Mathematics - 2022-09-15

Not Tfue

1:10 AM

But can't correct that because needed 6 characters.

Ted Shifrin

1:28 AM

@NotTfue Fixed.

Not Tfue

2:14 AM

Nice

Akiva Weinberger

2:41 AM

$(a+b)(ax^2+by^2)-ab(x-y)^2=(ax+by)^2$

Consequence: if $(a,b)=1$, then $(a+b,ab)=1$

Ted Shifrin

Cute!

Of course, it’s immediate anyhow.

PM 2Ring

2:56 AM

Also see

Brahmagupta's identity

In algebra, Brahmagupta's identity says that, for given n {\displaystyle n} , the product of two numbers of the form a 2 + n b 2 {\displaystyle a^{2}+nb^{2}} is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically: (...

There are a few more complicated related identities.

Also related: en.wikipedia.org/wiki/Chakravala_method

Akiva Weinberger

@PM2Ring Oh cool that's like Brahmagupta–Fibonacci but better

PM 2Ring

Indeed!

It's kinda weird that the simpler version has the more complicated name.

Akiva Weinberger

I wonder if it's actually standard to call them like that or if it was just a choice Wikipedia made to differentiate them

PM 2Ring

I think it's standard. Or at least reasonably common in recent decades.

copper.hat

3:12 AM

How nice to have an identity named after oneself.

PM 2Ring

It's just another example of Stigler's law en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy

Not Tfue

is there a link to download intro to download intro to analysis by wade

copper.hat

I always think there is a certain hubris to assuming one person is the first to discover.

Akiva Weinberger

@NotTfue Almost certainly b-ok.cc or else libgen will have it

copper.hat

I love books. Pdfs (or whatever format) just are not the same for me. Generational I suppose.

Akiva Weinberger

3:19 AM

The difference is $$$

copper.hat

Except my Rockafellar's Convex Analysis is falling apart. Even Ted's Abstract Algebra has the spine broken, and that's a recent acquisition.

Akiva Weinberger

though Amazon has a used copy of that analysis book for $25

copper.hat

There is a 2nd hand bookstore in Berkeley (Moe's Books) that has a lot of great mathematical texts. That is where most of my acquisitions have come from.

Akiva Weinberger

If it's for a class, 100% do not buy. Fully not worth it

If it's for intellectual curiosity, go ahead

Not Tfue

@AkivaWeinberger Thanks Math pi*@*e

PM 2Ring

3:21 AM

@copper.hat I know you like Irish pipe music. Here's a bunch of excellent musicians, mostly Irish, and at least one Englishman (Danny Thompson on bass), with a few American guests: Jerry Douglas, Allison Kraus, Sarah Jarosz. youtu.be/bB1V_zZ83Lc

Akiva Weinberger

Look, if you're pirating a novel or a film, there's an actual ethics discussion to be had. Artists deserve pay for their work. If you're pirating a textbook, you're not gonna break the system more than it already is broken

Ted Shifrin

That’s crap.

@copper.hat sorry. The publishers have shit quality …

Akiva Weinberger

Oh? I thought in your case the publishers were being unreasonable in regards to pricing

copper.hat

@TedShifrin My fault entirely.

Ted Shifrin

I’ve had issues with all three of my publishers.

They’re universally unreasonable, but why does an artist deserve royalties and I don’t?

Akiva Weinberger

3:25 AM

'Cause artists don't get research grants

Ted Shifrin

Nor do most mathematicians, dumb ass.

copper.hat

@PM2Ring I like the voice around 23:45.

Ted Shifrin

I last had one when I was a postdoc. Nothing to do with my work writing excellent books.

Akiva Weinberger

Why do I routinely find math textbooks offered for free online by the author?

You never find novelists doing that

copper.hat

@PM2Ring I'm related through a brother's marriage to the lad on the right in youtube.com/watch?v=DNKCJ5iKQJY. He used to play with De Dannan.

Ted Shifrin

3:28 AM

You’re using this to justify your rude logic?

PM 2Ring

A popular novel, film or song may get to the top of the charts & rake in lots of cash for its creators & publishers. Good textbooks take a lot of skill & effort, but there's no way they can sell like popular art.

copper.hat

If I own a book I am OK with having the pdf.

Ted Shifrin

Because most math authors know they won’t make money from royalties given the world.

@copper.hat of course! Me too.

Akiva Weinberger

So you're not dependent on it in the same way.

copper.hat

The mathematics library in Berkeley was good, and my advisor had a good selection as well. Albeit he would stamp his name all over the book before lending it to me. Always felt a little slighted when he did that as I was far from the first student to borrow the books.

PM 2Ring

3:34 AM

@copper.hat Nice.

copper.hat

I had a thicker skin back then.

Akiva Weinberger

It's a moot point anyway 'cause the last book I pirated, I'm pretty sure I have university access to it anyway

'cause I have university access to everything in Springer

So I wouldn't need to pay the (checks) $160 it actually costs

(or $137 if I'm renting it!)

copper.hat

moot means something different in Ireland/UK.

Akiva Weinberger

I may just be using it wrong

copper.hat

No, that is how it is used in the Good Ole.

leslie townes

3:42 AM

ted is writing from his yacht that he bought with some of the proceeds from 'multivariable mathematics.'

PM 2Ring

Gold plated yacht

leslie townes

yes

PM 2Ring

Tolkien uses the old meaning of "moot" = "meeting" in LOTR.

The gathering of the ents is called the entmoot.

Akiva Weinberger

Correction: the last book I pirated *not for class

I think the last book I pirated for class was Classical Mechanics by Taylor, which I only need so I can do the exercises for homework

leslie townes

when the USPS lost about half of my math books i went on a pirating spree

it joker-ified me

PM 2Ring

4:05 AM

Hatcher has taken an interesting approach with The Topology of Numbers. He makes the PDF of it available while he's still working on it. So he gets free proof-reading & feedback. pi.math.cornell.edu/~hatcher/TN/TNpage.html

Ted Shifrin

I’m glad Hatcher is still alive and kicking. I thought I had heard he had cancer…. But maybe I’m confused.

leslie townes

i think i'd heard that too? although i don't remember where

happy for everyone to be alive and kicking

Nick

9:27 AM

https://qr.ae/pvUhlw
Here's a nice story I hope you folks may or may not find soothing.

Not Tfue

9:57 AM

@Nick Strange Story

Nick

@NotTfue True

Not Tfue

I think it is normal to forget calculus when you are doing other field of maths.

Jakobian

10:53 AM

Good they didn't ask about the integral of $\sin(x^2)$

Maybe the mathematician just didn't want to bother because he was thinking about something.

Jakobian

11:24 AM

@AkivaWeinberger I constantly pirate books. I like to read in English and in my country they'd cost me too much.

Koro

The movie 'the road' is amazing :-)

Jakobian

The last book I bought was concrete mathematics by Knuth etc.

I haven't read through it so kinda feels like a waste. It just got too hard for me in the middle of the book

I was reading about some kind of algorithm that determines something about binomial coefficients there. That's where I just dropped. And the difficulty of the exercises discouraged me

Koro

I know regular manifold but not manifold.

user193319

1:08 PM

The following curve passes through $(3,1)$. Use the local linearization of the curve to find the approximate value of $y$ at $x=2.8$: $2x^2 y + y = 2x + 13$.

I just want to make sure I understand this problem correctly. So, implicitly differentiate to get the slope of the tangent line at $(3,1)$, and then plug in $x=2.8$ into the tanget line to get the linear approximation. Is that right?

Koro

I think no. I think you're supposed to find error in y as follows $\delta y\approx \frac {dy}{dx} \delta x$. Here, you may take $\delta x=3-2.8=0.2$ etc.

Then $y$ at 2.8 should be $\approx 1+\delta y$

Koro

2:44 PM

Suppose that f is from $R^m\times R^n$ to $R^n$, suppose $\phi$ is a function from $R^m $ to $R^n$. How do I find total derivative of: $f(x,\phi(x))$?

Jakobian

3:05 PM

Using chain rule

This is composition $f\circ (\text{Id}\times \phi)$

Koro

@Jakobian: How does Id $\times \phi$ make sense?

Jakobian

Chain rule should give $$D_x(f\circ (\text{Id}\times \phi)) = D_{(x, \phi(x))}(f) \circ \begin{bmatrix}
\text{Id} & 0 \\ 0 & D_x(\phi)
\end{bmatrix}$$ if I'm not mistaken

I am mistaken, the matrix on the right should look a little different

Thorgott

3:23 PM

it's $(\mathrm{id},\phi)$, not $\mathrm{id}\times\phi$

$x\mapsto(x,\phi(x))$ versus $(x,y)\mapsto(x,\phi(y))$, to be clear

Koro

0

Q: How to find the total derivative of implicit function in implicit function theorem?

Suppose that $f:U(\subset\mathbb R^m\times \mathbb R^n)\to \mathbb R^n$, where $U$ is an open set, is a continuously differentiable function such that for some $(a,b)\in U, f(a,b)=0$ and that $D_2f(a,b)$ is invertible. ($D_2f(a,b)$ is the total derivative of $f|_{(\{a\}\times \mathbb R^n)\cap U}$...

Jakobian

Why does $\overline{\text{conv}(e_n)}$ has non-empty interior in $l^1$, where $e_n$ is the standard basis

Koro

@Thorgott I think that's a wrong definition.

Thorgott

3:42 PM

well, I disagree

Koro

what if I want to find $D_x f(x,y)$ where $y$ is non zero?

By restricting to $\mathbb R^2\times\{0\}$, isn't this case being left out?

Thorgott

it's all the same

I'm restricting the differential, not the function

Koro

Oh yes!! You're right.

Thorgott

as for the question, consider Jakobian's remark and my correction

Koro

The said remark is confusing and seems evasive.

in that it doesn't clarify its symbols - what is $fo(id,x)$?

how to compose a map with a pair of two maps? :(

or, $fo (id,\phi) (t)=f((id,\phi)(t))=f(t,\phi(t))$

:-)

Oh, I got it.

Thanks a lot @Jakobian et @Thorgott

Thorgott

4:04 PM

a map $X\rightarrow Y\times Z$ is the same thing as a pair of a map $X\rightarrow Y$ and a map $X\rightarrow Z$. that's why I chose that notation.

ZaWarudo

I know that, in integration by substitution for definite integrals, invertibility is required. For instance, if we substitute $t=\tan(x/2)$ in $\int_0^{2\pi} \frac{1}{5+4 \cos x}dx$ we get an integral from $0$ to $0$ which is clearly wrong because the integrand is continuous and positive in $[0,\pi]$. Considering $\int_0^{\pi} \sin^2(x)\cos x dx$, if we substitute $t=\cos x$ we get the correct result $0$ because again it is an integral from $0$ to $0$.

But this time cos is invertible in $[0,\pi]$, so this latter use of the substitution should be valid. So, the question is: if after a substitution I get and integral from $a$ to $a$ with $a \in \mathbb{R}$, can I conclude that the integral is $0$ if the substitution I made is invertible? Or the notation $\int_a^a$ is not rigorous after substitutions?

Alessandro Codenotti

4:20 PM

@Jakobian doesn't it contain a ball around 0?

Jakobian

@AlessandroCodenotti I don't think it contains $-re_1$ for any $r>0$ for example

Thorgott

@ZaWarudo well if you make an invertible substitution, the two boundaries after substitution won't be equal unless the two boundaries before substitution were already equal anyway. in that case, the integral is 0, yes.

Alessandro Codenotti

Ah yes that's annoying

Jakobian

What's weirder, in $l^2$ it's supposed to be false

robjohn

@ZaWarudo the integral on $[\pi,2\pi]$ equals that on $[0,\pi]$. Since $\tan(x/2)$ is injective on $[0,\pi]$, do your sub there and double.

Substitutions should be invertible on the domain of integration. You may have to break up the domain to get invertibility on each piece.

Jakobian

4:35 PM

Every $c\in \overline{\text{conv}}(e_n)$ can be written as $c = \sum \alpha_n e_n$ where $\alpha_n \geq 0$. If we assume that there is a subsequence $\alpha_{n_k} > 0$ then since $\alpha_{n_k}\to 0$ we have $c-2\alpha_{n_k}e_{n_k}\notin \overline{\text{conv}}(e_n)$ but $c-2\alpha_{n_k}e_{n_k}\to c$. But $\text{int}\text{conv}(c_n) = \emptyset$.

I think this proves that the interior is actually empty? @AlessandroCodenotti

So basically, c is not an interior point, so interior points must be points of the convex hull. But that one is trivially empty

This was supposed to be an example when $e_n\to 0$ in $w^*$ topology but $\text{int }\overline{\text{conv}}(e_n) \neq\emptyset$. So I guess they really do want me to take $\pm e_n$ instead.

I think I'll be able to handle this one

I got the idea from this answer math.stackexchange.com/a/4426014/476484

Yeah. Now pretty trivially the unit ball in $l^1$ is contained in this closed convex hull

leslie townes

good enough to steal ideas from, but not good enough to upvote, apparently

Jakobian

Because it's not a correct argument.

leslie townes

:)

Akiva Weinberger

4:55 PM

In the class I'm grading, one of the homework problems was to describe the solution sets of Re(1/z)=C and Im(1/z)=C

I took off 0.5 points (out of 10 for that question, 60 for the whole p-set) from anyone who didn't exclude the origin

(The solution is punctured circles and, if C=0, punctured lines)

That got a lot of people

I didn't take off from anyone who wrote "radius $\frac1{2C}$" instead of "radius $\frac1{2|C|}$", because I'm nice like that

leslie townes

i might have taken off more.

Thorgott

same

Akiva Weinberger

I asked the professor, he seemed OK with it

leslie townes

fractional points, no thanks.

Akiva Weinberger

One student simplified it to $x^2+y^2-x/C=0$ and concluded that it was a circle "centered at $x/C$"

This student seemed particularly weak on their algebraic skills. They also had concerning answers for other questions

But he's an outlier; everyone else seems to mostly get the material

leslie townes

5:03 PM

that is the general problem with letting minor things slide. eventually you find someone who illustrates the slippery slope by doing like 25 individually minor things in the span of a single problem, and also, not a single problem in the whole set is worked out correctly.

i would sometimes scold people without taking any points off. just so they knew i was watching them.

Akiva Weinberger

I texted the teacher about him

"I’ll take a look and refer to [the TA] as needed. Please keep track of all such students and notify [the TA] with a copy to me."

leslie townes

"keep track of all such students" your next text should be a photo, taken through bushes, of a bedroom in somebody's apartment with "he's here right now"

Thorgott

@leslietownes yeah, I do that a lot

Akiva Weinberger

(nervously clicking the arrow to see what the referent of Thorgott's comment was)

leslie townes

hahaa

123

5:26 PM

Hello World.

I have posted very simple and elementary question related to euclidean synthetic geometry. But i no satisfactory answer i found.

robjohn

@Jakobian perhaps I’m looking at the wrong thing; what is an incorrect argument?

leslie townes

123: under the usual set of axioms those would be three names for the same thing, not three distinct things

123

@leslietownes Thank you very much.. It means all coincident lines are thought of one line.

Using this analogy two intersecting lines. The point of intersection is one point in common for both lines. Am i correct?

robjohn

@123 that’s sad. That’s all I can say from what you’ve said.

123

I am confused because incident geometry says each coincident lines or points are considered as different objects. Pls read this. en.wikipedia.org/wiki/Incidence_(geometry)

Incidence geometry

In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It...

Incidence structure

In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which points are on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane. Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine, projective, and Möbius...

Ted Shifrin

5:36 PM

@123 I don't even find the word "coincident" on that page. What are you talking about?

123

And the same topic incident structure/geometry says if we consider plane is a set of points then line in the plane considered as subset of plane , idea of containment is used. What is this. It is not clear. Pls help.

Ted Shifrin

Of course, a line is a set of points, points are the elements of the plane, and so the line is a subset of the plane. You can think of lines as abstract entities, but saying a point is incident with the line is saying that you can think of the line as being the union of all the points to which it is incident.

Your tone "What is this. Pls help." is unnecessary and annoying, by the way.

123

@TedShifrin Hello Sir. They give same examples as intersecting lines, Point on line, concurrent point. According to your comment i feel you are saying coincident and incident are different topics. Am i correct sir.

Ted Shifrin

As I said, they never use the word coincident.

123

@TedShifrin Sir , i am really sorry. My english is weak. So pls don't take it this way. My question is very humble.

Ted Shifrin

5:40 PM

In general, two lines are coincident if they are the same. Two lines are incident if they have a point in common.

123

@TedShifrin Pls suggest the topic with link. So i can read the topic according to my question. I am trying to find the topic since more than 3 weeks than finally i asked question here.

Ted Shifrin

I don't know links. An excellent, but sophisticated book on geometry is Dan Pedoe's Geometry: A Comprehensive Course. There's always Euclid.

I don't know what the topic is here.

123

@TedShifrin Thank you . I definitely download this book now.

Ted Shifrin

It seems like the Wikipedia page you linked has lots of references. I know none of them. I am a geometer, but not this sort of geometer.

123

After discussing here. I come to conclusion that coincident and incident are two different topics.

robjohn

5:57 PM

They are. Look up both on Wikipedia

123

Last question. Angle construction postulate restrict angle between 0 to 180. But angles are more than 180 and direction of angle clockwise (negative) and anticlockwise (positive) what postulate/idea allow us angle more than 180 degree. Also it is used in concave polygon.

Ted Shifrin

You can construct 240° by constructing 120°.

123

@TedShifrin you are right sir, but in concave polygon when we construct 240 degree the approach is same?

Ted Shifrin

What is your problem with it?

123

@TedShifrin My confusion is. When we defining angle on circle we have angle more than 180 degrees. But angle construction postulate allows only angle between 0 to 180. There should be another postulate which extend the angle construct more than 180.

Ted Shifrin

6:10 PM

I don't know anything about this. So I withdraw from the discussion.

123

@TedShifrin Thank you sir. Your and other members answers really helped me. I was searching for the answer for long time. Once again thank you all.

Jakobian

@robjohn Of the answer I got the idea from

123

6:28 PM

I don't find any wikipedia page for coincident in euclidean synthetic geometry. But other websites have this topic. Does coincident (points / lines) are part of geometry or not?

robjohn

6:41 PM

@123 Coincident Lines, as in the lines coincide or are the same. Compare with Incidence

Akiva Weinberger

Messing around

desmos.com/calculator/6sofv7cxog

The red curves are the loci of the local maxima/minima

123

@robjohn Hello Sir. I have read both topics and i compared both. I thought after coincidence and incidence are same after comparison. But incidence geometry talk about heterogenous relation means points, lines, plane all are different objects. If two lines l and m intersect at point O, these are three different objects.

robjohn

As Ted said, Coincidence means that two things consist of the same points, whereas Incidence means that some part of the two things are the same, e.g. a common point or a common line.

123

But from euclidean geometry , euclidean plane it made up points. Line on the plane is a subset of points of plane. According to this idea two coincident lines means it is one line with two different names.

Here two different approach which confuse me.

@robjohn Ookay sir. It means incident does not have overlap two lines on top of each other?

robjohn

correct

if two lines intersect, the point of intersection is a point of incidence (they both contain that point).

123

6:51 PM

Does incident part of eucildean geometry topic? If it is part of it, why we don't use axioms of euclidean geometry also for intersection of two lines. Because same approach also support intersection of two lines.

@robjohn Yes i agree. My confusion is that incidence talk about hetergenous relation. Means points and lines are different objects. It means two intersecting lines , point of incidence is third object.

From euclidean geometry axiom. point of intersection of two intersecting lines is one point in common for both lines. Means lines , points all are subset of points of plane.

robjohn

If you are talking about Incidence, that is part of geometry. It is usually approached from a more basic point of view, that is, removing many of the restrictions of Euclidean geometry, but still getting some useful results that will continue to be true when we add the rest of the restrictions of Euclidean geometry.

Sort of like topology looks at geometry from a less restrictive point of view.

123

@robjohn Thanks for much satisfactory answer.

Akiva Weinberger

One of the homework problems in the thing I'm grading is to show that the sum of the nth roots of unity are zero

123

For elementary geometry can we only use axioms for euclidean geometry. Means points lines are subset of plane? Using same idea we can say point of intersection of two intersecting lines is one point not two?

Akiva Weinberger

The question did not specify that $n>1$

I really want to give a bonus half point to the people who pointed it out but I'm not going to

robjohn

6:59 PM

@123 it is one point, but that point belongs to two lines.

Akiva Weinberger

I'll just write "Good observation that this fails for $n=1$" in the (digital) margin

(they were submitted online)

robjohn

@AkivaWeinberger aww...

123

@robjohn Okay sir. . . Thank you :)

robjohn

@AkivaWeinberger a pat on the back means so much more than a better grade ;-p

Akiva Weinberger

Also, no one is doing it through my preferred method (Vieta's formula for the sum of the roots of a polynomial: they're roots of $z^n-1$ and the coefficient of $z^{n-1}$ is zero)

123

7:02 PM

@robjohn Last it means three coincident lines AB , AC , BC from three collinear points belong to one line with three different names?

robjohn

@AkivaWeinberger Oh, that is how I was thinking of it. Are they using $e^{2\pi ik/n}$?

@123 If the three lines are coincident, they contain the same points; i.e. they are the same line. The points A, B, and C may be different, but they are colinear (they belong to the same line).

123

@robjohn It means we are just given different names of one line. Because they are coincident.

Akiva Weinberger

@robjohn Mainly they're using geometric series

robjohn

@123 The lines are being determined by possibly different pairs of points, but the lines consist of the same points.

Akiva Weinberger

Occasionally they're doing $\zeta(1+\zeta+\cdots+\zeta^{n-1})=(\zeta+\zeta^2+\cdots+1)$

so $\zeta S=S$ so $S=0$

(unless $\zeta=e^{2\pi i/n}=1$ (which happens when $n=1$), as some but not all pointed out)

robjohn

7:08 PM

That is a decent approach, as well.

But pointing out that $n=1$ is an exception is worth something.

123

@robjohn Thank you sir.

robjohn

you're welcome

Akiva Weinberger

From one student: "For fixed $n$, let $z$ be the first root of unity; that is, $z=e^{2\pi i/n}$. So then each root of unity is of the form $z^i$ for $i\in\{0\mathbin{..}n-1\}$."

My note: "Why would you use $i$ to mean something other than the imaginary unit in a complex analysis class?!"

robjohn

$i$ck

which of those $n$ values of $i$ are used in $z=e^{2\pi i/n}$?

leslie townes

conway, functions of one complex variable i

robjohn

7:17 PM

@leslietownes so make $\gamma$ an arbitrarily large circle, and $f(z)=z^n-1$

as long as $n\gt1$, the integral along the circle vanishes as the radius increases.

There again, we get the $n=1$ exception

Akiva Weinberger

Test $[a\mathrel{\ldotp\ldotp}b]$

Test $[a\mathbin{..}b]$

leslie townes

yes. i was just pointing out a use of i as a sum index in a complex analysis book

Akiva Weinberger

Huh, the spacing is slightly different

(Knuth recommends using .. to mean integer ranges)

robjohn

@leslietownes Oh! I completely missed that.

@AkivaWeinberger Yeah, I just updated an old meta post of mine regarding \mapsfrom and noticed that \mathrel does affect spacing.

leslie townes

rudin, real and complex analysis

i think the conclusion is that i is too useful to just be i

Thorgott

7:22 PM

@leslietownes in my experience, most people do that

robjohn

using $i$ as an index and as a complex unit make confusion that much more easy.

If there is no possibility of confusion, then it is probably okay.

Feynman_00

Tipical QM notation $\hat{x}\lvert x\rangle=x\lvert x\rangle$

leslie townes

not ahlfors apparently, he seems to be all about n and k

robjohn

I try to use $n$ and $k$ with an occasional $j$, but only use $i$ as an index if the question uses it or I run out of indices and the problem is not one about complex numbers.

Akiva Weinberger

$\newcommand{\mapsfrom}{\mathrel{\style{display:inline-block; transform:scale(-1,1);}{\mapsto}}}$ Test

$y\mapsfrom x$

Hah, can't believe I never noticed that was missing

Thorgott

7:30 PM

huybrechts does this the entire time

Akiva Weinberger

$(\cos(2\pi\rm{i}_{\textit{unit}}{\rm i}_{\textit{index}}/n)+{\rm i}_{\textit{unit}}\sin(2\pi{\rm i}_{\textit{unit}}{\rm i}_{\textit{index}}/n)$

leslie townes

every time that someone writes $\sqrt{-1}$ i die a little bit inside

Akiva Weinberger

What about $\Bbb Z[\sqrt{-5}]$

leslie townes

that's OK

robjohn

$\sqrt{-\frac1{12}}$

I can kill two birds

leslie townes

7:33 PM

goofballs

Akiva Weinberger

$\displaystyle\sqrt{\Sigma\Bbb N}$?

robjohn

indeed

Feynman_00

@robjohn Sometimes I run out of "nice" indexes in differential geometry and I have to resort to Greek letters, I'd like to avoid mixing those D:

Akiva Weinberger

What do I do with the students who wrote $z=1^{1/n}=e^{i\frac{2\pi k}n}$

robjohn

@AkivaWeinberger at least they defined which $1^{1/n}$ they meant...

@Feynman_00 I pretty much only use Greek indices for multi-indices.

Akiva Weinberger

7:36 PM

Time derivative of iota is the imaginary unit

$\dot\iota=i$

robjohn

ouch

Feynman_00

@AkivaWeinberger This one is gold

PM 2Ring

TIL that LaTeX has a dotless i for those who prefervit for the imaginary unit.

Akiva Weinberger

(I'm taking a physics class, I have to get used to the dot notation)

PM 2Ring

Was it $\iota$ (\iota) or $\imath$ (\imath)? ;) LaTeX, at the very least, clearly recognizes the latter as an alternative to $i$ :) — Danu ♦ Dec 4, 2016 at 10:33

Feynman_00

7:37 PM

The $\Delta$ operator is not scalar as triangles are not invariant under rotations

Ted Shifrin

@Thorgott Just use different indices and/or write $\sqrt{-1}$. Grr.

Akiva Weinberger

Yeah that's how you write $\hat\imath$ if you don't want $\hat i$ @PM2Ring

for the unit vector $(1,0,0)$

Ted Shifrin

Mathematicians often use the dot for time derivative.

PM 2Ring

Right. Hats & bars on dotted letters look terrible.

Akiva Weinberger

I suppose $î$ is also an option

except LaTeX isn't so good with unicode...

Feynman_00

7:38 PM

@AkivaWeinberger Math courses use it too sometimes

I'm afraid I don't understand the $1^\frac{1}{n}$ joke

robjohn

$\style{display:inline-block; transform:rotate(90deg)}{\Delta}$

2

@Feynman_00 definitely

Akiva Weinberger

Test: $\style{transform:rotate(90deg)}{\Delta}$

aw

robjohn

@AkivaWeinberger display:inline-block;

Akiva Weinberger

@Feynman_00 Raising to the 1/n power is multivalued in the complex field; if we need to fix a principle value, defining $1^{1/n}=1$ is the most sensible

so I don't like the people who wrote $1^{1/n}=(e^{2\pi ki})^{1/n}=e^{2\pi ki/n}$

Feynman_00

Ah alright you meant that the equal sign makes no sense

Akiva Weinberger

7:42 PM

Yeah

robjohn

and if you don't mean $1$, you need to say it, and they did.

Feynman_00

Thanks

Koro

I understand that there are various forms of Implicit function theorem like injective form and surjective form.

Akiva Weinberger

@robjohn for why

Feynman_00

Regarding the index mess above, I avoid mixing Greek and Latin indexes because in Physics Latin indexes are understood to range from 1 to 3 while Greek indexes range from 0 to 3

Of course in the context of relativity

robjohn

7:44 PM

@AkivaWeinberger otherwise it tries to format more than it should. display:inline-block; formats something that can be displayed inline. It's an HTML requirement.

Akiva Weinberger

$\style{display:inline-block; transform:rotate(45deg)}{\text{help}}$

robjohn

rotation does not increase the bounds, that needs to be done otherwise

Akiva Weinberger

♫ I need somebody

robjohn

$\style{display:inline-block; transform:rotate(45deg)}{\text{help}}\Rule{0em}{.9em}{.4em}$ $\leftarrow$ try selecting that

Thorgott

I only know one form of the implicit function theorem

Akiva Weinberger

7:48 PM

I only know it implicitly

$\style{display:inline-block; transform:rotate(45deg)}{\text{AAAAAAAA}}\Rule{0em}{.9em}{.4em}$

Koro

@robjohn it became a bottle!!

robjohn

@AkivaWeinberger you need bigger up and down extensions

PM 2Ring

A leading space improves it slightly, but the ascender of the 'h' still gets chopped.

$\style{display:inline-block; transform:rotate(45deg)}{\text{ help}}\Rule{0em}{.9em}{.5em}$

Akiva Weinberger

$\style{display:inline-block; transform:rotate(45deg)}{\text{AAAAAAAAAAA}}\Rule{0em}{2em}{2em}$

robjohn

@PM2Ring In my browser, the "h" looks fine. It has a sloped serif

Koro

7:52 PM

$\style{display:inline-block; transform:rotate(40deg)}{\text{ Oh my god }}\Rule{0em}{.9em}{.5em}$

robjohn

Crap. I've created monsters.

PM 2Ring

Ok. I'm on my phone, so it could be just a Samsung (or font) issue.

Akiva Weinberger

$\style{display:inline-block; transform:rotate(180deg)}{\text{g'day mates}}$

Hm

Can LaTeX do a for loop

Koro

Robjohn: selecting 'help' from your message gives a bottle shape.

robjohn

$\style{display:inline-block; font-family:fantasy}{\text{Don't see this font in chat often.}}$

Akiva Weinberger

7:54 PM

That's "fantasy"?

robjohn

@AkivaWeinberger it's not PostScript, unfortunately.

Akiva Weinberger

Wait woah

Koro

$\style{display:inline-block; transform:rotate(90deg)}{\text{ Implicit function theorem }}\Rule{0em}{.9em}{.5em}$

Akiva Weinberger

On my phone it's a fancy cursive; on my laptop it's Impact

robjohn

@AkivaWeinberger that is a generic font name, it gets filled in by whatever your system thinks is fantasy-like

Akiva Weinberger

7:56 PM

My phone's fantasies are of Narnia and Equestria; my laptop's fantasies are of a Cold War bunker

robjohn

This part of chat will look confusing to someone without ChatJax.

Koro

$\Huge{\ddot\frown}$

Akiva Weinberger

$\phantom{Here's a secret message just for them}$

robjohn

yep

PM 2Ring

PostScript is awesome, but it's too powerful because it's Turing-complete. So PDF was created as a safer limited dialect of PostScript.

geocalc33

7:59 PM

I'm on the clock

Koro

$\phantom{I didn't know this one.}$

robjohn

@geocalc33 I hope that is not uncomfortable. A cuckoo clock?

Akiva Weinberger

Was Turing complete?

PM 2Ring

In PostScript each glyph of a font is a procedure, so it can have arbitrarily long runtime.

geocalc33

@robjohn employer clock

robjohn

8:01 PM

@Koro It is useful for aligning things

Akiva Weinberger

I suppose he was not, in the end.

robjohn

@AkivaWeinberger I guess one would say he is now.

PM 2Ring

Did Turing complete himself, or was he terminated by someone else? His mum thought his death was an accident, and never accepted that he committed suicide.

Andrew Hodges maintains an excellent website dedicated to Turing: turing.org.uk/index.html

geocalc33

Are there existence problems for foliations that specify topological or geometric data on the leaves? For example, "does there exist a foliation of $X$ whose leaves are all homeomorphic to $Y.$"

Koro

8:37 PM

Let $U\subset R^{m+n}$ be an open set, $(0, 0) \in U$ and $f: \mathbb R^{m+n}\to \mathbb R^n$ $C^1$-map such that f(0; 0) = 0 and Df(0,0) is surjective. Then
there is an open neighbourhood V of (0, 0) and a diffeomorphism $G : V \to G(V )
\subset R^{m+n}$ such that
(1) G(0; 0) = (0; 0) and
(2) foG(x; y) = y (that is, foG is a projection map).

Does this statement make sense?

I don't understand why this follows from Implicit function theorem.

robjohn

What exactly does $Df(0,0)$ mean? With respect to what is the derivative taken?

Is it an $n\times (m+n)$ matrix?

Koro

8:58 PM

@robjohn It is the total derivative of f at (0,0).

@robjohn yes, the matrix of Df(0,0) is an $n\times (n+m)$ matrix.

robjohn

so how is surjectivity defined there.

Koro

rank of Df(0,0) = n

Df(0,0): $\mathbb R^{n+m}\to R^n$ is a linear map.

robjohn

Yes.

Koro

I'm writing up my question post to detail my confusion.

As I'm new to implicit/inverse theorems, my question may sound silly.

robjohn

Isn't that the statement of the inverse or implicit function theorem?

Ted Shifrin

9:06 PM

No, it’s not. But just apply the inverse fn thm as in the proof of the implicit fn thm.

robjohn

Okay. I need to look those up again to see where I'm jumping.

Ted Shifrin

implicit gives level set as a graph

robjohn

Ah, that reconciles the difference in dimension.

Koro

9:19 PM

0

Q: Help in understanding a **terse** proof of an alternate version of Implicit function theorem

Statement: Let $U\subset R^{m+n}$ be an open set, $a\in U$ and $f: \mathbb R^{m+n}\to \mathbb R^n$ be a $C^1$-map such that $f(a) = 0$ and $Df(a)$ is surjective. Then there exists a diffeomorphism $\phi: V\to \phi(V)$ of a neighbourhood $V$ of $0$ onto a neighbourhood of $a$ such that $f\phi(x_1,...

real-analysis multivariable-calculus

Derivative

does anyone here know how publishing a mathbit to the american mathematical monthly works?

Koro

@robjohn: I've posted my question.

@Thorgott This version (corollary?) of implicit function theorem is what I earlier referred to as surjective form of the implicit FT.

Ted Shifrin

10:19 PM

@Koro: Remember that row operations are multiplication on the left by elementary matrices and column operations are multiplication on the right by elementary matrices. The former means a change of basis in the codomain; the latter means a change of basis in the domain.

geocalc33

11:19 PM

Can anyone help with this vector function? $f(\mathbf x):=\bigg(\int k(\mathbf x)~d\mathbf x, \int \frac{y_1 k(\mathbf x)}{c_1}~d\mathbf x \bigg)$. I would like to try some examples for smooth functions $k \in K^2$ (class of smooth functions from $\Bbb R^2_+$ to $(0,1)$). And, considering the set of all 2-dim. non-negative vectors $\mathbf x:=(x_1,x_2)^⊤.$

Gwyn

11:57 PM

For $f: \mathbb{R} \to \mathbb{R}$, assume that $\lim_{x \to -\infty} f(x)=A$. Show that $\lim_{n \to \infty} f(x-n)=A$. I tried this: by hypothesis for any $\epsilon>0$ there exists $M<0$ such that for any $x \in \mathbb{R}$, $x<M \implies |f(x)-A|<\epsilon$. So, if I consider $N=\lceil{x-M\rceil}$, if $n \ge N$ it is $x-n\le x-N=x-\lceil{x-M\rceil}<x-(x-M)=M$, hence $|f(x-n)-A|<\epsilon$ and so $f(x-n) \to A$ as $n \to \infty$. Could this work?

leslie townes

looks ok to me. there is a missing quantifier (presumably the existence of the limit is to be shown for all x, which your argument does)

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