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12:25 AM
I just defended my bachelor thesis :)
12:35 AM
12:46 AM
Thanks, I thought I was going to be asked a ton of technical questions but everyone was kind to me
We can make up for that here, if you so desire.
I have 40+ years of experience! :D
I think Ill wait till I defend my phd to get pummeled
OK, up to you.
3 hours later…
3:31 AM
What's a good symbol for an alternative group addition? Kind of like \oplus but not \oplus because this already has a common meaning in group theory. :P
Hi everybody.
Good Morning.
i like oplus. people familiar with the direct sum are unlikely to be confused by it. i like $\star$ for an arbitrary operation but if you want some way of notationally signaling a commutative operation it may not be ideal for that purpose.
amsmath has \boxplus which is exactly what it sounds like.
i've seen people overset or underset symbols on plus. like dots or lines.
Hi, lone.
3:46 AM
@leslietownes I forgot about boxplus! Thanks!
4:04 AM
as in good evening...
lone had morning, so i was countering :-)
he's right, you know. it is evening.
in albany summer time, at least :-)
am curating a list of psqs to address in the following 4 exercise free days.
4:30 AM
A long list?
just kidding :-). exercise is my therapy, without it i will be climbing the walls and more cantankerous that usual.
Not to mention obstreperous.
wonderful words
trying to figure where i can rent a tuxedo for my son for his prom
@copper.hat I guess in your country now it is evening?
obstreperous, captious................the vocabulary one learns in this chat.
4:40 AM
@lonestudent in one of my countries, yes :-) in the other it is very early morning.
captious is my word of the month, i am waiting to use it in convo.
@TedShifrin Hi. Respects. $\ddot\smile$
but i am isolated at present, so not much opportunity
@copper.hat Here 08:41 ( 24 hours)
as in convo with real live people around?.............I forgot that this is a thing.....and it will remain only a thing as we are going to be remaining in a stay at home state up here..... :(
21:42 here (california)
@dc3rd yup. i need convo and go ever more insane without
4:43 AM
my arm is beginning to hurt
inoculation ?
sry to hear it.
it's better than the alternative.
plus, maybe an excuse to loaf around tomorrow. i hope i don't end up regretting saying that.
@copper.hat How are the situations there, are the workplaces closed? Can you make money?
4:46 AM
@lonestudent unfortunately i work remotely. but the work is the same.
Stay hydrated and be prepared for mild fever and headache, Leslie.
A Frenchman was in here earlier and I learned a French word I'd not known.
My wife had pfizer and felt bad for a few days. my son and myself had a mild headache but that could be hay fever.
What was the word :-)
to leave?
4:49 AM
@copper.hat Things are very awful here. You are in luck. Making money remotely is a good idea.
@lonestudent where is here?
But he paid me the ultimate compliment of saying he assumed I was French :)
:-), is that a good thing>
i was going to say.
oh, because of your french
4:50 AM
i guess if you're speaking french to someone it is.
I'm a Francophile, so yes.
mais, vous francais c'ete tres bien. es pour ca
Lone, you in India?
i don't think i can but two words of french together anymore, maybe je m'en something or other...
my sister is fluent. had a french bf for a while
@copper.hat I wish my English was good..I signed up for a some math site to earn some money. I got full points in mathematics (high school math but except probability). I failed because my English was bad. Those who are good at English are very lucky.
4:54 AM
@lonestudent i can only speak english, unfortunately. it is a very irregular language.
where are you?
@copper.hat Turkey.
nice. i hope to visit someday.
Your typed English is quite adequate, lone.
@copper.hat So hard
@TedShifrin I failed grammar exam .(5 times)
@lonestudent your written english is fine, as Ted mentioned. certainly better than my written Turkish :-)
4:58 AM
@TedShifrin I mean photomath
@TedShifrin I mean photomath for become math Expert haha ..I am not expert
@copper.hat do you know Turkish?
Most Americans would fail your grammar exams, too.
@lonestudent I know no other language. I do not even speak my 'native' language.
@TedShifrin Are you native speaker? Then you are lucky.
I know a little Bahasa Indonesia & a tiny bit of Mandarin.
Yes, and studied Latin, French, German, Russian.
5:02 AM
A polyglot
Well, sorta.
I have a few polyglot friends. Disgustingly accomplished.
i was really impressed with one of mine, and then i heard him speak spanish. the grammar was fine (as far as i could tell) but his accent was worse than mine ever was.
@copper.hat hepinizi seviyorum.
and i thought, what if he's like this in every language and people are just too polite to tell him.
5:04 AM
That happens, leslie, yup.
still impressive, but less so. i mean i enjoy a good accent myself.
@lonestudent :-)
at least i have no accent. everyone else does, though
that's my line.
you are mistaken
very often
We were teasing a friend the other night. He is French and every language he speaks is with a thick French accent.
5:07 AM
i like listening to little kids speaking other langauges
@lonestudent is it for students also to ask questions in app ? Is it free or paid ?
@Rover no, for become math expert. My problem " if and only if" is English. I can not progress.
@lonestudent ok
@leslietownes Is your native language american english or UK English? Can I ask?
@TedShifrin ok, my brain is resisting this question now , it's going on from 3 days ..
5:18 AM
i'll answer the second question first. yes. american english. :)
@leslietownes $$\ddot\smile$
@TedShifrin I missed your comment. "English is quite adequate" Thank you. I learned by answering math questions. I couldn't open my mouth before.
@rover That exercise I gave you is super cool. Generalizes to all dimensions.
@copper.hat are you here?
Would you please give it's solution, after trying out..
5:28 AM
@TedShifrin super-cooled exercises... do they float over magnets?
@robjohn Is your G5 reception going well?
Nope, Rover.
@lonestudent yes, but leaving for sleep shortly
Super-cooled phone
@copper.hat ohh I forget.."it was evening" well no problem. I wish well beings.
5:32 AM
Ah, not aware of that.
I was playing on 5G
@TedShifrin okay, then I have to sit and work it out .
That sounds like an excellent idea
but you could also stand. sometimes, pacing about helps for me, too.
i used to nap on math problems. it sometimes worked, and when it didn't, still got the nap.
5:44 AM
@leslietownes I sometimes get some of my best thinking in the shower. That has the added benefit of getting clean, if nothing else.
in places i've lived where there's a bathtub, that's also good.
it's a kind of mix of getting clean and lying down. i don't think i've ever napped in a bathtub and would not recommend it.
whats the equation of a cone with vertex through the origin?
we need to formulate a PDE for it
the axis can by anything. not necessarily the z axis
$x^2+y^2=m^2z^2$ axis is $z$-axis
@satan29 What does that mean?
Find the partial differential equation of all cones which have their vertex at the origin.
5:50 AM
Makes no sense to me.
Can you give an example?
for example, the previous question had the additional constraint that the axis is z axis
in which case, the equation is $x^2+y^2=m^2z^2$
more context is needed. an example would help immensely.
You said PDE
5:52 AM
so basically
we need a PDE with a solution z(x,y)= sqrt(x^2+y^2)/m
this is reminding me vaguely of those problems in calculus books where you're given a family of curves and asked to find an ODE having that family as solutions. but i don't want to guess.
$m^2(u\cdot p)^2=|p-(u\cdot p)u|^2$ where $u$ is any unit vector.
Let me show you the solution for this example
I know all sorts of differential systems stuff that sounds like this but isn't.
we get $2mz z_{x}= 2x$
dividing, $z_{x} /z_{y} = x/y$
So, the required PDE is
5:55 AM
I see.
but I'm guessing that has the $z$-axis as its axis
My book defines $z_{x}$ and $z_{y}$ as $p$ and $q$, and $z_{xx}, z_{xy},z_{yy}$ as $r,s,t$
so the answer is py-qx=0
@robjohn yes
But without specifying an axis you aren't guaranteed independent/dependent variables.
the next question doesnt have this constraint
@TedShifrin didnt get you...
@robjohn that simplifies to $p\cdot p=\left(m^2+1\right)(u\cdot p)^2$
6:04 AM
BTW your example gives all cones with z-axis as axis …. No control over Vertex.
hmm yes
You can't have that control with a differential equation.
yes, because you can always add a +c
i guess?
that wont change the derivatives
My other point is that if you have a different axis, then it won't be a graph of a function of $x,y$.
because it fails the vertical line test?
i.e you get 2 outputs for a single input?
6:10 AM
Right. So I don't know what your book expects
i mean they just want to repeat the procedure
with the equation of a general cone i guess
It needs to be implicit, not explicit.
3 hours later…
9:30 AM
Q: To find $\dim_B F$, why is it enough to consider limits as $𝛿\to 0$ through $\{𝛿_k\}$ such that $𝛿_{k+1} \ge c𝛿_k$ for some $0 < c < 1$?

epsilon-emperorTo set the stage, let me recall the definition of the box-counting dimension of a set $F \subset \mathbb R^n$. The lower and upper box-counting dimensions of a subset $F$ of $ℝ^n$ are given by $$\underline{\dim}_B F = \underline{\lim}_{\delta\to 0} \frac{\log N_\delta(F)}{-\log \delta}$$ $$\over...

I just posted a question here, but I have figured out the answer now
Should I delete the post, or post an answer? What is suggested?
Are we allowed to ask our questions from here ?
3 hours later…
12:40 PM
Anybody here?
@Thorgott Hm, I'm not currently reading through a book, so I figured I could use the definition which says that $F/K$ is purely inseparable if the separable degree of $F/K$ is $1$. Now, the separable degree I take as the degree of $S/K$ where $S$ is the separable closure of $K$ in $F$ (so it consists of all elements in $F$ which are separable over $K$).
3 hours later…
3:18 PM
@ShaVuklia Say the characteristic is $p$. Pick $\alpha\in L$ and let $P$ be its minimal polynomial. Write $P=Q(X^{p^n})$ with $n$ as large as possible. Then $Q$ is irreducible and separable (why?), so it is the minimal polynomial of $\alpha^{p^n}$ and $K(\alpha^{p^n})/K$ is a separable subextension of $L/K$, which forces $\alpha^{p^n}\in K$ by the inseperability hypothesis.
3:59 PM
@epsilon-emperor I was going to suggest that you answer your own question. Your question is well-posed and so should pass the PSQ test, the answer would be a benefit to the site.
I see that you've deleted it, but you can undelete it.
4:16 PM
@robjohn What is the PSQ site?
@robjohn Cool, let me see how.
@epsilon-emperor PSQ = problem statement question
4:40 PM
@epsilon-emperor PSQ = "Problem Statement Question" or "Poorly Stated Question" depending on whom you ask. There are site standards on what is a well-posed question.
@copper.hat beat me to it ;-)
need something to take my mind of mind numbing sprint meeting.
and the utter collapse of my equities...
I got sidetracked while looking up the links for my response, so I didn't see your response until after I hit send.
@robjohn Oh thanks haha! Could you take a look at my latest question? It's an inequality that needs some geometric insight that I'm missing
4:57 PM
@epsilon-emperor I have answered, but maybe I missed some key point of your question.
5:28 PM
Could someone teach me more about monoidal categories / string diagrams?
As I have been so generous as to tutor several newer mathematicians here in python / algebra.
I understand how adjunctions work, but not enough to fully recall the definitions.
odd request
I know, thought I'd throw it out there at least
@robjohn Saw it, but not sure how it justifies N_\delta(C) \le c\delta^{-n}
In particular I'd like to understand more of the content of this glorious paper: arxiv.org/pdf/1208.5205.pdf
answering specific questions is fair game, teaching is a bit of a reach surely?
5:31 PM
No, I don't think so.
it was rhetorical
There are a lot of people willing to teach
Like I tought a few people here some things
I would like to be the student for once
@epsilon-emperor You want to link to the question?
Q: Why is $\dim_{B}F \le n$ in $\mathbb R^n$? (Upper Bound on Minkowski–Bouligand dimension)

epsilon-emperor(Please skip to the end for a word on notation) For $F \subset\mathbb R^n$, where $F\ne \varnothing$, we have $$0 \le \underline\dim_B F \le \overline\dim_B F \le n$$ and hence $$\dim_B F \le n$$ where $\dim_B F$ denotes the box-counting dimension of $F$. $\underline\dim_B F$ and $\overline\dim_...

This one
If the length of a side of the cube is $a$, then you need $(a/\delta)^n$ little cubes to cover it.
5:44 PM
of side \delta, you mean? right?
so you're using the third one of the five equivalent definitions?
@TedShifrin Oh okay so the constant "c" stated by the author is actually the nth power of the side of cube C?
Right. It depends on $C$ but not on $\delta$.
Got it, thank you! Maybe @robjohn could add this to their answer for clarity, it really helped me
@robjohn Answered this, by the way
Yes, it was a reasonable question.
5:53 PM
By the way, just a general question
I've noticed on this site that users are quick to downvote questions which do not meet community standards, which is great
but hardly anyone upvotes questions which meet community standards :(
the issue, i think, is that it's pretty easy to spot questions that don't meet standards even if you don't know or have any interest in how to solve a well-posed problem.
On several occasions, I've asked questions with excruciating detail of what I've tried and all typed in beautiful MathJax but it just gets ignored. My last few questions are a good example of this. How do I improve my posts so as to get answers, etc?
so it's a numbers game.
as a practical matter, the upvote standard seems to be more a vote of 'i am interested in this' rather than 'this is a well posed problem,' so fewer people are likely to click that button. that isn't the actual standard you see spelled out if you hover over the upvote button, which is "this question shows research effort, it is useful and clear."
I confess that I usually look at a subset of my targeted areas.
i expect some people who are hair triggers on downvoting obvious junk, but don't understand the mathematical substance of question, would say that they don't know whether it shows research effort or is useful or clear if they can't understand the question
5:58 PM
I agree with Leslie. I almost never downvote without first commenting, and then more often vote to close if no response.
@leslietownes I think that is sad, because if questions are upvoted then they are more likely to reach the target audience.
yeah, but if i read a post on higher category i wouldn't be able to tell if it was beautifully written word salad or an actual question, so i probably wouldn't upvote even if the mathjax was clear. although i tend not to downvote unless something is borderline abuse of the site. i'm a fan of commenting.
Ah, that makes sense too
By the way, robjohn did answer your question with $m$ instead of $\delta$, didn’t he?
He did! I just didn't realise where the side "s" came into picture from, until you pointed it out (the fact that its nth power is exactly "c")
6:02 PM
I typically upvote only when it's a good question that I understand :)
Should I avoid writing LONG questions? In that case, people might not read, I think.
for the proof of equivalence of box-dimension definitions, I wrote a post so long that I myself wasn't able to read it in one go later
i like if the main point of the question is made clear in a paragraph or two. i don't see supplemental information like definitions and such to contribute to 'length'
Too long is likely to be ignored. But it's important to define non-obvious notation and to indicate what you've tried and where you’re stuck. I thought your question was reasonable.
i am unlikely to read more than several paragraphs without being given a firm idea of where the poster is going with their question
OK, I upvoted, after all that :)
6:07 PM
I have a quick question about the site. Does upvoting old answers or commenting bump a question in some sense? Idk if this is an issue but people seem to yell at each other a lot for "necrobumping" stuff on the internet at large.
@leslietownes Got it! I just edited one of my questions and started it with "Main Theme of the Question" summarized in a few lines. Left the actual details for later paragraphs.
@TedShifrin Thanks :)
@Quin i think people are making trivial edits to 5-yr old questions. Very annoying. Upvotes don’t bring it to the forefront. Not sure about comments.
okay, thanks! I dare not edit anyones post. But there are some very helpful things that are older but I havent upvoted for this fear
Most high rep users are a lot more upvote friendly
although there's a small group of super downvoters
I get repeated upvotes on my stupidest answers.
6:14 PM
although many of their downvotes don't appear on their totals
since they also delete many questions
Yes, complicated answers do not seem to get a lot of upvotes
most user's "greatest hits" are easy stuff
that observation is what inspired the creation of tik tok
I’ve written a few good answers with lots of votes.
my highest upvoted answer, aside from what i would characterize as answers to soft questions, is pretty dumb.
the second highest is good, though. then more dumb.
all of my answers are dumb
I check before submitting
im a big fan of really simple stuff so im probably the target audience for a lot of the dumb posts. maybe it feels dumb because it is so simple, but a lot of people (in particular, myself) are no so clear in their thinking so the simple stuff is a great help
simple stuff has a larger target audience, and a larger portion of that audience is going to be willing to read it, since it's usually also quicker
6:21 PM
i would not equate clarity and simplicity with dumb.
I think it makes perfect sense
@copper.hat, thats my point!
there's also some high rep users who do amazing stuff
Ron and Wofsey are some great examples I think
e.g. i've read a lot of ted and leslies posts and have yet to read something dumb
Keep going :)
6:28 PM
the site wasn't as easily searchable way back when, so there were a lot of duplicates floating around. and no end to the PSQs.
wait what
has the number of dupes and psqs been reduced?
well now you can use google to pretty quickly locate duplicates. before google indexed that stuff you had, maybe the internal search function? which wasn't as good
so you'd have many more people answering versions of the same question. it was harder to validate that gut feeling of "this has to be a duplicate."
i do think the site is better at closing duplicates than it used to be.
glad to hear it
It still relies on people who've answered before seeing the dupes and remembering, no?
I think that's the bulk of the closed questions
6:35 PM
people on the site are better at closing duplicates, i should say. it's easier to find them if you suspect they are out there.
maybe it's easier because now there's a lot of duplicate targets
i'm not sure i get the big worry about duplicates.
there's a lot of high iq search engine experts
who assure me dupes are bad
although I am too smooth brain to understand :'(
hi i have a doubt regarding crammer's rule (Determinants).If D=0 then either there is infinite number if soln or no soln. But what if D=0 and one of the other determinants(D1,D2,D3) is zero
That's the infinitely many solutions case, isn't it?
6:39 PM
So if one of the three is zero infinite soln is guaranteed/
I haven’t thought about it.
I have never seen a case like that while doing probs.
Cramer's rule is mostly useless, anyhow.
we agree again.
it was used for a prank in my university in the seventies
they told people they needed to bring their cramer ruler for class
It makes more sense in spanish
6:43 PM
It is theoretically important and ok for 2-variable problems.
Found this in a site
"When D=0 and either Dx or Dy is nonzero then the system is inconsistent."
Maybe you should try to understand/prove it.
That's the 2D case.
s & 0 \\
0 & 1/s \\
e^s & 0 \\
0 & e^{-s} \\
The first matrix can be used to do a "squeeze mapping"which is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping
The second matrix can be used for the same purpose? Which one is more preferable?
6:59 PM
preferable for what?
the second family is just the subset of the first family with positive coefficients, parametrized slightly differently. so if you care about mapping the first quadrant to itself, i guess you would want to limit to that
that's the only difference i notice. absent context i have no view on what is preferable
yeah that's what I was thinking too, that they are different parametrizations of the same mapping
7:12 PM
Also I read that, $\bigg\{\begin{pmatrix}
e^{s} & 0 \\
0 & e^{-s}
\end{pmatrix},s\in\Bbb R\bigg\}$ is the one parameter matrix flow of the geodesic flow
Does that mean $\bigg\{\begin{pmatrix} s & 0 \\ 0 & 1/s \end{pmatrix},s\in\Bbb R\bigg\}$
is also the one parameter matrix flow of the geodesic flow?
geodesic flow on what?
without getting into what 'the geodesic flow' is, note that you can't have s equal to zero in that second parametrization, and that the first parametrization gives you a group homomorphism from the additive group of reals to a group of invertible matrices, and the second does not
this may be an obstacle to your second map having nice properties, such as arising from a 'flow'
but this would be buried in the definitions of what that flow is
it can't be answered at the current level of detail
okay thank you
7:41 PM
Does anyone know of cool applications of multivariable calculus in pure math?
I need to choose a topic for a project
8:13 PM
tough question.
well, i can't think of a good answer. i assume 'multivariable calculus' tops out at approximately dimension 3 (i.e. while it may introduce more abstract tools it does not expressly bill them as such). that's mostly where i'm getting stuck.
i'm stuck on 'pure math'.
that too.
@leslietownes no it's fine if it has however many dimensions
or infinitely many
@copper.hat well I have to do a presentation and I want it to be interesting to an audience of math undergrads and professors
8:18 PM
if you flow a simple closed curve in the plane by its curvature, it eventually becomes convex, if it wasn't before, and it converges to a point in the appropriate sense, and if you scale the flowed curves so that they have constant area, the resulting family of curves converges to a circle. i think that is a cool and intuitive low dimensional result.
and a lot of it can be motivated via multivariable calculus, but as far as i know, eventually you need to appeal to PDE (which you can mostly use as a black box) and some pretty gross formulas.
that sounds interesting
what does it mean to flow a curve by its curvature
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature...
what you'd expect. differentiate the family of curves appropriately and you get the curvature
or mean curvature i guess in the general case
that you can do this might be where PDE comes in, at least in general
ah, extrinsic curvature
and the flow gives an isotopy, cool
there used to be a website where you could draw a curve and watch the flow. i can't find it
8:33 PM
check out the end of Ted's youtube lecture series for applications of multivariable calculus
9:31 PM
i found it. math.berkeley.edu/~sethian/Applets/java_files_curve_flow/… it's java. good luck finding a browser that will run it
i can't find where the app is on the page, i keep going in circles...
at least you are visitor number to this page
9:47 PM
i think the server still has that stuff, modern browsers just refuse to load it. some of math.berkeley.edu/~sethian/2006/Applications/Geometry/… still works.
my computer doesn't know what to do with that mpeg. maybe a more widely understood format, such as RealVideo, would have been better.
or .FLC.

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