Mathematics - 2017-07-14 (page 1 of 4)

12:26 AM

Hi @Demonark

Glad your talk went decently. I warned you it was technical.

Thanks! And yeah, definitely

There was actually this one part of it that I only truly figured out because someone asked me then

I was like, yeah the symbol pushing checks out and whatnot, but someone asked a question of why we were allowed to take the orbit closure instead of just the orbit

CivilSigma

12:41 AM

Hello, can any one please help me verify if I solved an integral correctly ?

V dc/dt = QC_i - QC_e - kVC_e , were V,C_i and k are constant

the variable is Ce

You can enable mathjax if you use the link in the room desc

That's a good question, Demonark. What did you say?

hi Semiclassic

wonders what "greater company" means

chat.stackexchange.com/transcript/message/20671002#20671002 (hmm, so there is some patterns on determining the existence of a closed form for something that has no antirato)

But how to control the chaotic dynamics of nonlocal operators...

I mentioned that we didn't need $\omega$ to be a shift of $\xi$, just that the recurrence theorem and the fact that the distance between two elements being less than 1 implies that they agreed at the 0th position would suffice

Do that enough times and you get your arithmetic progressions, even if the other terms didn't match

Well, I'll be a better coach/critic when you get to diff geo :D

Of course, that might be when I'm traveling again.

12:54 AM

We've done about a week, so likely in 3 weeks we'll start diffgeo

Yeah, I'll leave on 8/16.

Hopefully that stuff is less technical?

Well, to me it is ... although there are a few sections that are heavy. It's all much lighter with differential forms as opposed to classical stuff.

I mean I'm alright with some technicality as long as it's possible to extract some kind of idea, but I'm not totally sure if that resonated all too well. For all this, all this, all this, all this, there exists this, this, and this

Ah, nice

There are pictures for most things in diff geo, but not all.

12:58 AM

What's an example in that latter category?

The non-pictures?

ya

Things like the Gauss and Codazzi equations. They're basically integrability conditions, but hard to give pictures per se.

hmm

(They come from structure equations of $SO(3)$ from the forms viewpoint, from equality of mixed third-order partials from the classical viewpoint.)

1:01 AM

first one sounds neat, second sounds tedious.

indeed.

Hmm, how would you contrast diffgeo to difftop?

(Also, general question to the chat, do you get the vibe that working in groups on math ends up being at least 75% banter?)

You need more discipline in the groups.

Rather different flavor, Demonark. I don't know how to answer briefly.

Hmm

And lol that's probably true. The monoids are hopeless but maybe the groups can be improved

smacks Demonark

1:11 AM

And who can tell about the groupoids?

yo chat

heya Eric — perhaps you'd like to weigh in on my comment to Semiclassic

Lol @Semi, Zorn vs Choice vs Well-Ordering reference?

heh, you caught me

the comment abt the Gauss-Mainardi-Codazzi eqs?

@Daminark when I work in groups it tends to be like at least 80% math

those i really do not think about in terms of a picture

1:17 AM

Oh, sick. Lol usually what happens with us is that we start strong and then get stuck or really don't feel like working out some very necessary details, a few snide remarks happen, memes all around

And that tends to continue until we just decide to get down to it or someone has divine revelation. Which takes quite some time

I think Demonark's crowd is mostly goof-offs.

I had problems with my Multivariable Math students ... Too often group work turned into copying from one person who had a solution and mostly socializing. I tried to make "rules."

Oh copying is not good

like you said i think of it as integrability conditions coming from the structure eqs of SO(3)

i think my crowd is just more serious than daminarks

well, more elementarily, integrability conditions come from equality of various orders of mixed partials.

yeah clairaut's theorem or w.e. it's called

1:28 AM

Likely. How many physics people do you guys have?

Only Stewart's book seems to have used that name. I have no idea what's accurate historically.

I've actually heard it claled that a lot

@Daminark idk more than a few

Sort of like the confusion between Poincaré's lemma and its converse. I don't know whom to trust.

ah yeah that deall

relevant, but not decisive: en.wikipedia.org/wiki/Symmetry_of_second_derivatives#History

1:31 AM

Hmm, half the time it starts when we turn to whoever the physics people are and say "Hey so we can we become physics majors and just handwave this stuff? Like look at the picture"

It's downhill from there

Of course, that article said "None of ... were ..." — so the bad grammar kills it for me.

Find a smaller group, Demonark.

To be fair, what else is there which we'd associate Clairaut with

this doesn't happen with the physics people i know @Daminark

There's a nice theorem about geodesics on surfaces of revolution, @Semiclassic.

@Semi there's Clairaut's relations with geodesics

1:34 AM

True.

that's good stuff

idt symmetry of mixed partials needs a name personally

nah

I think a lot more people are liable to see Clairaut in Stewart than in diff-geo, though, simply due to the size of the audiences involved.

that's true

of course ... but the source you cited suggested it was due to someone else.

1:36 AM

Yeah, that kinda undermines it.

ive never read Stewart

not a great book, Eric, but very popular ... as with most calc books

But, eh

Bernoulli, Euler, Cauchy, Lagrange, Schwarz...

i read one calc book (I think by larson and edwards or something) and that was waaaay back and then after i started spivak i realized i learned like nothing from reading it lol

To me it comes down to the question of what the purpose of associating names to theorems, given how many of them go back to Euler.

1:39 AM

BTW, ERic, Spivak still hasn't answered my email. Sigh.

I did Stewart for my undergrad.

:(

Stewart ripped off a lot from Edwards & Penney.

But I also have no point of comparison.

Yeah don't use Stewart everyone should just use Buck /s

1:41 AM

yes, Buck that trend :P

idk how one would write a good calc book

I do not like Buck at all.

Bye for now.

i'm out for now too

bye Ted and Semi

Lol I don't either, it was annoying

See you guys!

@EricSilva shrugs

1:47 AM

having a cold suuuuucks

Oh meh, hope you feel better

I couldn't give my lecture bc i lost my voice :/ I was looking forward to it, I prepared some cool stuff

Is it gonna be postponed or will you just post the lecture notes and move on?

it's postponed

Ah, at least we've got that

Actually did anyone who was set to present in the bootcamp ever get sick last year? If so, what happened?

1:54 AM

one time no one prepared anything i think

(Also woo next week we're gonna get to the complex analysis)

havent you been doing stuff

I mean we've been doing chapter 1 on infinite products, integrals, all that

Yesterday's lecture was on Sterling's formula, for example

oh yeah

oh shit sterling's formula is cool

Actually tomorrow is when we'll get to chapter 2

1:59 AM

cool

i probably wont be able to go

unfortunately

i doubt ill feel better already

user356448

Hello

Hola

2:18 AM

Fair @Eric

Hey @Akiva

Cómo anda

How goes it

Muy bien, gracias, y tu?

Muy bien

Caminé mucho hoy

user356448

Anyone likes primes?

@user356448 I know little NT but I /like/ them for sure

@Akiva es una buena actividad

user356448

2:23 AM

I have a question

Does there exist a gemometric sequence with infinite sequence illiteration, that will never produce a prime?

@Daminark Sí

pero si Google Maps dice que va a tener 45 minutos, no tratá hacerlo en 35

@Akiva Lo siento si no hablo español bien, en el año pasado no hablé mucho en español. Pero tienes una oportunidad para corregirme, es bien si quieres practicar.

Yeah I was just saying that if Google Maps says it takes like 45 minutes to get somewhere, don't try to do it in 35

Running is also a good activity but I'm not very good at it

Entendí, la última cosa que dijo es en general

¿Alguna matemática interesante pasó?

2:29 AM

Yo tambien, pero no soy muy... atlético, necesito practicar más

@EricSilva Stirling is one of those results I liked until I used it so much that I got sick of it.

En la casa en que estoy viviendo hay una perra enferma

que hace pipí en todas partes

This paper might interest some here

An Editor Recalls Some Hopeless Papers

@Akiva :(

Also, I just saw someone use Axler's tendency to refer to rank-nullity as "the fundamental theorem of linear maps"

shrug

Hm, that's not a bad choice

if I had to give that name to some theorem

@user356448 I dunno what is illiteration

2:39 AM

The proof that $\Bbb Q(\alpha)=\Bbb Q[\alpha]$ for algebraic $\alpha$ goes through that theorem I think

Well no

Never mind

Actually, in funny things

Laci has these two theorems he refers to as the first and second miracles of linear algebra

First miracle is that if you take $k$ linearly independent vectors, each of which is in the span of some $m$ vectors, then $k\le m$

Second miracle is that row and column rank are equal

I'm actually blanking on how we prove row and column rank are equal

Agh, I should know this

Row and column operations preserve both row and column rank

I think morally it comes down to "you can do Gaussian elimination using either row ops or using column ops"

Oh right OK that makes sense

2:47 AM

So you can perform those operations until you get a matrix with at most one entry in each row and each column

All hail the elementary matrices

and Gaussian elimination

Wait what is Gaussian elimination anyway?

row reduction to echelon form

I've heard about it but never knew what theorem it was referring to

Oh that thing

Maybe I did hear it but have been suppressing that memory

Just messing with a matrix until it looks pretty

2:50 AM

I'm blanking a bit, now that I think of it.

elementary row operations basically amount to taking linear combinations of equations, and that obviously doesn't change the solution set

but what do elementary column operations correspond to?

linear substitutions of variables?

Row space is the orthogonal complement of the null space, right?

yes, DogAteMy

and vice versa

yes, change of basis in the domain, Semiclassic.

makes sense.

Oh OK so doesn't change the solution set either I guess, just changes what we call the stuff in it

it's all in my book, DogAteMy :P

row rank = dimension row space, column rank = dimension column space.

2:54 AM

Are you talking about the linear algebra one or the multivariable analysis one

Yes, DogAteMy ... in both :P

Oh whoops

The number of pivots determines both of those. So they're equal.

Right right

I can't remember what a pivot is, if I'm honest.

2:55 AM

Now that you mention the word "pivot" it's coming back to me

nonzero leading entry in a row of echelon form

gotcha.

How's your Spanish immersion/embedding going, DogAteMy?

I'm not quite sure if it self-intersects

but it's going well

I hope it's going smoothly, lest Ted's description not apply

2:56 AM

Would a Spanish immersion done according to best education standards and practices be a canonical immersion?

It's an undecidable problem, I'm sure

I always put a quote from Oscar Wilde on the syllabus of my differential topology course, Semiclassic ... because the Reverend Chausible referred to how immersion of adults was quite canonical.

lol

There's no algorithm that, given the list of students involved, always produces the optimal teaching strategy

well, optimal teaching is ill-defined, I guess

2:58 AM

Teaching is an undecidable problem

^I think that was the title of a paper I read once

Another good reason for me not to care about logic.

reasoning is interesting, logic is boring

Not like a rigorous paper :P

Just someone talking about their experience as a teacher

Wait what? Logic seems fantastic

^

3:00 AM

@Semiclassical what about logical reasoning then?

On that note ... bye for now.

cya

See you!

\o @nitsua60

@LasVegasRaiders o/

3:11 AM

Has your summer vacation started yet?

@LasVegasRaiders are you talking to me?

Yup

:-)

How do you know whether or not I get a summer vacation?

I thought you teach senior high school math?

Spywork

3:14 AM

^that too :-D

@LasVegasRaiders I do. I'm just curious how you knew that?

@AkivaWeinberger Nah. Spies don't get a good enough summer vacation =)

I think you mentioned it.

I was responding to your question on "How do you know I get summer vacation"

@AkivaWeinberger (I know--just riffing.)

(removed)

3:17 AM

(Oh, OK, I thought you misunderstood what I meant)

Okay. Fairy 'nuff.
Yes, summer vacation's started. I actually teach at a school that ends its year early enough that I've been on vacation for 5 weeks now =)

Nice.

@LasVegasRaiders The flip-side is teaching classes every Saturday of the school year.

Right, I was gonna say they must be gifted students.

Annelise Toft

sorry for the interruption but I'd like to ask something about an algebra question. If I have a proof that holds for Z, what changes do I have to do to make the proof work on euclidean domain?

3:21 AM

@LasVegasRaiders Nope, just captive!

@AnneliseToft could you give an example?

Annelise Toft

@LasVegasRaiders Yes, p is prime iff <p> is maximal

that is the exercise

I already have the proof in Z

A: Norms of Quadratic Rings in General and their Factoring

Go through the proof line-by-line, see if each step works for Euclidean domains

Annelise Toft

ok

Typhon

4:05 AM

1

This is not true in general. For example consider the case $a=0, b=4$, so $R = \Bbb Z[2i]$ $N(c+dx) = |c+2di|^2 = c^2+4d^2$. $x = 2i$ has norm $4$, $2+x = 2+2i$ has norm $8$, but there is no element of norm $2$ (because that would require $d=0$ and then $c^2=2$) However, your conjecture is tru...

@AkivaWeinberger mind explaining what they mean by integrally closed? They say the ring is integrally closed, but within what set? Itself?

4:17 AM

$P(X = x)=\binom{n}{k}p^{k}(1-p)^{n-k}=1 + (1 - 2p)^{n}$

Is that true?

none of the equalities are true

hmm, ok I have a question

I was asked to write down the probability that k errors occur in n bits using the binomial distribution, so I got this:

$P(X = x)=\binom{n}{k}p^{k}(1-p)^{n-k}$

just the normal binomial right?

should be, yeah.

The next part of the question asked for the probability that an even number of errors occurs so I got this:

$\frac{\binom{n}{k}p^{k}(1-p)^{n-k}}{2}$

Why?

4:24 AM

since the binomial distribution sums to one, and even number of errors is half of all errors that could occur divide by 2?

is that wrong?

well, for one: What's $k$ doing in there?

number of errors I was thinking

Sure, but you don't care about which particular number of errors you get.

All you care about is that it's an even number.

yeah, so k = 0, 2, 4, ...

i thought that would be half the numbers up to n

Sure, but you need to include all those possibilities.

4:26 AM

oh right, I need to sum them eh?

Right. And you wouldn't include a factor of 1/2 by hand.

${\binom{n}{2k}p^{2k}(1-p)^{n-2k}}$

Is it something more like tha?

Yeah, that'll do it

well the sum of that i mean

Right.

4:34 AM

from 0 up to n

or is it n/2?

Depends on whether $n$ is even or odd.

If it's even, then yeah, $k=n/2$

cool thanks for the help, ill go do some more work on it

mmkay

\o @MikeMiller

5:34 AM

@Abcd No, that's different, $2n\pi+$\pi$ means you only have odd multiples of $\pi$ added to $\frac{\pi}{2}$, $n \pi$ on the other hand means you add any multiple of $\pi$. Thus if you wrote $2n\pi+$\pi$ you actually missed out some solutions, instead of including both sets of solutions as you expect. You can easily check that by plotting those points on the sine curve and compare

you will find you missed out exactly half of the solutions

It seems that $a^2+b^2$ would never have a single factor of a prime in the form of $4n+3$

How's your fong bong stuff going?

yesterday, by Leaky Nun

@Secret and the results turned out to be fine, if you care enough

oops I must be asleep when you replied me

Nice

@Secret but you talked just afterwards

5:39 AM

yeah very sorry, I am sure my carelessness and missing messages will get better soon now that my calcualtions are fixed so my brain can stop worrying about it

you don't need to be sorry

I cannot wait to tell my supervisor the result he had been asking me to check. After 3 weeks of back and forth with technicians to fix that program, I finally get his question solved

5:58 AM

@Waiting @SimplyBeautifulArt @robjohn Here's an integral for you to play with that I saw in my dream last night:

$$\int \sin \theta \sin\left(\sin\left(\frac{\pi \tan \theta}{2}\right)\right)d\theta$$

Wolfram alpha is unable to find any solution in terms of its library of special functions. From the output of wolfram alpha which rewrote the integrand in terms of a difference, the hard bit of this integral is the following companion integral:

$$\int \cos \left(\theta - \sin \left(\frac{\pi}{2}\tan \theta\right)\right)d\theta$$

[Integral project: Preliminary] $\textbf{Definition:}$ Given an integral with some integrable integrand $f$

$$\int f(x) dx$$

the $\textbf{canonical integral}$ is the integral obtained after a combination of change of variables and rewriting of the integrand $f$ by the family of functional identities that each function of the integrand belongs to, such that the number of steps in evaluating the integral is minimised

$\textbf{Example:}$ The following integral: $$\int \frac{\ln 2}{1+W(u)}$$ can be converted into the canonical integral $$\int (e^{x \ln 2})' dx$$ by the change of variables $W(u)=x\ln 2$, which can be evaluated in one step

Currently open problems:
1. Conditions for the existence and uniqueness of canonical integral of some given integral

$\textbf{Definition:}$ Two integrals are said to be $\textbf{canonically equivalent}$ if they share at least one canonical integral

6:24 AM

Volterra's function

In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties: V is differentiable everywhere The derivative V ′ is bounded everywhere The derivative is not Riemann-integrable. == Definition and construction == The function is defined by making use of the Smith–Volterra–Cantor set and "copies" of the function defined by f ( x ) = x 2 sin ⁡ ( ...

O, so we can have directional things to happen, where we can differentiate something but cannot integrate it back

6:47 AM

The Story of the Equals Sign

6:57 AM

> A factore added to a quantitie of thryeye is equalle to a dyffyrynte factore frome whyche is takene awaye a quantitie of foure

why the hell is "three" spelled "thryeye"...

@LeakyNun Well, the same reason the rest of it looks terrible. Back then people spelled however they felt like

@TobiasKildetoft no, the other words conform in some manner to French orthography

having a silent "e" at the end

and "y" makes an "i" sound

but "thryeye"?

@LeakyNun Think of ye as a single vowel

@TobiasKildetoft it isn't

@LeakyNun That might be due to the transcription

7:01 AM

@TobiasKildetoft what transcription?

just like the letter that used to be pronounced as "th" tends to look like a "y"

from whereever this was originally written to a computer

they are still using the Latin alphabet

@LeakyNun Who?

@TobiasKildetoft writers of the Middle English

This is clearly transcribed from some old text

7:02 AM

the man who wrote the sentence above

Given that they had at least one more letter than we use now in English, I see no reason they could not have had more

@TobiasKildetoft they didn't.

@LeakyNun As I said, the letter that looks like a Y on many old signs is really a different letter that is pronounced like "th"

as in "Ye old inne"

I know this

hence they had one more letter

7:03 AM

but how can "ye" be a single vowel?

@TobiasKildetoft yes, they had the letter "thorn".

1 min ago, by Tobias Kildetoft

This is clearly transcribed from some old text

I think the text is probably Middle English, where Latin alphabet is used.

@LeakyNun Well, was it written ye in the actual physical text?

Also, it is entirely possible for a combination of two vowels to be put together in a way that is then considered a single vowel. Danish had this until like the 1940's

And it still occurs in names

@TobiasKildetoft e.g.?

The Danish letter å used to be written aa but treated as a single letter

alright, I do not know any tradition that considers "ye" to be a single vowel, to even double it to represent its longer version

English is a mixed up language :-)

7:33 AM

$\lim_{x \rightarrow 0} \frac{cos(x)}{x}$ does not exist i think?

its infinity right!

@BAYMAX No, it just does not exist

can I get a quick proof :)

@TobiasKildetoft

@BAYMAX You can make it as large as you like by having $x$ be close to $0$ and positive and same for as small as you want by making is negative

nice

but not with $\lim_{x \rightarrow 0}\frac{\sin(x)}{x}$

right, because there both numerator and denominator tend to $0$

7:48 AM

gotcha

Hey there people!

hi daminark

we the people!

Perturbative

8:06 AM

Howdy @Daminark

retractionwatch.com/2017/07/12/… Interesting story in itself. But especially noting that the paper has not been cited a single time in the past 16 years despite being published in Annals

3

If $(X,d)$ is a metric space then it cannot have exactly 3 dense subsets?

but $X$ can have exactly 4 dense subsets

I don't see any relation here?

To be fair this does feel almost random

I see that they must be a power of 2 :)

Oh that'd make sense actually