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12:00 AM
No, I love Mathematica, but I find that it is terrible at trigonometric simplifications, even when I tell it to simplify.
@Semiclassical I love to learn it but only if it is worth it.
well, take a look at the top of it at least
also, their picture is sorta cute
EulerAngles[r] gives Euler angles $\{\mathrm{A},\mathrm{B},\Gamma\}$ corresponding to the rotation matrix r.
@Semiclassical is this carried out by Mathematica?
@robjohn right
12:01 AM
Oh cool, robjohn.
the default version assumes rotation order z-y-z
I was thinking to illustrate them by Tikz.
EulerAngles[r,{a,b,c}] allows you to specify rotation order a,b,c
let's see if i can verify that, though
it may or may not work for symbolic matrices
actually, it probably can't: how would mathematica know if your matrix is actually a rotation matrix?
@Semiclassical solving symbolical equations I guess
EulerMatrix[{alpha,beta,gamma}] does verify their equation 2.72
12:04 AM
I'm not sure to be honest if the software is smart enough to simplify the trig identities.
i'm not too worried about that, tbh
mathematica may not be great at simplifying trig functions in the way you want it to, but it's pretty good at verifying when two things are equal
do you know any commercial software though? i've heard about maple
mathematica is commercial
I've only used Mathematica
also, no dice on mathematica being smart enough
assuming i've put everything in right of course
12:12 AM
Well, may be sometimes hand-calculations is the destiny.
But my university provides mathematica to students. I guess I should give it a chance
one does have $r_{33}=c\beta$ at least
@Semiclassical I know how to do it by hands
12:15 AM
@CroCo Wolfram: hey, kiddies, want a free dime bag?
@robjohn what's a dime bag?
@CroCo drug pushers used to get kids hooked with free drugs (dime bags)
Then when they are hooked, start charging
@robjohn lol. So true. Well, in my case, the university will pay.
I can see this with other software and apps.
Apple is a prime example.
12:37 AM
$\exp_3(S^1)$, defined as the set of subsets of $S^1$ containing between $1$ and $3$ points, is homeomorphic to the 3-sphere
$S^1$, as a subset of it, is a trefoil knot
2 hours later…
2:17 AM
Hi All..
@CroCo Yes, they will pay for the license while you're at the university, but once you're hooked, and you're somewhere else...
2:44 AM
Feb 3 '17 at 20:47, by Akiva Weinberger
@TedShifrin I love how one can smoothly transition between "not knowing anything about the problem" to "teaching Socratically"
Random memory
Klein bottle you can put in 3d
From the image, I can see that identifying 1 and 3 gives a part of klein bottle but I can't see why identifying 3 and 2 gives part b
you kinda push 2 through 3 and then it meets 1
Maybe but I think the orientation should be reversed
2:57 AM
2's orientation is backwards
Your first picture is wrong
You should have one pair of opposite edges oriented the same way, and the other pair of opposite edges oriented the opposite way
@AkivaWeinberger this
the orientations match up tho
The above should be write. The fundamental group of the above space is $\langle a,b,c|aba^{-1}b^{-1}cbc^{-1}\rangle$ if I let $c$ be a line segment connecting the inner circle and bottom left vertex
3:01 AM
'it isn't the klein bottle" i think is the resolution
What if you took your first picture and just reversed the orientation of the innermost circle
matching 2 and three would be tough
I mean, it doesn't say which way to identify the boundary circle with the longitudinal circle
true, could "slide inwards"
It seems the same problem happen when I identify 1 and 3 but twisting would give the space
I don't know this is the right one
3:04 AM
Reverse 3, identify it with 1 by leaving 1 static and "pulling 3 inside" and over, and then bring 2 around
No wait
Do what I just said but don't reverse 3
It's unnecessary
3 mins ago, by Akiva Weinberger
I mean, it doesn't say which way to identify the boundary circle with the longitudinal circle
^This choice doesn't matter
I can't understand the part 3
OK, can someone help with this (it's different than what I posted before).
If partial sum of $\sum_{k=2}^{\infty} a_k$ (starting at 2) is given by $S_n=(n-2)/(n+6)$, find the sum (from 1) of $\sum_{k=1}^{\infty} a_k$.
@love_sodam me neither
still 3 is weird
3:14 AM
How so
I don't understand where the hole is. like it is "pushed in"?
In Sodam's third picture, 3 is the dotted lines. Erase them, and it looks like a torus
I see what you're doing
yeah I saw it now
Thanks a lot @AkivaWeinberger !
3:17 AM
No problem
@Jeff Interesting. What's the formula for $S_n-S_{n-1}$? (That should be $a_n$)
If we assume $a_1$ follows the same formula, we can add it on to the rest of the series
Not given (that's a different part of the problem). But it comes out to $a_n = 8/((n+5)(n+6))$. However, it does not hold for $a_2$. @AkivaWeinberger
I don't think we have enough information
$a_1$ could be anything without affecting the partial sum of the sum starting at $a_2$
@AkivaWeinberger Not what the boss says.
@AkivaWeinberger Good point
3:39 AM
@Jeff Are you still going on with this? Nuts.
@TedShifrin It's not quite the same. I misunderstood the problem (there were no typos).
You’re still typing garbage that I corrected before.
Given the sum starting from 2, and the formula for its partial sum, find term a_1, and find the sum from 1.
Yes, but that's how I was given it.
It's crap.
OK. It's still how it was given.
3:43 AM
You are not typing partial sum when you go to infinity, and there's no way to know what $a_1$ could be, Be a critical thinker and explain to your teacher why it's not rightf. Stop telling us crap was given ..
There are bad teachers, but you have to hold them accountable. We have no power to mindread.
I say this having taught more than 40 years.
Yeah, I tried telling her. She didn't reply until I told her that I gave students credit for the problem because it had a type. Then (and only then) did she reply "There were no typos". But she said nothing else. She is not what you would call helpful.
@TedShifrin :D
In fact, she gave me a problem the week before that which was (that conditionally convergent series) that couldn't be done with the tools given and she insists it's right.
Well, that one might have been right. I need to see screen shots of what she gives before I can say definitively.
She and I don't get along very well. She is very micromanaging, so she looks at all of the papers I grade and critiques all my grading (which, I might add, is fine for all the other professors in the department).
Oh, you're a grader, not in the class?
Not in the class. Recitation instructor.
3:51 AM
So you are a grad student? You know better than this.
Complain to the graduate coordinator or head of undergraduate studies. Don't complain to us..
I wasn't complaining here. I was asking for help on a problem I didn't understand. She is who she is.
She's in charge of the course and you have to make her responsible. As I said, if she is not responsible, complain to the department .
I spent 8 years being associate department head and dealing with such fun.
4:39 AM
Where is the issue in the reasoning that $$0 = \Im(-1) = \Im(i^2) = \Im(i)\Im(i) = 1^2$$, where $$\Im(z) = \frac{z - \overline{z}}{2i}$$ is the standard imaginary part.
imaginary part isn't multiplicative
ok cool. I think I just remembered that both z-> z and z-> \overline{z} are, and decided to assume or something
comments from a fan
Thanks, DogAteMy.
4:58 AM
Can I edit this question? The edit will turn the given answer useless...but it doesn't answer the question.
What if $\theta$ doesn’t evenly divide 360?
5:20 AM
> If $n$ is in decimal then only the integral part is taken.
I don't believe that.
Uhm Why? If $\theta$ doesn’t evenly divide 360 then an incomplete image would be created...
2 hours later…
7:10 AM
never thought I would dig so deep in stats
for the sake of my new project...
I am making a scientific python project
I am first making statistical functions
I don't know much about stats so I may ask silly questions...
is there any nice sort of algorithm to calculate variance...?
8:12 AM
@LeakyNun thanks for that, made it so easy :)
in a separable metric space, is it always possible to find a sequence of uniformly continuous functions that converge pointwise to the indicator on an open set?
@LeonhardEuler You mean like the mean of the squares minus the square of the mean?
nvm I already coded it
now I am working on some methods to visualize data
@LeonhardEuler Ah, you mean you're learning to lie ;-)
8:21 AM
How to Lie with Statistics is a book written by Darrell Huff in 1954 presenting an introduction to statistics for the general reader. Not a statistician, Huff was a journalist who wrote many "how to" articles as a freelancer. The book is a brief, breezy illustrated volume outlining the misuse of statistics and errors in the interpretation of statistics, and how these errors may create incorrect conclusions. In the 1960s and 1970s, it became a standard textbook introduction to the subject of statistics for many college students. It has become one of the best-selling statistics books in history...
I am going to burst out laughing
So many people have lied with statistics that this book was written to show people how to avoid it.
Is it possible to code analytical functions like zeta function?
I mean approximating the value is easy but I am talking about the exact value
Well there is a formula for zeta function at even integers
The exact values are only known for a countable number of real points.
8:27 AM
and infinite sums?
@LeonhardEuler and negative integers.
I wonder which algorithms mathematica uses
@LeonhardEuler for approximating?
@LeonhardEuler this answer does that
@robjohn for getting the exact value
I think mathematica uses a database where many standard infinite sums' values, integral, etc. are stored
it simplifies the given integral or sum and then uses the values from the database
I think this is it
Okay I am rewriting my code in c++
have fun
8:45 AM
Dear @robjohn I am truly thankful to you for not banning me for 30 min.-Your Sincere Audience William
under what circumstances do two measures on metric spaces coincide if they agree on all open or closed sets of the metric space? (here the metric space has the borel sigma algebra)
if the measures are sigma finite, then certainly I can see that this is true
but does it necessarily hold in general?
If $X = S^1\times S^1$ then how can I presentation of $\pi_1(T_f)$ in terms of the induced map $f_*:\pi_1(X)\to \pi_1(X)$ where $f:(X,x_0)\to(X,x_0)$ is a continuous map. $T_f := X\times I/(x,0)\sim (f(x),1)$ for $x\in X$ ?
9:07 AM
@porridgemathematics ack! don't make me think about non-sigma finite measures.
Feb 21 at 21:48, by robjohn
@Jakobian I guess I am saying that non-sigma finite measures are very weird creatures.
@eryceriousmatherfunker what did you do that might require that?
9:33 AM
wow the problem is too hard to me
@robjohn Hehe some things should be kept secret. Apologies for wasting your time my lord.
9:58 AM
@porridgemathematics No, there are easy counterexamples
There are arguments with the monotone class theorem or Dynkin's $\pi-\lambda$ stuff that give sufficient conditions but I always forget how these work
Look at the counting measure on $\Bbb R$ and the measure that is $0$ on $\varnothing$ and $\infty$ everywhere else for example
If you try to prove that they agree on all Borel then you see that the issue is with complements, you can't say anything when a set and its complement both have measure infinity
10:42 AM
@eryceriousmatherfunker just don't get too happy.
10:58 AM
@robjohn thanks for saying that
I don't always use measures, but when I do, I use sigma-finite.
11:31 AM
Hi I am stuck at this integral.
Here, m is non negative. I tried to take E as a complex variable, complete a contour and integrate, but that didn't work out as the integral doesn't seem to converge at infinity for any non negative t.
Any other suggestions?
I am interested in the t-> infinity limit. So if I solve the integral, or justify limit, both are okay for me.
11:49 AM
hi, could someone help me figure out why 'it is clear' that $\mu(K) > 1 - \epsilon$ in this proof? imgur.com/a/75EJMDK
@porridgemathematics Let $A$ be open, so that $A^c$ is closed, take $f_n(x) = \min\{1,n\cdot d(x,A^c)\}$. If $d(x,y) < \varepsilon$ then $d(x,A^c) \leq d(x,y) + d(y,A^c)$ and vice versa so $|d(x,A^c) - d(y,A^c)| < \varepsilon$, and capping at 1 does not hurt this
oh you're answering an old question of mine, yeah I found one, I went with $(min(1,d(x,A^c)))^{\frac{1}{n}}$
of course, you're works just as well
@porridgemathematics I guess geometric series
so you're saying if we put $A_n = \cup_{k=0}^{N_n} B(r_k,\frac{1}{n})$, then $\mu(A_1 \cap ... \cap A_k) > 1 - (\frac{\epsilon}{2^1} + ... + \frac{\epsilon}{2^k})$, but why?
Is that really what I'm saying
12:00 PM
idk, what are you saying
I think I'm saying $\mu(X\setminus K) = \mu(\text{union of sets of measure less than eps/2^n}) < \epsilon$
or <= whatever
ah okay, yeah that makes sense
thanks :)
one can probably also phrase this in a direct way, but like this I find it the easiest
you're welcome!
it's stuff that I have to revisit anyways
12:19 PM
@NamanAgarwal are you sure this converges
@user2103480 No. Unfortunately, I am not sure about that.
exchange i with $(i + \varepsilon)$ for any $\varepsilon > 0$ and then this should at least converge
2 hours later…
2:06 PM
How change decimal in binary?
two methods are explained here ^
2:36 PM
Does anyone know of any interesting diffusion equations where the diffusion matrix is singular?
2:59 PM
Conjecture: for any sequence of $1$s and $-1$s, there exists a $c$ such that $(-1)^{\lfloor c3^n/2^n\rfloor}$ realizes that sequence
Hello, can anyone prood these two well-known facts in functional analysis by constructing inverse maps?
Q: Prove that for $W$ is closed vector subspace of $V$, $V/W\cong W^\perp$ as well as $V^*/W^\perp\cong W^*$ by a different method

MikeThis is a well-known result in functional analysis, in terms of dual of quotient spaces and annihilators of subspaces. Let me formulate the problem first and a new attempt to prove it: Let $W$ closed vector subspace of $V$, $V/W$ the quotient space, and $W^\perp$ the annihilator of $W$. (1) Show ...

@AlessandroCodenotti define a counting measure to be a measure on $(\Bbb R, \mathcal{B}(\Bbb R))$ that has values in $\Bbb N$ and is finite on every finite interval. This is equivalent to a an increasing sequence in $\Bbb R^{\Bbb Z}$ that goes from $-\infty$ to $\infty$, where we assign to every interval the number of points in the sequence lying in said interval.

We can define a kind of "weak" measurabability on the space of counting measures $\mathcal M$ by choosing the sigma algebra generated by evaluation functions $\mathrm{ev}_C \colon \mathcal M \rightarrow \Bbb N, m \mapsto m(C)$, a
3:28 PM
@RyanUnger what have you been up to? Are you still doing Riemannian stuff?
3:47 PM
@TedShifrin hi!
4:08 PM
@user2103480 do you know that there must be such a topology?
@anakhro no
4:26 PM
@AlessandroCodenotti i mean, one could possibly take the topology generated by exactly the same functions
(Not sure if that isn't bigger than the sigma algebra we aim for)
But expand
Not every sigma algebra is the Borel sigma algebra of a topology is my concern
Oh, I thought that was a "do you know that this is actually a fact"
No, I dont know whether such a topology exist
I see
I'm in a seminar right now but I'll take a look later
Ok! Its not important anyways
Consider a non-linear diffusion equation $ \partial_t u_t = \div( A ( \nabla p(u) + u f ) ) $ where $p$ is thought of as the pressure, $f$ is some forcing term, and $A$ is some matrix. The atypical example is the Porus medium equation. What examples are there of this equation when $A$ is singular ? I cant find ANY interesting examples online
sorry I should of wrote $ \partial_t u = \div( A ( \nabla p(u) + u f ) ) $, $u$ is the object of interest function of time (and say space and/or velocity)
4:38 PM
it's funny to me that LaTeX assigns the least-used symbol of all time to \div
aha agreed In my macros its set to \divv.
i'm also annoyed that LaTeX doesn't have something simpler than langle and rangle for the inner product stuff. my macros have it as \< and \> but i shouldn't need macros.
@leslietownes Is it the symbols you don't like, or the amount of typing?
i love the symbols. it's just weird that you need such strange text to call what many would regard as a very standard thing. more common than a lot of other things, anyway. yet without macros, it's langle and rangle.
@leslietownes I guess it depends on your particular focus. I use them once in a while, but I don't find their macro length annoying. I wish \mathrm was a bit shorter, though.
and \operatorname
4:51 PM
i worked in functional analysis. everything was langled and rangled to high heaven
the weird thing is you'd get tex from your collaborators where they didn't introduce a shorter macro for it, and it was literally langle and rangle written out everywhere. weird way to find out that you're working with a psychopath
@leslietownes then, yes, a set of macros would be useful. There are some people who preface even the shortest posts with a HUGE set of macros.
i worked with a guy who would verbalize langle and rangle (rhymes with angle) while writing on a chalkboard
@leslietownes They probably had a keyboard macro for those.
complete nutcase
I believe I wrote a macro \ip#1#2 for that.
4:56 PM
@TedShifrin yep
it's surprising how many (smart) people write $<x,y>$
Of course, in most of my math writing I just used dot for dot product, anyhow, but still ....
That just looks horrid, spaces wrong, etc., etc., etc.
Smartness and LaTeX-awareness and laziness are probably all uncorrelated.
yeah, i don't know how you can look at that and think this is the way tex is supposed to look
When I wrote it on the blackboard, I certainly didn't write < and >.
how do people manage spacing with dx in an integral? I usually toss in a \, before the dx. It feels kludgy and not the right way to do it
that's what I do
5:00 PM
I usually do that, unless there are parentheses or other things to spread it out. And I always do dx\,dy in iterated integrals. But I do not do \mathrm d for dx.
mathrm d is for nerds
I actually don't like the way it looks.
that too
Have we turned into the Nitpicky LaTeX morning?
I am procrastinating a calculation
5:01 PM
@TedShifrin Ah, yes. That is why I don't often use $\langle\dots\rangle$
so why not
@RyanUnger <kick>
In your prior mathematical days, @robjohn, I would suspect you had lots of inner products in your work.
mathrm d is the devil
@leslietownes $d\mathrm{evil}$
5:03 PM
i feel like when people do that they're trying to tell me that they're better than me. i'm working through it. therapist says it will take some time.
Oy vey. Now we need LaTeX therapists.
@TedShifrin There were, but mostly, they were expressed as integrals, and we just wrote out the integral for clarity (with $\mathrm{d}x$).
What if you were in a Sobolev space, @robjohn?
@TedShifrin I would try to be as smooth as I can.
Uh huh.
5:05 PM
I think lots of people use (,) for Sobolev inner products
<> is usually for the Riemannian metric
I do remember Hörmander using a lot of $\langle\dots\rangle$s
you're right but the people who model their writing after hormander's are probably the same people who insist on mathrm d
But back then, there was only $\TeX$, and that was hard no matter how you worked with it.
I started learning (AMS)TeX in 1989, a full ten years after my Ph.D.
I did my thesis on a Hermes portable typewriter and then paid a secretary to type it officially on an IBM Selectric. What a pain (especially with all the integral geometry and indices).
i worked with a guy who would not depart from troff. he had twenty-year-old macros. troff or get off. he should have learned at least one dialect of TeX.
5:09 PM
$\LaTeX$ is so much nicer to work with.
About the third revision of my first textbook I realized I needed to learn LaTeX. Manually chasing down changes in Theorem and Exercise numbers got old really fast.
@TedShifrin I did a lot of work on an Olivetti typewriter which had a changeable ball.
i had a conversation with dick kadison once about how he had an array of stamps he would use to annotate his typewritten manuscripts with sigmas and integrals and such. stamps, with a pad and ink. it felt so prehistoric.
OMG. And I just used my Parker 51.
I feel like I'm in a math retirement home.
Luckily, when I started writing things up, I had a Mac and Word would allow a bit of mathematical typesetting.
5:11 PM
You're a young'un, @robjohn.
@TedShifrin lots of people still use fountain pens
Before, it was all handwriting (mechanical pencils)
I didn't mean the fountain pen, @Ryan, just this discussion as a whole.
I still have—and use daily—the Parker 51 I bought with hard-saved $10 when I was about 10.
trying to flag a question and it only lets me choose from 3 alternate stack exchange websites, none of which are the one I would put down?
@BigSocks flag for migration?
5:13 PM
Just flag for execution instead.
Someone at google should invent a latex search engine.
@robjohn I don't know how to do a migration (probably can't because low point count)- I was just doing a flag and hoping something like "stackoverflow" would show up, or the one for theoretical cs
@DanielAdams Try Approach0
would have to be smart enough to not distinguished between different symbols used to denote the same thing.
Well, TeXShop's LaTeX spell checker works pretty well.
5:14 PM
@TedShifrin possibly a good option, thanks
@robjohn ahh its in the works :)
@DanielAdams It searches MSE and AoPS (probably others, too)
and I guess it doesnt distinguish between different variable names?
say I searched for $x^y=y^x$ I would get the same as $z^w=w^z$
@DanielAdams yeah, it does well with different variable names.
@DanielAdams yes, I believe so
and ordering as well I guess. cool stuff
5:21 PM
@DanielAdams Yep. As you go down the list of posts, it gets more and more broad in how it interprets the search patterns.
I have trouble seeing how some of the later search results even match what I've given.
cool concept
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