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20:14
@r9m Sure. Well, this is not a big deal ...
Proposed Problems or Solutions

Proposed problems or solutions should be sent to:

Doug Hensley, Monthly Problems
Department of Mathematics
Texas A&M University
College Station, TX 77840

In lieu of duplicate hardcopy, authors may submit pdfs to [email protected]
@r9m this is only for problems and solutions, but for articles things are a bit different.
@r9m Did you see my last problem on main?
@r9m I'm sure you knew how to do it, but that was your way of telling me you have stuff to send? That's great! ;)
Anyone with some background in statistics/probability?
@r9m I feel myself very close to Ramanujan after the last tools I developed. It's just a feeling ... :-)
Ramunujan always makes me think of this joke:

"I went to visit him while he was lying ill at the hospital. I had come in taxi cab number 14 and remarked that it was a rather dull number. 'No' he replied, 'it is a very interesting number. It's the smallest number expressible as the product of 7 and 2 in two different ways.'"
4
The story was about 1729 I think
@Chris'ssis Yes, that's why it's funny.
20:26
@DavidWheeler Ah, I missed the part with the joke. LOL, indeed.
Only a mathematician would "get it".
@DavidWheeler I initially thought he wanted to tell the real story.
A poet, a priest, and a mathematician are discussing whether it's better to have a wife or a mistress.

The poet argues that it's better to have a mistress because love should be free and spontaneous.

The priest argues that it's better to have a wife because love should be sanctified by God.

The mathematician says, "I think it's better to have both. That way, when each of them thinks you're with the other, you can do some mathematics."
Note the bit about 14 being expressible as the product of 7 and 2 in two different ways is a joke-within-a-joke: $\Bbb Z$ is a UFD, and mathematicians regard the two factorizations as "the same".
Nothing can beat the comathematician joke.
20:29
Paste it in here!
A comathematican is a way of making cotheorems from ffee?
yeah, that one
@DavidWheeler Almost, turning cotheorems into ffee
(you also need to reverse that arrow)
"What's the value of a contour integral around Western Europe?" "Zero. All the Poles are in Eastern Europe."
oh, right, contravarient.
lol
@Tobias I heard the version which included "using a rollary".
20:31
@BalarkaSen Ahh, cool addition
Was totally perplexed for a while thinking about what it stood for :P
Comathematicans ugh when they're ill.
hahaha
"The number you have dialed is imaginary. Please, rotate your phone by 90 degrees and try again..."
Told that one in my Complex Analysis class and that got a few laughs.
20:47
Maybe not. Let's remove it then.
Chris's sis laughs alone
that's a sign that you're getting mad doing too much integration.
@BalarkaSen No, it was just a good joke. I couldn't refrain from a ROFL.:-)
rehi, @Ted. how's california?
21:04
@MaryStar Are you still there ?
@r9m I'm exceptionally creative these days. I just created another amazing series (a few seconds ago). :-)
r9m
r9m
@Chris'ssis So all I have to do is mail the solution pdf to that mail address?
@Chris'ssis which series? :)
@r9m Well, I suppose you can say a few words about what that mail is about. Basically, yes. :-)
r9m
r9m
@Chris'ssis ah! many thanks!! :) I was planning to send my full solution to Knuth's problem :)
@Chris'ssis I haven't checked main in a while .. lemme check :)
@Chris'ssis ah!! nice :D
Hi again, @Balarka .... Finally being lazy and doing nothing for a bit. How be you?
21:08
@r9m I mean to do it in the art's way! :D
r9m
r9m
@Chris'ssis okay :)
@TedShifrin I am thinking about, like, ten problems at once. each of them seems to be as fun as the other, so I can't really fix which one I should be thinking about.
So you're getting nowhere?
@r9m Great! By the way, your full solution is the way you posted on your blog plus some more explainations or? It's worth to send it, it's a nice solution!
you've set up your home finally, then?
21:09
It's past your bedtime again ...
@TedShifrin I have made infinitesimal progress, but yeah.
No, no, don't move until end of July.
r9m
r9m
@Chris'ssis yes! basically my blog solution + some details I left out :)
22
Q: What are some examples of awkward sounding but grammatically correct sentences?

eventualEntropyWhat are some examples of awkward sounding but grammatically correct sentences?

Hi @Clarinetist
r9m
r9m
21:10
@Chris'ssis you think so? :D Thanks :D !!
@r9m All your solutions are nice, not that I wanna be nice, but this is the reality. Each solution is precious, anyway.
Morning, @TedShifrin. See this. I figure you're probably the person here who can answer it.
Goodnight, Mike.
r9m
r9m
@Chris'ssis r9m blushes and runs off to a corner :P
@TedShifrin i am trying to use my knowledge of linear algebra to compute fundamental groups of linear groups, like GL_n(C), etc.
21:12
@r9m lol :-)
Morning @Ted
@Mike: Essentially, jets are involved. You want to define higher-order tangent (or osculating) spaces of a submanifold. I don't really want to type all this on my iPad.
It's not for me in any case.
To what compact subgroup does it deformation retract, @Balarka?
yes, U(n), I know.
21:14
@r9m I'm a bit overwhelmed by the last tools I developed, they make me feel very special ... :D
r9m
r9m
@Chris'ssis what kind of tool?
it will work just fine?
is GL_n(R) even connected?
@r9m Some real analysis tools for solving some crazy hard integrals and series.
no. so?
r9m
r9m
21:15
@Chris'ssis awesome!! :)
whoops, no, I meant something else.
I am fixing my argument for path-connectedness of GL_n(C), not my def. ret. argument, which is just fine.
r9m
r9m
@BalarkaSen it has 2 connected components right?
I guess I am sleepy and can no longer do any mathematics for today
21:17
See above ^^^, Balarka :)
?
@r9m yes, you're right : it does.
r9m
r9m
@BalarkaSen which argument is your favorite? :-) (I mean for showing path conn. for Gl_n C)
I am doing it by myself, so haven't seen much of an argument yet.
r9m
r9m
ah! Nice :D .. it was a midsem question for us in topo course :)
@TedShifrin are you referring to the "way past your bedtime" thing again? :P
21:19
he's referring to when he asked what group it d.retracted to
well... U(n)
pick a matrix in GL_n(C).
the rows form a basis for C^n. construct another matrix whose rows form an orthonormal basis for C^n by Gram-Schmidting.
the latter is in U(n) by construction. now linearly slide the first matrix to the other (i.e., make the map F : GL_n(C) \times [0, 1] \to U(n) defined by F(A) = A_t where A_0 is the original matrix I picked, A_1 is the one obtained from G-S'ing and A_t is t * A_0 +(1 - t)* A_1)
@abel Thank you for the explanation!! I will think about it and tell you if I have understood that...
it's not hard to verify that this lies in GL_n(C).
@Hippalectryon Yes
so there, we have the desired deformation retract.
21:25
What in the world is viXra?
$\lim_\limits{|t-\text{BedTime}|\to\infty}\text{@BalarkaSen}=\text{sleepy}^2$
I didn't expect such lame joke from you, @Hippa
No, I was referring to the bedtime.
smirks at @Mike
Can anyone tell me for f(z) = 1/(z+1)(z+3) in 1<|z|<3, how do I know should I expand in laurents or taylor?
21:28
I think you should generically expect much lameness from Hippa, Balarka.
@MaryStar I've looked at your question (math.stackexchange.com/questions/1306826/concept-of-a-continuum), what's the link with stress tensors ?
@BalarkaSen All my jokes are lame :(
you're still cross at him for those memes, @Ted?
@ailhahc: The taylor expansion would give you a series valid in $|z| < 1$. You need to calculate the Laurent expansion in the neighborhood you gave (calculate the coefficients by integrating).
0
Q: Find $\lim_{x\to 0}\frac{\sin x-x\cos x}{x^3}$

RobChem$$\lim_{x\to 0}\frac{\sin x-x\cos x}{x^3}$$ How do I go about doing this? I can see no simple way of doing this. Application of l'Hopital's rule would be very laborious. A Taylor expansion seems feasible but is that the best way? It seems like it may be also very laborious.

I would honestly just use L-Hopital
Hmm let me try to understand what to mean
21:32
Absolutely, use Taylor. Super efficient and powerful.
@Clarinetist: I can easily do it with Taylor in my head. This is no exaggeration ....
I should really learn what those $o$, $O$, $\Omega$, etc. symbols mean
@Hippalectryon This question is not related to the stress tensors... Reading the notes I had several questions,among others the question of the link and the stress tensors, where I hadn't understood the meaning...
@TedShifrin: Do you remember the proof that every curve in the base manifold of a bundle $P \to M$ (with a connection) has a horizontal lift? I want to do the same thing in the setting where the fibers are just manifolds, and a connection in this setting is just a decomposition $TP \cong TM \oplus \mathcal H$.
Apparently lifting uses compactness in this setting and I don't know why.
gah, no, too sleepy to think about the problem right now. i guess i'll be heading to bed.
@MaryStar Regarding that question (the one I linked), what are your thoughts so far ?
21:37
@TedShifrin even Artin talks about intersection homology :( I want to learn it so badly.
@Mike: For curves it's just an ODE. In higher dimension, curvature gives the obstruction to integrability.
I only care about curves today, I want to define parallel transport.
Oh wait, younstill want to,lift curves?
hi all :)
I guess I should just figure out what the ODE is and I'll see where I use compactness of the fiber there.
Thanks.
21:40
No, a connection gives you a $\mathcal H$ which projects iso to $TM$.
r9m
r9m
@skillpatrol Hello skull :) haven't seen you in a while :)
@r9m hi pal
You meant $M$ to be the fiber?
Skill = skull?
that's so... unskullful
21:41
Hi skull!
No. I now have a bundle $F \to P \to M$. A connection in my setting is just defined to be a decomposition $TP \cong TF \oplus \mathcal H$. No connection form here (not even clear how I would define a connection form).
@BalarkaSen haha
Sorry, I see your point. Typos galore.
r9m
r9m
hmm ... skill to replace skull !!
@Hippalectryon As far I understood, according to the concept of a continuum we consider that the fluids have an ideal continuous structure, although the matter is not uniformly distributed in the volume of the fluid... Is that correct?
21:42
@r9m yep
r9m
r9m
@skillpatrol :D awesome :P
OK, Mike, we are on the same page now. Let me think a moment.
@TedShifrin Hi professor :)
Out of just plain curiosity, have any of you seen Pitch Perfect 2? I am considering going with my gf
@TedShifrin I guess the billinear form on homology naturally comes from the dual pairing $H_n \times H^n \to \Bbb Z$ and going Poincare?
can we actually visualize this beast?
21:46
Intersection homology comes about with singularities where PD fails.
It's called intersection theory for a reason. Why not just learn it if you're so obsessed?
I can't : I don't know enough differential topology.
Start with classic diff top first. Good grief.
"Whereof one cannot speak, thereof one must be silent."
6
sigh. OK, I'm just thinking out loud.
21:48
stays silent
:)
I am merely curious because I am learning linear algebra and trying to connect that with the (very little) algebraic topology I know. I'm sorry if I am being a pain in the neck by asking silly questions.
@Mike: I usually do the principal bundle case with the Lie algebra valued 1-form. I'm not sure how to write down the system of ODEs here. Patching may take compactness?
@MaryStar Hm it's correct but it seems a bit too 'obvious'. We're just saying that's it's a continuum when it can be considered continuous.
@MaryStar (sorry for the delayed answers, my connection is horrible)
@Balarka: Poincaré thought about the intersection pairing of $H_k$ and $H_{n-k}$ quite geometrically/simplicially. It amounts to pl transversality.
@Hippalectryon No problem...
@Hippalectryon How else could we explain it?
21:56
Maybe, @Ted. Do you have a (local) picture of what the ODE should be?
@TedShifrin
hi
@MaryStar I believe that the expected answer lies in the possibility of defining a mesoscopic scale
what do you think of apostol analysis
I am reading it atm I find it nice but some of his proofs could actually be shrinken down
You write down your bundle splitting in coordinates, @Mike.
Hi @Karim
Apostol is a nice book.
I would get Apostol, but too expensive
I haven't read Tao much, but I really like Tao's text
22:00
@Hippalectryon What is a mesoscopic scale?
lol @Clarinetist
you could always go to libgen
and type its isbn and get the book
Nah, always will be a hardcover/softcover person
I don't know I mean those books are quite expensive
I bought DF though
but if I start buying all books I find interesting I would spend like huge amount of money
I have mostly what I need now to be satisfied. When I was working actuarial, first thing I did was build up my math library. Still could use a few stats books, but I'm pretty satisfied
I see
I have bunch of books too in math and physics but not up to my satisfaction yet
22:03
It is, however, going to suck to move these
since I'm working on relocating
I was thinking about that too @Clarinetist
I will have to go to waterloo in 1 year for grad studies
1 year and half
One of my professors probably has about 4-5x as many books as I do in his office
have to move all of those and will have to pay good amount of money
22:04
Which is absurd for me to think about, since I have quite a few
here air canada you know I pay like 100 $ per 20 kg
lol
I used Amazon Prime for all of my books
Saved a lot of money
Best of luck at Waterloo. :) It's a good school from what I've heard. Waterloo also houses probably the #1 actuarial program in the world (not that it matters).
I would go there if it meant I could get a decent job in academia in the U.S., but politics in this country isn't making things any better for academics.
oh I see
22:08
So @KarimMansour, remind me, are you finishing up an undergrad right now?
my last year yes
Ok, @Ted.
@KarimMansour Congrats, how are you feeling about leaving?
@Mike: compactness gives you the tube lemma to get a single nbhd in the base.
I am pretty excited to move on to grad school but I want to improve alot before I attend waterloo that is why I am reading alot of stuff this summer and also will do that next summer @Clarinetist
22:10
Hi @DanielFischer!!!
I wanted to ask you if $e^{i(kx-\omegat)}, k, \omega>0$ is a periodic function..
@KarimMansour Nice. Any idea what you want to do in grad school?
yeah I want to do mathematical physics
Tube lemma = defined for $\varepsilon>0$ implies defined for all time?
@Clarinetist waterloo has really good program in mathematical physics
Oh, I get it
22:11
@evinda Yes, the period in $x$ is $2\pi/k$, and the period in $t$ is $2\pi/\omega$.
Nice. I wish I had the mind for mathematical physics. All way above my head.
yeah mathematical physics is very interesting
@DanielFischer In order to determine it, do we check if $e^{i(kx-\omegat)}=e^{i(k(x+2 \pi)-\omega(t+2 \pi))}$ ?
No, getting a single nbhd over which you can trivialize the bundle. The splitting is upstairs, not downstairs.
Hi @DanielF
this year I am planning to finish DF along with excerises and some analysis book and get into more advanced algebra and analysis I want to be able to do category theory by the end of next summer
22:13
@evinda The periods aren't $2\pi$, they are $2\pi/k$ and $2\pi/\omega$.
@Clarinetist
Hi @TedS.
@KarimMansour Very, very nice. I still recommend Tao for analysis :P
And hi @MikeM.
@DanielFischer So do we check if there are $T_1, T_2>0$ such that $e^{i(kx-\omegat)}=e^{i(k(x+T_1)-\omega(t+T_2))}$ ?
22:14
I will read apostol and tao I heard good stuff about tao too
like his analysis book
@KarimMansour I will say I learned from Bartle and Sherbert, but really, if I could've done it all over again, I would've learned Calc I - III via Spivak
Hi @DanielF.
and that basically would have replaced the watered-down Analysis class I had
@evinda You should just see that. And what the periods are, after all, you know $z \mapsto e^{iz}$ has period $2\pi$. Then you can verify what you saw.
@KarimMansour Well, Spivak covers Calc I - II, but I would've learned from @TedShifrin's book for Calc III
22:16
@MikeMiller On the Starboard it looks like you have become a Wittgensteinian?
I never learned from spivak anything I did calc 1-3 from stewart calculus
Thanks for the plug, @Clarinet.
I did calc 3 in my 1st year got 95 but now I forgot most of it
Hola @AntonioVargas.
Hey @DanielFischer
22:17
change of variables and greens function I forgot all of that
@TedShifrin: I think what I just said matters too. Pick a neighborhood of each point; existence says that near each point there's a solution for $\varepsilon > t > 0$. Without compactness it's not clear I can patch these together into a globally defined solution for time up to $\varepsilon > 0$. But if I do have such a thing I know that there's a solution for all time.
@KarimMansour When you do take the Math GRE Subject Test (IDK if Waterloo requires it or not), make sure you know all of the standard vector calc theorems. When I was trying to review the vector calc stuff, I struggled
Multivariable analysis is essential for math ohysics, @Karim.
I definitely can't recommend Stewart for vector calc
yeah I am gonna learn it @TedShifrin from apostol I think he covers it
22:18
Not sure what you mean by the tube lemma though.
lets see if he does
canada doesn't require gre exams @Clarinetist
Do you mean this?
I don't even see how you get a DE locally without what I said, @Mike.
I'm not trying to write down a DE yet. I don't know how to yet.
22:19
yeah @TedShifrin apostol does it
That's why I'm asking about your tube lemma.
Once you have an ODE, I don't see why compactness is needed.
@DanielFischer Substituting $x+\frac{2 \pi}{k}$ and $t+\frac{2 \pi}{\omega}$, we get exactly the same, $e^{i(kx-\omega t)}$, right?
Because of the discussion above?
Hi @AntonioVargas
22:20
Oh, I see. Sorry.
anyway brb guys I will go for a walk
But you have a parametrized path, coming from $[0,1]$, Mike.
For individual curve lifting I don't need compactness. To define parallel transport I do.
@evinda Yes, but you should see that the function is separately periodic in $x$ and in $t$, so if you replace only one, you still get the same value.
I see your point. Thanks.
22:21
oh, ok.
Yo @evinda
I think I was just wrong. I have a path, I pick a point in the fiber, I use your proof to lift that path. No need to lift "everywhere at once" or something.
In any case, I still don't know what the tube lemma is :)
right, that's why I was confuzled.
@DanielFischer Because of the fact that $e^{2k \pi i}=1$, right?
@AntonioVargas What's up?
22:23
@DanielFischer Nice, thank you!!! @DanielFischer
Tube lemma? Any open set containing $\{x\}\times Y$ contains a tube when $Y$ is compact.
@DanielFischer Just that one. I hope you don't identify me with the tractatus.
@evinda not much, just reading some papers. I saw lots of people in here so I wanted to drop in :)
hi @Antonio
@AntonioVargas Aha... :) What is the subject of the papers?
22:25
@MikeMiller Well, one could do much much worse than Wittgenstein ;)
@TedShifrin: So you want to use that to do my business over $\Bbb R^n \times F$? But this is a fiber bundle, I already have local triviality.
I don't need to trivialize over a neighborhood of $F$ or something...
hey @Ted
@evinda Mostly asymptotics for orthogonal polynomials. I don't know much about it but I think it's pretty cool.
The splitting is upstairs, not down.
Am I wrong?
Still don't parse. You mean the connection splitting $TP = TF \oplus \mathcal H$? But when I locally trivialize this just looks like $T\Bbb{R^n \times F} = TF \oplus \mathcal H$
I also still don't see what diffeq I want to write down but that's my own problem, so maybe that's the issue :)
That splitting gives a local splitting upstairs. It doesn't make sense down on $M$.
22:31
@AntonioVargas Interesting...
@MaryStar Basically between the macroscopic scale (the fluid like we see it every day) and the microscopic scale (looking at the molecules in the fluid) we can define an intermediate scale called the mesoscopic scale.
@MaryStar In that scale, the elementary particle is composed of lots of molecules (hence it's not on the microscopic scale anymore), but it's small enough to be considered as having its own mass, temperature, behavior.
22:44
@MaryStar (my connection is really bad today, i'll continue tomorrow)
@Hippalectryon Ok... I will think about that again... See you tomorrow!! :-)
I someone of you familiar with Dispersion equations?

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