Mathematics - 2021-04-16 (page 1 of 3)

I wonder what @feynhat will decide at the 11.8th hour!

Hawk

Question about part of a proof. Let $e\in R$ be idempotent and R is a ring with 1. then (1 -e)e = 0 and since e\neq 0, 1 -e is not invertible.

why does this show 1 -e is not invertible? If e\neq 0, are they implying 1 - e = 0?

Thorgott

a zero divisor is not invertible

except in the zero ring

Hawk

ah okay wow. thanks.

the zero rng

12:25 AM

ring my bell

Hawk

Just a question, are "regular modules" just the "normal" modules i am thinking of? I am reading AoPS and the definition of "regular (left/right)" looks exactly the same as a R-mod in most books.

Sayan Chattopadhyay

@Ted it must be crazy for feynhat, I will have to go through all this shebang in a year, and it gives me serious anxiety

12:40 AM

Do your homework @Sayan. There was no internet when we old guys did it. Search faculty, publications, contact a few grad students to ask how their experiences have been. Ask your profs who know your strengths and ability for advice and recommendations.

i do not miss that time in my life.

I did visit a few schools. On my own dime. These days a lot of departments fly in some prospective students. UGA took this seriously.

when i was in school the department would fly people in, but ask that they stay with grad students. this was thought to be a good idea. i thought it made the department look cheap, and put way too much recruiting emphasis on social interaction with students with a spare room or couch.

1:01 AM

Well, I helped recruit a few people with my cooking :)

i believe it. in our case it was just, every year the same people volunteered. i wondered, what about students who didn't 'click with,' or already go to undergrad with those people (pretty common).

they were fine folks, but i wouldn't want a grad program populated only with people who like hanging out with me.

That brings up a good question. How did you find out what potential advisors were researching what or what interest you could pursue with respect to institutions. Now of course we could do this all online and there is a vast amount of information available. I guess back in your guys' days it would mostly revovle around the good local universities and some of the "big names" I guess?

whoah, there's no "your guys'." the internet existed when i did it.

Mine were a few random people. Maybe some geometry interest. I truly have no recollection.

i also cheated by going to the same grad as undergrad.

1:07 AM

Yes, you're a baby, and you didn't venture anywhere. I consciously decided not to stay at MIT even though I got in.

so, like the kids these days, i looked at web pages, papers, etc. i visited other schools and talked to some people.

stanford did not offer to reimburse me for my $5 train ticket. i should send them an invoice.

I applied in 1973.

Well there were obviously journals then so I guess you would read the journals and see what profs were doing what and kind of zero in from there?

I did not do that.

if x is an integer, does its residue class modulo n contain itself too?

1:11 AM

Although visiting Berkeley and seeing Steve Smale lecture was not a huge plus :) I got to know him much better later, but avoided classes.

What do you think, shintuku?

looked at web pages, but over dial up right @leslietownes ?

yes. i stayed true to dial-up until 2005.

I recognize that name from dynamical systems texts....

@TedShifrin it would depend on whether we consider that x -> x is a modulation by n

Surely I switched in Athens, GA, before that!

Is $0$ divisible by $n$?

1:13 AM

we may have a question in search of a definition here. what is the residue class of x modulo n?

if x is an integer, it is the set of integers whose distance from x is dividable by n, no? @leslietownes

leslie's being facetious with me....dial up was long gone by 2005

With leslie you can never be sure.

dc3rd, i was definitely a late adopter. it was definitely still there. at universities, anyway, if not commercially.

2005?..........I refuse to believe that.....that's beyond late adopter...lol

1:15 AM

what I wonder is, do we consider 0 to be dividable by n. if this is true, x \to x is a modulation by n, which means we consider x to be part of its residue class modulo n

I was 10 years late getting an answering machine, and got my first smart phone in 2015.

shintuku: yes, although this introduces 'distance' and 'divisible' which themselves arguably, in some crazy world, require definition. there are simpler to work with definitions. but forgetting that, ted asked the key question.

ted and i tied for 'year of first smart phone.'

Of course, shintuku. $0=0\cdot n$.

as a bonus: i'm still using my first smart phone.

I just upgraded a few months ago. The battery in my original was about to explode a year ago.

1:17 AM

I would've still been using my Samsung S3 if it didn't just die, so....I'm on an S5 now.....

i'm on my third battery.

ahhh... that makes sense, thank you!

Confirmed Apple person here. First computer 1988.

It cost me a small fortune!

but now you buy Apple gadgets, so you have to deal with their famous planned obsalescence

i think 'is a multiple of' is often an easier term to think with than 'is divisible by.' because you get out of dealing with division.

1:18 AM

Agreed, leslie. Yet again.

noted!

i also got my first computer in 1988. or, my parents did, and mostly let me use it. it wasn't cheap.

My Mac II + RAM + laser printer was something like 12K.

20mb hard drive on that thing. mostly used floppy diskettes.

@shintuku The residue class of $x$ modulo $n$ is a subset of the integers. It contains all integers $k$ so that $k\equiv x\pmod{n}$. Since $x\equiv x\pmod{n}$, $x$ is in it's own modulo class mod $n$

1:20 AM

our "pc clone" (people were still saying that) was in the $2000 range plus peripherals sold separately.

Now I get computers for 1.8K or something.

You missed the point of contention, @robjohn.

@TedShifrin I usually do. :-(

Awwww ....

It's about 0.

@TedShifrin modulo $0$?

No!

1:22 AM

good

it's the "Since x = x (mod n)"

i don't like a = b mod n language in general but it does make that very easy to see.

\equiv is fine

@leslietownes you'd rather see $n\mid a-b$?

or is it the $=$ vs $\equiv$?

it's the (mod n) you have to write every time. it makes students who know something think you have to compute a 'mod' function, that everything is {0, 1, 2, ... , n-1}.

i like equiv and then you define it in the normal way.

oops

Unoops

1:26 AM

it's not symmetric notation. you have mod n on the right. it's ugly and i hate it.

$\bmod{(notation)}$ okay $\bmod{(notation)}$ <- symmetric

Writing $[a]_n$ gets tedious too.

quick proof using the self-containment property: show that for two integers a, b; if their equivalence classes are the same, they are modulations of one another. Proof: trivially, if their equivalence classes are the same, the two integers can be found in either of their equivalence classes, which means one is a certain number of modulations by n away from the other

does this work?

I use bar notation once $n$ is tixed.

robjohn, i was thinking more along the lines of $n \mod a \equiv b \mod n$

1:28 AM

are you familiar with Lowess models @robjohn ?

Show subset both ways, shintuku.

I prefer equivance classes instead of mod mud.

@dc3rd not by that name

maybe even $n \operatorname{dom} a \equiv b \mod n$.

if i could mirror the n, i would.

but the concept you do know ? I'm only asking based on the "mean squared" user name so I figure you would know a thing or two about it.

1:29 AM

You're fired.

@TedShifrin Hi

Howdy.

No need to ping.

sorry

if $[a] = [b]$; $a, b \subset [a]$. therefore: $a \equiv b \mod n$

I am used to it

haha

Ted I finished my course in Thermodynamics got 95%.

1:31 AM

I should say Lowess method rather....

I will put these certificates in my C.V.

wow, well done.

Nice! I took two chem thermo classes in college and loved them.

$(w\, \text{pow})\ x\equiv y\ (\text{mod}\, m)$

as long as the symmetry in the notation has order 2.

1:32 AM

Typical math departments won't be interested.

Yeah it is so cool. My next Physics course will be Electrodynamics.

@robjohn hm, what is \rm{pow}?

I want to also work in mathematical physics it won't interest them ?

@shintuku tried to rotate $\rm{mod}$

incredible

1:33 AM

Not listing classes/certificates. Do you list every math course?

This is not how adults operate.

@shintuku are you running ChatJax?

If a particular school asks, then you tell them.

Yeah I agree.

@robjohn ohhhh that's what I'm missing

@robjohn Am I still blurry?

1:35 AM

@TedShifrin I apologize if my pings annoy you. I do them in case the room has many conversations at once.

I though people here got so used to writing mathjax that they didn't even care anymore for the actualy latex representation

where do I get chatjax?

@TedShifrin as soon as I am not picky, you will not be blurry

You don't generally ping for just a “hi.”

@robjohn Could be a while!

2

LOL

1:40 AM

square identify 00 with 11 and 10 with 01

is this called a squaar?

cube identify opposite diagonal vertices

name?

No names that I've ever heard.

do you treat the square and cube to be homeomorphic to a circle and sphere here?

alright, second attempt. Show that, for $a,b \in \mathbb{Z}$, $[a] = [b]$ implies $a \equiv b (\mod n)$. Trivially, $[a] = [b]$ implies that $a, b \in [a]$. Let Let $k$ be an integer some integer s.t. $a + kn = b$. Therefore, $a \equiv b (\mod n)$.

The first sentence is tautology to me.

If $a\in [b]$, then $[a]\subset [b]$.

huh, right

thanks hehe

1:57 AM

@shintuku In your defense, it's almost impossible to do trivial proofs.

I have always considered all that is left to the reader is to despair if a trivial proof isn't self-evident

pro is all that is left when you take away the of

@shintuku look in the upper right of the chat screen. There is a description of the room which includes a link to the installation page for ChatJax.

let me know if you have trouble getting it going.

2:12 AM

seems like edge isn't liking it

no reaction

@shintuku you have to run the bookmark when you're on this page.

I'll try on chrome

no reaction on edge

I don't know how many use Edge here, but I have only heard people having trouble with Chrome, but not everyone using Chrome has had problems.

@Ted: do you use Chrome?

Talking to shadows again.

alright, works on firefox

yes, it definitely works on FF

2:17 AM

thanks for the plugin! both for the reference and making it hehe

It's a lot easier to read the room when the MathJax is rendered.

okay I have refined my question

consider projective 2-space but with only finitely many antipodal points identified

this would be a subspace of projective 2-space?

@geocalc33 How do you identify only a finite set of points? I'm not sure I understand.

2:34 AM

@robjohn consider a unit disk with 4 boundary points s.t. the boundary points define a square. Identify pairs of boundary points that are furthest apart

@geocalc33 okay, but then there are other points that need to be identified, points near one of those points will be identified with points near the point identified with that one, right?

no

@robjohn I wasn't thinking that

okay, I am misunderstanding

@robjohn what about consider a (filled in square). And identify only (0,0) with (1,1) and (0,1) with (1,0)

does that make more sense?

So the edges are not identified?

just the corners?

so it has a boundary

2:41 AM

edges are not identified

correct

I was (visualizing) it as taking a filled in unit square and identifying a pair of opposite diagonal points by "pulling them downward" and identifying the other pair of opposite diagonal points by "pulling them upward"

@geocalc33 yes. I get that.

@robjohn Yes. I do MSE on Chrome on my computer, use Chrome for chat on my iPhone (rarely) but Safari on my iPad. I forget why.

feynhat

2:57 AM

I accepted IU's offer.

i use lynx

@geocalc33 Absolutely not.

@TedShifrin Okay, just checking that at least someone is using Chrome and that ChatJax works there.

i use chrome (among others)

it works on my moto g power and my laptops.

and my bidet screen

@copper.hat that's good, so it's not chrome that is the problem that people are having

feynhat

2:59 AM

does lynx have vim bindings?

@robjohn: Works just fine. I think I told you that a few weeks ago, too.

Yeah, just checking. I was telling shintuku

@feynhat: Congratulations on making the decision.

@feynhat i was kidding. i used to use lynx a few decades ago

when i was a paid up member of eff

@geocalc It's a perfectly orientable surface, and the projective plane is not. You can picture it and construct it easily enough. Take a handkerchief and put the two opposite corners together; then do it again (except do it underneath the first glued point).

3:01 AM

do folks nowadays know what a handkerchief is?

like a pay phone.

why is orient used for direction but occident not?

@copper.hat Shaddup!!

i was waiting for it :-)

3:16 AM

it's a shame i was playing with my daughter 15 minutes ago or i would have made that joke

i have an idea

on balance, probably better to be parenting than wisecracking on the internet. she's in bed now, though, so i would have said something like "take a handkerchief, you know, from the set of the nearest silent movie. if they're out, ask jeeves to bring you one."

my daughter's learned to say that. "i have an idea!" she hasn't learned to have great ideas but she does have them.

i want to make a website (similar to proofwiki) of mathematical proofs

please use plain unstilted english.

sry

3:19 AM

i mean in your website

i wasn't implying anything about your english above.

and remember your good friends langle and rangle. there, two good ideas already.

people can sign up and add proofs

would someone like to help

make sure there are really large margins

yeah

but i am worried about spam

despite my best efforts, my offspring love spam.

what would be different from proofwiki?

3:26 AM

a bit different

one of the o's in the domain name would be a 0, and one of the i's would be an l. also a lot of really dodgy websites would be advertising on it.

'men's health'

proofbase.org

Or something like that

still need a value proposition...

yes...

3:27 AM

if it was tightened to one area and not just a junkyard of proofs, that could be something. i think the category people have something like that.

nonsense.org

i would try

making such a website with html css js is difficult

reinventing the what now?

can't anybody make their own wiki? i thought some version of that was in some sense open.

you would just use an existing framework surely>

Well yes

@leslietownes i was thinking of a different design

3:31 AM

you will spend your days tweaking that then

@copper.hat occidental disorientation

maybe i will not use html css and use angular

disoccidentation surely?

i was completely disoccidented

I was completely dichotomized

do we need to pick a direction, won't the woke pc police be after us?

i'll take Ted's advice now.

3:34 AM

does anyone know if, by definition, the inverse element of a non-abelian group is commutative?

what does that mean?

i meant, operations with the inverse element of a non-abelian group is commutative

every element of a group is an inverse element.

there is no "inverse element" of a group. there is an inversion operation, but it is unary, not binary, so it has no hope of being commutative.

whew, I missed something there. back to reading!

3:36 AM

each element of a group does have an inverse element. but you wouldn't be able to single out any one of them as "the" inverse element of the group.

ah, yes that makes perfect sense

unless your group is remarkable small. a grp maybe. $G=\{e\}$. the search for a non abelian trivial group continues.

ok, suppose I have $a, a^{-1}$ in the group $G$. I don't know whether $a*a^{-1}$ is commutative unless G is an abelian group, right?

the cyclic group generated by an element must be abelian

Inverses are always two-sides inverses. Learn definitions!

3:40 AM

ahhh, there you go! I thought so as well, but found it incredible for groups to have that property

this level of group discussion i can handle.

thought there was an error in the wikipedia article

there has never been an error in a wikipedia article.

i would not rely on wiki, there are many mistakes.

wikipedia never has errors

3:41 AM

my great grandfather has a wikipedia page. only one in our entire genetic line.

shintoku, again just for clarity of thinking, you wouldn't say $a \star a^{-1}$ (which is an element of $G$) is "commutative." Commutativity is a property of a binary operation, or (by common usage) of the group with that binary operation. not of individual elements.

a pair of elements of a group can be said to "commute."

does that form a group

yes. i think copper.hat is the identity element of that group

i am the perverse element.

@leslietownes thanks for that added info!

so $a, a^{-1}$ always commute

3:43 AM

now we're talkin

by definition

that is very, very swell

the wiki articles for fairly elementary stuff tend to be fairly good in terms of accuracy of information, but often not good in terms of quantity. too much or too little. too many perspectives.

i should have known that an hour ago when I started filling out this cayley table

even fairly bad textbooks at least try to have some consistency from one page to the other in terms of what follows what. that's somewhat incompatible with wikipedia, so you have notions and definitions from all over sort of swimming around with one another. even if none of them are "wrong" it gets confusing.

3:44 AM

if it is any consolation, i am still grinding through elementary group theory texts after many decades.

there's a dover reprint of rose's group theory. i got pretty well into it, then it broke my brain

has anyone here ever created a website

not as people would mean it today.

many times

i created some very crummy websites back when just anybody could do it. they were functional but people couldn't put very much into them. strings of text maybe but no creating pages or anything like that.

3:58 AM

i made a rough design of how it would look like

this is a very rough design

the website can be different from this

4:09 AM

the website can be useful for many people

my main goal is that the website should contain the proof of almost every well known theorem

not to be snarky but this sounds an awful lot like proofwiki

i obviously don't know every theorem and proof so this is why people can add proofs themselves

@leslietownes can you tell why

just the first like of their site description: "ProofWiki is an online compendium of mathematical proofs! Our goal is the collection, collaboration and classification of mathematical proofs. "

they don't require theorems to be well known apparently, and they don't explicitly state that they want almost all of them. but it seems similar in spirit.

well it is meant to be similar. my website will be a compendium of proofs, and proofwiki is also a compendium of proofs

Why duplicate? Seems a bad idea.

4:23 AM

anyone can give me ideas to make it better

and it will not be a complete duplicate

only the purpose is bit same

you need a better value proposition.

offer NFTs. when someone submits a proof, give them a page they can print out that says "hooray, you did it! this is non fungible. it's all yours. [MD5 hash of whatever they put in]"

ok

and the website will not have ads

i'm joking. i should be clearer about that.

i don't want to earn anything from it

4:28 AM

that is sweet, but everything needs money to keep it going.

yes

i will try myself

and donations (if someone wants to give)? is it a good idea?

not to rain on your parade, but no.

okay so yes i need ads

i mean the idea is not distinct enough. why not add proofs to proofwiki?

well actually because i had some more ideas

i can add proofs to proofwiki but i cannot change the website

4:46 AM

Perhaps it's hubris to want to create a new website when a perfectly good mousetrap is already there.

(My "perhaps" is more than gentle.)

Rajorshi Koyal

I have a doubt in certain form of applied math..It is a request if somebody knows please answer.Ted Shifrin you can ignore if you do not like it

I read somewhere that econometry is defined by, "The field that seeks to apply statistical and mathematical methods to economic analysis. The sophisticated analyses of micro and macro sub-fields would not have been possible without the major advances made in econometrics over the past century or so."
I did some research and found out that,"Quantitative Analysis of Actual phenomena based on concurrent development of theory and observation related by appropriate methods of inference." I would like to get a detailed insight into how both of these are related.

What is meant by quantitative analysis of actual phenomena?

Based on concurrent development of theory and observation,related by appropriate methods of inference?

Please give me a detailed insight into the same..

Why are you asking mathematicians about economics and econometrics?

Please give us detailed insight into why you keep bothering us with things that are not mathematics.

@RajorshiKoyal its just statistics applied to economics

@RajorshiKoyal Economics might be a better site.