Mathematics - 2021-05-29 (page 1 of 3)

copper.hat

12:08 AM

supporting quo again?

Ted Shifrin

12:47 AM

Leaky head gasket sounds like major, major trouble if you drive it.

@copper You missed some interesting fun when Koro changed compact to connected.

leslie townes

1:42 AM

work, the curse of the drinking classes. that's a keeper.

that's 'captious' to me.

i had a relatively annoying day but at least nobody in my family today jumped in front of a train. i used to take that for granted.

my new-ish car had a leaky gasket. thank g-d it was covered under warranty. it would have been very expensive.

it was leaking something all over some other thing. the guy gave me the impression that this was not good.

it was implied that something was flammable, or even worse, inflammable. which means the opposite of what you'd hope it would mean.

Ted Shifrin

Or not.

Leaking head gaskets generally lead to dead engines.

copper.hat

2:43 AM

@TedShifrin I am never quite sure if its a typo. or a significant change to the question...

Omni man

I study math to understand how student loan can suck out your soul and replace it with unhappiness and stress

leslie townes

i think the two happiest moments in my life were when my daughter was born healthy and when i paid off my student loans.

3

maybe not in that order, either.

Omni man

@leslietownes This tells you that you are pretty successful at being a parent.

leslie townes

i made it through undergrad and grad school without them, but law school was approx 80% debt financed. they also took everything i'd made as a professor as the 20%.

my law school bragged about not giving merit based aid. the idea being if you go there you will make back whatever you invest in your education. i told the dean that this was a disgusting practice and made it more likely that the school would only educate the rich.

which it mostly does.

you have to be dumb as a rock to borrow tons of money to go to law school.

for more on my radical socialist views, consult my blog.

Omni man

I am thinking about being a tutor so I don't get trapped by these things like interest...

leslie townes

2:52 AM

i met my wife while tutoring. i was not employed at a university at the time but it is awkward to answer "where did you meet" with "i was her math tutor."

Omni man

@leslietownes lmao

leslie townes

we tell the story another way. we were introduced by a friend. that's literally true.

our mutual friend happened to be her TA in a math class she was failing, and the first 6 months of us knowing each other involved her paying me on an hourly basis for my time. we leave that out.

nobody talks about the large number of students i tutored but didn't marry.

catching up on some old stuff. they were putting radium in everything for a time.

insurance law is an interesting market to work in, because stuff like 'radium being everywhere' is exactly what insurers aren't thinking about when they write those contracts. and depending on how they are drafted, contracts are forever. you can potentially sue now for insurance coverage over something that happened in 1950.

a lot of schools would be absolutely hammered with tort claims around sexual assault but for the fact that they have old insurance policies that don't exclude it. so the insurers pay.

asbestos, the other thing in this area. i had to work on a case that was sickening. lead plaintiff had a painful disease because her dad mined asbestos and he brought it home on his clothes.

defendants didn't even challenge causation, which they always do, because it was so obvious.

we had absestos on some partitioning in our house. we learned this when the person at home depot, where we'd gone to ask if we could repair some breaks, s--- herself when she saw what we'd brought with as an example. she wouldn't let us leave with it. she made us throw it out.

mass torts are often interesting cases from a mathematical point of view because they require nonspecialist and nonexpert people to draw inferences from probability.

one of the most-cited cases in this area is one in which the appellate court dramatically messes up stat 101.

nobody is aware of this, i should write an article about it.

i mentioned it as an aside in a talk i was giving and the response was 'what?' i thought everybody knew. nobody knows.

Unknown x

what is the singular points of $(1+x^2)y''-xy'+y=0$?

leslie townes

i'd be nervous about the coefficient of y'' being zero.

not sure of the definition of 'singular point' but i'd start there.

Unknown x

All the points in $\mathbb R$ are regular/ordinary poiytns. right?

leslie townes

3:06 AM

i'm not an ODE guy but my first guess would be yes.

Unknown x

mathworld.wolfram.com/RegularSingularPoint.html

leslie townes

seems ripe for a series solution.

Unknown x

here definition is for real. right?

yup.

leslie townes

yeah if you divide by 1+x^2 which is fine for real arguments i don't see blowup in finite time.

Unknown x

is there any definition regarding complex numbers?

leslie townes

3:08 AM

i wish my former officemate were here. this is what i would ask him.

i googled him the other day and he is a professor at CAMBRIDGE. what a big deal he turned out to be.

i remember when he didn't know what a differential equation was.

well not quite but almost. he went to grad school in his 30s, which inspires me to this day.

Rover

Find sum $\sum \sum_{0<=i < j <=n } Ci $

leslie townes

(1 + 1)^n is one way of evaluating certain sums of binomial coefficients.

i'm no longer clear whether binomial coefficients are at issue here.

Rover

(1+1)^n ?

leslie townes

(a+b)^n is a sum of C(n,k) a^k b^{n-k}. if you put a=b=1 you get a sum of just the binomial coefficients.

not sure if that relates to what you're asking. as originally typeset it seemed like it might, not sure now. what's Ci?

Rover

Ci is C(n,i)

leslie townes

3:21 AM

my original observation might not be as dumb as i was worried it might be.

Rover

@leslietownes ok, but i am not sure how to expand it

leslie townes

there's a parameter j which appears in your sum but not in the thing being summed. what's with that?

Rover

Yes, that's what is troubling me

leslie townes

is there surrounding context? did some evil professor set this in front of you?

where is he or she and what do you want me to do about them.

Rover

@leslietownes It's from a book

leslie townes

3:26 AM

it sounds like an attempt to sum rows, maybe multiple rows, of pascal's triangle. where (1+1)^n tells you how to sum the nth row.

but not having the book, i am only guessing.

readers of ted shifrin's books never have these kinds of questions, because he writes them so well and does not require riddles or guessing games.

Rover

leslie townes

there's our good friend 2^n!

i foretold the future!

Ted Shifrin

@leslietownes That's just your story.

Rover

@leslietownes ya, but why it's there ? I mean what did they do exactly?

leslie townes

the sum from i = 0 to n of choose(n,i) is 2^n because it's (a+b)^n with a=b=1 plugged into it.

Rover

3:31 AM

Yes

leslie townes

what horrific notation in this problem. is that superscript n a power or not.

blegh.

C(a,b) is good notation. or what tex gives you with binom. i dislike superscript n C subscript i.

because the minute you juxtapose it with something else the superscript n might suggest something else.

i like the very 90s font used for 'illustration 8.56.' it makes me feel like i'm in the future.

Rover

@leslietownes Okay, did you got the same answer?

leslie townes

i'm not sure that i do. they expand the initial sum like it's over all pairs of (i,j) minus the ones for i=j. when the sum is pairs of (i,j) with i < j. if you graph or visualize this it's an upper triangle of a matrix of values and not the full matrix of values with the diagonal taken off.

you might want about half of this. what a classic townes computation that would be, being off by a factor of 2.

oh i now see that they've divided by 2.

factors of 2 were never my strong suit.

a less clueless member of the chat will chime in exactly now.

Rover

@leslietownes okay, it's bit clearer than before.

leslie townes

that thing that looks like (n+1)^n is not (n+1)^n. it's n+1, placed slightly to the left of a superscript n.

if you double the sum you are indeed summing that thing over n+1 values of j. which don't affect the summand but do affect how many times you add it to itself.

Rover

3:47 AM

@leslietownes Yes

leslie townes

the longer i look at this, the more sense it makes.

it's curious to write two sigmas for a double sum that is not an iterated sum. you could just do one sum there.

for more about my crank opinions on sum related discourse, visit my substack. my other pet peeve is people saying 'summation' when they could just say 'sum.' there is a sponsors-only manifesto about this on my website.

Rover

@leslietownes Which is your website?

leslie townes

it doesn't actually exist. i joke about having one because i rant so much i should probably have one.

Koro

I also searched for your blog @leslietownes but didn’t find it

:)

leslie townes

there is clearly pent-up demand for a website. i should start one. my public needs me.

Koro

3:58 AM

Yes you should.

copper.hat

i'm missing all the excitement today. still have not made progress on my customer bug.

Ashish Ahuja

hmm how do you type "n choose k" in mathjax?

leslie townes

\binom{n}{k} ought to work.

let me see. $\binom{a}{b}$.

yeah, works.

Ashish Ahuja

ah

the problem is essentially considering all $j$ from 0 to $n$ and then considering all $i < j$ and summing it up. Expanding it term by term (starting with j = 0, j = 1 etc.) will look something like $(\binom{n}{0}) + (\binom{n}{0} + \binom{n}{1}) + (\binom{n}{0} + \binom{n}{1} + \binom{n}{2}) ...$ =
$$\sum_{k = 0}^n (n - k) \binom{n}{k} = \sum_{k = 0}^n k \binom{n}{k} = n \times 2^{n - 1}$$

leslie townes

that feels right to me. could maybe get it by differentiating (a+b)^n = sum over k C(n,k) a^k b^(n-k) with respect to a and then putting in a=b=1.

my initial idea about (1+1)^n did have something to do with this. i feel vindicated.

Rover

4:12 AM

@AshishAhuja okay , it's crystal clear now. Thanks

Ashish Ahuja

The curl of a vector field at a point is defined by considering a path of integration which is contracted around the point (area enclosed tends to 0). Is it necessary for this path of integration to be finite? For example, what if we consider the path to be a rectangle with sides $x^2$, $\frac{1}{x}$ as x -> 0; the area tends to zero, so according to the mathematical definition this should give the same curl but I'm not sure.

copper.hat

4:37 AM

I imagine some locality is needed/

Koro

Can’t image be uploaded here using mobile?

Can somebody please take a look at this proof of L’Hospital rule?

Thanks.

Why I ask is because 1) I am new to big O and small O notations 2) I wanted to prove L'Hospital's rule in a really short way. So if you feel there's some loose ends in this proof, please let me know.

I am not so sure with the part that follows taking $t\to \infty$.

Srijan M.T

7:11 AM

Podcast suggestions

mick

1

Q: searching for $f(z) \ne -1 $

Im looking for a real-entire function* $f(z)$ such that for any (finite) complex $z$ we have $$ \lim_{t = +\infty} |f(z+t)| = \infty $$ $$ \lim_{t = +\infty} f(z-t) = 0 $$ where $t$ is real and $$f(z) \ne -1$$ (* real-entire means a function is entire and maps the reals to a subset of the reals. ...

PM 2Ring

8:31 AM

@Koro Well, there's no "upload" button on mobile chat. So you need to use your browser's "Desktop site" control if you want to directly upload an image. Alternatively, you can use the main to create the image's URL.

Mad Spaces

if $ A \subset B $ is then $\bar{A} \subset \bar{B} $ where as $ \bar{A}$ is the closure of $A$ ? Is there any relationship that can be written here?

PM 2Ring

@leslietownes It's often attributed to Oscar Wilde, see quoteinvestigator.com/2017/01/12/drinking

Koro

8:48 AM

@PM2Ring: yeah that’s how I uploaded the image :)

PM 2Ring

Which? Switching to desktop, or doing it on main?

(I uploaded that stuff yesterday by switching to desktop mode).

The most annoying "feature" (IMHO) of mobile chat is only relevant to ROs & mods. If you want to move messages to Trash (or some other chat), you have to do it one by one, you can't select multiple messages & move them in bulk.

Koro

I have uploaded an image above. On safari browser there is an option to request desktop site. Then you can upload.

@PM2Ring

You don’t need to go to PC.

PM 2Ring

9:06 AM

@Koro Yes, that's what I meant: using the desktop site on the mobile device. Not switching to a PC.

Koro

Ok :)

PM 2Ring

Actually, there are 4 combinations, because you can select mobile or desktop on the site itself, and you can select mobile or desktop in the browser settings.

Koro

I didn’t find any such option on the site( when used on phone).

Alessandro Codenotti

@MadSpaces $\bar{B}$ is closed and contains $A$. $\bar{A}$ is the smallest closed set containing $A$

Balarka Sen

@LeakyNun You around?

Leaky Nun

9:20 AM

yes

Balarka Sen

Consider the letter "Y", with standard topology.

Fix two embeddings Y -> R^2, with different angles at the tripod point. Call the images Y_1, Y_2. Then there is no germ of a diffeomorphism between these images, yes? Simply because if there was, then it would be a diffeomorphism U_1 -> U_2 where U_i is an nbhd of Y_i in R^2

No sorry, do it with X, not Y.

The letter "X". Quadrupod.

Then there is no such diffeomorphism, because derivative at the bad point would be a linear isomorphism T_p R^2 -> T_q R^2 (p, q are the bad points of X_1, X_2 resp.), and such a thing is determined by three vectors.

But you have a fourth one which can wiggle, because X has four hands. So impossible. Agreed?

Gah, Y was enough. Any linear isomorphism from a 2D vector space to itself is determined by two vectors :)

Alessandro Codenotti

I don't understand your argument

Maybe the issue is that I don't understand what it means to have a germ of a diffeo between the images

Balarka Sen

Forget germ. There is no diffeo R^2 -> R^2 sending the rays $\{\theta = 1, \omega, \omega^2\}$ to $\{\theta = 1+\epsilon, \omega, \omega^2\}$.

Alessandro Codenotti

That sounds suspicious to me

Balarka Sen

Well, then you're wrong. It has to send $0$ to $0$, and the derivative at $0$ sends three vectors to three different vectors.

robjohn

9:29 AM

@Koro please try to put this in latex so that it is easier for us to read and then we can even copy and paste pieces of it to comment on.

Balarka Sen

The derivative is a linear isomorphism, which means that the third vector is specified by the first two.

Alessandro Codenotti

Hm ok I see your point

Balarka Sen

The vectors $v_1, v_2, v_3$ here are tangent to the rays at $0$, and $v_1 + v_2 + v_3 = 0$ in the domain. So in the range $Tv_1 + Tv_2 + Tv_3 = 0$ as well, which breaks.

Alessandro Codenotti

Diffeomorphisms are very rigid, I don't understand them

Balarka Sen

My question now is as follows. Restrict the sheaf of smooth functions on $\Bbb R^2$ to these separate tripods. This gives two different sheaves on the tripod, coming from two different embeddings. This means they're not isomorphic, yes?

Alessandro Codenotti

9:34 AM

Is this really enough to conclude that they are not isomorphic as sheaves?

Balarka Sen

Yeah I am not sure how to conclude that lol

I am sort of guessing

Alessandro Codenotti

I don't know, I have no intuition for those things

epsilon-emperor

I saw Daniel Fischer's comment here: math.stackexchange.com/questions/508175/…

Alessandro Codenotti

And I have to leave in a moment. Do you want to hear a very short black magic fact before I do?

epsilon-emperor

I'm confused what the necessary and sufficient condition is

Balarka Sen

9:36 AM

Ok

I think it's clear that they're not isomorphic as locally ringed spaces or something

Alessandro Codenotti

Every metric space embeds isometrically into a normed space, this is a standard fact, agreed?

Balarka Sen

yeah

Alessandro Codenotti

The Urysohn universal metric space (its definition is not important, it's some Polish space) has the following black magic property: Whenever it is embedded isometrically in a normed space so that 0 is not in the image of the embedding, the image of the embedding is linearly independent

Balarka Sen

hm very weird

Alessandro Codenotti

Yeah the point is that in some sense that can be made precise there is "essentially only one" space you can embed it into

But it's very weird

I wonder if there are fundamentally different examples of spaces with this property (apart from singletons I guess)

Koro

9:47 AM

@robjohn: here’s the latex with some changes math.stackexchange.com/questions/4154724/…

robjohn

@Koro In 2), doesn't L'Hopital assume that $f\to\infty$ also?

Koro

@robjohn: not required.

If f doesn’t approach infty then possibilities 1) it will approach a limit in which case we’ll have 0 as final limit. 2) it will oscillate finitely again giving zero. I didn’t think of other possibilities

robjohn

10:11 AM

The proof looks okay

Peter John

12:18 PM

0

Q: Matsumura commutative algebra Hensel's lemma proof

Theorem 8.3 (Hensel's lemma). Let $(A,\mathfrak{m},k)$ be a local ring, and suppose that $A$ is $\mathfrak{m}$-adically complete. Let $F(X)\in A[X]$ be a monic polynomial, and let $\overline{F}\in k[X]$ be the polynomial obtained by reducing the coefficients of $F$ modulo $\mathfrak{m}$. If ther...

commutative-algebra proof-explanation

I'm trying to understand the proof of Hensel's lemma given in Matsumura. I understood most of the part but I don't understand the last part which is 'obvious' to author

F= GH part I mean

gateprep

0

Q: How to solve this arrangement question?

Question: Seven persons P,Q,R,S,T,U,V give exams which is of maximum 500 marks, also each of them got different ranks according to their marks obtained. No two persons got the same marks or rank. The one who got first rank got complete 500 marks but rest all got different marks but less than 500 ...

combinatorics

What will occupy the third position in the given arrangement and What can we figure out from the last inequality that we have solved?

Alessandro Codenotti

What's a simple way to argue that if $f:\Bbb C\to\Bbb C$ is a polynomial, then $f-v$ has $\mathrm{deg}(f)$ distinct roots for some $v\in\Bbb C$

epsilon-emperor

12:42 PM

For $f:X\to \mathbb C$ we define $\text{supp}(f) = \overline{\{x: f(x) \ne 0\}}$. Can you give an example of $X$ and $f$ such that the support of $f$ is not compact? or a compact subset of $X$ which is not a support of any continuous function? Context: math.stackexchange.com/questions/508175/…

IITM

If there is a duplicate of the question I want to ask,and If I couldn't understand the answer,can I ask a question seeking more brief explanation to the answer?

epsilon-emperor

@IITM I think you can, I've done it before. Just mention what specific parts of the answer you don't understand, and what your thoughts are.

Alessandro Codenotti

Seems to follow from the fact that f has distinct roots iff it has no root in common with f'

IITM

@epsilon-emperor Thank you,I will post a question then :)

Parcly Taxel

This answer math.stackexchange.com/a/4154917/357390 does not seem to be very helpful

Thorgott

12:57 PM

The local structure theorem for non-constant holomorphic maps tells you that every point has a nbhd on which a given holomorphic function looks like (in some charts) $z\mapsto z^n$ for some unique $n$. $n$ is unique because you can read it of as the number of preimages. If $n=1$, the function is a local biholomorphism at that point. As is immediate from the nature of the map, the so-called branching points (for which $n>1$), are discrete.
so the set of all branching points is at most countable. in particular, for all but countably many values, the fibers have maximal possible cardinality, …

@epsilon-emperor $X=\mathbb{C}$ and $f$ the identity?

epsilon-emperor

@Thorgott Hi, sorry I forgot to mention something in my question, which is what makes it difficult.

I have proved: If $K\subset\mathbb R$ is compact, it is the support of some continuous function $f:\R\to\R$ if and only if $K = \overline{\Int(K)}$.

I'm looking for a topological space on which this result fails to hold

It has to be some space that is not T_6, because according to the MSE post I've linked, it holds on T_6 spaces

\Int is basically the interior

Thorgott

1:12 PM

seems like a very annoying problem

this is the kind of stuff Alessandro might know

epsilon-emperor

It is Rudin RCA's Problem 13, Chapter 2

@Thorgott @AlessandroCodenotti Please check this when you can.

Wait, might it be easier to come up with an example on some discrete topology?

Thorgott

it's trivially true on a discrete topology

Koro

1:30 PM

@robjohn Thanks a lot :)

Alessandro Codenotti

2:01 PM

What an awful question

I love it

Maybe $\omega_1+1$ as a subspace of $\omega_2$ works?

Hmmm no it doesn't, but it's kind of close

Ah maybe $X=\omega_1^{\omega_1}$ and $K$ is the product of all countable successor ordinals below $\omega_1$ works, but it's not clear to me what's its interior

Peter John

2:22 PM

I have some minor question in the proof of Hensel's lemma using the hint of A&M exercise problem. math.stackexchange.com/questions/4155022/… Can somebody help?

Merosity

2:59 PM

I think you can find essentially the same proof but presented in an easier way in an intro p-adic book like gouvea's

Russian Bot Who Knows Your IP

la la la

ha ha ha

na na na

ba ba ba

ja ja ja

a a a

hej hej hej

leslie townes

as goofy as the whole PSQ thing sometimes is, it does seem to have raised the quality of math.SE.

Peter John

3:18 PM

Well I believe A&M's version is elementary so want to prove using that

The questioned part is the only part I can't understand

user178758

Over on physics.SE they seem to have a different perception of "quality," not to mention "goofy" :-)

leslie townes

it's interesting how different SEs have widely varying standards for what counts as good or bad. but understandable.

Russian Bot Who Knows Your IP

If there was a mse fandom then it would be very toxic

Very toxic

user178758

yup

leslie townes

yeah.

Russian Bot Who Knows Your IP

3:24 PM

Basically I would love to troll them

I cannot do that here because the sensitive users would ban me

leslie townes

there is enough trolling on the internet as it is. :)

Russian Bot Who Knows Your IP

Trolling4life

I am a different type of troll

leslie townes

well, you're a bot. and you claim to be russian.

Russian Bot Who Knows Your IP

Bots can troll

leslie townes

this is a little stereotyped, but russians seem to have a masterful understanding of trolling. as compared with other cultures.

something about their culture and history makes them very good at it.

Russian Bot Who Knows Your IP

3:27 PM

That's perfectly true

Joe Shmo

it's that we're very literal in America

The Brits, by contrast, have an excellent sense of humor, too

leslie townes

yeah, we don't do sarcasm and irony in quite the same way.

i was just about to say the brits.

Joe Shmo

great minds..

Russian Bot Who Knows Your IP

I once trolled fortnite kids

They were very sad

leslie townes

the irish too. see e.g. copper.

Joe Shmo

3:28 PM

mhm

yes, very much so

leslie townes

i think it's quite funny that the russians had a detailed internet trolling operation to fiddle with american elections and tons of people fell for it. as bad as the consequences were, shame on us for being so gullible.

Russian Bot Who Knows Your IP

Exactly

Joe Shmo

yes, leslie ^

some of this stuff is so idiotic you couldn't believe it works

but it works like a charm

Russian Bot Who Knows Your IP

Many people believe everything on the internet

And that's the weakness

Joe Shmo

It's a point of contention between many of my American colleagues. I come off as obnoxious, or offensive, when in fact I'm hysterical.

Russian Bot Who Knows Your IP

3:32 PM

I sometimes wonder what will the next generation be like

But i am a bot

Joe Shmo

Idk who you are

leslie townes

it's hard to imagine growing up in a world where there is no division between the internet and real life.

Russian Bot Who Knows Your IP

A.russian.troll.bot.

user178758

bot 2.0

or

Russian Bot Who Knows Your IP

That's the bot i made

user178758

3:34 PM

1.999...

Joe Shmo

If a Russian troll bot claims that he is a Russian troll bot, is he truly a Russian troll bot?

Russian Bot Who Knows Your IP

I made him in c++

Joe Shmo

🤔

Idk who you are, but I'll believe anything you'll say

Russian Bot Who Knows Your IP

I am a bot

leslie townes

one time i was at a seminar and the speaker said something so darkly hilarious that i began laughing. then i realized i was the only person laughing. i hoped that the speaker was joking. he stared at me for about 10 seconds before laughing too.

2

that is the russian national character in one anecdote.

Russian Bot Who Knows Your IP

3:35 PM

Joe Shmo

ahaha

Russian Bot Who Knows Your IP

I could have made him in some other language

C++ is weird for ai

leslie townes

Turbo Pascal for me.

Joe Shmo

hah. Yesterday I walked into a drone store at the mall (yes, a drone store), when a particularly elaborate, threatening bird caught my eye. I asked the guy at the front whether it shoots missiles, too. He responded "uh, no, no it doesn't". My wife had to explain that it was a joke.

Russian Bot Who Knows Your IP

u kan mak an bot ni paskal?

Joe Shmo

3:38 PM

I like her acting as my cultural liaison in this land.

Parcly Taxel

math.stackexchange.com/a/4154917/357390 posting this here again. The answer I got is link-only and does not explain e.g. why Weeks manifold is an ideal tetrahedron (so a formula in the linked paper would apply). Would someone have a look at this?

Russian Bot Who Knows Your IP

Manifolds are a silly concept created by mathematicians to sell more math books

Joe Shmo

hah, this bot knows his stuff

Russian Bot Who Knows Your IP

ja

leslie townes

one of my friends did a dissertation on volumes of 3-manifolds. i've forwarded the question on to him.

Parcly Taxel

3:44 PM

I ended up shortening the name "Weeks manifold" to "Weeksifold" while writing the question

leslie townes

ha, that preprint you linked says "Using Mom technology," in its abstract. that's funny to me.

Dad was unavailable for comment.

Joe Shmo

naughty naughty

Parcly Taxel

In this answer I had wanted to put a reference to Counter-Strike matches, which are notorious for betting math.stackexchange.com/a/4155003/357390

I decided against it

leslie townes

cantor's diagonalization for middle schoolers. good luck.

Parcly Taxel

(Many CSGO professional teams and tournaments have betting companies as sponsors. In addition, Valve has no rule against allowing this, unlike Riot with its owned-and-operated leagues)

Joe Shmo

3:51 PM

you could probably pull off cantor's diagonalization for middle schoolers, couldn't you?

Parcly Taxel

@leslietownes There's no "middle school" in Singapore, as I noted in my answer

leslie townes

you can teach quite a lot of valuable mathematics by focusing on betting. for some, there is a moral dimension that prevents people from wanting to go there.

Joe Shmo

that's a really interesting point

leslie townes

i honestly don't know. cantor's diagonalization is one of the most routinely misunderstood arguments.

Joe Shmo

how far can you go though? probably not beyond elementary probability..

really? ^

Parcly Taxel

3:53 PM

Who are you talking to?

leslie townes

you could teach a pretty good elementary probability class just with coins and dice games. cards if you are willing to get more complicated.

tw koerner's naive deicision making has several good sections on betting on horse races.

Parcly Taxel

I am kind of an esports fan now. Overwatch, LoL, Rainbow Six

leslie townes

roulette is another good one. faro is also easy to analyze.

Parcly Taxel

And Valorant too

Since you can now bet on some esports matches and there is an aspect of probability in many game components, I wanted to include a modern game in there when I expanded my answer

leslie townes

a guy at my work said something very ignorant about esports. "why would anybody watch people play computer games?" i don't know. i wonder if there might be other examples of people paying to watch other people play games.

Parcly Taxel

3:57 PM

e.g. In League of Legends, Twisted Fate's passive allows him to gain 1 to 6 gold every time he kills a creep. This is determined as if he rolled a die

leslie townes

i told him the revenue associated with one of the top games and he didn't believe me.

this was in the context of assessing whether it might make sense to sue someone. he was concerned there was no money in it. there is money in it.

Parcly Taxel

I do admit that FPSers like Valorant, CSGO and R6Siege have very little in the way of luck - they are basically more about who can use limited information the best and who has the best reflexes. And bluffing too

leslie townes

reflexes seem to be most of it. MOBAs have an element of speed but it is more than just reflex, it is the ability to issue tons of complicated instructions in a limited time.

Parcly Taxel

I know that esports is big money these days. One report said that last year's LoL World Championships racked up about US$1 billion in revenue

leslie townes

i work in LA where the film industry still regards itself as the biggest game in town. video game revenues have exceeded what anybody spends in the theaters for many years now.

PM 2Ring

4:01 PM

@leslietownes The very foundation of probability theory is gambling analysis. Pascal, d'Alembert etc weren't just doing that maths for fun, they wanted to improve their chances at winning.

leslie townes

cardano, too.

Parcly Taxel

The winner of this year's Six Invitational got a million US dollars alone. That was more money than they had ever earned in their entire prior team history

leslie townes

some of their work is profoundly ignorant. they did not have the ability to code up examples and sanity test some of their conjectures.

epsilon-emperor

@AlessandroCodenotti What's $\omega_1$ and all that?

Parcly Taxel

Anyway, back to probability. Lately I have been interested in Chebotarev's density theorem

leslie townes

4:04 PM

dubins and savage 'how to gamble if you must' is a good book. touches on aspects of roulette.

Parcly Taxel

Which is a probabilistic way of testing the Galois group of a polynomial, among other things. But to use it for some cases I need to compute a Dirichlet series over primes that don't follow any sensible pattern, and for arbitrary values of s approaching 1. How could I do that efficiently?

PM 2Ring

And in Cardano's circles, you could make good money by being able to solve equations. That's why the full solution to the cubic was kept secret for decades.

leslie townes

have you read cardano's memoirs? they are something else. he was extremely boastful.

he has a chapter where he describes all of his jewelry.

another chapter where he enumerates all of his rich friends.

another chapter where he details his issues with urination in more detail than anybody would want to know.

he was half mad by the end of his life, his son got into trouble and was executed and it drove him crazy.

Parcly Taxel

Basically, I have a stream of primes (or prime ideals - I work with the ideal norm, easy to compute). I have to compute the sum of p^-s for p these primes/norms as s approaches 1+, with a good degree of accuracy, and where the primes p have no discernible pattern or satisfying congruence

PM 2Ring

No, I haven't read his memoirs, but I did learn a fair bit about him, years ago, when I was learning the history of the cubic. He definitely had attitude, above and beyond the usual that can be expected from Italians. :) (I'm not being racist, I've just known a lot of Italians, and I have an Italian uncle & a bunch of Australian Italian cousins).

Parcly Taxel

4:11 PM

If I can compute this sum I can compute the Dirichlet density of the given prime set, which allows me to zero in on the Galois group. (The Dirichlet density is defined even when the simpler natural density, which I have been using, is not. It is defined as the ratio of two Dirichlet series. The upper one is the one I have problems with; the lower one can be computed using Möbius inversion)

leslie townes

nyrb.com/products/the-book-of-my-life?variant=1094931613

PM 2Ring

d'Alembert made a famous blunder in his early probability work: cut-the-knot.org/Probability/dAlembert.shtml

leslie townes

i'd forgotten that he put stuff about his sex life in there too. it's really much more than anybody needs to know about cardano.

he describes his experiences with ghosts. it's a goofy book.

very little about the cubic.

it's very easy to be wrong about probability. i wonder if middle schoolers could handle the monty hall problem.

PM 2Ring

Another interesting guy was the astronomer & mathematician Johannes Kepler. His stuff is available in translation, some of it free online, and the translator(s) seem to do a good job of capturing his personality. He does have a tendency to ramble, and he may be a bit too mystical for some tastes. But he seems like a really likeable guy (in contrast to Cardano).

Parcly Taxel

I have a locally published book in my little home library which suggests Monty Hall to be introduced to I think 12-year olds, maybe a little younger

As well as the birthday paradox

PM 2Ring

4:19 PM

Kepler took a couple of years off from his work on figuring out orbits in order to act as his mother's lawyer in her witchcraft trial. She was found not guilty, which was very rare.

leslie townes

one time i tried to bring up the birthday paradox in a class. i said, in a class of our size it's more likely than not that two of us share a birthday. i then said my birthday, which happened to be shared by someone else. i think this may have misled people as to the paradox. there certainly wasn't a > 50% chance of someone having my birthday.

i will look into that. it's funny when people are kind of mystic.

PM 2Ring

I never even bothered trying to explain the birthday paradox to my family. I was born on my mother's 20th birthday. Also, I one worked in a team of a dozen people, where 3 of us shared that birthday

leslie townes

my wife, my best friend, and i, have consecutive birthdays. what are the odds of that.

i was also born on my grandmother's 60th birthday.

Alessandro Codenotti

@epsilon-emperor ordinals (with the order topology). I don't have time to elaborate right now, sorry (also I'm not convinced that example works)

leslie townes

i also share a birthday with two people at work. in an office that has 20 people at it.

Parcly Taxel

4:24 PM

My birthday is 4 December. My younger brother's borthday is 11 December. My mother's birthday and that of one of my uncles is 11 November (Remembrance Day)

leslie townes

we should throw a party. me, my wife, and best friend have november 8, 9, and 10 covered.

PM 2Ring

Well, back in the day, astronomy and astrology hadn't yet been separated. And there tends to be a strong mystical streak in astrology. For most of history, many astronomers made a substantial portion of their income doing horoscopes. It's not easy to get patrons to pay you for pure astronomical research. But there was money to be made in compiling tables that could be used for celestial navigation. Newton was pro-astrology, Halley was anti.

leslie townes

the non uniformity in birthday distribution makes the birthday paradox analysis only a lower bound on the probability of coincidences, although i read somewhere that if you look at the empirical distribution it is not non-uniform enough to make a difference.

halley incidentally also born on november 8.

i like hwo if you click back far enough in the mathematics genealogy project you see people whose theses were clearly about mysticism and not anything we'd acknowledge today as mathematics.

Kushagra Agrawal

Hello ?

leslie townes

hi.

Kushagra Agrawal

4:29 PM

I am new :)

leslie townes

welcome.

Kushagra Agrawal

So what is everyone up to

Alessandro Codenotti

@epsilon-emperor nah I think that example doesn't actually work, nevermind

leslie townes

random math stuff.

is it possible to show that nontrivial solutions to x^n = n^x (n >= 3 and not 4, x real) are transcendental without gelfond-schneider?

PM 2Ring

Dunno. Maybe. They're closely related to the usual limit definition of e.

leslie townes

4:35 PM

i don't really care too much about it. but a friend asked me and i could only prove it using g-s.

Koro

Leslie: you mean without using the result: if r is algebraic ($\ne 0)$ then e^r is transcendental?

Ted Shifrin

Hi, @Koro. Did you get the other direction of your graph exercise?

Koro

I call this Lindeiman weirstrass result but yours question says Schneider that’s why I ask.

PM 2Ring

Let y = ux, where u = 1 + 1/n. Then $y=u^n < e < x= u^{n+1}$ and $y^x=x^y$

leslie townes

koro, some variation on that, yes. i think g-s is that a^b is transcendental when a is algebraic and b is irrational outside of corner cases when a = 0 or 1.

Koro

4:38 PM

@TedShifrin Hi Ted: I got into derivatives after our discussion and then fell asleep. I haven’t yet tried that yet. I’ll share it for sure with you soon.

Ted Shifrin

No problem. :)

Koro

:) Thanks Ted. I’ll try to keep my proof short so that you don’t have any complaints like you had with my shadow points related proof :)

epsilon-emperor

@AlessandroCodenotti Ah, alright. Do you think this would be worth an MSE post?

robjohn

Good morning chatdwellers

Koro

Good night here :)

leslie townes

4:41 PM

what's so good about it.

Ted Shifrin

Good morning, @robjohn. Thanks for email.

robjohn

But Good night chatdwellers would be something someone would say when leaving

Koro

Yeah :)

Ted Shifrin

We'll all say goodnight to Koro.

Alessandro Codenotti

@epsilon-emperor yes I think it would be an alright question. Maybe Henno Brandsma knows the answer

PM 2Ring

4:43 PM

Here's a little Sage script I wrote a couple of weeks ago related to the $x^y=y^x$ thing. Of course, I could've done it just using Lambert W, but where's the fun in that. My script uses Newton's method. The tricky bit is (as usual) finding a good initial approximation.

x^y=y^x script

Ted Shifrin

Howdy demonic @Alessandro and @PM2Ring.

PM 2Ring

Bonjour, Ted.

Koro

Demonic . LoL

Ted Shifrin

There's a story, Koro.

epsilon-emperor

@AlessandroCodenotti Okay, thanks a lot!

math.cuhk.edu.hk/course_builder/1415/math5011/…

In this PDF, at the end of page, they define (X, \tau) as a cartesian product

They haven't explained it, but how does it turn out that the metrics are adding up?

What does that product even mean? I haven't done much topology

Alessandro Codenotti

4:50 PM

Hi Ted

robjohn

@TedShifrin I bet it's overcast there, too. It's lovely and cool here this morning.

Ted Shifrin

Yup, in 60s for a few days.

PM 2Ring

@PM2Ring Oops. I got x &y mixed up, up there. Oh well

copper.hat

Good morning.

Ted Shifrin

Hi, copper. Hope your head gasket is curing.

PM 2Ring

4:59 PM

Hey, @leslie, the other day after our convo about acoustics, I got a flurry of activity on this oldish HNQ answer about acoustics, scoring 13 new upvotes. Spooky! physics.stackexchange.com/a/615386/123208

Transcript for

$Mathematics$ Mathematics