Mathematics - 2022-07-20

user4539917

01:09

Distinguished moment.

4

shintuku

01:20

Feed a fish some man, he meals one eat.

user4539917

Evolution selects out only the most able fishermen.

Unknown x

Suppose $x=f(t),y=g(t),z=h(t)$ be a curve. Is it always true that there exists surfaces $\phi(x,y)=0$ and $\psi(y,z)=0$ such that curve will be the intersection of the following surfaces. If $f'(t)\neq0$, I can deduce that for some open set such that $f$ is invertible and $t=p(x)$. Therefore $y=g(p(x))\implies \phi(x,y)=y-g(p(x)).$ What if $$f'(t)=0$?

Ted Shifrin

01:37

You assuming $C^1$ functions?

Unknown x

But in the textbook, they didn't give anything

Ted Shifrin

What sort of course is this?

Unknown x

Partial differential equation

user4539917

What sort of textbook is this?

Ted Shifrin

But you know the implicit function theorem, say?

You need at least one of the derivatives nonzero or else nasty stuff can happen. And you can’t argue $f$ invertible locally without continuous derivative.

@user4539917 I like that!

user4539917

01:41

Thanks professor.

NormalsNotFar

Hi all, is there a name for a product <,>: V X W -> R for two distinct vector spaces V and W? When V=W this would be a bilinear form

Unknown x

@TedShifrin thank you

polite proofs

Hello professor

How is your evening going?

Ted Shifrin

Almost dinner time.

Did you finish that problem?

polite proofs

Sorry professor I've been cleaning

I'll do it now

Ted Shifrin

01:46

Oh, you can come clean here!

polite proofs

Only if I learn math by cleaning...

Ted Shifrin

You do not need to call me professor.

polite proofs

Dr. T. Shifrin

Ted Shifrin

Yikes, London fires and Spain fires. Stupid politicians everywhere.

I am fine with just Ted.

one potato two potato

@Thorgott The map you're talking is $H^p(X;R)\to Hom(H_p(X;R),R)$ by $[f]\mapsto [f\circ i]$ where $i:Z_p(X;R)\hookrightarrow C_p(X;R)$ right?

Leaky Nun

01:57

@NormalsNotFar I think it would just be called a pairing

polite proofs

@TedShifrin I have cooked up that $\frac{a}{a+b}y + \frac{b}{a+b}x = \overrightarrow{AD}$

Where $b = ||y||, a = ||x||$

This seems oddly symmetrical

one potato two potato

Anyway for the injectivity part, by assumption, $f:C_p\cong Z_p\oplus B_p\to R$ has zero on $Z_p$ and $B_p\hookrightarrow C_{p-1}$, defining a map $g:C_{p-1}\to R$ by letting $f$ on $B_p$ and extend to $C_{p-1}$ then $C_p\xrightarrow{d_p}C_{p-1}\xrightarrow{g}R = f$. Yes? @Thorgott

NormalsNotFar

02:21

@LeakyNun Thanks!

polite proofs

Oh my god this problem is so simple.

What have I been doing?

Let $\overrightarrow{AB} = x$ and $\overrightarrow{AC} = y$. Observe that
\begin{align*}
\overrightarrow{AD} &= \overrightarrow{AC} + t\overrightarrow{CB} = y + t\overrightarrow{CB}, \\
\overrightarrow{AD} &= \overrightarrow{AB} + s\overrightarrow{BC} = x + s\overrightarrow{BC}.
\end{align*}
Note that $s\overrightarrow{BC}= \overrightarrow{BD}$ and $t\overrightarrow{CB} = \overrightarrow{CD}$. Also, clearly $\overrightarrow{BC} = - \overrightarrow{CB}$, so $y + t\overrightarrow{CB} = y - t\overrightarrow{BC}$. Using problem 1.1.10., we find that $t = 1- s$. Therefore, $$\frac{||\overrightar…

I honestly don't know what I was doing before.

I'm stressed out now, that I took so long

(I skipped stating an application of 1.1.10. in the middle there but it's obvious)

(there probably is a shorter solution, though)

Ted Shifrin

02:43

Ok. It’s still wordier than necessary. Just use one expression for AD. $t$ alone. You get there way faster.

polite proofs

How can I use just $t$?

Maybe $t\overrightarrow{CB} = \overrightarrow{CD}$ but $(1 - t)\overrightarrow{BC} = \overrightarrow{BD}$, is that right?

polite proofs

03:33

Answered two questions but neither got accepted by the OP so far. I'm so sad.

(and no other answers were posted)

ryang

05:53

Hi all, what is the correct practice when editing closed Questions for reopening? There is a "this edit resolves the closed issue" checkbox on the editing page, and, separately, the usual "reopen ([#])" option beneath the closed Question. Which to opt for, or both? Please ping me when replying.

robjohn

08:43

34

Q: Guidelines for context edits and rewrites

Suppose you come across a question that has been closed for lack of context, has high quality answers, and on its way to being deleted. Can you save the post by editing it to include more context? Here is the deciding question: Can the question be salvaged without changing the author's intent? ...

@ryang: This might help

Then you might want to post a reopen request here.

ryang

09:52

@robjohn Thanks, but where on that page (I skimmed the post but skipped the comments) does it address my query? I'm just wondering about (1) ticking the 'reopen queue' checkbox prior to edit submission, versus (2) clicking on that "Reopen" thingy in that "Share Cite Edit Follow Reopen(4) Delete(1) Flag"" lineup. Does one pathway invalidate or interfere with the other, OR are they complementary separate pathways, OR are is one pathway a strict subset of of the other? Thanks!

SHASHAANK B.H.

13:01

What does the statement if these derivatives do not vanish simultaneously mean in Cauchy's theorem?

Xander Henderson

You would have to remind me of the statement of Cauchy's theorem, but my guess would be that there is no point where all of the derivatives are zero.

SHASHAANK B.H.

Let f(x) and g(x) be two functions continuous in the interval [a,b] and have finite derivatives at all interior points of the interval. If these derivatives do not vanish simultaneously and f(a) not equal to f(b) the there exists a c belonging to (a,b) such that g(b)-g(a)/f(b)-f(a)=g'(c)/f'(c)

This is Cauchy's theorem

Koro

it means that there is no d in (a,b) such that f'(d)=0 and g'(d)=0

Xander Henderson

So, in other words, there is no point $c$ in the domain such that $f'(c) = g'(c) = 0$.

SHASHAANK B.H.

Oh

Thank you

But there could be individual points in either f(x) or g(x) where their derivative could be zero

But not simultaneously ryt?

Xander Henderson

13:15

That is correct.

SHASHAANK B.H.

Thank you :)

Jakobian

@NormalsNotFar dual pair

there could be like some thing here that makes math questions appear better in the chat

Xander Henderson

tinyurl.com/cfqcvpc

Jakobian

so I can search for them better while browsing

Xander Henderson

See the room topic.

Jakobian

13:21

you misunderstood me, I meant it as in, stand in from the crowd better

Xander Henderson

I don't understand.

Jakobian

well, when I'm looking at chat logs, I'm mostly just browsing with my eyes and it's possible I'd skip something I could be helpful at

so if the questions were standing out more, that'd be convenient for me I guess. But that's pretty minor

Xander Henderson

I'm still not sure that I understand, but if you are asking for a chat feature which highlights "math questions", I don't know how that would ever be implemented, and it is kind of not the purpose of chat, anyway. :/

Jakobian

yeah, something like that

cardinality is the measure of how large is your set. If two sets have the same cardinality, that means there is a bijection between one and the other.
$\aleph_1$ is the smallest cardinal strictly larger than $\aleph_0$. It's not necessarily the cardinality of continuum (in usual axiomatic take to math which is ZFC, this is known as continuum hypothesis, and it can't be proven or disproven).

Continuum here means the cardinality of real numbers

SHASHAANK B.H.

13:44

Does the function ln(sinx) satisfy Rolle's theorem condition in the interval [pi/6,5pi/6]?

I think yes If not please let me know

Jakobian

@SHASHAANKB.H. It's not defined for $x = \pi$

SHASHAANK B.H.

That's not in the interval in the question [pi/6 to 5pi/6]

Jakobian

Yeah, my bad

Then yeah, it satisfies Rolle's theorem

SHASHAANK B.H.

Thank you

Does the RHL equals LHL for limit x-->0 x^3

I meant the function x^3

Jakobian

right-side limit and left-side limit?

as x goes to 0

SHASHAANK B.H.

13:52

does limit exist in the neighbourhood of zero for x^3

yes @Jakobian

Jakobian

we say, at 0

yes, the function is continuous on $\mathbb{R}$

so $\lim_{x\to t} x^3 = t^3$ for all $t\in\mathbb{R}$

SHASHAANK B.H.

But for a tiny number less than zero its cube would be negative and a tiny number above zero the cube would be positive how will they be equal @Jakobian

@Jakobian ohh

Jakobian

It would be negative/positive but the limit is still 0

SHASHAANK B.H.

Oh ok..

Jakobian

Yeah, because negative numbers and positive numbers, you're right they don't have common points

but with negative numbers you can approach 0 in the limit, and same with positive ones

SHASHAANK B.H.

13:56

so you say that the cubes get closer to zero on a number line?

Jakobian

yes

SHASHAANK B.H.

Hmm...

Jakobian

If $|x|$ is small, then $|x^3| = |x|^3 \leq |x|$ for all $|x|\leq 1$ so it's even smaller.

SHASHAANK B.H.

14:15

Thank you @Jakobian

SHASHAANK B.H.

14:29

Oh yeah....

How to prove if a function is one to one( for Cauchy's theorem) f(a)not equal to f(b)

eq let's have a function x^2-2x+3 prove that in [1,4] f(a) not equal to f(b)

In Cauchy's theorem can g(x) be many to one ( I have already given the statement above )

leslie townes

this might be overkill but on that interval you could certainly use the MVT.

SHASHAANK B.H.

mean value theorem?

MVT?

Can't I use x1 and x2 and prove that equality wouldn't be possible for f(x)

leslie townes

f(b) - f(a) is f'(something) times (b - a) and here f' is nonzero on the interior of that interval

SHASHAANK B.H.

Yeah

That was Lagrange's theorem ryt?

Do the functions f(x)= e^x and g(x) =x^2/1+x^2 satisfy the conditions of the Cauchy theorem in [-3,3]

How do you solve f'(a) not equals f'(b) in that interval

That becomes more lengthy :(

is there a shorter way of proving it?

Koro

14:50

hi leslie

@SHASHAANKB.H. what is g(x)? Did you mean $g(x)= x^2 +\frac 1{x^2}$?

SHASHAANK B.H.

Let f(x) and g(x) be two functions continuous in the interval [a,b] and have finite derivatives at all interior points of the interval. If these derivatives do not vanish simultaneously and f(a) not equal to f(b) the there exists a c belonging to (a,b) such that g(b)-g(a)/f(b)-f(a)=g'(c)/f'(c)
This is Cauchy's theorem

g(x) is defined here

Thorgott

15:13

@onepotatotwopotato not sure what the equivalence class on the right is supposed to be $f\circ i\colon Z_p(X;R)\rightarrow R$ factors over the quotient $Z_p(X;R)\rightarrow H_p(X;R)$ to yield the desired map

@onepotatotwopotato yeah, that's the correct argument. to be precise, convince yourself that you understand at which part $f$ vanishing on $Z_p$ and at which part $R$ being a field are used in the proof

SHASHAANK B.H.

Using Lagrange theorem can someone help me estimate the value of ln(1+e)

Also using Lagrange theorem please give me some ideas on how to prove x/1+x<ln(1+x)<x at x>0

I can apply L Hospital's rule only if the function is differentiable in a certain neighbourhood of a and at 'a' itself right? also Limx-->a f(x)=0/0 or oo/oo right?

Adil Mohammed

15:30

Hyello

Quick question, what's Boolean algebra for?

I have a set of formulas that are applicable for a,b,c... in set B. What are the elements of this set B? Are they binary numbers?

SHASHAANK B.H.

I think that deals with Binary

Adil Mohammed

Ah so set B is the set of all Binary numbers?

SHASHAANK B.H.

16:23

Is ln(0) an indeterminate form?

Bob

Hi

is this question off topic?

-1

Q: What is the minimum net worth of a family needed to be in the Top 1%?

Problem: a) How large does a family's net worth have to be to be in the top $1$ percent in 2019? b) a) How large does a family's net worth have to be to be in the top $2$ percent in 2019? Answer: I will work in units of thousands of dollars. My answer is based upon data from the following website...

statistics

shintuku

i think since your title doesn't look mathematical/statistics enough someone might have downvoted

but your content is not off-topic

Bob

can you suggest a better title?

shintuku

Calculating the minimum net worth of a family needed to be in the Top 1% using insert theoretical name of what you're doing here

even more precise: Unable to find my error in calculation of the minimum net worth of a family needed to be in the Top 1% by using insert theoretical name here

SHASHAANK B.H.

limx-->0 ln(x) tends to?

limx-->0 x^00 is?

shintuku

16:30

@Bob yes the title you just updated to is good

Bob

@shintuku Do you think yours is better?

shintuku

yours is good, but it is even more efficient if you have the space to add, what exactly is your problem in the title. e.g., in this case, the fact that you need help to find a mistake in your calculations

Bob

I will go with yours

thanks

shintuku

so people know you're lost and have done work but can't figure out by yourself, without giving the sense that you're expecting an answer handout

these sort of questions, e.g., "help me with my calculations, i don't know where i've gone wrong", are necessary to newbies but some assholes always think they're not precise enough

it is hard however to evaluate whether someone has put enough work on these

anyways, your question is fine with the updated title

Bob

@shintuku thanks

shintuku

16:34

np

Bob

@shintuku bye

shintuku

good luck!

leslie townes

22:58

well let's not all talk at once.

Xander Henderson

@leslietownes SHHH!

leslie townes

what's up, xander? any hot takes?

Xander Henderson

@leslietownes Epsilon-delta arguments are highly overrated.

Undergraduate analysis should be all about sequential limits.

Fewer quantifiers to phaff about with.

pfaff?

phfaff?

ffphaphphff?

How do you spell that?

leslie townes

beats me. don't drag me into your drama.

Xander Henderson

You asked, jerk! :(

leslie townes

23:08

it's tough. i think a lot of undergrads definitely do better with sequential stuff, but for teaching future teachers, i dunno. epsilon-delta can be given a day or two in a high school calculus class, there's a picture that goes with it.

harder to 'visualize' collections of all sequences convergent to a point.

Xander Henderson

Yeah, but why do we care at all about epsilon-delta?

You don't need it unless you are dealing with more complicated spaces.

leslie townes

because i guess i dispute that it's fewer stuff to deal with. conceptualizing the set of all sequences converging to 0 is a whole ton of mental stuff to deal with.

but epsilon and delta are just epsilon and delta.

ted's going to pop up and say i mean less stuff and not fewer stuff.

Xander Henderson

Epsilon and delta are both quantified over all positive real numbers.

And, again, I think that it is harder to work with two quantifiers (all epsilon, and there exists delta). And if you look at what calc students actually do when they want to compute a limit, they often just plug in a sequence of numbers into their calculator.

That is the underlying intuition that they seem to have. Why not follow that intuition as far as you can?

leslie townes

i dunno, man. when i taught this, i used sequences

kenneth ross, elementary analysis, the theory of calculus. it's not a superb book, but it is good.

it doesn't throw rudin-level hurdles in people's way.

Xander Henderson

Oh, you wanted a hot take? Baby Rudin isn't very good.

How is that for a hot take?

leslie townes

23:21

that's not the hottest of takes.

shintuku

@XanderHenderson hi pls give alternatives

if you will so shamelessly desecrate the only authority i know, what idol must i replace it with

Jakobian

@AdilMohammed complemented distributive lattice

Ah. You asked what is it for. Sorry

Well it's a very specific kind of lattice

It's completely proper to think about it in terms of subsets of some bigger set

We can define filters and ultrafiters on Boolean algebras

There's a lot of useful constructions associated with those

Xander Henderson

@shintuku Almost any other book on elementary real analysis.

leslie townes

shin for multivar any one of spivak's books is a better choice, for integration bartle has a good book. i do think rudin's treatments of sequences and series and riemann integration are pretty good.

Xander Henderson

If you want something that is of a similar vintage, I like Courant.

The five volume set by Stein and Shakarchi(sp?) is, I think, a reasonable option.

Tao's Epsilon of Room.

Royden and Fitzpatrick is okay, if a little wordy (personally, I like wordiness, but it is not everyone's cup of tea).

shintuku

23:39

for multivar i'm doing ted's

slowly but surely

i was thinking about the parts before the 9th chapter, or something, before the multivar kicks in

leslie townes

baby rudin ch 3-8 is great

shintuku

i'll check those out, thanks people

Xander Henderson

I was taught real analysis out of two books: one by Folland, and another by Dangello and Seyfried.

Neither is terribly remarkable, but Dangello and Seyfried's text is the only one I know of (off the top of my head) which includes a proof of the Riemann rearrangement theorem, which is one of my favorite results in undergraduate analysis.

So it has that going for it.

shintuku

i'm looking for Stein and Shakarchi 5 volumes

i wonder what you can fit in so many volumes

Xander Henderson

Undergrad real and complex, Fourier analysis, measure theory(?).

It is published by Princeton University Press, I think.

And all of the volumes are relatively short.

I think that the idea is that each book is approximately one semester's worth of material.

shintuku

23:45

Princeton Lectures in Analysis

The Princeton Lectures in Analysis is a series of four mathematics textbooks, each covering a different area of mathematical analysis. They were written by Elias M. Stein and Rami Shakarchi and published by Princeton University Press between 2003 and 2011. They are, in order, Fourier Analysis: An Introduction; Complex Analysis; Real Analysis: Measure Theory, Integration, and Hilbert Spaces; and Functional Analysis: Introduction to Further Topics in Analysis. Stein and Shakarchi wrote the books based on a sequence of intensive undergraduate courses Stein began teaching in the spring of 2000 ...

?

Xander Henderson

Yup, that's the set.

I guess there are only four volumes.

I thought there were five.

:/

Counting is hard. :(

shintuku

the 5th one is the secret volume

it contains the dark secrets of analysis

Xander Henderson

@shintuku Oh, the umbral calculus?

shintuku

yes

leslie townes

the fifth one is the necronomicon

Xander Henderson

23:47

@leslietownes Oh, yeah, Spivak is good. And I think that Apostol has a book on analysis... anything Apostol is likely to be alright.

@shintuku Also, "desecrate"? Please. And don't idolize people. Read widely.

shintuku

i don't like reading

i need arguments from authority

"so and so said such and such, thereby i am right"

simple, fast, efficient

the economy is going badly, need to make appropriate cuts

Xander Henderson

Then I suggest you choose a field other than mathematics.

Perhaps political science, with an eye towards a talking head position on Fox News?

leslie townes

they should put the net worth of the mathematician on the back cover of each book

i don't wanna learn from some guy who cant even get his paper together enough to afford a good car or decent jewelry

rudin had a frank lloyd wright house, that's a pretty good flex

Xander Henderson

@leslietownes That is a phenomenal flex.

shintuku

i would also like their lineage and their relative increase in real income relative to their forefathers

leslie townes

23:56

like, fuck what you think about chapter 9, i'm living out here

Transcript for

$Mathematics$ Mathematics