Down with standards! Make math great again!

@Thorgott sure

But you want $|S/G|\leq |S|-|G|$

You don't want $G$ to be an equivalence class because then you'll just get that $1$ is the only invertible element of $S/G$

@ShaVuklia i missed this but whenever you see this, yes :)

funana?

Is that like an extra fun banana?

@amWhy I have Mid-February first

@robjohn Hope PT went well. Don’t forget Abe’s and my birthday!

@TedShifrin If we say, "Make Analysis Great Again", we can repurpose a bunch of stupid signs.

We need a color change, however.

Ok, now I need to work on that explanation.

@TedShifrin I had assumed that $f'$ is Riemann integrable, so I have added that as an assumption.

@TedShifrin You share a birthday with a penny faced president?

@robjohn I’m cheap?

@TedShifrin No, just have a birthday equivalence

mod 1 year

Or were you talking about two different birthdays?

Nope, same birthday.

Darwin too.

They’re lucky I honored them by joining.

OK, I get the proof now. It was definitely too telescopic before.

Telegraphic?

@TedShifrin I had forgotten to add that $|f'|$ is assumed Riemann integral

@TedShifrin Wow! I never realized that Darwin and Lincoln were born on the same exact day: Feb 12, 1809.

I assume there’s an obvious non-BV counterexample.

@TedShifrin I haven't seen one. I should look for one.

Amazing, eh? I think we think Darwin is later.

is the evaluation map from C[t] to C(t) injective?

I want to show that the set of all finite subsets of $\mathbb{N}$ is countable, is this an injection from $S$ to \mathbb{N}$: eg. I am given {4,3,1}, cardinality is 3. I take the first 3 primes and do $p_1^4 p_2 ^3 p_3 ^1$, and do this process for all the sets?

@userunkown what do you mean?

@nickbros123 Not well-defined.

@TedShifrin yeah, im not able to formally write out this process

I do know another approach to solve this problem, ie

Explicit counting by arranging the sets based on the largest number in the set

I was wondering if this prime no idea can work

Sure. But you need to order your subset.

yeah, you want a map that is defined on subsets, not arbitrarily ordered lists. or equivalently, as ted is suggesting, you might want a unique way of representing each finite subset as an ordered list. there is one "natural" ordering that suggests itself, pun intended

The seniors say it's good to do internships and projects in the summer and winter breaks (I may not believe in its usefulness, but at a superficial level, it is useful for future applications I believe), but there's catch: I don't know crap. I know half baked linear algebra and calculus / analysis, and nothing much else.

I can't obviously go up to a professor and ask them "I wanna read a book on linear algebra, but I wanna do it under you instead of doing it at my home, so that I can plaster it on my cv" that would not look good. Seniors are saying read an advanced topic under a prof, like lie algebras etc, but again, it circles back to the problem of lack of prereq. I personally think self studying at home till I build the prereq is good, but I would like to also know the opinions of actual profs on this one

Is every Lie bracket over a vector space of the form XY - YX?

one: uh, that question maybe needs some clarification to even make sense (the lie bracket being simply different data from a product on the space), but have you seen ado's theorem

ado's theorem is a good enough answer. Thanks!

nick: opinions of actual profs at wherever you are would probably be more helpful than anything here because the expectations around things like "internships and projects" type stuff varies so much from place to place and person to person. but at the schools i have been at, reading courses for undergrads and the like tended to "sound" as though they required more as prerequisites than they actually did.

e.g. a reading course from a prof a topic like "lie algebras" (using as an example only because that seems to be in the air right now) would start at the absolute beginning of some textbook type treatment of that, and not assume much at all from other areas (or indeed teach much at all about other areas). even if knowledge of the usual undergrad "abstract algebra" class would help and/or make sense as a prerequisite.

the extent to which prerequisites listed in a course listing are 'real' vs. simply desired is really individual to an instructor/supervisor but it would not surprise me if the name of a potential opportunity made it sound more exclusive and 'advanced' than it actually was.

if you are thinking about grad school (what kind of cv are you trying to build with all of this might matter), "research" experience is probably a mild plus, but most undergrad math research is definitely closer to being taught than it is closer to what people do once they begin grad school. e.g. it is often highly curated (so you don't need a ton of prerequisites to dive in) and also it is not expected to go anywhere

to use an analogy from the lab sciences, it is more akin to washing test tubes than it is to designing/conducting an experiment whose outcome matters to anybody's long term research program

often, anyway

Thanks @leslietownes for the input. I'm currently a 2nd year undergrad physics student, but in a few months I will change my major to math, onto an Ms in math

nice choice

@leslietownes cools :D

I made a post a week ago on SE, but got no response: math.stackexchange.com/questions/4842442/…

so I want to show that if $\mu$ and $\nu$ are positive measures on $X$, and we define the representations $\pi_\mu:C(X)\to B(L^2(X,\mu)),\phi\mapsto (f\mapsto \phi f)$, and likewise for $\pi_\nu$, then $\pi_\mu\oplus\pi_\nu$ being cyclic implies that $\mu\perp\nu$

I wrote down three attempts as you can see in the post, but I had no luck

in the highlighted part..

what does the author mean when he says "if a function follows 1 and 2, it uniquely derermines the distribution of *a* random variable"? does he mean the function obeying 1 and 2 describe the pmf of some random, random variable ? if so, does he mean, by "uniquely" , that this pmf only describes one random variable?

@nickbros123 it means that there is a discrete random variable with $p_X(x) = P(X = x)$

and the distribution, meaning the function $A\mapsto P(X\in A)$, is unique

more or less this is what it means

@Jakobian it seems rather trivial then, why does he say it needs an advanced course??

Hey, I was solving an integral and got -ln(1-x-x^3) at the end. when I check on an integral calculator it turns it into -ln(|x^3-x-1|). can anyone explain why? thanks

@MathHorse probably cos integral of $x^{-1}$ is $ln(\vert x \vert)+c$

Yes but, I have -1 * integral (u^{-1})

so it should be = - ln (|u|)

and u = 1+x-x^3

how did it become - ln (|-u|)

@MathHorse Why don't you show us what you have done? I suspect that you lost a sign in your change of variables, but it is quite hard to tell from what you have written here.

I've done exactly this in my notebook

then went to check on the calc to compare results

and on the website it says:

@MathHorse isn't $-\log(|u|)=-\log(|-u|)$?

@nickbros123 existence of random variable with given distribution is generally something not discussed in more basic probability courses

@ShaVuklia i can give more detail later (i'm about to get my car fixed) but one conceptual way of seeing this is that representations like pi_tau are "multiplicity-free" and the representation pi_mu + pi_nu will not be if mu and nu are not mutually singular

if mu and nu are not mutually singular there is some nonzero sigma with sigma << mu and sigma << nu, and the rep pi_mu + pi_nu will have two mutually orthogonal copies of pi_sigma in it. in this circumstance pi_sigma is equivalent to a subrepresentation of pi_mu [resp. pi_nu] by writing d sigma = h d mu [radon nikodym derivative] and looking at the subset of X on which h is positive. pi_sigma is equivalent to pi_mu acting on L^2(that subset)

@robjohn I found the formula integral of x^-1 dx = ln(t) + c in my course book and went by that, not sure about log

i don't know how much of this in conway (it may not be the easiest way to see it from what is there) but it is a very high level way of 'seeing' it

Car fixed? I thought you never drove it!

@MathHorse What does $|u|$ mean? Note that your course book should say $\ln(|t|)$. You cannot take ln of a negative number.

@TedShifrin u = -x^3+1+x

I don’t care about that. What is $|u|$ if $u$ is a real number?

the absolute value of u

Right, which means what?

should I split it into two cases? x>0 and x<0?

@MathHorse In most math books, $\log$ represents natural log. It is often written $$\int\frac{\mathrm{d}x}x=\log(|x|)+C$$ This formula gives the impression that $\int_{-1}^1\frac{\mathrm{d}x}x=0$, when in fact the integral does not converge (except in principal value, but that is probably beyond the scope of this problem).

@MathHorse That would be the way I would handle this.

What I want you to realize is: The correct answer has absolute value in it, and $|-u| = |u|$.

@TedShifrin The absolute values make it seem as if one can span $0$ in the integral.

@TedShifrin ohh

so its just a "cosmetic" change?

Aw come on. In standard integral calculus exercises we write the antiderivative as $\ln|u|$. Hopefully the teacher explains what’s going on if you try to include $0$ in your interval.

hello guys

@MathHorse Yes.

i am in very difficult

can someone help me :(

I'm studying inner product spaces and I'm working the following basic exercise:

> Are the functions $1,x,x^2,\ldots,x^n,\sin x, \cos x, e^x$ linearly independent, considered as vectors in $C([0,1])$? ($n$ is some positive integer)

$(1+(sinx)^2)$^(1/x) , Taylor-McLaurin

They do not specify any inner product, which confuses me. Because my attempt would be to determine if the functions are orthogonal, then they would be linearly independent, right? They did mention in an example that $\int_a^b f(x)\overline{g(x)}dx$ is an inner product for $C([a,b])$, where $C([a,b])$ is the space of continuous, complex-valued functions defined on $[a,b]$, but I don't get that the functions are orthogonal with respect to this inner product.

They aren’t, and inner product is irrelevant here.

how can i do the developments of this expression?

@TedShifrin by what means can I then determine if they are linearly independent or not?

Your teacher must have done an example like this. You need to take the log and write the Taylor expansion of the log. @Alessandro

i need to transform in exponential , like: $e^(ln(1+sin^2x)/(x)

like this?

@psie @psie This is a pain, but you must have some techniques. Like write a linear combination equal to $0$ and differentiate a bunch of times.

@AlessandroTerminiello Is this the whole problem, or is it part of another problem?

is only a part

what is the bigger problem?

I'll try to do the rest myself

yeah, sounds like a pain

@robjohn i dont know how to do these compound developments

It’s not so bad if you think about what I said carefully. Suppose the monomials weren’t there. Then what?

I know that in the limits it must be developed until there is an indeterminate form in the limit, and I think that when I make the developments, then I can't even rewrite the terms that I don't need.

there is also $$\left(1+\sin^2(x)\right)^{1/x}=1+\frac1x\sin^2(x)+\frac{1-x}{2x^2}\sin^4(x)+\dots$$

How far out do you need the expansion?

third grade

How many terms

what do you mean

do you mean third order?

yes

He speaks Italian :)

yes

Then the first two terms will do... $$\left(1+\sin^2(x)\right)^{1/x}=1+\frac1x\sin^2(x)+o\!\left(x^2\right)=1+x+o\!\left(x^2\right)$$

i dont understand ...

how did you do that

You mean $o(x^2)$.

@AlessandroTerminiello $\sin(x)=x-\frac16x^3+o\!\left(x^4\right)$

yes

i know this

sinx = o(x^2n+2)

why here is 4?

Hang on, I'm checking my exponents.

okok

why now it is x^2?

@TedShifrin yes. I need to check things I do in my head.

im not understanding

$\sin^2(x)=\left(x-\frac16x^3+\frac1{120}x^5\right)^2=x^2-\frac13x^4+O\!\left(x^6\right)$ so $\frac{\sin^2(x)}x=x-\frac13x^3+O\!\left(x^5\right)$

@Jakobian tbh I can totally see why such a thing might appear intuitive or obvious but is actually a technically involved proof

@robjohn can you explain what are you writing please:(

(I don't know if it is, btw, but a little experience tells it may be)

@AlessandroTerminiello You know the binomial theorem?

or how to square a polynomial?

n! / k! * (n -k)!

that is the value of a binomial coefficent

@robjohn From prior experience, we know that Alessandro’s algebra skills are weak. Opt for the simplest and most basic.

@robjohn yes

@TedShifrin this is why I wanted to know the bigger problem. Otherwise, it is hard to know how to proceed.

Surely the question is a limit question, as he has had indeterminate forms recently.

I don't want to know how to do the whole exercise, but how to do the things you sent, in general the developments of composite functions

@TedShifrin okay, you seem to have a better understanding of where he is coming from. I will withdraw and get back to work.

??

It’s best when @Sine helps in Italian.

i can use translator

Prosciutto

turn gesturejax on for best results

@leslie I thought your car never went anywhere.

@TedShifrin ok now i try

My idea is that all of mathematics is only vaguely related, so each case requires distinct treatment. There is no way to gain true confidence or experience that might allow you to be exceptionally good at every question in the field you're interested

ted: precisely for this reason there was a bit of deferred maintenance

Ah, the car threw a tantrum?

@TedShifrin i did sinx -> x-x^3/3! then i square like you said and i obtained $sin^2(x) -> (x^2) - (x^6/(36))$

@Jakobian Not universally, but experience is knowledge.

@AlessandroTerminiello Bad error. How do you do $(a+b)^2$?

do i have to square together all the factor?

Not universally, yeah. So you need to keep yourself on your toes if you want to be correct

bc i squared individually

You have to learn basic algebra.

What is $(a+b)^2$?

a^2 + 2ab + b^2

Aha. So do that here.

why do i have to do that?

Because it is correct.

I want to be correct - this is something fundamental for me. So its also something I try to think about

the result is : x^2 - x^4/3 + x^6/36

No. Do it carefully.

i missed the +!!

@robjohn This is down your alley.

now its correct?

Yes. OK.

But we can throw out the terms with exponent >4.

because it was degree 4?

How to be correct, and at the same time useful?

This is a rhetorical question by the way

Yes. So let’s call $u=x^2-x^4/3$. What is the TP of degree 4 in $x$ for $\log(1+u)$?

So what is the TP for $\log(1+u)$ in terms of $u$?

okok

wait

@leslietownes gads! this would be an obscene room if everyone's gestures were translated.

robjohn: gesturejax

It is already?.

Leslie gestures with his little hands …

Like a T-Rex?

Better than like a well-known idiot …

It would be funny if the T-Rex had an orange head.

That should be your next icon!

it is $x^2 - x^4/3 - (x^2-(x^4/3))^2/(2)+(x^2-(x^4/3))^3/3 - (x^2-(x^4/3))^4 / 4$

is right?

Cool. But keep only terms with power 4 or less.

so

i have to calculate all this terms?

Only three terms will show up. Think about exponents.

ok!

x^2-x^2/2-x^4/3???

I couldn’t recall the fancy word avatar, @robjohn.

Yes, Alessandro. Simplify.

Oh, no. The second term is wrong.

the second term is x^4/2???

-x^4/2

Right.

Now simplify.

x^2-5x^4/6

That looks correct

did i finish?

No, you need to multiply by 1/x

then you need to evaluate $e^u$ where $u$ is the thing you got after multiplying by 1/x

You finished that step. Remember, we wanted $(\log(1+u))/x$. And then we exponentiate.

sorry, I fade into the shadows

but are you considering that I rewrote it as $e^(ln(1+sin^2(x))/(x))$?

Your parentheses are not quite right.

now?

$e^{(\log(1+u))/x}$

Right.

wasnt u -> x^2 - 5x^4/6?

No, that is $\log(1+u)$.

Write everything in very clear, organized steps.

okok

so its e^(x^2-5x^4/6)/x?

We’re almost at the end, but you will need a TP for $e^v$.

Simplify and then put in the exponential.

what i have to do?

calculate u?

Divide by $x$.

x-5x^3/6?

right?

Right,

les go

Call that $v$ and then put in the TP for $e^v$.

okay!

do i have to keep only terms with power 4 or less?

Now, power 3 or less, because your goal is degree 3.

why now is degree 3 and before was 4?

You told us 3 at the very beginning. I was saying 4 earlier because I looked ahead and saw you would be dividing by $x$.

You have to learn to think like that. :)

I can't read TP without thinking about toilet paper

@TedShifrin so if it was u/x^2 , u terms were max 5 degree?

and if it was like cosx, sinx , cosx * logx , x^3 * logx , How should I choose the grade?

if there had not been only the single x^n

That’s too complicated. Yes, to the first. I cannot follow the rest.

okok

so

now i have

@Jskobian Even if I put in the periods, your thoughts won’t change.

1+x-5x^3/6 + x/2 - 5x^3/12 + x^3/6

@TedShifrin I'm not sure why, but this pinged me instead of the intended Alessandro

It pings both. First four or five letters does it.

Ah I see

You skipped way too many steps and did it wrong, Alessandro.

-

Demonic Alessandro … too popular a name in Italia.

The curse of Alessandro

wait

i made a mistake

getting pinged when Alessandro gets pinged

Yes, you do. So $1+v+v^2/2+v^3/6$, terms up to power 3.

It is indeed, I once attended a small workshop in France with 14 partecipants, we were 6 italians out of which three of us were called Alessandro

LOL … all interchangeable?

1+x-5x^3/6+x^2/2+x^3/6

?

Looks right.

les go!

Now practice lots (without me)!

okay ! last thing

so with Taylor I always have to develop all the things I have?

Yes, you have to build it up from pieces you know how to do.

Years ago, I posted a handout I wrote for my students on this, i’ll post it again and you can save it and study.

ok, for example in e^sinx I must first develop sinx to the degree I want and then make e^u where u are all the developments of sinx?

Right.

thanks !

clearer now

@TedShifrin where i can find your posts?

I'm going to try to post it here. Give me a minute.

I can no longer upload the .pdf file here, sadly.

If you give me your email, I'll send it to you.

Cool. Just a second. (You can erase it now.)

@TedShifrin were you able to send pdf's here before?

Yes. Up until a few years ago.

I've posted bunches of my class handouts here.

They clearly changed it to allow images only.

@TedShifrin For future reference, deleting a comment in chat does not completely delete it---lots of people can look at the history of a comment (I believe that anyone with sufficient XP can do it).

You can ping a moderator to purge the history of a comment (which I have taken the liberty of doing here).

Though I am going to have to play around with it to completely eliminate the PII

Yeah. I guess I could have just posted mine (which is public) and asked him to email me. I'll do that in the future.

→ 1 message moved to Trash

Thanks, Xander.

NP.

Do you know why we used to be able to upload .pdfs and no longer can?

You can also just ping a moderator. The best think to do is Edit to Remove PII -> delete -> ping a moderator.

@TedShifrin No idea.

@AlessandroTerminiello Sent.

I can remove things from the room but not from trash.

@TedShifrin maybe because of sites like libgen

There's a size limit to what could be uploaded, though.

Ugh... I'm tired. This has been such a brutally difficult week. :(

hmm... maybe its about viruses then?

All this asynchronous teaching sucks, eh, Xander?

@TedShifrin thanks!

@TedShifrin Yes, and my mother has been in the hospital all week (they sent her home today).

Oh, I'm sorry. I send her my best wishes for a speedy recovery. I went through 15 years of stress with mom and such.

It took three (minimally invasive, laparoscopic) surgeries over the course of the week to get the clots out of her lungs. It has been super stressful. But they did get them, so all seems to be well, now (though it is going to be a few weeks before she is back to status quo).

Any idea what caused the issue?

Being old?

(No idea.)

Ah, that's why I've been through hearing aids, multiple dental surgeries/implants/crowns, and soon the second cataract surgery. :) ... Back operations in the future.

Yup, age.

@XanderHenderson That's good

Howdy to bionic @copper.

@TedShifrin Yeah. At least this time, I heard about it before the treatment was over. When she went through breast cancer 10 years ago, I didn't learn about it until after the treatment was done. And even then, it seemed like it was something she let slip on accident.

Did your sister know?

@TedShifrin So far as I know, none of my brothers or sisters knew. My father knew, and (I found out today) one of the deans at the college knew. But no one in the family.

Wow.

Ah, I only knew about one of each, I think.

There is a sister who has done conservation work (she is currently "funemployed"), a sister who is a dietitian (part-time at the moment, to take care of the toddler and the baby-on-the-way), a brother who is a public defender, and the brother who lives with me (he has Downs; I'm his legal guardian).

Oh, I guess I did know about both brothers. I think my context for sister was the "new" car.

They are all younger---the one with the kid is the youngest (14 years younger than I---she has no memories of me ever living in the same house).

Or my memory is not as good as it once was (duh).

the result is -2/3, I get -1/3

:(

Close. But you never told us the rest of the problem.

Hrm... I have no limes. This saddens me. Gin and tonic without lime is hardly worthwhile. :/

yes because I would have liked to try to finish it myself

now il will write the full problem

Lemon works, Xander. If you're using Hedrick's gin, so does cucumber :D

it is (1+sin^2x)^(1/x) - e^(sinx)

all

/ x^3

(e^sinx) -> 1+x-x^3/6+x^2/2+x^3/6

OK, so simplify both of these before proceeding. Then subtract.

wait wait wait

You actually have everything correct.

I had miscalculated in the end

@TedShifrin I don't have either of those, and the gin I have is either Safeway's house brand, or St George (the one with the red label, which I haven't opened yet).

Sorry, Xander. There's only so much I can do for you. Skip the tonic and do a martini. :D

everything ok, I solved it!

les gooooooooooooooooooooooo

There are lots more problems in the text I sent you, Alessandro. You can practice :)

@TedShifrin I do have olives... and a really good dry vermouth.

What's your really good dry vermouth?

last doubt. A tip to always find the right grade for each development.?

My favorite I can only find by going all the way to Costa Mesa (I complained to you about that before).

@TedShifrin Vya. I get it off the internet.

and then it doesn't matter that I didn't use o(x^n)?

You have to think backwards about what you need at the end, Alessandro. The stuff I sent you will give more examples.

okay!

We are writing the Taylor polynomials themselves. If you need a limit $/x^3$ at the end, then the $o(x^3)$ will tell you that the polynomials of degree 3 give the correct result. I have stuff on that in my text, too.

@XanderHenderson I've never seen that. My favorite is Boissière. Dolin in second place. I'll have to look for that.

okok thank you so much

I'm going now, bye everyone, good evening!

@TedShifrin I like Dolin. I've never tried Boissiere (not doing the accents, too lazy).

thanks !

Wow. Your favorite is more than twice as expensive as mine.

Good evening, Alessandro.

@TedShifrin Sucks to be me. :D

I might try it, but maybe I should just order from my place in Costa Mesa and have them ship it. ... I've cut way back on my drinks, anyhow, trying to lose some weight and help B.P.

Oh, @Xander, at HiTime, your favorite is way cheaper than from their own website.

Still almost twice the cost of my favorite, but $7 less (so 28% cheaper).

Interesting. I usually grab a bottle when I put in an order through wine.com (if you order a lot at a time, it can be pretty cheap, so I tend to order all my wine for the year in one big order).

Well, I'll try one when I order.

MATH EXERCISE: how much does xander's favorite vermouth cost at hi-time wine cellars?

Give that to Munchkin (censored to remove alcohol).

This is cool.

Oh! I just found some elderflower tonic in the back of the fridge! G&T, sans citrus, will be fine.

@TedShifrin Hi Ted, today is week 4+2days. Walking semi-normally, albeit not quite at a point where I would declare the exercise worthwhile. Much of the day it taken up with a litany of exercises from the PT lady and some procrastination thrown in. I was able to go for a short level bike ride yesterday which improved the spirits.

@copper.hat I really need to get my bike back into ridable condition. There really isn't anywhere to ride here, but even going from one end of town to the other would be good for me.

fwiw it might not be possible

Why define the function on $\mathbb{R}\cap(0,1)$? and not just $(0,1)$? Are these not the same thing?

You're correct

@XanderHenderson yeah, the bike is (& was) a life saver for me for both exercise and just getting out.

Since $J(x)$ is really close to $f(x)=x^{1.7}$ I think it's a stretch for $J(x)$ to be real analytic.