Mathematics - 2022-07-22

copper.hat

00:34

@Bob Did you try plotting the points?

Bob

@copper.hat What points would you plot?

@copper.hat I think my model is wrong.

copper.hat

Percentage vs log of net worth maybe. I would look at the data before creating a model.

leslie townes

now's the time to hawk my cryptocurrency, the only proven way to get into the top 1% in 2019 or any other year

Bob

@copper.hat I think that is a good idea

copper.hat

i heard lesliecoin was plagued by insider trading

leslie townes

00:49

i've got the defamation suit against you and ted ready to go. you just can't do this to my currency.

Bob

@copper.hat The reason I think my model is wrong is if you use it to find the wealthiest family in America, you get a very large number.

copper.hat

i don't think much about wealth these days. i think more about its absence.

Bob

That is a depression thought. You have a PhD. You should be well paid.

copper.hat

hopefully i will have one member of my family back tomorrow night. my 19 yo son is flying back from croatia via frankfurt & boston on his own, his first international alone. i hope it all works out.

living in the bay area is a good antitdote

Bob

it almost certainly will

copper.hat

00:52

i'm a worrier

Bob

I understand

Ted Shifrin

My return from Croatia to SD 5 years ago was well over 24 hours.

Bob

what does SD stand for?

copper.hat

san diego?

Ted Shifrin

@leslietownes I will sue you for defamation.

Bob

00:53

@TedShifrin any other comments about my post?

Ted Shifrin

Indeed.

Model seems unmotivated. I already was skeptical.

Bob

I tend to agree

I may post an answer stating that.

Ted Shifrin

Copper’s suggestion to investigate things more graphically takes me back to high school science labs.

Bob

and I think it is a good suggestion

I am going to sign off for the evening

thanks for the help

good night

Ted Shifrin

Take care.

copper.hat

00:59

good luck

@TedShifrin Were you in Croatia for work?

Ted Shifrin

Work? Hell no. Long after retirement. Beautiful country.

copper.hat

:-)

Ted Shifrin

Glued to hearing #8.

copper.hat

i think the main effort of the hearings is to legitimately remove trump from running, so his followers will vote for another republican instead.

Ted Shifrin

I think the point is to try to save democracy from seriously flushing down the toilet. The guns are all out there.

This is infinitely bigger than coward Tromp.

leslie townes

01:20

josh hawley, track star.

copper.hat

i wish i had your confidence in the intent...

leslie townes

i'll be happy if it gives people funny video clips so they can humiliate each other in their primaries. that hawley clip is one for the ages.

of course, whoever primaries him is going to be worse, and win.

copper.hat

tucker carlson

polite proofs

03:08

In Spivak chap. 14, problem 1 which consists of finding derivatives of integrals using FTOC Part I and other tricks, why don't we first need to check that the integrand is continuous before applying FTOC?

For example, find the derivative of $F(x) = \int_a^{x^3} \sin^3 t ~ dt$

We let $f(x) = \int_a^x \sin^3 t ~ dt$, and then since $\sin^3 t$ is clearly continuous on the reals, we can apply FTOC to $f$.

leslie townes

sure, yes. it's not so much that you don't 'need' to check any hypothesis, it's just easy to check the hypothesis and so maybe not the focus of attention.

polite proofs

And note that if $g(x) = x^3$, then $f(g(x)) = F(x)$ so $f'(g(x))g'(x) = F'(x) = (\sin^3 x^3)(3x^2)$

@leslietownes I don't think it is though..

I mean, for the other examples

leslie townes

maybe it isn't. i don't have have spivak in front of me.

polite proofs

My point is, when given exercises like this am I supposed to check that it satisfies FTOC, even though it's clear that we need to use FTOC to solve the exercise

Or is it just an exercise in applying FTOC without worrying about if we can even apply it

leslie townes

for that example, sin^3 t being a continuous function plus some theorem tells you it has a differentiable antiderivative and then the FTC applies.

that kind of thing might depend on the instructor, to be honest.

polite proofs

03:12

@leslietownes Instructor being Spivak

What does he want?!?

leslie townes

we don't have the option of asking him. which book is it?

polite proofs

Just his Calculus book, chapter 14

leslie townes

i'd assume it is mostly an exercise in symbolic manipulation, with a background understanding that because the inputs are not 'ugly' the theorems will apply to them. but i've never taught out of that book.

polite proofs

I would assume so as well (and so I didn't bother proving the integrands are continuous), however the dilemma I keep thinking of is that when you do apply FTOC 'in the real world' i.e. not computing symbolic manipulation out of a chapter in a textbook, you will not always know if the integrand is continuous

So should I be preparing for such a scenario by making sure that the integrand is indeed continuous or... what exactly?

copper.hat

03:27

you would need to check that the hypotheses of the relevant result apply...

leslie townes

there's something beyond this, polite proofs, which is, the FTC holds under very general and subtle conditions, if you replace the riemann integral with another integral more suited to the purpose. the conditions in spivak are sufficient for FTC to apply but not necessary.

but in practice, the functions people run into 'in the real world' do not test these extremes.

and the spivak result will work just fine.

Ted Shifrin

He wants you to observe that all the integrands are continuous. No big deal. On an exam, I want you to tell me that without belaboring it.

There are later problems where subtleties are pursued.

polite proofs

03:52

@TedShifrin Without belaboring it as in, without proving it?

Ted Shifrin

Yes. We know sums, products, compositions, etc.

As I said, where there are subtleties later, you should prove details.

polite proofs

04:10

@TedShifrin So for example, to prove we can apply FTOC I to find $F'$, where $F(x) = \sin \left( \int_0^x \sin \left( \int_0^y \sin^3 t ~ dt \right) ~ dy \right)$, we would go something like: Define $g(x)$ by $\int_0^x \sin(f(y)) ~ dy$ where $f(y)$ is defined as $\int_0^y \sin^3 t ~ dt$. Then $F(x) = \sin(g(x))$. $\sin$ is continuous, so we need to check if $g$ is continuous.

Then I suppose we need to check if $\sin(f(y))$ is integrable

leslie townes

errythang in site is continuous, inheriting its continuity from sin(t)

polite proofs

Well, $\sin$ is continuous, so we need to check if $f(y)$ is integrable

And $f(y)$ is integrable because $\sin^3 t$ is continuous

@leslietownes Hmm, can we say that?

Ah yeah I guess

No, wait.

@leslietownes I don't think we can say that so easily. $f(g(x))$ is only continuous if $f$ is continuous at $g(x)$ and $g$ is continuous at $x$

leslie townes

g is continuous because g(x) is the integral of a continuous function from 0 to x. the FTC tells you that g is even differentiable.

and the thing being integrated is continuous because . . .

it's layers of versions of the same theorem. i don't know the specific hypotheses used in spivak.

polite proofs

@leslietownes But you still need to reason through $\sin(f(y))$ being continuous

(above is how I did it)

leslie townes

because sin is and f is

polite proofs

04:20

I know, but you can't conclude that without making sure that $f$ is

But you said they inherit it from $\sin(t)$

leslie townes

yes. this is what i meant by, it's layers of applications of the same thing.

sin(t) being continuous tells you that f(y) is continuous tells ou that sin(f(y)) is continuous

Ted Shifrin

The integral of an integrable fn is continuous. Proved in previous chapter. You’re belaboring.

polite proofs

@TedShifrin But my point is that I am checking that the fn is indeed continuous, without proving it

Ted Shifrin

shrug

polite proofs

:(

Houshou Rattengod

04:39

I’m on mobile, pardon the lack of proper annotation.

but which is more proper to write? 3 = k(mod0) or 0|(3-k)

or do I have that wrong… now I’m confusing myself…

polite proofs

You wrote that $0$ divides $3-k$, which is probably not what you want.

Houshou Rattengod

Yeah that’s what my brain just clicked…

polite proofs

Did you mean to write $3 \equiv 0 ~ \pmod k$?

Houshou Rattengod

basically I found a pattern in a problem. That deals with mod3

polite proofs

Okay, so meant to write $k \equiv 0 \pmod 3$, which says that $k$ has remainder $0$ after being divided by $3$

We can write this like so: $k - 0 = 3c$ for some $c \in \mathbb{Z}$, or using the other notation: $3 \mid k$

Houshou Rattengod

04:50

Yes? I’m sorry… pure mathematical notation like that is hard for me to read/comprehend.

polite proofs

$a \mid b$ means $a$ divides $b$, or into other words there some integer which we can call $c$ such that $\frac{b}{a} = c$, so multiplying by $a$ we get $b = ac$.

leslie townes

it's genuinely confusing, especially at first. particularly as computer languages have a "mod 3" function and saying a = b mod 3 might actually refer to computing a value of that function in such languages.

whereas in the pure math world it's just a relation, __ = __ mod __ having a meaning without __ mod __ necessarily being given a meaning.

Houshou Rattengod

Yeah. But when talking about infinity and finding a pattern that dis includes 1/3 of all infinity when that third is a multiple of 3…wait it might be a 1/6 of infinity…

3*1, 3*3, 3*5… 3, 9, 15… that’s effectively every 6 numbers…

polite proofs

No it's not

Houshou Rattengod

Yeah it is, they e just been shifted 3 numbers

polite proofs

04:59

I think you might be interested in reading up on what are called equivalence classes

Houshou Rattengod

But it doesn’t matter because the Even counterparts can’t be reached either, so… yeah. 1/3 of infinity

But back to the original question. Which method for writing a formula for Mod3 is preferred?

Prithu Biswas