Mathematics - 2021-12-29 (page 1 of 2)

sashas

00:06

0

I had asked a question earlier about multiple definitions for soundness of a formal system. Following were the two definitions A theory $T$ is sound iff its axioms are true (on the interpretation built into T’s language), and its proof system is truth-preserving, so all its theorems are true. T...

any help appreciated

Unknown x

Can anyone help me to understand the physical meaning of $G(x,\xhi)$?

-1

Q: Can you explain what does $G(x, \xi)$ do? What is $f(\xi)$ over the interval $[a,b]$?

Reference: How to figure out the free body diagram? Question 1 What is the physical meaning of vertical reactions are proportional to the displacements at the end point. This leads to the boundary conditions. Question 2 Can you explain what does $G(x, \xi)$ do? What is $f(\xi)$ over the inter...

newtonian-mechanics string boundary-conditions greens-functions

This was the example for motivating reader to study green's function. after reading, all the motivation has gone :P

what is deflection*force?

summary of question: what is deflection*force?

What is $f(\xi)$ over the interval $[a,b]$

PenAndPaperMathematics

00:40

Adrianne Lenker - Full Performance (Live on KEXP at Home)

►

Prepare for your mind blown

@Unknownx you'll have to fix your post, make it more accessible per the comments

Or should I say purrr... the comments

@dc3rd what r u up to?

How many replicants did you get today, etc

Bysshed

can anyone help me find a copy of this article?

Shanks, D. Euclid's primes, Bull. Inst. Combin. Appl., 1, 33-36 (1991).

i've tried searching for it but had no luck

PenAndPaperMathematics

@Bysshed same, maybe try a university lib

@Bysshed I know an exact formula for counting twin primes in $(p_n, p_{n+1}^2)$ as $n$ grows. If interested, I can link you to a proof for it

Bysshed

Thanks, unfortunately i don't have access to a university library at the moment

PenAndPaperMathematics

@Bysshed: scholar.google.com/…

Related articles to that one might be of use

@Bysshed sadly, I'm doubtful of that article's existence on the web

dc3rd

@PenAndPaperMathematics In the Caribbean....telling myself every day I'm going to do some math and my notebooks haven't left my suitcase the whole time.....I don't know whether to laugh or cry...

PenAndPaperMathematics

00:52

@Bysshed what proof specifically are you interested in? Perhaps there's an article quoting / discussing it

@dc3rd the greatest support of learning is sleeping / taking breaks. Without it, the mind will not remember what it needs to. So this break from studying will actually be a boon once you get going again

You needed to reset your brain's math crystals, in layman's terms

Bysshed

it contains a heuristic proof that euclid's proof generates every prime

it's referenced on page three here

math.dartmouth.edu/~carlp/nothingtalk2.pdf

PenAndPaperMathematics

@Bysshed don't these top search results help?

google.com/…

dc3rd

Truthfully I do agree because I do feel rejuvenated and excited to get back into the mud. It had been a long 2 yr slog with no breaks and I was feeling really burnt out last month

things weren't being recalled without excessive effort and no enthusiasm behind it.

PenAndPaperMathematics

@dc3rd take ample breaks, that is more difficult than it sounds, some people go the opposite way, thinking it will help - turns out they get more burned out

dc3rd

I was definitely in the latter group....my internal "guilt" of trying to "make up" for squandered time.

PenAndPaperMathematics

00:56

Take a 5-10 minute break for each hour of studying, and you'll go longer

dc3rd

Been getting a lot better at being at peace with it and just bing cool with my journey.

PenAndPaperMathematics

Also, don't study at your bed, you'll fall asleep lol

And maybe have a daily time / ritual / location / lamp, etc.

But that's also difficult

dc3rd

yea those are all in place. It was a matter of actually enforcing the breaks where things were getting hazy

PenAndPaperMathematics

@RyanUnger that's a neat av

Bysshed

01:14

@PenAndPaperMathematics Thanks for your help, but they only reference Shanks' argument without repeating it, from what I have read

PenAndPaperMathematics

@Bysshed make a post on MSE, perhaps someone has that article on their shelf and can photograph it for you

Maybe Ted does, he was born in 1908 probably :P

Ted Shifrin

01:36

1909, sorry.

PenAndPaperMathematics

He tutored Ramanujan

leslie townes

ted's so old, some of his books are out of copyright. [this one is a little 'inside']

PenAndPaperMathematics

He tutored Gauss

He tutored the same soul of my cat

leslie townes

ted's so old, uh, his phd advisor was a brontosaurus

PenAndPaperMathematics

01:51

Ted's so old, he predates the known universe. He was here before the local big bang

leslie townes

nice to see that the law of diminishing returns is still in full effect

PenAndPaperMathematics

Ted's so old, that his phd advisor was a prokaryote.

Leslie is so old that her phd advisor was a string

lol

robjohn

@leslietownes sounds like some confusion from PenAndPaper.

PenAndPaperMathematics

I'ma work on this math website...

Got Django to find static files, finally, lol

leslie townes

robjohn is so old, the blues guitarist is named after him

PenAndPaperMathematics

01:59

Leslie is so old, that she ask her phd avisor about category theory and their response was an electrical sine wave

Ryan Unger

Raccoon Plays with Colorful Abacus

►

PenAndPaperMathematics

@RyanUnger looks like they're computing something feverishly

Ted Shifrin

02:16

Do raccoons get fevers?

copper.hat

04:08

hi

Koro

Hi !

copper.hat

Hi Koro

Koro

Hey copper!

It's about time to do multivariable calculus :)

copper.hat

:-)

What are you studying now?

Koro

I just studied sequence and series of functions (chap 7 in PMA) skipping proofs of Stone -Weierstrass theorems. I know the statements for of Stone-Weierstrass theorems (weak and strong form) without proof though. :)

Now, I'm studying multivariable calculus.

copper.hat

04:14

are you studying as part of a course or for fun?

Koro

For both actually. I completed my graduation in 2018 and then took up a job at a company. And currently I am employed at the same company. The job is not related to Maths at all. I wish to leave the job and join some college.

copper.hat

cool! good luck

Koro

thanks. :)

copper.hat

i wished i could have found a job where maths was more important, but software always paid more (and product dev was interesting in its own right)

what are you studying in multivariable calculus?

Koro

@copper.hat what about being a professor ? :)

copper.hat

04:21

too many politics. my advisor wanted me to, but i knew i would not have the right termperament

Koro

In multivariable calculus -I'm studying Jacobian matrix after visualizing derivative (total) as a linear transformation.

copper.hat

also, the pay was not as good :-)

as in frechet derivative, i presume?

Koro

For probably the first time, I think I studied maxima and minima of two variable functions in a way that I won't ever forget again I think. Earlier I memorized it as I didn't understand what was going on (Criteria for extrema based on determinant of Hessian matrix).

Frechet? I'm afraid, I don't know that term. :(

copper.hat

essentially the derivative as a linear operator rather than coordinate wise or directional.

Koro

Ah

Ted Shifrin

04:34

Do not do multivariable from Rudin. Bad treatment.

Koro

I'm doing that from Apostol :)

Ted Shifrin

Not comparable to Rudin in analytic depth, more just calculus. A good tough first course.

Koro

Ted, are you talking about vol 2 of Apostol's? I referred to Apostol's 'Mathematical analysis' in my last message.

Ted Shifrin

Ah. OK. No differential forms, though.

leslie townes

04:50

no worries. just write omega and say "d" a lot

copper.hat

and talk about formal sums

leslie townes

omega, chains, omega, complexes, d, no simplices, d, closed, omega, exact, d, omega

Koro

omega and d? These are species living in differential forms? I'm not there at differential forms yet :(

leslie townes

can i just leave now

dead silence? ok, i'm staying forever

copper.hat

i'm stoked now

Leslie is showing good form

leslie townes

04:59

our power was out for several hours last night. i apologize for not being here

copper.hat

no tesla battery?

Koro

A Mathematician's Apology

A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy, which offers a defence of the pursuit of mathematics. Central to Hardy's "apology" — in the sense of a formal justification or defence (as in Plato's Apology of Socrates) — is an argument that mathematics has value independent of possible applications. Hardy located this value in the beauty of mathematics, and gave some examples of and criteria for mathematical beauty. The book also includes a brief autobiography, and gives the layman an insight into the mind of a working mathematician. == Background == Hardy felt...

leslie townes

hardy should have told that book to a therapist

copper.hat

no wonder funding is slow

Koro

@Koro ?

copper.hat

05:04

i think there is only one Koro here...

leslie townes

you'll go blind doing that

copper.hat

lost in the gulf of onan

Koro

Ah, I wanted to ask what were omega and d that you were referring to. Because I haven't seen them (yet) in multivariable calculus.

leslie townes

koro no substance. people often use the letter omega to denote a differential form. d is involved.

copper.hat

differential forms

leslie townes

05:06

that's all you need to know, and as far as i understand it, all there is to the subject.

some people make sad little lives out of this, but i'm not one of them

copper.hat

my sister's entire family have been omicroned

leslie townes

hope things stay light and mild.

copper.hat

so far no big impact for 8 family members (5 double vaxed, others too young)

hoping our 92yo aunt either does not get it or it is mild for her

leslie townes

the vaccines have done a pretty good job, it was a lot darker before then

copper.hat

i suspect that highly transmissible and low impact are related

except for infowars

Koro

05:29

@copper.hat I see. :)

Thanks a lot. :)

copper.hat

It really irritates me when someone describes a non obvious fact as obvious.

Obviously.

robjohn

Fergus D O'Mega. He had several wee bits o property in the north. Each was pretty small, but together...

leslie townes

i sometimes upvote answers to questions where they're not, like, stellar answers that i'd hang on a christmas tree of answers, but i don't want the OP to delete the question

i think he ran for albany school board along with sarah stalin

copper.hat

there is the other conclusion which i prefer to avoid

robjohn

that it really is obvious?

copper.hat

05:42

yes, computational dysfunction

robjohn

the fear that anything one asks will end up being trivial.

copper.hat

the new pfizer thinkagra, even donald has clear thoughts with it

dole did one, why not tripe?

i guess erdos used amphetamines

robjohn

erdos was a doser?

leslie townes

yeah he was buzzing

copper.hat

depends on how you pronounce doser

leslie townes

05:45

he was a device for turning speed into theorems

copper.hat

i am looking at wasted pages filled with $\sum$, $p^{n-1}$, $\binom{p^{n-1}}{k}$, $p^k$.

robjohn

@copper.hat All work and no play make copper a dull boy. All work and no play make copper a dull boy. All work and no play make copper a dull boy. All work and no play make copper a dull boy. All work and no play make copper a dull boy. All work and no play make copper a dull boy.

REDRUM

Alex

Hi everyone

copper.hat

@robjohn time for an episode of narcos mexico

good morning Alex

robjohn

Hi one

@user69608 can we tell your favorite site from your name, or was that just the first site you joined?

Koro

06:00

@copper.hat I watched that long time ago!

copper.hat

@Koro its a bit disturbing...

Alex

Good morning. We have a comfortable temperature now. I have to go for some fruits. On the other hand, there are some good movies on Netflix.

Koro

Lost in space is a good one too. :)

Alex

I agree.

I was watching a crime miniseries, where they do interrogations. Very interesting, it's a miniseries.

Koro

I was watching the Witcher last night. :)

user69608

06:03

@robjohn i have bookmarked math.stachexchange and chat.stackexchange tabs

leslie townes

i enjoy watching non-US crime stuff to compare with US criminal procedure.

copper.hat

vice

leslie townes

i watched one episode of the witcher and found it extremely confusing. i assume i have to watch 20 novels for it to make sense. or play the game.

Alex

@Koro Many talk about that series. It is advisable? I think the plot is similar to "The 100".

@leslietownes You should see the miniseries I'm watching is very good.

copper.hat

i like bbc stuff, they usually have the sense to chop series at a small number before it becomes focus group fodder

robjohn

06:07

@user69608 However, Puzzling was your first account, I believe

user69608

@robjohn it was math.stackexchange

copper.hat

i am still amazed there there are no equivalents of the profumo affair, philby, cambridge five in the usa

robjohn

@copper.hat what are they?

leslie townes

there's something about upper class englishmen and irresponsible sex stuff

copper.hat

various relationship, foreign agent scandals

leslie townes

06:12

also a media culture that seeks that stuff out. here, i don't know that most newspapers could be bothered.

Koro

@Alex I am not sure if you'll like that. But I tend to like shows on witches, dragons, monsters etc. with the time setting of historic times. I have not seen the 100 show.

leslie townes

we had the rosenbergs. that was pretty boring.

copper.hat

surely fox news

leslie townes

if they weren't interested in trump, they're not interested in other stuff.

copper.hat

bill clinton blipped a bit

i mean what happened to the russia tape??

i've even forgotten the escort's name

who cares about his taxes, i want the dirt

leslie townes

06:17

it's honestly not that interesting if someone like trump or clinton does it. of course they do that. it's only interesting if some upper class british twit does it.

and they always do weird stuff, not just what you'd expect.

copper.hat

still, would be good to know what the daily mail had on melania

leslie townes

i took a great set of videos today of my daughter behaving responsibly with the cat. they played for three minutes, uninterrupted, everything was cool. i leave the room and immediately my daughter pushes it too far and gets a 2" scratch on her face. the skin was not broken.

copper.hat

i presume your daughter has had tetanus shots

leslie townes

yes, although the cat doesn't go outside. no risk of that.

robjohn

@copper.hat I don't watch fox news, and don't call me Shirley!

copper.hat

06:30

@robjohn fox and news are a little disjoint

robjohn

I agree; more of an editorial station

copper.hat

hmm, i'm struggling with a problem in Ted's book, and while searching for answers discovered that i had added an answer in february (admittedly a piggy-back answer) when working through dummit & foote.

that is a bit disturbing

leslie townes

more like dummy and foot in your mouth, am i right?

copper.hat

indeed :-)

i am not smart enough to give up

leslie townes

i heard that ted was on some kind of triple secret probation for giving too many low grades and assigning homework that even he didn't know how to do.

copper.hat

06:43

:-)

dtn

07:37

Perhaps the question is a little strange, but I need to ask it. I'll break it down into two parts:
1. Can the product of the n-th number of polynomials with real roots give a polynomial with complex roots?
2. Can the product of the n-th number of polynomials with complex roots give a polynomial with only real roots?

robjohn

09:44

@dtn I don't quite understand what you're asking: what do you mean "n-th number of polynomials"?

Do you just mean a product of n polynomials?

the roots of $p(x)q(x)$ are the union of the roots of $p(x)$ and $q(x)$.

dtn

@robjohn Yes

@robjohn ok, that's what I want

Prithu biswas

10:07

How do we check something like:

$∀ε > 0$ $∀a,b ∈ ℝ$ $∀d_1,d_2 ∈ ℝ$ $(|d_1| < ε/5 ∧ |d_2| < ε/5 ⇒ |(a+d_1)(b+d_2)| < ε)$

Using graphs?

robjohn

@Prithubiswas why not just give a counterexample?

$\varepsilon=6$, $d_1=d_2=1$, and $a=1, b=3$

Prithu biswas

@robjohn But isn't finding the counterexample sometimes hard?

robjohn

it can be, but I don't see an easy way to check that with a graph, there being so many variables. You'd have to know a counterexample ahead of time to know what to graph.

Prithu biswas

@robjohn sliders for $a$ and $b$ in desmos?

robjohn

You said using graphs, not using desmos.

Prithu biswas

10:20

well , desmos seems to be a online graphing calculator.

robjohn

okay, so you can find a graph using desmos by having all the variables as sliders.

Prithu biswas

@robjohn Because I don't know how to implement it. I can put $a$ and $b$ as sliders. But how do I check it for all $ε$ ?

The reason why I am asking this is so that to have a first check that the initial bound I am working with is bad or not.

robjohn

as I said: "find a graph using desmos by having all the variables as sliders"

then graph $(a+d_1)(b+d_1)$ and $\varepsilon$

doesn't desmos allow you to have more than 2 sliders?

graph $(a+d_1)(b+d_1)$ as a function of $b$ and graph $\varepsilon$ as a constant function of $b$

if the graph of the former stays below the graph of the latter, then your claim is verified, at least as far as the graphs can tell you. If the graph of the former ever exceeds the latter, then you have a counterexample.

Koro

11:40

Given that $f:\mathbb R^n\to \mathbb R^n$ is differentiable on an open set $S$ then I want to show that given $x\neq y\in S$, for every $a\in \mathbb R^n$ there exists a $z$ in $\mathbb R^m$ such that $a.(f(x)-f(y))=a. (f'(z)(y-x))$

I don't understand the proof given in the book.

The proof given is along the following lines: Let $u=y-x$. Since S is open, there is a $\delta\gt 0$ such that for every $\color{red}{t\in (-\delta, \delta+1)}, x+tu\in S$. I don't understand the red highlighted part.
I think all we can say is that there is an $r\gt 0$ such that $B(x,r)\subset S$. So $||x+tu-x||=|t|||u||\leq r\implies |t|\leq \frac r{||u||}$. So there is no reason to believe why $t$ has to be greater than $1$.

I missed the following information in my statement of the theorem: $x,y$ are such that $\{y+t(x-y): t\in [0,1]\}\subset S$.

But still I don't understand the red highlighted part. :(

Thorgott

ok, that hypothesis is crucial, and it also explains away your doubt

except this is confusingly written cause the variable names are switched

the hypothesis tells you that $x+tu\in S$ for $t\in[0,1]$. openness allows you to choose $t$ a $\delta$ smaller than $0$/larger than $1$ without leaving $S$

Koro

11:59

Ah, so $x+u \in S$ (that is, at t=1) means that there is a ball around $x+u$ of radius $\delta_1\gt 0$ which lies completely in $S$. So $x+u+\delta_1 u=x+u(1+\delta_1)$ is in S. There is a $\delta_2\gt 0$ such that ball of radius $\delta_2$ around $x$ lies completely in $S$. In particular, $x-\delta_2u$ lies in S. So $\delta$ can be chosen as $\frac 12\min(\delta_1,\delta_2)$.

Thanks a lot @Thorgott. I think my confusion is clear now. :)

Adil Mohammed

tiny doubt, if n is the no of elements in set A then what will be the elements of the power set of A? is it $2^n$ or $2^n -1$?

Thorgott

@Koro This is almost the right argument, but you if you want to do the estimates explicitly, you have to take into account that $u$ need not have norm $1$

@AdilMohammed what do you think?

Adil Mohammed

12:15

yes we must include null set in the power set, I remember vaguely seeing {phi,{1},{2}... } so the answer is $2^n$?

Thorgott

yes

Adil Mohammed

thankss

Koro

12:31

Thor: I’m not assuming u to be a norm 1 vector.

Odestheory12

If $A\subset{\{(x,y,z)\in\mathbb{R}^3:0\leq{z\leq{1}}\}}$ and $A\cap{\{(x,y,z): z=t\}}=\{(x,y,t): (x-t)^2+(y+t)^2\leq{t^2}\}$ for $0\le t \le 1$ then $A=\{(x,y,t): (x-t)^2+(y+t)^2\leq{t^2}\}$ with $0\le t \le 1$ right?

Thorgott

then knowing a radius $\delta_1$ ball around $x+u$ is in $S$ only allows you to deduce $x+u+tu$ for $|t|<\frac{\delta}{\lVert u\rVert}$ is in $S$

Koro

13:06

Ah, I meant to state only the idea earlier. Of course, there is no harm in choosing $\delta_1'=\delta_1/||u||$

user193319

I am trying to show there exists a holomorphic function $f$ defined on $U := \{z \in \Bbb{C} : |z| > 4\}$ whose derivative is $$\frac{z}{(z-1)(z-2)(z-3)}$$. Doing partial fraction decomposition, we obtain $$\frac{z}{2(z-1)} - \frac{z}{z-2} + \frac{z}{2(z-3)}.$$ So, I think I just need to find a holomorpic function on $U$ for each summand. However, I think this will involve logarithms, and any branch cut of the logarithm won't be defined on all of $U$, right?

Leaky Nun

13:44

@user193319 q implies p, not q does not imply not p

user193319

15:30

@LeakyNun I don't understand your remark.

Leaky Nun

@user193319 it's a fancy way of saying, just because you can't solve for each summand, doesn't mean the question is wrong

user193319

Ah, I see. Yes, you're right. I wasn't saying that the problem was equivalent to solving it for each summand. I was just stating what I tried and why I don't think it works.

Leaky Nun

@user193319 and actually if you bear with the un-defined-ness of each summand, then combine them together, you might get something that works :)

user193319

So, you're saying that the "undefined" pieces will cancel or something like that?

Leaky Nun

yeah

user193319

15:40

So, e.g., naively antidifferentiating $\frac{z}{z-1}$ gives $z + \log(z-1)$, and just do the same for each summand; then add them all up and this should suggest something to me?

Astyx

I find it weird you keep z in the numerator

when you could have summands of the form $a/(z-\alpha)$

Leaky Nun

@user193319 symbol pushing is important, but you also need to understand the big picture (i.e. meanings of stuffs)

both are important

but in the meantime yeah add them up first

user193319

Hmm...so, if I am not mistaken, I get $3z + \log [(z-1)(z-2)^2 (z-3)^2]$...

Oh, wait, I might have screwed up some signs and scalars

Leaky Nun

wolfram alpha is your friend lol

(for checking)

user193319

Lol yes, that's how I know I screwed things up

monoidaltransform

15:57

If $H$ is a subgroup of a finite group $G$, and $g\in G$ and $D$ is a conjugacy class of $H$. How do I show that $|\{ x\in G : xgx^{-1}\in D \}|$ $=$ $|\{ x\in G : x^{-1}gx=g \}|$ $|D|$

?

user193319

Okay, I end up with $\log [(z-1)^{1/2} (z-2)^{-2} (z-3)^{-3/2}]$

Does that seem right?

Leaky Nun

@user193319 does WA agree?

@user193319 the last exponent is wrong

user193319

Ah, okay. I see. But isn't there a problem with log not being defined on all of $U$?

leslie townes

monoidal: you only need one pair of $s around all of the stuff that's in math mode. does g bear any relation to D? do you know various homomorphism theorems?

monoidaltransform

16:15

no relationship between $D$ and $g$

@leslietownes

leslie townes

monoidal: is there any relation between |D| and the conjugacy class of g in H?

monoidaltransform

well they coincide

Leaky Nun

@user193319 and this is the point when symbol pushing ends

do you understand why $\log(z)$ (the simplest example) is not defined on all of $\Bbb C$?

leslie townes

monoidal: when we say "conjugacy class of H" we mean what exactly? conjugacy class of an element in H, where we form conjugates using other elements of H? or conjugate of H in G?

Leaky Nun

@leslietownes conjugacy class of an element in H

user193319

16:24

Hmm...it isn't entirely clear to me...

Leaky Nun

@user193319 well then that's something for you to think about :)

also, is entirely a pun?

monoidaltransform

@leslietownes first one

user193319

Lol no it wasn't meant to be a pun

leslie townes

leaky: i'm being socratic but thanks

monoidal: what happens to the left hand side in an abelian group, if g is in G but is not in H? is g supposed to be in H?

user69608

17:12

string "bcc" is lesser than string "bcccc" lexicographically right?

leslie townes

usually, yes

Fargle

@user69608 Gonna echo Leslie and say yes, in most cases. One can think of it as though you're padding "bcc" with a couple of "empty characters", and you set the convention that "empty" < any other character.

user69608

ok

Fargle

One can of course do this convention the other way too, where "empty" > every other character, but I've never personally seen it.

Astyx

17:33

dictionaries use this convention, hence the name

leslie townes

can you imagine what the yellow pages would have been like (back in the day) if they had used the opposite convention

AAAAAAAAAAAAAAAAAA LOCKSMITHS

Fargle

A veritable aaaaaaaaaaaaaaaaaaaaaaaaaaarms race.

leslie townes

for the younger folks, can you imagine what the yellow pages would have been like, full stop

Fargle

I am just barely old enough that I still had occasion to use one in my house for at least some part of my childhood.

And they even taught in school how to do phonebook lookup, which is in one way very funny, and in another might be more of an expression of how outmoded the educational system can be than of how old or young I am.

leslie townes

haha, i remember my elementary schools would have a yearly trip to the library with an explanation of how to use the card catalog.

even at the time i knew that was a waste.

the people who know, know. the other ones won't be back until the same time, next year, when they again won't be listening

Fargle

17:48

The thing I always found very funny was how much time was spent teaching the Dewey system in public school, but when I got to college all the libraries used LoC.

Less than useless.

Thorgott

the right formula is $|\{x\in G\mid xgx^{-1}\in D\}|=|\{x\in G\mid x^{-1}gx=g\}||\{D\mid g\in D, D\text{ is conjugacy class of $H$}\}|$

leslie townes

fargle: haha, i forgot about dewey. our local public library did use it, so it was practical at the time. and in college i didn't read books.

i had a feeling something was 'off' about that formula

maybe use a different D on the right?

i saw stuff that seemed to be reaching outside of H, and other stuff that seemed to be only within H, and it didn't compute

Thorgott

18:43

my notation sucks, but I'm lazy

leslie townes

i'm just lazy

Ted Shifrin

Indeed.

AMDG

I've heard about "infinite norm" before. Is there a well-defined meaning for $\mathbb{R}^{\infty}$?

leslie townes

you sometimes see people say that about an element of some space, if a supremum in a recipe defining a norm fails to exist for that element

sometimes people refer to the sup norm or essential supremum norm as the 'infinity norm' by analogy with L^p spaces

without more context i don't think that phrase singles out any one thing

AMDG

Interesting

Ted Shifrin

18:58

I've never heard "infinite norm." Leslie is correct that it's the "infinity norm" or $L^\infty$ norm.

AMDG

Ah yes, that's what I meant. infinity norm

My bad

Could it be valid to say that some relation $0y(x)$ in some way can be interpreted as a function of time/arc length?

AMDG

19:17

Perhaps put differently. From a geometric perspective, $y(x)$ describes a point, $(x, y(x))$ in $\mathbb{R}^2$. $0y(x)$ describes the point $(x, 0)$ which is a point on the real x axis. Now at least for any function of time $f(t)$, the point $(f(t), f(t))$ is on the line $x = y$. If the inverse of $f$ exists, then $x = y$ is the cartesian equation of any $(f(t), f(t))$ since $(f(f^{(-1)}(t)), f(f^{(-1)}(t))) = (t, t) \equiv (t, f(f^{(-1)}(t)))$.

Since a function of time does not map to $y$ per se, $f(t)$ and $0y(x)$ here are virtually equivalent in nature because $0y(x)$ gives us a point on the x axis which can also be described as $x(t) = t, (x(t), 0)$.

Neither

Hi, I wonder when an n-dimensional integral can be written as a product of n separate integrals and I find this answer. It mentions Fubini's theorem so I look it up and all I see is being able to change order of integrals, not writing them as products. What did they mean & when can I separate an n-dimensional integral?

So that it's not XY, I'm looking at $\int_{\mathbb{R}^n} e^{\frac{-1}{2}x^T A x}\,dx$

$A$ is positive definite and I'm trying to justify to write this as an n-product of integrals.

Ted Shifrin

@Neither The first question is the shape of the region. My interpretation os product is that you must have $\left(\int_a^b f(x)\,dx\right)\left(\int_A^B g(y)\,dy\right)$.

So then the point is that the function you're integrating must be a product of functions of the individual variables.

You can do that in your instance by changing variables to diagonalize $A$, so that $x^\top Ax$ becomes a sum of squares.

This is a very standard computation.

Neither

Yes that's the product I meant, sorry for not being clear. Region is $-\infty$ to $\infty$. Since A is symmetric I diagonalize it and am able to write it as products.

Oh you wrote that last part already :)

@TedShifrin So as I understand, as long as I can write it as a product of individual variables, it's fine to separate it to n integral products. Fubini's theorem isn't in play then, is it?

Ted Shifrin

19:35

Sure. Fubini is used any time you compute a multiple integral as an iterated integral.

UnderMathUate

Fubini

Ted Shifrin

In addition, in this setting, in principle you need to justify convergence of the improper integrals, but it's no big deal.

Neither

Thanks.

Alex

Hi, here there are an interesting question math.stackexchange.com/q/1399431/941425. Can I use the same idea for to calculate $\displaystyle \int_{0}^{+\infty}\frac{\sin(ax)}{x}e^{-ax^{4}}{\rm d}x$? I had already asked about this problem here before and was working on one approach but it didn't work. Now I was trying to use a series approach but I'm still stuck.

Ted Shifrin

Yuck. Pass.

Vrouvrou

19:50

Hello, please how to find the barycenter of the homogeneous plaque P determinate by y=x and y=6x-x^2

Ted Shifrin

Your course has given you the definition in terms of double integrals, @Vrouvrou, so do them.

Alex

@TedShifrin but 😞

leslie townes

alex, might be worth its own question on math.SE (if you can assure yourself that it's not a duplicate of an already-asked question). there are perverts out there who like that kind of stuff.

not like this this wholesome chat, which is populated with upstanding citizens.

Ted Shifrin

Like our former chat denizen Chris'ssis from Romania, who published a whole book on it.

For once in a blue moon, leslie and I agree on something.

No one has given me an answer to my $2^m+3^n$ question yet :(

leslie townes

i have it on my to-do list. i'm just finishing up some actual work.

Ted Shifrin

19:57

I forget who brought up the brachistochrone in here, but this is an interesting twist.

Isn't munchkin doing your "actual work"?

Alex

@leslietownes I offer my sincere apologies. I will try to be a better citizen 🥺

Vrouvrou

@TedShifrin we don't see it, how can I find the definition please?

UnderMathUate

@TedShifrin I didn't realize that question was open to all. I thought it was solved already.

Ted Shifrin

You can find it on wikipedia, for sure, @Vrouvrou, but if they assigned that to you, your text or lecture must have given you the formula.

UnderMathUate

I guess that doesn't mean I couldn't have tried it anyway.

leslie townes

19:58

ted: we did respond phhtbphbptpbhtpht to someone last week. she did do most of the work on that response.

Ted Shifrin

Ah, that's good to hear, @leslie.

@UnderMath I gave it to all to work on. I know how to do the problem :)

UnderMathUate

Alright, let me give it a go :D

My first thought was maybe try induction of some kind, but I didn't write anything down yet.

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