Mathematics - 2018-05-27 (page 3 of 4)

Leaky Nun

15:00

@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

15:16

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

15:24

@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

15:30

@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

15:38

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

15:41

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

15:49

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

15:55

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

16:01

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

16:07

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

16:11

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

16:18

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

16:34

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

16:40

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

16:51

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

17:15

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

17:30

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

tatan

17:41

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

geocalc33

18:03

$f(x)$

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

18:23

@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

18:36

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

18:40

@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

18:44

@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

18:54

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

18:57

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

19:14

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

19:29

I am being stupid, fixed it.

geocalc33

does anyone want to see a mathematical structure i made

GFauxPas

20:09

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

sure geocalc

geocalc33

20:23

@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

20:50

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

21:02

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

$f(x)$

MatheinBoulomenos

@user193319 it's a $R$-module which is just the direct sum of $n$ copies of $R/\mathfrak{m}$

What exactly do you mean by $R^n/\mathfrak{m}^n$?

When I see $\mathfrak{m}^n$, then I think of the product of ideals, but that's probably not what you mean

Hey @Antonios-AlexandrosRobotis @AlessandroCodenotti

Antonios-Alexandros Robotis

hi @MatheinBoulomenos!

user193319

Not exactly sure. I was thinking it was the same thing as $R^n/\mathfrak{m}R^n$, but you seem to be saying that's wrong.

MatheinBoulomenos

I think you mean the same thing as $R^n/\mathfrak{m}R^n$, but the notation is confusing

Daminark

21:12

Aw man some nerds

How's it going?

MatheinBoulomenos

@user193319 more generally, if $I$ is an ideal in $R$ and $M$ is an $R$-module, then $IM$ which is generated by elements of the form $im$ with $i \in I,m \in M$ is a submodule of $M$. So one can also look at the quotient $M/IM$. In this case we just have $M=R^n$ and $I=\mathfrak{m}$

Hey @Daminark

user193319

So $\mathfrak{m}R^n$ denotes the direct sum of $n$ copies of $\mathfrak{m}$?

MatheinBoulomenos

it turns out to be that, yeah, but by definition I'd say it denotes the submodule in $R^n$ which is generated by elements $mx$ with $m \in \mathfrak{m}$ and $x \in R^n$

@Daminark Pretty well, thanks. I will give a talk in a seminar organized by my advisor on tuesday

When I said that $R^n/\mathfrak{m}R^n$ is a direct sum of $n$ copies of $R/\mathfrak{m}$, that wasn't the definiton, but more what it turns out to be, if you know what I mean @user193319

user193319

Yes, I do. Thanks!

Mohammad Areeb Siddiqui

0

Q: Analysing a sum of products

I am trying to understand that what could be the possible outcomes of this following expression as $k \to \infty$ and as $|t| \to \infty$. $$\sum_{k=0}^{k=n}\dfrac{(\prod_{1}^{t}k)^{-i}}{\sqrt{k}}$$ Considering the numerator first: $$(\prod_{1}^{t}k)^{-i}=\prod_{1}^{t}\frac{1}{k^i} = (\prod_{1}...

summation infinite-product

MatheinBoulomenos

21:16

@user193319 np. I'm always happy to talk about commutative algebra :)

How's it going for you? @Daminark

@Antonios-AlexandrosRobotis How are you doing? Do you still have 2 TA jobs?

Antonios-Alexandros Robotis

just finished those when the semester ended, @MatheinBoulomenos . Teaching linear algebra the summer, and trying to get towards being able to write a compelling master's thesis.

MatheinBoulomenos

Sounds good. By teaching you mean actually giving the lectures?

Antonios-Alexandros Robotis

yeah and the grading and so on

MatheinBoulomenos

wow, I don't think the let grad students do that here

certainly not masters students

Antonios-Alexandros Robotis

I impressed the right people somehow LOL

I'll be able to feed myself properly now rofl

MatheinBoulomenos

21:21

for LA (which is a mandatory Freshman course here), they always have full professors or at least some experienced post-docs which permanent positions etc.

Jacksoja

Hi

How to find the max of this function

10 ln (x^2+2y^2+1) -x^2-y^2

f(x,y) =

I did the partial derivative w.r.t both x and y , it did not give me much to work on

x^2+2y^2=9 from I) and x^2+2y^2=19 from II)

geocalc33

Anyone know a little about manifold theory

Jacksoja

on the circle x^2+y^2 =16

forgot to add that part

MatheinBoulomenos

@Antonios-AlexandrosRobotis here's a fun suggestion for treating determinants (it's acutally what one of the profs here did) Let $k$ be a field
1) Define $\Bbb Z/2\Bbb Z$-graded algebras over $k$.
2) Define the twisted tensor product of $\Bbb Z/2\Bbb Z$-graded algebras and show it's again a $\Bbb Z/2\Bbb Z$-graded algebra
3) Define the Clifford algebra associated to a bilinear form by a universal property
4) Use twisted tensor products to show the existence of Clifford algebras (uniqueness follows from abstract nonsense)

Anderson Felipe Viveiros

Is the set of eigenvalues of a $\Bbb R$-linear operator a compact subset of $\Bbb R$?

MatheinBoulomenos

21:31

@AndersonFelipeViveiros in finite dimension, of course, as it is finite. In infinite dimensions, no

Anderson Felipe Viveiros

That's right

Hmm

MatheinBoulomenos

It could be all of $\Bbb R$ even

Anderson Felipe Viveiros

And my space is not finite-dimensional ^^'

MatheinBoulomenos

Consider the space $V=C^\infty(\Bbb R)$ of all smooth functions and the derivative operator $d:V \to V$. Then every real number $\lambda$ is an eigenvalue with the eigenvector $x \mapsto e^{\lambda x}$

Anderson Felipe Viveiros

Exactly. The space I'm working is indeed $C^\infty(\Sigma)$, with $\Sigma$ a surface embedded in a 3-manifold $M$ ^^'

The operator is the Jacobi operator

Is there any hope? ^^'

MatheinBoulomenos

21:35

I know zero geometric analysis

Anderson Felipe Viveiros

Hahaha I'm not an expert either

Thanks anyway

MatheinBoulomenos

@AndersonFelipeViveiros if you know that your operator has some super nice properties (e.g. compact self-adjoint on a Hilbert space), then there might be stuff like spectral theorems

geocalc33

@MatheinBoulomenos do you know any manifold thoery

Anderson Felipe Viveiros

@MatheinBoulomenos Hm, thanks. I'll check it

MatheinBoulomenos

@AndersonFelipeViveiros In that case, you know at least that the set of eigenvalues is countable and as a sequence it converges to $0$

but that's pretty specialized

If you have a bounded operator on a Banach space, then the set of eigenvalues is bounded by the operator norm at least

Lozansky

21:40

@Jacksoja Lagrange multiplier maybe?

MatheinBoulomenos

But I don't think $C^\infty(\Sigma)$ is such a nice space

@geocalc33 "any" manifold theory? yes, I took courses in diff geo and they also came up in alg top. I'm far, far from an expert, though

Jacksoja

@Lozansky thanks but I remeber doing this in diff way, i did this long time ago

geocalc33

@MatheinBoulomenos do you know if it's possible to take a set of curved geodesics and build up a representation of the manifold that results from the geodesics?

Jacksoja

@Lozansky is chaning to polar coordinates something that can be done here?

Lozansky

Yes might be a good idea

MatheinBoulomenos

21:43

@geocalc33 I don't understand the question

geocalc33

like for example if you have a graph of straight lines going across a 2D surface

Semiclassical

I think you mean: Given a family of 2D curves, find a 3D manifold whose geodesics project to the plane curves

geocalc33

you know that you're dealing with flat euclidean space

@Semiclassical I think that's what I'm getting at

Lozansky

@Jacksoja You have $10\ln(r^2(1+\sin^2\theta )+1) - r^2$, set $r^2 = 16$ so maximum for $\theta = \pi/2$ and is $10\ln(33)-16$

MatheinBoulomenos

@geocalc33 Certainly your set of geodesics has to have some properties for this to work. (e.g. resulting from uniqueness) What do you know about your geodesics? Do you know how they intersect etc.? Is the union of all those geodesics a manifold?

geocalc33

21:53

Here's a picture for reference

MatheinBoulomenos

In any case, I'm really the wrong person to ask for this. I "know" manifolds, but it's a fleeting acquaintance

geocalc33

They do intersect

Lozansky

@geocalc33 It reminds me of four-leaved rose ie $r=a|\sin 2\theta|$

geocalc33

@Lozansky yeah i can see the similarity

It also resembles a penrose diagram rotated

Daminark

@Mathein sorry I was out but yeah things are going quite well!

What are you giving the talk on?

MatheinBoulomenos

22:02

local class field theory

It's the main result of the seminar

I'll prove some version of local class field theory, building on all the results from the previous talks. But it doesn't quite look like the usual formulation, I'll prove another result the week after that to fix this

geocalc33

@Semiclassical can you help with the manifold picture

MatheinBoulomenos

that's pretty cool considering that class field theory is probably the most imporant result in 20th century number theory and a lot of modern research is about generalizing it

philmcole

Hi, can I choose the order in which I compute an iterated integral freely in general, I mean if I integrate first over $x$ and then over $y$ or vice versa?

Lozansky

@philmcole Sounds like Fubini's theorem?

MatheinBoulomenos

@philmcole not in general, no. The conditions for Fubini/Tonelli are not superfluous

philmcole

22:09

Ok thanks. Indeed I was assuming that the function is reasonable such that Fubini can be applied.

MatheinBoulomenos

in that case, this is precisely the statement of Fubini

philmcole

Okay. While we're at it. Do you know if this is true?

7 hours ago, by philmcole

58 mins ago, by philmcole

For a bounded real-valued function $f(x,y)$ (not necessarily continuous!) Fubini's theorem says that one can compute the double integral as iterated integral first over $y$ and then over $x$ for almost all $x$. Not necessarily for all $x$ since the inner integral over $y$ must not exist for every $x$. This means the problematic points $x$ are a Lebesgue null-set.

I was wondering if this poses any problem in real life when computing iterated integrals. Is it true, that we can just ignore the problematic points since the integral over a null set is zero anyways?

MatheinBoulomenos

To see what kinds of things can go wrong, consider $f(x,y)=\frac{x-y}{(x+y)^3}$. You can compute that $\int_0^1\int_0^1 f(x,y) \mathrm{d}x\mathrm{d}y=-\frac{1}{2}$, but $\int_0^1\int_0^1 f(x,y) \mathrm{d}y\mathrm{d}x=\frac{1}{2}$

Jacksoja

@Lozansky thanks !

@Lozansky how to find the max inside the circle ?

geocalc33

@MatheinBoulomenos could i ask my geodesic/manifold question on math stack exchange do you think?

MatheinBoulomenos

22:15

@geocalc33 I don't find your question very precise as you stated it, to be honest. It might get closed for "unclear what you're asking"

@philmcole as the example shows, you do have to be careful when you interchange the integrals. The function $f(x,y)= \frac{x-y}{(x+y)^3}$ looks innocent enough, but it actually doesn't satisfy the conditions for Fubini

Daminark

That sounds fantastic

Lozansky

@Jacksoja Try solving $\dfrac{\partial f(r,\theta)}{\partial r}=0$ with $0 \leq r^2 \leq 16$. Then plug it back in together with $\sin^2 \theta = 1$

MatheinBoulomenos

@Daminark yeah, I'm stoked

philmcole

@MatheinBoulomenos What happens here that this goes wrong? In our theorem the assumption is that the inner integrand $f_x: X \to \Bbb R$ must be Riemann-integrable. Why can you actually compute both versions of the integrals if the inner integral doesn't exists?

Mohammad Areeb Siddiqui

0

Q: Analysing a sum of products

I am trying to understand that what could be the possible outcomes of this following expression as $n \to \infty$ and as $|t| \to \infty$. $$\sum_{k=0}^{k=n}\dfrac{(\prod_{1}^{t}k)^{-i}}{\sqrt{k}}$$ Considering the numerator first: $$(\prod_{1}^{t}k)^{-i}=\prod_{1}^{t}\frac{1}{k^i} = (\prod_{1}...

summation infinite-product

MatheinBoulomenos

22:18

@philmcole the integral is improper here

philmcole

Ah right, the left bound is a limit to zero.

We didn't talk about improper integrals in the context of multivariable calculus yet.

MatheinBoulomenos

well, if you do iterated integrals, then it's just single variable calculus

philmcole

We discussed in the context of Fubini's theorem that Fubini is true "almost for all" points in the domain. The problem points are just those where the inner integral doesn't exists. But those are a Lebesgue-null-set (apparently).

MatheinBoulomenos

In general, your argument that removing finitely many points doesn't change the integral doesn't work, I think. When you remove points from the (closed) interval you're integrating over, then you end up with an improper integral for which the theorems about integrals on closed intervals don't necessarily apply

You could also say in the example that $(0,0)$ is the only problematic point

philmcole

Okay, I was being too optimistic :P

MatheinBoulomenos

22:23

I only learned Fubini in a measure theory context, so I can't really comment on what you might do in multivariable calc

Waiting

I didn't see @hippalectryon for a long period of time ...

MatheinBoulomenos

(never even took multivariable calc)

philmcole

Wait how's that possible? For me it's mandatory (part of second semester)

MatheinBoulomenos

I'm from Europe, we do things quite differently

philmcole

Lol ich doch auch.

MatheinBoulomenos

22:28

Du bist Deutscher?

philmcole

ja

studier halt in Zürich grad

MatheinBoulomenos

meinst du Ana 2 mit multivariable calc? Da haben wir Lebesgue-Daniel Integrale, Hilberträume, Fourierreihen und Differentialformen gemacht

aber unser Prof war auch verrückt

JessicaMendoza

Hi guys

philmcole

Ja genau. Wow wir machen nicht so viel aber bis zu Jordan-messbaren Mengen gehts schon.

Skript sind auch 800 Seiten

Ana 1 und 2 zusammen aber

MatheinBoulomenos

Okay

Ich kenne Fubini für Riemannintegrale nicht, also kp

philmcole

22:31

Okay, ja ich habs jetzt schon besser verstanden auf jeden Fall!

JessicaMendoza

Can anyone please tell me where the "1" in "P + bP − dP = (1 + b − d)P" comes from in this math example?

Let us imagine that we wish to predict the population of a country in 20 years’ time.

One very simple model we might use represents the entire country as a pair of numbers (t, P(t)).

Here, t represents the time and P(t) stands for the size of the population at time t.

In addition, we have two numbers, b and d, to represent birth and death rates. These are defined to be the number of births and deaths per year, as a proportion of the population.

MatheinBoulomenos

@philmcole es wird wahrscheinlich klarer, wenn du das nochmal nächstes Semester in Maßtheorie siehst

philmcole

Ich glaub der Physik Studiengang den ich mache hat keine Maßtheorie. Aber wir haben noch noch Funktionentheorie glaube ich.

Lozansky

@JessicaMendoza Factor out $P$?

MatheinBoulomenos

@philmcole echt? Also die Physiker hier in Heidelberg hören alle Ana 3, wo man Maßtheorie macht. Aber Funktheo ist wahrscheinlich sogar nützlicher

philmcole

22:35

Ja ich denk die haben das wichtigste aus Ana 3 schon in Ana 2 gepackt. Jedenfalls kenn ich noch Unis, da kommt die Hälfte was wir in Ana 2 machen erst in Ana 3 dran. Bei euch scheint aber Ana 2 schon krass gewesen zu sein :D

MatheinBoulomenos

@philmcole Wir hatten halt so einen Prof, der total genial ist aber alles immer auf seine Weise macht. Wir haben absolut konvergente Reihen als Lebesgue-integrierbare Funktionen auf $\Bbb N$ bezüglich dem Zählmaß definiert ^^

Semiclassical

$P=(1)P$

MatheinBoulomenos

Bei andern Profs ist das hier auch nicht so heftig

JessicaMendoza

@Lozansky is it because a multiple cannot be 0?

philmcole

@MatheinBoulomenos Gut, da versteh ich noch nichts von :D

Aber bin schon gut beschäftigt mit unserer Analysis VL. Die legen so ein Niveau vor... Vor allem fehlt irgendwie jegliche praktische und anschauliche Herangehensweise. In den Übungen macht man 90% Beweise und vielleicht mal ein Rechenbeispiel. Gut für die Mathematiker aber nicht für die Physiker...

MatheinBoulomenos

22:40

@philmcole grob gesprochen, wenn du den Begriff von "Integral" allgemein genug fasst, kannst du für eine Funktion $f:\Bbb N \to \Bbb R$, die Summe $\sum_{n \in \Bbb N} f(n)$ als Integral auffassen. Das macht auch irgendwie Sinn, du könntest dir ja vorstellen z.B. du machst aus der Funktion $f:\Bbb N \to \Bbb R$ einfach eine funktion auf $\Bbb R_{\geq 0}$ wenn du immer auf Intervallen der Länge 1 konstant bist

(das ist nicht wirklich der Grund, warum das als Integral zählt, nur als Motivation)

philmcole

Ok cool

MatheinBoulomenos

Also wenn es dir um Anschaulichkeit geht, würde Maßtheorie auch nicht helfen. Eher im Gegenteil, das ist dann noch mehr abstrakte Theorie

philmcole

Ich glaube ehrlich gesagt auch nicht mehr dran dass es irgendwann besser wird :P

Lozansky

@JessicaMendoza What do the terms in $P+bP-dP$ have in common?

MatheinBoulomenos

Hier ist auch so, dass viele Physikstudenten die paar Mathevorlesungen, die die so hören müssen, als ziemlich hart empfinden

(LA1 + Ana1-3 wäre das)

philmcole

22:44

So geht's mir auch

MatheinBoulomenos

Ich hab sogar von Physikstudenten gehört, die LA1 bis kurz vor die Bachelorarbeit aufgeschoben haben und dann daran gescheitert sind

JessicaMendoza

@Lozansky P

philmcole

Ja ich bin so ein Kandidat. Bei uns ist LA1 halt auch wieder super anspruchsvoll weil jedes Thm gleich in seiner allgemeinsten Form gegeben wird, also z.B. zwischen endl. und unendl. dim. VR unterschieden werden muss. Dann kommt z.B. eine MC-Frage in der Prüfung ob irgendwas allgemein gilt was in endl-dim. VR immer gilt und dann kreuzt du fröhlich "ja" an und liegst falsch...

Lozansky

@JessicaMendoza Now apply the distributive law

MatheinBoulomenos

@philmcole so sind wir Mathematiker halt :P

philmcole

22:48

Die ETH macht es den Physikern echt schwer :P

JessicaMendoza

@Lozansky P + bP − dP = b * P + b * P

Semiclassical

What?

MatheinBoulomenos

@philmcole aber Funktionentheorie wird höchstwahrscheinlich wieder besser: man schränkt sich da von vorneherein auf eine (wie sich herausstellt) sehr enge Klasse von Funktionen ein, die überraschend viele schöne Eigenschaften hat. Also es wird mehr "Wow, gilt das wirklich? Das ist doch zu schön um wahr zu sein"-Momente als "Wtf, ohne 50 Annahmen gibt es immer ein seltsames Gegenbeispiel"-Momente geben.
Und man kann mit der Theorie tatsächlich wirklich schön einige knifflige Integrale ausrechnen, das finden oft die Physiker viel cooler als die Mathematiker

Lozansky

@JessicaMendoza I mean $a(b+c) = ab+ac$, apply this (but from right to left)

JessicaMendoza

@Lozansky P(b + d) = bP + dP.

philmcole

22:56

Das ist cool. Man kann wirklich paar schöne Dinge berechnen. Ich fand z.B. Teilmannigfaltigkeiten in R^3 ganz cool z.B. den Schnitt zweier Zylinder. Hab die dann immer in Mathematica geplottet und mir ein bisschen Intuition verschafft.

Semiclassical

@JessicaMendoza so what if you add P to both sides?

MatheinBoulomenos

Ja, Mannigfaltigkeiten ist für Physiker ganz cool

JessicaMendoza

@Semiclassical what do you mean?

Semiclassical

What happens to the expression you just had if you add P to both sides of it?

JessicaMendoza

P(b + d) + P = (bP + dP) + P

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$Mathematics$ Mathematics