Mathematics - 2022-08-20

leslie townes

1:24 AM

pbhtpohbtphtbpbtphbtptb

Ted Shifrin

3:58 AM

@leslietownes The mouthful of thistles again!

copper.hat

4:54 AM

Sounds like a place name in one of those vowel deficit $n$th world countries.

of course, part of me is very proud that Ireland is a real $3$rd world country in the original sense.

austere1993

6:48 AM

Hey guys can I get some help understanding the answer to this?

2

Q: The formula to calculate the average cost of winning a game of chance

As the title says, I'm trying to find the formula that calculates the average cost for a game of chance. Here are the rules of the game - There is a 25% chance to win a game The player wants to win exactly 3 games in one day. After that he stops playing. There is a maximum of 10 games per day Th...

austere1993

6:59 AM

I don't understand the reasoning behind the third line

Ajay

9:20 AM

I am contemplating asking a question on MSE, but i'm not sure if it will be closed. As part of a research project, I am working on evaluating the integral of $\sin(t^7) \,dt$. Is there anywhere in math where this integral arises(over any bound) that is needed to solve some real world problem? I was thinking there may be something in Fourier analysis but I still haven't found anything. Would this be an appropriate question to ask on MSE or MO?

i'm aware that when I am looking for applications in engineering, physics etc to ask on other sites, but since it is about math application I'm not sure if it should be MSE or MO.

So to summarise: Can i post this without it being closed? If so, which site should I post it on?

leslie townes

12:55 PM

MSE seems like an OK place for it. MO does not seem like an OK place for it. you might mention where you have looked, or tried to look. and (if there is a useful story) where the focus on that integral came from. people on MSE like to see surrounding context and a search for answers, even if the search didn't lead anywhere.

as an attempt at answering the question, it would surprise me if that integral has ever come up anywhere in a physical model (but, i have been surprised before). i would expect that people would ask in the comments, where did this come up for you?

why 7 and not 12 or 6.5? exponents in physical problems tend to come from somewhere.

Ajay

Well, it can really be any odd number, I just chose 7.

Pretty random actually.

This is what i'm planning to do @leslietownes

But it seems kind of dry just to have all theory and no real world application.

leslie townes

oh. often for something like that, the only significance of the integrand is that it's something where you can't write down a nice formula for the antiderivative and thus evaluate the definite integral 'explicitly' using the FTC.

i agree that any given example of such a thing would be somewhat contrived, although the general phenomenon of 'function that doesn't have a nice antiderivative' is not contrived.

the integral of sin(t^2) dt somehow comes up in optics. the 'fresnel integral.' you might peek around the physical origins of the 2 exponent in that and see if there's some way of varying the hypotheses (perhaps with a modified physical law on a different exponent) that would incorporate sin(t^p) for other p.

Ajay

That seems quite advanced though...

leslie townes

you could do the same exercise (pictured above) with sin(t^2), really. it doesn't have a 'nice' antiderivative, and does have some physical meaning.

Ajay

1:14 PM

hmm... I would be able to find the Maclaurin series and approximate sin(t^2) however, what purpose does approximating the Fresnel integral serve?

leslie townes

i don't know the physical origin of it, i just know that it has one.

this is awfully close to 'what purpose does anything serve.' which is above my pay grade :)

Ajay

Sorry if i'm being confusing.

There is this: downloads.hindawi.com/journals/mpe/2018/4031793.pdf

But I don't understand much of the math.

I mean, I guess what I really want to show is that we can take an impossible integral and compute its true value to some accuracy by hand.

But then, what's the whole point of doing something like that?

What consequence does it have on the real world?

leslie townes

none? i don't know? the broad message is just, you can numerically compute definite integrals without antiderivatives. that's a useful message.

this particular example, or any particular example, who cares.

if anything, it tells us that 'having a nice formula for an antiderivative' is not that significant of a property of a function, although a lot of the toy examples in calculus textbooks enjoy that property.

you might think about int 0..1 cos(x) dx for example. it's not "impossible" in the sense that there's a "nice" formula for it (namely sin(1)). but what is sin(1)? how would you compute it to three digits?

that's a question that comes up even if you do have a formula for the antiderivative.

Ajay

1:40 PM

Sorry Leslie, give me a little time. I'm trying to figure out what I really aim to achieve here.

leslie townes

2:28 PM

1960s hippie voice we're all trying to do that, man.

Koro

The set of all irrationals in an interval [a,b], a<b is a Borel set.

leslie townes

this is true

Koro

Because $[a,b]=\cap (a-1/n,b+1/n)\in \mathfrak B$. For any rational r in the interval, $r=\cap (r-1/n,r+1/n)$. So the set of rationals in the interval(being a countable union) is Borel set.

leslie townes

yes, that's a very good way of looking at it. any countable or co-countable set is borel, and [a,b] is borel.

Koro

Let the set of rationals in the interval be E. So [a,b] intersection E^c is also a Borel set.

@leslietownes :-)

It’s so sad how some big companies make fool of customers. Like a purchase is made by a customer and the product being covered under warranty is not repaired due to arbitrary reasons by the company. In such cases, many customers just decide to never ever purchase anything from that company. They don’t file lawsuit which I’m sure they’ll win if they do, because filing lawsuits will take time and they don’t have time.

leslie townes

2:40 PM

time and money, at least in jurisdictions where by default you pay your own legal fees even if you win.

in such places, you can basically harm a customer as much as you want, as long as you don't want to harm them more than the cost of hours of attorneys' time

Koro

yeah, someone should start a ‘startup’ which helps the oppressed customers.

leslie townes

in the US we have a 'class action' mechanism designed to deal with this, but it doesn't really work

Koro

I’m sure such startup will become multibillion very quick.

@leslietownes :(

@leslietownes This is so unfair

leslie townes

it sometimes works. it deters a lot of the worst abuses.

it can succeed in punishing the wrongdoer even if it doesn't succeed in compensating people who were affected. also, attorneys get paid, so there's that.

Bob

3:08 PM

Hi

from Google I see that the general from of a plane is:

Ax + By + Cz = D

but it seems to me that it should be:

Ax + By + z = C

that is we need only three constants

is there something I am missing?

Jakobian

You're missing some planes with that definition

And for the fist one you need to assume one of the A, B, C are non-zero

Bob

isnt 2Z = 5 a plane?

Jakobian

what you could assume is say, A^2+B^2+C^2 = 1, then you have 4 constants but additional constraint

leslie townes

yes, the 'number of constants' is not a useful focus.

putting a 1 in front of the z is an assumption that the coefficient of z is nonzero. which is a fine assumption, but does rule out some planes.

Jakobian

and every plane should be uniquely described by some constants A, B, C, D with A^2+B^2+C^2 = 1

Bob

3:12 PM

I have not seen the additional constraint of: A^2+B^2+C^2 = 1

I claim ever plane can be uniquely described by:

Ax + By + z = C

Jakobian

Well, instead of assuming that one of A, B, C is non-zero we can just divide by sqrt(A^2+B^2+C^2)

and it leads you to the constraint I said

Bob

so I think we agree

Jakobian

@Bob then you're wrong

Bob

then I do not understand

Jakobian

something like x = 0 describes a plane

and isn't of that form

Bob

3:14 PM

oh

Jakobian

to uniquely describe planes you'd need something like I wrote

Bob

that makes sense

I thank you very much

leslie townes

one of the coefficients of x,y,z has to be nonzero but you don't generally know which one

Jakobian

np

Bob

have a nice day

HelpMeToUnderstandContours

3:45 PM

Hii, I'm stuck on a question today.
Question is to find $\dfrac{d^2x}{dy^2}+ 20$ when $y(x) = (x^x)^x$, $x>0$.

I started it by differentiating $y$ wrt $x$ and got $\dfrac{dy}{dx} = xy(2\log(x) + 1)$. And so, $\dfrac{dx}{dy} = \frac{1}{xy(2\log(x) + 1)}$.

I know I can now differentiate $\dfrac{dx}{dy}$ wrt $y$ so as to obtain $\dfrac{d^2x}{dy^2}$ but I think there might be a clever technique to solve it.

Ted Shifrin

4:13 PM

Why do you think that?

HelpMeToUnderstandContours

@TedShifrin It became very cumbersome.

leslie townes

i think it's just gonna be very cumbersome

Ted Shifrin

Not so bad. What is so cumbersome? Product rule + chain rule

It’s a rather artificial problem, anyhow.

leslie townes

well, the result is not something i want to hang on the christmas tree.

but that's not a sign that anything is going wrong. ask a goofy question, get a goofy answer.

HelpMeToUnderstandContours

Ohh wait, my textbook gives a hint on last pages of textbook.

The hint is $\dfrac{d^2y}{dx^2} = -\left(\frac{dy}{dx}\right)^3\cdot \dfrac{d^2x}{dy^2}$.

But how is it derived?

Ted Shifrin

4:24 PM

I don’t think that makes it any less cumbersome.

You derive it the only way you can, pun intended.

HelpMeToUnderstandContours

@TedShifrin Badly sad :(

Ted Shifrin

As I said, it does not help. You have the same issues differentiating $y$ twice.

HelpMeToUnderstandContours

But I think finding $\frac{d^2y}{dx^2}$ is much easier than $\frac{d^2x}{dy^2}$.

Ted Shifrin

4:55 PM

I’m done arguing.

Summer Child

Does anyone have any advice on this interpolation problem?

0

Q: Finding $a\in(0,10)$ so that the linear spline for $f(x)=e^{-x^2}$ is sufficiently accurate and $n_1+n_2\le 12$

Suppose we're given a function $f(x)=e^{-x^2}$ and we want to make a piecewise linear interpolation on the interval $[0,10]$ under the following constraints: We divide the interval $[0,10]$ into two intervals $[0,a]$ and $[a,10]$ and make equidistant partitions into $n_1$ and $n_2$ intervals re...

optimization interpolation numerical-calculus

Koro

@HelpMeToUnderstandContours here’s one approach that you might like: Suppose that you want to find dy/dx and you know that y= g(x). Write f=y-g(x) to have f=0. Use chain rule to get: $f_x+ f_y dy/dx=0$.

Ted Shifrin

@Koro No partial derivatives allowed. HelpMe is in Calc 1.

It's just chain rule/implicit differentiation.

Koro

This makes calculations simpler as it only requires you to differentiate while keeping one variable fixed at a time.

@TedShifrin I see.

@HelpMeToUnderstandContours it was asked in JEE exam once.

Ted Shifrin

5:13 PM

I continue to be unimpressed by the JEE. One can test understanding of important concepts without introducing complicated nonsense.

leslie townes

but how do you filter trillions down to millions, smart man.

Ted Shifrin

Are we filtering to the right millions, crytoman?

leslie townes

i dunno, i mostly sell cryptocurrency on here.

Ted Shifrin

Let's just give PhDs to all the people who win the Putnam and be done with it.

After all, that's what research in mathematics should be.

Wordle has turned into complete psychological games.

HelpMeToUnderstandContours

@TedShifrin I think I can use them. Since I just have to choose only correct option from given 4 options.

@TedShifrin And yes I think the question is intended to be solved using chain rule/implicit differentiation only.

@Koro Yes. Thanks. I answered a question on MSE in the morning using same concept. here

Ted Shifrin

5:39 PM

It’s no different at all from usual implicit differentiation; it just sorts your algebra slightly. Of course, the implicit function theorem gives hypotheses and proves that it is valid.

HelpMeToUnderstandContours

Yes.

Xander Henderson

Well... we are going through my father's sports memorabilia. We just found a 1960 Johnny Unitas card. Fun.

leslie townes

wow

robjohn

@TedShifrin one derives on the right side in the US and on the left side in the UK.

Xander Henderson

Oh, sh*t Sandy Koufax and Jackie Robinson from 1955.

Ted Shifrin

5:55 PM

@robjohn Just bring in the Polish in reverse!

Even I had a few baseball cards in my childhood.

leslie townes

at this rate, xander will retire on baseball card proceeds before lunch time.

Xander Henderson

Heh.

HelpMeToUnderstandContours

All the options wrong or what?

Graphing all the functions shows that none of the answer is correct :|

Ted Shifrin

What is the correct answer?

HelpMeToUnderstandContours

@TedShifrin I don't know the answer. My friend sent me a link of the question.

Ted Shifrin

6:09 PM

Can’t you differentiate correctly?

leslie townes

i think i agree that the answer is not on that list

HelpMeToUnderstandContours

Yes I got $\dfrac{1}{\sqrt{1-\sin x}}\cdot \dfrac{\cos x}{2\sqrt{\sin x}}$

Ted Shifrin

It is up to sign. We have to think about domains and whether their sign is correct. I claim it is.

OK, now what might you do, Help?

Oh, no, the sign is an issue.

For these sorts of problems you have to manipulate further.

HelpMeToUnderstandContours

@TedShifrin How? At first, I thought I've to do more work to simplify $\frac{1}{\sqrt{1-\sin x}}\cdot \frac{\cos x}{2\sqrt{\sin x}}$. But that doesn't seem so.

Ted Shifrin

Rewrite $\cos x$.

Yai0Phah

6:17 PM

@LukasHeger I am not sure whether I understand correctly, but there is also analysis in GAGA.

Ted Shifrin

If $0<x\le \pi/2$, their answer is right.

HelpMeToUnderstandContours

@TedShifrin as $\sqrt{1-\sin^2(x)}$ may be?

Ted Shifrin

Yes, up to sign.

Bart Michels

The number of answers and questions is about half that in the 2021-2022 academic year than it was in previous years. Why?

Is there a meta post about this?

HelpMeToUnderstandContours

@TedShifrin Yes 3rd option.

Ted Shifrin

6:20 PM

@Yai0Phah Agreed. He was just saying that to get the RS as an algebraic variety took some analysis too. Maybe we can get to RR and avoid Serre duality :)

@HelpMeToUnderstandContours Yup.

leslie townes

bart: interesting observation. i'd guess that a lot of MSE stuff is driven by coursework in schools, and that as the pandemic persisted, more people found avenues outside of historical ones for asking/resolving questions.

HelpMeToUnderstandContours

@TedShifrin But I feel like I've been scammed. They didn't mention to find derivative in $[0, \frac{\pi}{2}]$. :p

Ted Shifrin

It felt like we were buried in homework and exam questions. Maybe your data exclude all the closed/deleted questions.

Bart Michels

I found this answer about it (but doesn't answer the question I asked here in chat) math.meta.stackexchange.com/a/34623/43288

Ted Shifrin

@HelpMeToUnderstandContours Yes, that’s a bad error, but maybe they did it intentionally.

Bart Michels

6:22 PM

It's the data from site analytics

Ted Shifrin

@BartMichels I don’t know how they count.

HelpMeToUnderstandContours

@TedShifrin Hmmm... Inverse trig. functions really suck.

Bart Michels

12 years of Math.SE

leslie townes

it's the general decline of all things. don't you feel it?

Bart Michels

:)

Ted Shifrin

6:25 PM

I’m in rapid decline.

Bart Michels

This post has some speculative answers: meta.stackexchange.com/questions/336925

That was about data from before covid

leslie townes

we're all making a beeline for the drain, like waste water.

gradient descent

Bart Michels

less answers -> less search engine results -> less traffic -> less questions -> less answers
I suppose

Some very active members quit/moved on and that started the feedback loop or something

Ted Shifrin

I’m answering far, far fewer than a few years ago.

The question quality has declined markedly.

Heading to the neighborhood farmers market to get some lunch ;)

Koro

you don't cook?

:)

@leslietownes :(

@BartMichels some accounts were banned also :(

leslie townes

6:44 PM

just my alt accounts. they can't get rid of me

user4539917

7:05 PM

the royal "they"

Not watching the fight? @copper.hat

Offical Usyk-Joshua 2 weights:
Oleksandr Usyk: 221.5lbs – 0.5lbs heavier
Anthony Joshua: 244.5lbs – 4.5lbs heavier

Ted Shifrin

@Koro I cook all the time. But not Chinese dumplings!

Koro

8:18 PM

:)

The set of all nilpotent elements in a commutative ring with unity is same as intersection of all prime ideals of the ring.

copper.hat

@user4539917 unfortunately not. not sure who i would bet on.

user4539917

I think Usyk should win, but if it goes to the judges...who knows

Jakobian

8:48 PM

Jacobson radical

In mathematics, more specifically ring theory, the Jacobson radical of a ring R {\displaystyle R} is the ideal consisting of those elements in R {\displaystyle R} that annihilate all simple right R {\displaystyle R} -modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by J ( R...

this right

Koro

@Jakobian I've not yet studied Jacobson radical. I'll study this soon.

The proof that I know relies upon Zorn's lemma.

Thorgott

intersection of all prime ideals vs intersection of all maximal ideals

Jakobian

oh sorry, it's this en.wikipedia.org/wiki/Nilradical_of_a_ring

ring theory is too much for me

Koro

I'm thinking about maximal ideals of Z[$\sqrt {-5}]$.

I think that if I consider the natural map $\phi: Z[x]\to \frac{Z[x]}{(x^2+5)}$, then ideals of the Gaussian integer ring are same as the ideals of Z[x] which contain the principal ideal (x^2+5).

This is by correspondence theorem (one of the isomorphism theorems).

Thorgott

that is correct

Koro

8:54 PM

But I don't know how to show that every maximal ideal of the Gaussian ring contains a prime no.

correspondence theorem talks about ideals to ideals but doesn't say that maximal ideals also "correspond".

or Z[x] is a commutative ring with 1. (x^2+5) is irreducible so (x^2+5) is a maximal ideal of Z[x] and therefore, the quotient ring by (x^2+5) is a field. What is a maximal ideal of a field? There is no maximal ideal of a field. So vacuously the statement is true. But this looks wrong.

Thorgott

(x^2+5) is not a maximal ideal of Z[X], so that's incorrect

it's also incorrect that there is no maximal ideal in a field. every non-trivial ring contains a maximal ideal. in any field, the zero ideal is the unique maximal ideal.

@Koro think about how the correspondence works and whether it plays nicely with inclusions

Koro

@Thorgott oh yes. the zero ideal.

@Thorgott I think yes. The inclusions are respected in the following sense: If I, J are two ideals containing kernel and satisfying I$\subset J$. Then, $I \mod K\subset J\mod K$, where K:= kernel.

Thorgott

9:11 PM

alright, so the correspondence preserves inclusions. what does that tell you about maximal elements?

Koro

9:24 PM

I don't think it tells anything about maximal ideals. :(

Thorgott

what is a maximal ideal

Koro

a proper ideal M of a ring R is said to be a maximal ideal if the only ideals containing M are M and R.

I take R as commutative with unity.

copper.hat

@XanderHenderson do moderators get some notification if a question has a clot of comments?

Thorgott

9:40 PM

ok, so think about why maximal are called maximal

Xander Henderson

@copper.hat Yes, but is there a particular example you are worried about?

(the threshold for auto-flagging is 20 comments, I think)

copper.hat

Not worried, just curious.

Xander Henderson

@copper.hat Yes, if a comment thread gets long, autoflags are raised. I tend to be pretty brutal with moving those threads to chat, particularly if it is a back-and-forth between only two or three people.

copper.hat

I guess in some instances I would rather prune my comments than have it moved. Rarely.

If the comments drag on, I usually try to update my answer accordingly.

But I was really just curious about the process.

Koro

@Thorgott by definition of maximal elements. If M is maximal ideal, and I is a proper ideal containing M, then M=I.

Ted Shifrin

9:52 PM

Sometimes I've done my best teaching on this site in such comment streams. And I don't think it's bad. Usually, I ask the OP to write an answer based on our discussion once he/she's understood. Sometimes that happens.

Xander Henderson

@TedShifrin I don't think that it is bad, but the model is that SE is a Q&A site, and any content which is meant to be preserved should be put into either a question or answer, and not comments. Comments are supposed to be ephemeral.

Koro

yeah. Ted, I think if you look up your old comments on mse posts, you'll find some of them deleted.

Thorgott

is this the same as maximality in some other sense you might be familiar with

Koro

yes, I know only this maximality so far. This is in line with maximal elements definition in partial ordered sets.

Thorgott

ok great, what partial order on the ideals are we considering here?

Koro

9:58 PM

we may consider set inclusion.

Thorgott

indeed

so now we have correspondence between two partially ordered sets (ideals of $R/I$ and ideals of $R$ containing $I$, each partially ordered by inclusion) and you have observed this correspondence preserves the partial order

Koro

true.

Thorgott

now prove the proposition that an order-preserving bijection between partially ordered sets maps maximal elements to maximal elements

Ted Shifrin

@Xander That's why it's good for the OP to write up the proof once he understands it. I used to get very upset when someone who wanted rep would post the solution while the OP was clearly interacting with me and figuring it out somewhat for himself.

Anyhow, I think I'm mostly phasing myself out of this place.

Koro

@Thorgott it doesn't seem to be true in general.

But particularly for the ring above, probably yes. I'm thinking about it for this ring.

oh wait, you said bijection.

Koro

10:18 PM

At of now, I visited mse 1009 days, 547 consecutive.

:)

Ted Shifrin

It seems I'm ahead of you.

Koro

of course.

you joined this site probably around the time this website came out.

Ted Shifrin

I'm also annoyed that someone who posts zillions of questions about Spivak's text and problems has not accepted the answer I gave him. It was a rare situation where Spivak just messed up the exposition in the text, I believe.

No, quite long after that, Koro. It was Pedro Tamaroff who dragged me into chat and that's how I got to know him. I don't remember how I landed at MSE in the first place.

copper.hat

It is good that you lend your expertise, it helps many.

Koro

Stack exchange came out in 2009 as per wiki. Ted, you have been contributing for more than 9 years now!! That's so great. It helps many.

Thorgott

10:26 PM

@Koro it is, however

Koro

After talking to customer care, when someone says - I understand how you feel. It doesn't sound right.

copper.hat

Visited 3699 days, 370 consecutive

scary