Mathematics - 2021-09-24 (page 1 of 2)

Avra

12:07 AM

@Semiclassical. So, what is the logic here please of changing functions repeatedly like writing $R(n) = S(2^n)$

Semiclassical

to make the argument simpler, probably

there's nothing stopping them from doing it

Avra

So they are assuming that input changes though or the form of input, does not that restricts our domain please?

Like when you are taking another form of input $2^n$ instead of original form $n$, this won't affect?

I see that only input form keeps changing

Are we saying that all original input can be formulated using new input form $2^n$?

I am not sure if I am beging correct herre please, so correct me if I am wrong

Semiclassical

if they're defining R(n)=S(2^n), then there's nothing much to say

though it's an odd chioce to make

like, you'd only be describing how S acts on powers of 2

Avra

So, my question is $R(n)$ takes one part of the domain of $S(n)$ is that correct please?

Semiclassical

right

can't say i know why they'd be doing it

Avra

12:12 AM

Exactly, so when you finally conclude that growth rate of the function is $O(n^2)$ presumably, they inferred that original function is also the same!

This is what confuses me. Have you got my question right please?

Semiclassical

i guess. i'll admit that i've actively avoided learning this stuff

Avra

:/

Thanks, I am just trying to understand logic behind it nothing more

Appreciated

Ted Shifrin

Your question had a typo or two to start .

Avra

@TedShifrin. Yes! I guess you refer to one of them which is $(1/2) R(n / 2) + (1/2) n

$

$$
R(n) = S(2^n)\\

≈ (1/2)S(√2n) + (1/2) lg 2^n\\

= (1/2)S(2^{n/2}) + (1/2) n\\

= (1/2) R(n / 2) + (1/2) n\\
$$

Should not the last equation be $=(1/2)R(n^{1/2})+(1/2)n$

Akiva Weinberger

1:33 AM

Fun fact: In any solved Sudoku, the digits in the blue region are the same as the digits in the orange region, just rearranged

(if you watch a particular YouTube channel you will be very familiar with this)

For example, in that example, there are two 5s in the blue and two 5s in the orange.

Puzzle: prove why

(Reminder that the rules are you can't repeat a digit in a row, column, or 3x3 box)

For what it's worth, the established convention for referring to a particular cell in writing is for example "r3c5" (i.e row 3 (third from top) column 5) which in this case is a 4

Boxes are numbered so box 1 is top left, box 3 is top right, box 4 is left middle, etc

Ted Shifrin

Just shows I’m not observant or curious enough. I do Sudoku every day.

leslie townes

the digits in the first row are also the digits in the 7th row, just rearranged

i rarely do sudokus but the hard ones are really, really hard.

Akiva Weinberger

@leslietownes Also the digits in row 1 and column 7

leslie townes

i call this the "1-7 principle." it's my secret weapon for solving sudokus.

Ted Shifrin

Ah, all matrices are symmetric.

leslie townes

1:38 AM

or sudoku, as the case may be.

Akiva Weinberger

1:52 AM

@leslietownes (Nudge: row 1 and col 7 intersect. Can you get a smaller pair that are the same?)

Ryan Unger

2:29 AM

@TedShifrin this is a nice theorem

Xander Henderson

@TedShifrin I must be even less observant / curious. I generally hate puzzles, and don't have the patience for them.

Ted Shifrin

2:45 AM

I started doing Sudoku when it was clear my mom had Alzheimers.

Akiva Weinberger

I highly recommend the YouTube channel Cracking The Cryptic, in which two British people film themselves solving sudokus

(They prove the thing I mentioned in quite a few of their videos, though, so potential spoilers)

@RyanUnger All matrices are their transpose, up to rearranging their elements

leslie townes

3:20 AM

my mother-in-law does sudoku for a similar reason, ted. although, she doesn't like challenging ones, which would seem to defeat the purpose. like doing the USA today crossword puzzles. what's a three letter word for "thing that meows" starting with C and ending with T?

Ted Shifrin

Munchkin doesn’t fit.

leslie townes

i picked her up from day care in my wife's car today, which really bothered her. she kept asking why i wasn't in the passenger seat. "i need to sit here to drive" didn't satisfy her.

Ted Shifrin

3:39 AM

No fair using the wrong car!

leslie townes

that was about half of the drive home. she remembered that i'd driven her in that car once before, to a checkup. i don't like that she's developing a memory. eventually she's going to catch me out on something.

Ted Shifrin

Guilty as charged.

Euler 2

3:59 AM

hello

Koro

@Euler2 Previet :)

Euler 2

I hate youtube

leslie townes

4:14 AM

my daughter likes watching cat videos on youtube. we have to ration them out. no more than two or three a day.

Akiva Weinberger

C for cat is quite large in YouTube's card index

K for kitten outranks it, but…

Koro

Leslie, I'm still stuck on yesterday's question. :(

leslie townes

koro, in the moment i saw some of that notation it looks like you might be blending matrices whose entries are scalars and matrices whose entries are vectors. this can be done but only carefully. it might be simpler to think purely in terms of boxes of numbers.

Koro

I'll try that question a bit more first.

leslie townes

so you'll need some notation like, if B = {b_1, ..., b_n} is a basis for a space V and v is in V, then [v]_B is the n-tuple of scalars c_i for which v = sum c_i b_i.

Koro

4:21 AM

@leslietownes yes, I'd realized that I had mixed up vectors and scalars. :(

leslie townes

when you've got multiple bases floating around it can be confusing to just stick something like "Tv" (T an operator on v, v a vector) into a matrix. what Tv is isn't going to change. the tuple of scalars will and the [ ]_B notation reflects that

the change of basis theorem is something like, if B = (v_1, ..., v_n) is a basis for B and B' = (b_1, ..., b_n) is a different basis and v is in V, then [v]_{B'} = [[b_1]_B | [b_2]_B | ... | [b_n]_B]^{-1} [v]_B. don't take my word for it, but it's something like that. might be stuffed into a textbook somewhere.

the 'row/column operations' perspective is just such a 'box of numbers' perspective that i think some kind of number-focused, as opposed to vector-focused, notation, is helpful.

Koro

@leslietownes Leslie, I know this theorem. I read it recently in Hoffman and Kunze's but it was not there in 3.C section (that's where I found the exercise problem) of LADR. I'll try with this. Thanks a lot.

P-addict

4:44 AM

hi, i have a question about vector bundles. the definition i most commonly see is that a real vector bundle on a topological space $X$ is a surjection $\pi:T->X$ with a finite dimensional vector space on each fiber and the local homeomorphism condition. i guess i'm wondering why we need the vector space condition given the local homeomorphism - does this not already attach a vector space to each point? i'm not sure why this is confusing me so much

huh, did i really just use $->$ instead of $\rightarrow$? lol

Node.JS

6:53 AM

I am wondering how to combine set-builder notation with existential and universal quantifers: $E = \{(v_i, v_j) \mid \forall X_i \in p (\exists p' (\forall v_i, v_j \in R(p') (\exists r (\texttt{LHS($p$)} = X_i \wedge r \in R(p') \wedge \texttt{LHS($r$)} = v_j \wedge v_i \in \texttt{RHS($r$)} )) \}$

I asked it as a question: math.stackexchange.com/questions/4258887/…

PM 2Ring

7:15 AM

@Avra That's impossible. Given $T(n) = 2(T(\sqrt{n}))^2$, when $n$ is 0 or 1, $n=\sqrt n$ so $T(n) = T(\sqrt n)$. Then $T(n)=2T(n)^2$ has two solutions, 0 or $\frac12$.

There's another Latin square puzzle like Sudoku called KenKen that uses arithmetic, which makes it more interesting, IMHO. There are free KenKen apps, and sites with collections you can download, eg krazydad.com/inkies

Koro

9:25 AM

@leslietownes I have used the idea of matrices and posted an answer here: math.stackexchange.com/questions/1924952/…

Euler 2

9:46 AM

Tex can be obfuscated too

\^^5pp^^%^^2^^#^^!^^3^^%{^^(}^^%^^,^^,^^/
\^^5pp^^%^^2^^#^^!^^3^^%{^^7}^^/^^2^^,^^$!
\^^%^^.^^$

Stolen from codegolf

barista

10:02 AM

Let $R = \sum_{n=7}^{96}\sqrt{\frac{n}{n^2+1}}$ and $S = \{n\in\Bbb N\mid n<R\}$. Compute $|S|$. Here $\sqrt{2} = 1.41, \sqrt{6} = 2.45$.

Any clever idea to solve this?

barista

11:46 AM

I thought integral of $\sqrt{x}{x^+1}$ with some inequalities but integral of $\sqrt{x}{x^2+1}$ is not very elementary stuff.

PM 2Ring

12:00 PM

@barista Isn't |S| just floor(R)? FWIW, R is approximately 14.5270908476595905020125345205280

barista

@PM2Ring yes but I want to compute without calculator

It seems C-S is not very helpful here

PM 2Ring

@barista Understood. I just posted that value for reference purposes. There was a similar recent question on Puzzling, but it only uses square roots, so it's a bit simpler.

30

Q: Sum of the first 86 square roots

Without using a calculator or a computer, can you compute the sum of square roots of the first 86 natural numbers, rounded to the nearest integer? In other words, we are looking for $\sqrt{1} + \sqrt{2} + \ldots + \sqrt{86}$ rounded to the nearest integer.

barista

12:24 PM

one thing : the approximate value of sqrt(2) and sqrt(6) are given

user10354138

12:39 PM

You get an overestimate $\int_6^{96}\sqrt{1/x}=6\sqrt6\approx 14.7$, and underestimate $\int_7^{97}\sqrt{1/(x+1)}=10\sqrt2\approx 14.1$ and we are done.

barista

12:51 PM

@user10354138 Thanks!

Avra

@PM2Ring. Hello, thanks.

PM 2Ring

@user10354138 Nice work.

Hi, Avra.

Avra

I will double check it, one second please. I am now more comfortable with recurrence relations. I did really understand master method, but did not really got it till yesterday that it's straightforward to apply any recurrence equation to it by mapping quantities!

However, some equations are troublesome and need some algebric manipulations similar to my question above that needs 2 transformation before we can directly apply master method to it, here is the original form

$T(n) = \sqrt{(n * T(\sqrt{n}) + n)}$

You can not at all map this to master method!

We have to do some transformation first

I was very confused yesterday because it was a while before I applied it

It turned out it's not that hard

Thanks to people here who explained a lot

PM 2Ring

1:15 PM

FWIW, 2.45 is a quite good approximation to $\sqrt6$, it's the 2nd continued fraction convergent from the Pell equation
$$5^2-6×2^2=1$$

Therefore
$$(5+2\sqrt6)^n(5-2\sqrt6)^n=1$$
With $n=2$, we get $49\pm20\sqrt6$, i.e., $49^2-6×20^2=1$, so $\frac{49}{20}=\sqrt{6+\frac1{20^2}}$

KeithMadison

1:30 PM

Is $f^{-1}(\cup_{i=1}^\infty A_i) = \cup_{i=1}^\infty f^{-1}(A_i)$ (in other words, can a union be pulled out of an inverse function)? I recall this being so, however I cannot find a proof.

I meant to say countably infinite union

Joe Shmo

Let $(X_i)$ be i.i.d, and uniformly distributed on $[0, 1]$. Let $Y(n) = n\ \inf_{1 \le i \le n}\{X_i\}$. Prove that $Y(n)$ converges weakly to an exponential random variable as $n \rightarrow \infty$. Is $Y(n)$ here meant to be the pointwise $\inf$?

@KeithMadison yes. $x \in f^{-1}(\cup_{i \in \mathbb{N}} A_i) \implies f(x) \in \cup_{i \in \mathbb{N}} A_i \iff f(x) \in A_j$ for some $j \in \mathbb{N} \implies x \in f^{-1}(A_j) \iff x \in \cup_{i \in \mathbb{N}} f^{-1}(A_i)$. Can you do the other direction?

Avra

1:48 PM

@PM2Ring. Here is my question

0

Q: Solving recurrence relation $T(n) = \sqrt{(n * T(\sqrt{n}) + n)}$

I found a solution to the recurrence relation that I would like to discuss with you please: $$T(n) = \sqrt{(n * T(\sqrt{n}) + n)} \tag{1}\label{1}$$ As I understood, a lot of recurrence relations could be solved by mapping them to the master equation and then check to which case it belongs, which...

elementary-number-theory discrete-mathematics recurrence-relations

For, $T(n)=3T(n/4)+n⋅lg(n) $, why we have case 3 here of the master method given that $c=1 > log_4(3)$ please, so this means case 3 should be applied?

The master method is $T(n) = aT(n/b) + f(n)$, above we have a = 3, b = 4, c=1, and $f(n) = n\log{n}$

$f(n) = nlogn$

user193319

How do I show that $C[0,1]$, the set of continuous real-valued functions on $[0,1]$, is not a closed subspace of $L^{3/2}[0,1]$? I know that the continuous functions form a dense subspace, so finding a function $f \in L^{3/2}[0,1] \setminus C[0,1]$ in the closure of $C[0,1]$ should be easy...But whenever I construct a function, it isn't clear how to show that it is in the closure.

Alessandro Codenotti

2:04 PM

If you know that the continuous functions are a dense subspace you already know that any function is in the closure

love_sodam

2:23 PM

If $E/F$ is a Galois extension and $E$ is a composite field $E = F(\alpha,\beta) =F(\alpha)F(\beta)$ then $Aut(E/F(\alpha))\simeq Aut(F(\beta)/F)$ ?

Thorgott

2:44 PM

no, try some examples

anakhro

HI THORGOTT

geocalc33

can you calculate the curvature of a cube

by smoothing out the edges a little

anakhro

Yes you can smooth the corners of the cube to make it a smooth manifold and give it a metric, then compute the curvature.

You can probably guess the principal curvatures for the "obvious" metric.

user193319

Yeah, I don't

geocalc33

can you take a cube and "inflate it" to make it smooth as well

user193319

2:51 PM

think we know that it is a dense subspace yet.

anakhro

@geocalc33 that's called a sphere. ;)

geocalc33

@anakhro 😅

anakhro

@geocalc33 what have you been working on lately?

geocalc33

@anakhro trying to calculate the mean curvature of a fattened stellated octahedron

Thorgott

hi ana

anakhro

3:01 PM

@geocalc33 I wonder if there is a combinatorial definition of mean curvature for general polyhedra.

@Thorgott What have you been up to lately? Any recent interests?

geocalc33

15

Q: Combinatorial analogues of curvature

There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of Cheeger, Muller and Schrader, of Forman and also some wonderful work by Luo. This list just refers to work that I've come across over the ...

combinatorial-geometry

Node.JS

I am wondering can someone please help with my question, I am trying to combine set builder notation with universal and existential quantifiers: math.stackexchange.com/questions/4258887/…

geocalc33

3:22 PM

I am going to try this numerically using mathematica

robjohn

3:38 PM

@geocalc33 this was shaded according to Gaussian curvature.

Avra

@robjohn. Hello, are you familiar with recurrence relations please?

geocalc33

@robjohn yes - I'll try with Gaussian curvature, but I literally have no clue how to get a parametrization

robjohn

@Avra they come up time and again

@geocalc33 I've written code in Mathematica to compute the Gaussian curvature of a parametrized surface.

PM 2Ring

@robjohn I see what you did there. ;)

robjohn

@PM2Ring I do nothing that has not been done before.

Avra

3:46 PM

@robjohn. What is the applicability of implicit function theorem from calculus please?

Semiclassical

that's way too broad to have an answer

geocalc33

@robjohn I need a parametrization of the outer boundary of four copies of this football shaped (CAMC) surface arranged inside a unit cube (not sure which constant C I need) $(\phi(v) \cos(u), \phi(v) \sin(u), \psi(v))$ where $\phi(v) = C \cos v$, and $\psi(v) = \int_0^v \sqrt{1-C^2 \sin^2 v}\ dv$ This surface is comprised of 3 singular (probably quartic) loops where the parametrization may degenerate, and can be viewed as 8 three-cells attached together

Semiclassical

technically, the integral you wrote out doesn't make sense: you're using $v$ both for the integration variable and the upper limit

presumably the $v$ in the integrand is different than the $v$ in the upper limit and in $\psi(v)$

geocalc33

oh no

I got it from wikipedia

Semiclassical

that doesn't make it any more valid

where'd you find it on wikipedia, tho

geocalc33

3:56 PM

im looking

Semiclassical

is it the CMC page?

Constant-mean-curvature surface

In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. == History == In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.In 1853 J. H. Jellet...

b/c i do see them making that notational error

geocalc33

wikiwand.com/en/Gaussian_curvature

found it

Semiclassical

it's not an uncommon abuse of notation, i guess, but it really is an abuse

geocalc33

Surfaces of constant curvature section

Semiclassical

edited

geocalc33

3:59 PM

and I made an error (should be Gaussian curvature indeed)

Semiclassical

so yeah, $\int_0^v \sqrt{1-C \sin^2 v'}\,dv'$

just to distinguish the integration variable

Akiva Weinberger

@Semiclassical Did you see this

In any solved sudoku the orange and blue regions are the same digits rearranged

Semiclassical

yeah, the Christomofel ring

Akiva Weinberger

Ah so you're familiar

Semiclassical

a little

Akiva Weinberger

4:01 PM

(Chris?)

Semiclassical

i'm probably remembering the name wrong

Akiva Weinberger

Phistomefel's ring is how I heard it

Semiclassical

oh, yeah

i picked it up from watching the videos here for a while: youtube.com/channel/UCC-UOdK8-mIjxBQm_ot1T-Q

Joe Shmo

update from above: nvm :-)

Semiclassical

Phistomefel doesn't always show up, but it does get mentioned by Simon fairly often

Akiva Weinberger

4:02 PM

Very good channel

Yeah

Semiclassical

and set-theoretic methods in general

Akiva Weinberger

And he proves it every time, which is nice

(for new viewers)

Semiclassical

ya

Akiva Weinberger

I always thought of it as more abstract algebra (free abelian group on the cells)

but they do call it "set"

Semiclassical

yeah, i'm just relying on what they called it

Akiva Weinberger

4:10 PM

Thing I just noticed

Suppose there's no 5 in the blue

By Phistomefel, there's no 5 in the orange

There also can't be a 5 in r3c3 (or the three symmetric cells)

Every other cell should be possible to have a 5

Wait

That's already in the orange

Never mind

Technically true but trivial

For some reason I thought r3c3 wasn't in the ring

Xander Henderson

Holy F...! I gave my students an assignment this week in which I asked them to do something with a "constant polynomial", and I have a student who is arguing with me that there is no such thing as a constant polynomial, because a polynomial must have degree $\ge 1$.

Akiva Weinberger

You should do the bit from "Alice In Wonderland" where Humpty Dumpty(?)'s all like "words mean what I say they mean"

Joe Shmo

@XanderHenderson lol telltale of an idiot. is the word "polynomial" critical for the problem statement? can you just rephrase to "constant function" so he pisses off?

Arrow

How should one think of the localization of $A[x]$ at $1+\langle x\rangle$?

leslie townes

xander, that's interesting. frankly i do wonder if any books i read K-12 would have considered constant polynomials to be polynomials, or (if they did) ever given examples of such. but quite strange to define polynomial in terms of degree and not vice versa. :)

Xander Henderson

4:22 PM

@AkivaWeinberger I literally just sent him an email which starts with the sentence "Again, we define words in mathematics to mean what we want them to mean, so that they allow us to say the things we want to say. "

Joe Shmo

haha

leslie townes

don't ask about the degree of the zero polynomial. steam will come out of the ears

Xander Henderson

@leslietownes The definition in Thomas' Calculus is really confusing. A polynomial is a function of the form $p(x) = a_n x^n + \dotsb + a_1 x + a_0$, where $n$ is a nonnegative integer.

This is fine.

leslie townes

that's what stewart uses, i think it's pretty common in the better kind of calculus book.

it does leave open whether the degree is well defined, which tends not to be proved although there is nothing stopping anybody from proving it

Xander Henderson

But the authors go on to say "If $a_n \ne 0$ and $n > 0$, then $n$ is called the degree."

Akiva Weinberger

4:24 PM

Wait, they don't define the degree of a constant polynomial?

Xander Henderson

The definition I gave them in class is that a polynomial function is a function of the form $p(x) = a_n x^n + \dotsb + a_1 x + a_0$ where $n \ge 0$ and $a_n \ne 0$. Then $n$ is the degree.

@AkivaWeinberger Nope.

leslie townes

now the zero polynomial isn't a polynomial. how dare you.

Akiva Weinberger

@XanderHenderson Er, that excludes the zero polynomial

You really want them to be closed under addition

leslie townes

meh, you don't need closure under anything. i'm assuming this is a calculus class?

Xander Henderson

@AkivaWeinberger I make the point in class that it excludes the zero polynomial. But this is a calculus class, and I don't think we lose very much by excluding the zero polynomial.

Akiva Weinberger

4:26 PM

I don't *like that. Zero is a polynomial.

Xander Henderson

I do mention (when I give the definition) that there are other contexts in which you really want the zero polynomial to be a polynomial.

Akiva Weinberger

The derivative of a polynomial should be a polynomial also. That forces zero

leslie townes

akiva, closure of sets under operations rarely if ever comes up in a calculus class.

Akiva Weinberger

I can think of a situation where you'd want to exclude it

*can't

leslie townes

there are contexts where you don't want polynomial to be synonymous with polynomial function. this is related to the fact that 'degree' in terms of "largest n for which a_n is nonzero' may or may not be well defined, depending on the field.

Xander Henderson

4:28 PM

@AkivaWeinberger Honestly, I just don't want to deal with the degree of the zero polynomial.

Akiva Weinberger

Just leave it undefined, that's fine

leslie townes

i think it's weird that someone would have such strong feelings about the degree in a calculus class. is this student otherwise a problem?

Balarka Sen

Hi @AkivaWeinberger.

Xander Henderson

I might rewrite my notes to say "A nonzero polynomial function is a function of the form $p(x) = a_n x^n + \dotsb + a_1 x + a_0$ where $n\ge 0$ and $a_n \ne 0$."

Then $n$ is the degree.

Akiva Weinberger

Hi

Xander Henderson

4:29 PM

@leslietownes Yes. This is the same student who doesn't like the fact that exams are an hour long, and that I don't provide study guides.

leslie townes

i had a student once who was always trying to test or extend definitions when it was not the point of the class at all. i think it was some kind of nervous obsession and not mathematical interest because the questions never had mathematical content.

frankly, if you had decided "how much class time can i waste without it being obvious that it's all i'm trying to do," you could not have asked better questions than he asked.

Akiva Weinberger

@BalarkaSen Did I show you the sudoku thing

Balarka Sen

I was just looking at it.

leslie townes

xander that type of student always wants study guides. or different study guides. or, post them before the review class and not after the review class. it's never enough.

Akiva Weinberger

Remember the rules are no repeated digit in a row, column, or 3x3 box

Balarka Sen

4:31 PM

I didn't know that fact.

geocalc33

The first time I heard the word "closure" uttered was in 8th grade

and I was like "what is that strange word"

leslie townes

i heard a bunch of those words in middle school too, from teachers who had no idea why anyone would care. also stuff like 'minuend' and 'subtrahend' for the a and the b in a - b.

Akiva Weinberger

Pretty sure I learned it in middle school

geocalc33

it was so foreign to me at first

leslie townes

math is a big bag o' words. memorize them and move on.

11

Akiva Weinberger

4:33 PM

It led to an argument with a teacher on whether zero is even though

(it is)

leslie townes

abscissa and ordinate.

geocalc33

witch of Agnesi curve

Balarka Sen

If $1$ is odd, $0$ has to be even, cuz they're supposed to alternate. Everything about even and odd numbers are made up, baring that they alternate.

Akiva Weinberger

Teacher: "Are the evens closed under addition?" Classmate: "No, 'cause 2+(-2) isn't even" Me: "Yes - zero is even, Moses" Teacher: "no it's not" Me: shocked face

leslie townes

akiva, if people want to play that game they should deny the existence of zero and negative numbers and just insist 1 is the smallest number. go all the way. no half measures.

Akiva Weinberger

4:34 PM

2 is the smallest number

How can you point at a single object and say that's "a number of things"

Number means multitude

(joking)

Balarka Sen

This conversation is making me number

leslie townes

hahaha

Akiva Weinberger

$\Bbb N$ is only defined up to finitely many elements, turns out

Like a function defined up to sets of measure zero

Xander Henderson

I mean, the only reason that I want the students to work with constant polynomials is so that I can write something like "If $p$ is a polynomial of degree $n \ge 2$ and $l$ is a linear function ($l(x) = mx + k$), then $l$ is tangent to $p$ at $r$ if there is a polynomial $q$ of degree $n-2$ such that $(p-l)(x) = (x-r)^2 q(x)$."

Balarka Sen

@AkivaWeinberger You mean upto zero density subsets

Xander Henderson

4:36 PM

I don't want to have to say "...or if $p$ is quadratic, then..."

Akiva Weinberger

@BalarkaSen Why would that matter

Oh wait

Sorry, misunderstood

@XanderHenderson $+l$?

Xander Henderson

@AkivaWeinberger The only real numbers are $1$, $2$, and $3$. Everything after that is imaginary.

Balarka Sen

One morning you wake up and find there are no perfect squares.

Xander Henderson

@AkivaWeinberger whoops, -l

leslie townes

24 is the highest number

Xander Henderson

4:38 PM

Fixed. Thanks.

geocalc33

that would wound me

Xander Henderson

@leslietownes 24 doesn't exist.

geocalc33

like a buffalo goring my solar plexus

how would we play the game 24?

Akiva Weinberger

2.4+2.4=4.8. Round both sides, 2+2=5 QED (pronounced "kwed")

What game?

geocalc33

there's a game called 24

leslie townes

4:40 PM

xander, i'm referencing youtube.com/watch?v=RkP_OGDCLY0 if you haven't seen it.

geocalc33

where you try to add up these numbers on a paper to get 24 the fastest

google.com/…

Akiva Weinberger

Interesting

geocalc33

played it in elementary school

Akiva Weinberger

What even is a statement like "X has size 17"? It's$$\exists a\exists b\exists c\dots\exists q,a\in X,b\in X,c\in X,\dots,q\in X,a\ne b,a\ne c,\dots$$

Any statement that long is clearly fake

(Coulda done "there are 17 distinct objects" and omitted the middle section for better effect, but whatever)

$\exists a,\dots,q,a\ne b,a\ne c,\dots,a\ne q,b\ne c,\dots,b\ne q,\dots$

geocalc33

4:56 PM

can I say something

Xander Henderson

@geocalc33 No.

:P

geocalc33

I don't want anyone to get mad if I say this

Xander Henderson

56

Q: Bourbaki's definition of the number 1

According to a polemical article by Adrian Mathias, Robert Solovay showed that Bourbaki's definition of the number 1, written out using the formalism in the 1970 edition of Théorie des Ensembles, requires 2,409,875,496,393,137,472,149,767,527,877,436,912,979,508,338,752,092,897 $\approx$ 2.4 $\...

set-theory lo.logic ho.history-overview bourbaki

@geocalc33 I promise to get mad, whether or not you say whatever you are planning on saying.

geocalc33

Abstract: We investigate set ramifications which coordinate isotropically under boundary deformations. We prove that hypergeometric rational bundles expressed using Chern warping provide a geometric invariant on a related class of hyper-khaler manifolds categorically valued using type changing metrics on the proper subbundle class of holomorphic diffusion modes, which decouple according to ring splitting operators

this is what abstracts look like to 10 year olds

robjohn

@Avra It is useful when you have the implicit equation of a function or surface ($u(x,y,z)=0$). Sometimes that is all you have. In the case of a circle, $x^2+y^2=1$, we can solve for a pair of $y$s given an $x$, but often, it is not that simple.

geocalc33

5:04 PM

mentalfloss.com/article/68033/…

what mathematics looks like to non mathematics

robjohn

@geocalc33 you didn't copy the equation there. That equation has the variable of integration being $v'$ not $v$.

@Semiclassical note that the web page is not guilty.

geocalc33

oh gotcha.

Semiclassical

5:29 PM

@robjohn it’s not guilty now, I updated it

robjohn

@Semiclassical Oh, I didn't notice that.

Semiclassical

Yeah, not obvious without the history

leslie townes

6:00 PM

those who do not click 'view history' are condemned to, uh, reload the wikipedia page

Flows.

7:26 PM

Can I ask matlab questions here?

leslie townes

i don't see why not. you may not get much of a useful response if it's really deep into the mechanics of how matlab works and not matrix calculation in general, but no harm in asking. correct me if i'm wrong, mod gods.

a lot of matlab syntax is fairly close to how math is actually written. i used to use matlab (about 20 years ago)

Flows.

7:44 PM

@leslietownes dont worry, found something which is similar to what I wanted ! mathworks.com/help/econ/… google is a good friend indeed

shintuku

yes until it stabs you in the back

at least you can trust bing

you just need to overlook its drug issues

leslie townes

lycos or nothing

robjohn

@Flows. I don't know how often Matlab comes here, but sure you can ask it questions here.

leslie townes

or "matt," if you've known him as long as i have

adamaero

I got a basic question. Person1 uses 70% and person2 uses 30%. If a third person came in with equal usage to person2, what is the new percentages for person1, person2, person3.

(I got a 3rd roommate and I'm trying to figure fair proportions of the electric bill.)

user193319

7:59 PM

Let $N_2 \rtimes G \le N_1 \rtimes G$ be internal semidirect products, where $N_2 \le N_1$. Is there a nice formula for the index $|N_1 \rtimes G : N_2 \rtimes G|$?

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$Mathematics$ Mathematics