5(-1-i) and 0.5(-1+i), and I want to find the length of the vector between each pole at 0Hz, and I get ~ 1.5, but the mark scheme says it should be ~0.707 (sqrt(2)/2) -- can anyone help me?

Lewis

7:29 PM

I am being stupid, fixed it.

geocalc33

does anyone want to see a mathematical structure i made

GFauxPas

8:09 PM

proofwiki.org/wiki/Laplace_Transform_of_Complex_Power finished product! look how pretty it is :3

sure geocalc

geocalc33

8:23 PM

@GFauxPas I made it on desmos

click on it to enlarge

I'm trying to determine the manifold that results from these geodesics (the black curves). A very tricky problem

quallenjäger

8:50 PM

Does a bounded variation path has to be continuous almost everywhere?

Because it is differentiable almost everywhere.

GFauxPas

Well it's pretty but I don't know manifold theory so good luck

user193319

9:02 PM

If $R$ is a commutative ring, then it has at least one maximal ideal, cal iit $\frak{m}$. What does $R^n/\mathfrak{m} R^n$ denote and how does it differ from $R^n/\frak{m}^n$?

geocalc33

$f(x)$

Transcript for

$Mathematics$ Mathematics