« first day (2605 days earlier)      last day (2714 days later) » 

19:00
@NV-US: If you look at the composition of two linear maps, what can you say about the its image?
@Semiclassical this is what my assignment looks like so far imgur.com/a/NpR9w
missing a couple things right off the bad.
bat*
and atypo.
great
I'm writing pedantic notes from the discussions on the ODE reading course yesterday
sigh
Why the sigh, Balarka?
latexing is a pain in the neck :P
The ODE reading course?
19:04
I like latexing.
It's not fun when you have to Tex matrices
@Alessandro Right, I'm doing one of those
if $E_{i} : V -> V_{i}$, then i can prove that $E_{i}E_{j} = E_{j}E_{i} = 0$ @TedShifrin also, if the maps are linear then their composition is linear.
How's that going?
oh true.
19:04
Matrices are easy. I write lots and lots of my own macros.
if you're going to latex, latex properly @Dodsy
$7.43x10^{-4}$ is an abomination
Well, then you're done, @NV-US, eh? If you really have it.
@BalarkaSen nice, at a university?
@LeakyNun what's a better way?
@Dodsy $7.43\times10^{-4}$
19:06
ah thank you
@Daminark yesterday was the first actual lecture. we talked about some basic slash technical theorems on ODEs and linear systems
and we talked about Poincare-Hopf because yolo
Like Picard stuff? That and the notion of Lyapunov are all I know
@LeakyNun I'm very new to texing, but math teacher wants us to do it so I'm getting practice out of it by texing my physics assignment.
Picard-Lindelof, yeah
@Dodsy alright
19:07
I dunno Lyapunov yet but I suspect we'd arrive at that as we progress
@Dodsy is d=(vi+vf)/2 one of your big 4?
and @Alessandro well, yeah but more of as with a professor than at a university
check if i understanding right. $E_{i}(v) = v_{i}$ (where $v = v_{1} + v_{2} +... + v_{k}$) ), u meant this, right? @TedShifrin
I didn't mean that, but it's right. :)
@Semiclassical I don't even know, to be quite frank. I just googled "the big 4"
Also you missed t.
19:09
dur, I keep doing taht
d/t = (vi+vf)/2
@LeakyNun not being defensive just letting you know what's going on.
(i know that's not quite the version you wrote, but it's how I remember it)
That's equivalent, Semi.
@Dodsy "alright" is just my response when I don't know what else to respond with
19:10
@LeakyNun alright.
I like that version because it says that the average velocity can either be expressed as d/t (change in position over change in time) or as (vi+vf)/2 (average of initial and final velocities)
oh that's true.
however, if that's not one of the equations you're given to use, you probably shouldn't use it directly
that's more intuitive.
then obviously $E_{i}E_{j} = E_{j}E_{i} = 0$ :)
19:11
Lyapunov is about determining stability of equilibria
But i think that's one of the big 4.
let me look
@NV-US: Once you've argued that $V_i\cap V_j =\{0\}$.
it may well be; it's a natural enough result to remember
@Balarka I dunno if it's standard or if that was just Schlag's thing
19:12
yeah we're gonna do dynamics sooner or later
mmm dynamics
@Semiclassical jesus he doesn't even mention the equations in the handout...
Lol when we did dynamics in the summer, none of it was ODE stuff
there are various flavors of dynamics
19:13
@Balarka: What book are you using? I told you to look at Hirsch/Smale.
allegedly
the big 4 mentioned here includes the one you wrote: physicsclassroom.com/class/1DKin/Lesson-6/Kinematic-Equations
over here, we tend to talk only about the 'big 3' and don't include that one
@Semiclassical this is where I got it from :)
@TedShifrin I have H/S saved. Arnol'd is my staple food but the professor who's teaching me doesn't have any specific text in mind.
19:14
Like it was there in the book but we focused on topological and symbolic dynamics, Ergodic theory, and some hyperbolic dynamics
Oh, OK.
In my own brain I tend to only think about x(t)=... and v(t)=...
because those are the two which follow directly from definitions.
the others can be derived from those definitions, of course.
(I file the v^2 equation as a special case of conservation of energy, which is why I find it a bit distateful for it to be part of the big 3)
if i let $v$ belons in $V_{i}\cap V_{j}$, then i can prove that $E_{i}v = E_{j}v = 0$, how can i get to v =0?
@TedShifrin
That's where you need to use the trace hint.
@Semiclassical the force that the tree exerts on the bullet, it is a net force?
@Semiclassical First we find the acceleration of the bullet by using the relevant equation: $v_f=v_i+a(t)$\\
$0=350+a(7.43\times10^{-4})$\\
$a=\frac{-350}{7.43\times10^{-4}}$\\
$a=-471153.85$\\

The negation is due to our convention that up and right are positive, it is accelerating $471153.85 m/s^2$ to the left.\\

Using N3L, we know that any force has an equal and opposite force. This means that the bullet will exert a negative force on the tree (due to our conventions) and the tree will exert a positive force on the bullet (again, due to our conventions.)\\
19:21
well, that'd presume that the force of the bullet on the tree is the only force which is acting on the tree
Wow, Nate is buried in physics more than in mathematics.
which isn't true for most trees I know :)
Does the tree go skidding off into the sunset?
lmfao
I have a lot of mathematics to do as well, Ted!
the trace of a projection is 1, right?
19:22
NOOO, @NV-US. Depends on what dimensional subspace you're projecting onto.
Work this out.
I gave Leaky a hint above if you need it.
But my mathematics teacher says we cannot talk to anyone about our math assignments, whereas my physics teacher has not brought it up.
ok :o
@Dodsy Do you see where I'm going with that?
@Semiclassical yes, so I need to say that we assume that the bullet is the only force acting on the tree?
19:23
o
Is my writing too convoluted?
I agree with not allowing us to do your homework for you, but I guess he needs to draw the line and I support that.
my point was really that the tree is already experiencing other forces---wind, contact force from the earth, gravity
Hurricanes.
so i wouldn't call the force of the bullet on the tree the net force
so $F_t$ would be a better convention than $F_{net}$
19:24
Or $F_{t,b}$
i.e. force on the tree from the bullet. (I'm not sure which order people usually use for that, tbh)
I like it!
@TedShifrin I also only have until 10pm to finish this assignment because I need to go to the library and print it off.
see for instance the force diagram at the top of this page: sun4.vaniercollege.qc.ca/VirtualMentor/physics/…
in which case they actually do the opposite of what I wrote :(
e.g. $F_{BA}$ is the force of B on A.
Do you think I need an fbd for this question?
ehhh. couldn't hurt, but if you do I'd just sketch it in by hand once you do a printout
latex tip: when doing scientific notation, use \times ($\times$) rather than x ($x$) @dodsy
so $3\times 10^8$ rather than $3x10^8$
first definitely looks nicer than the second
Thanks, Semi.
19:30
@Dodsy I thought I already told you ><
You did, but I didn't want to be rude to semi and tell him that.
If CH fails, is there a subgroup of $\Bbb R/\Bbb Z$ of order $\omega_1$?
@TedShifrin If $X$ is a vector field on $\Bbb R^n$ and $\gamma$ is a maximal integral curve of $X$, saying that for all intervals $(a, b) \subset \Bbb R$, $\gamma((a, b))$ is contained in some compact subset of $\Bbb R^n$ suffices to guarantee $\gamma(t)$ is defined for all time, right?
@BalarkaSen
And what I was commenting on was what I saw in the image
19:31
@Leaky I don't care.
who in this room cares about infinities
@Semiclassical yes, no worries Semi. I appreciate the advice.
In particular, I don't know.
apart from secret?
@LeakyNun puts up hand
19:32
I think you just need to take an $\omega_1$ subset and have it generate a subgroup
not sure what its order would be
I think I should ask this on main
you're doing finite combinations
@LeakyNun What do you mean of order $\omega_1$? That is an ordinal, not a cardinal.
@TobiasKildetoft oh, lol
$\aleph_1$
but they're the same set
@Semiclassical so much talk happens in this chatroom that its hard to tell what one person has already said, and I'd rather somebody tell me something is wrong than to worry that somebody else has already said it before. Hence why I wasn't going to say that leaky already told me.
I consider that good practice.
19:34
:)
@AlessandroCodenotti so it's still $\aleph_1$?
@LeakyNun Anyway, you must be nearly through the exercises by now or what?
@TobiasKildetoft oh I haven't started 10
my productivity is very high
it's equal to the sum of all natural numbers
which is -1/12
rolls eyes towards self
Well, you basically already did 10.1 I suppose
$E_{j}E_{i}$ live in $V_{j}$ and $E_{i}E_{j}$ live in $V_{i}$, but how does that help me ( if this is right ), @TedShifrin
19:37
@Semiclassical I don't want to bother you too much today.
So I'll skip to a question that is really freaking me out.
@Semiclassical imgur.com/a/11RMa
I'm taking enriched math and I don't even know wtf this is about.
Presumably what they want you to do is prove certain common derivatives directly from the definition
From the definition of a derivative using the formula?
no
for instance, they want you to show $\frac{d}{dt}t^n = n t^{n-1}$
@LeakyNun lol.
19:44
How did your instructor define the derivative?
@Semiclassical using infinitesimals :P /s
Hyperreals!
@Semiclassical he did not.
...
has stuff like $x+\Delta x$ shown up in the course?
I know exactly where it should, namely in the course of explaining how the derivative $f'(x)$ of a function $f(x)$ is obtained
hm let me check my notes.
19:49
either you have not been paying attention or that's really shit of him to tell you to compute derivatives before even defining them
in either case the scenario is not good
So he said that the instantaneous speed was the first derivative, that the acceleration was the second derivative. And then he said $a=\frac{dv}{dt}=\frac{d}{dt}(\frac{dx}{dt})=\frac{d^2x}{dt^2}$

He defined momentum as $\frac{d}{dt}(mv)=m\frac{d}{dt}+v\frac{dm}{dt}$

But he said since the object does not lose mass then:
$\frac{dm}{dt}=0$

That's really been it...
back up. is this a physics problem or a math problem?
Physics, ofc.
I will never ask a math problem on here.
19:52
@Dodsy which book are you using?
I ask because there's absolutely no physics in this problem.
unless it's linear algebra.
@Dodsy you will never ask a math problem in a math chatroom >_>
I am not allowed.
You're not allowed to ask about problems from your enriched math course.
19:53
@Semiclassical uhhh wat.
(just to reiterate)
Right.
why not?
@Dodsy yep.
b/c the instructor has willed it so.
19:54
@Dodsy here's what they're getting at. Suppose you wanted to approximate the derivative $\frac{d}{dt}f(t)$ at a time $t$.
@LeakyNun University Physics with Modern Physics.
right.
What you'd do is pick an interval of time, $[t,t+\Delta t]$, and compute the average rate of change over that interval
that'd be $\frac{f(t+\Delta t)-f(t)}{\Delta t}$
change in f over change in t
if you make $\Delta t$ smaller and smaller, you'll get better and better approximations to $f'(t)$.
right, like the secant method?
right.
so let's test that on their first example
with $f(t)=t^n$.
the difference quotient would then be $\frac{(t+\Delta t)^n -t^n}{\Delta t}$.
But if you say $\Delta t$ is small so that you can use their approximation, what do you get?
then $f'(t)=nt^{n-1}$ ?
20:00
yep
Hey, would anyone know a fun quote about integration, by any chance
literally all they're having you do is the secant method.
It's integral to a calculus education?
@Semiclassical hm...
That's... pretty lame though
20:01
@Semiclassical he hasn't even talked about the secant method.
:'(
Lol jk
@Dodsy Are we talking about the math course or the physics course?
With the physics one, I doubt he would.
Physics.
Okay but why then have the problems requiring the secant method.
The idea may have been that, if you're taking calculus based physics, you'll have seen enough calculus at that point that you can do these problems.
hm, alrught.
alright*
20:02
...which seems pretty silly, given that your enriched math course evidently hasn't gotten that far
what makes this problem fairly dumb imo is that what I elaborated on above was that $f'(t)\approx \frac{f(t+\Delta t)-f(t)}{\Delta t}$
but you can rewrite that equivalently to $f(t+\Delta t)\approx f(t)+f'(t)\Delta t$.
right, I've seen the "first principles" derivative thing before.
That's basically what this si right?
yep
so you can read off $f'(t)$ from their list of approximations: ignore the f(t) term and look for what's in front of $\Delta t$. that's literally all it requires.
hence the substance of the problem is really quite thin.
(and, as I indicated, there's absolutely no physics in sight.)
Am I the only one who sees a variation of Gödel's incompleteness theorem here? "Any amendable constitution is either incomplete (i.e. does not mention anything about protections that cannot be removed) or inconsistent (contradicts its own amendability).Mehrdad 2 days ago
mention of incompleteness theorem... in law.SE.
@Semiclassical really weird.
i'm not impressed, no
20:14
I think I need a break.
I'll be back later.
thanks semi.
20:26
@Daminark do it
Also, evening
@TastyRomeo bonjour
I'll get around to it eventually
20:57
is anyone here also a teaching assistant?
@Semiclassical I went to this free food thing and my gf ditched me so now I'm sitting drinking alone instead...
@SoumyoB @Semiclassical is a physics ta, I believe.
Z_6 has two maximal subgroups: {0,2,4} and {0,3}
ahh good to know at least one person here is undergoing a similar torture as I am
@TobiasKildetoft for 10.2 you might want us to show that |G| is prime... a stronger version of "G is cyclic"
TBH my torture this semester as a TA is minimal: upper-division quantum is so much much less work than intro physics
21:03
conjecture... a finite group has exactly one maximal subgroup iff the order is prime
But instead im torturing myself with "I need to be writing I need to be writing gdi why am I not writing"
I may try to TA for discrete math
Though probably not
@Daminark any idea?
I'm inclined to say yes
If it's a product of more than 1 prime, I think it's right via some vague Sylow considerations
@Daminark right, I'm also considering Sylow
21:18
If it's p^n I'm not sure, you might have a normal subgroup of order p^{n-1}
Actually wait shit
If it's abelian and of order p^n you might be dead
Let $|G| = p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n}$ with $n>1$.

Let $P$ be a Sylow $p_1$-subgroup of $G$. It is a proper subgroup of $G$. Then, we can enlarge the group to form a maximal subgroup containing $P$. Its order divides the order of $G$ by Lagrange. However, its order is not equal to $|G|$, since it is a proper subgroup of $G$. Hence, there is a prime power $p_i^{k_i}$ with $i \ne 1$ that does not divide $|P|$. By Sylow theory, we can construct a proper Sylow $p_i$-subgroup of $G$ and expand it to another maximal subgroup, creating another maximal subgroup.
Cyclic group of prime power order also has unique maximal subgroup
@TastyRomeo oooh
and non-cyclic groups?
Well in that case I think you can use generators then?
right
so we can prove that it's cyclic and order prime power
21:21
Wait does prime power plus abelian imply cyclic?
@Daminark no
think of V_4 as the smallest counter-example
@BalarkaSen SNIPED
Balarka: rkt
@Daminark Z/5xZ/5
21:22
@Daminark fundamental theorem of finitely generated abelian group basically gives you the answer
for any prime power you can break it down into products
Oh come on my connection is so slow I got sniped twice
does this make sense: a non-cyclic group has two elements that do not generate each other
no, this hardly makes sense
what's the difference between a cyclic group and a non-cyclic group?
But wait then if you have a group of prime power order which is abelian, and you have a subgroup of order p^{n-1}, you might still be in trouble
Oh wait no fuck
21:25
:P
@Daminark we already gave you two counter-examples
Sorry normality only implies uniqueness in the Sylow case
so you're sniped thrice
Nope
Wasn't a snipe
Let $G$ be a group with $M$ the unique maximal subgroup. Then for any $g \in G\setminus M$ you must have that $\langle g\rangle = G$ hence $G$ is cyclic.
21:26
You gave counterexamples to my old wonder, this was a different one
why are we doing algebra all of a sudden? let us do actual mathematics, that is to say, topology and geometry
@BalarkaSen flagged
It can't be infinite cyclic, cause then it would be $\mathbb{Z}$ which has plenty of maximal subgroups, hence it's $\mathbb{Z}_n$ for some $n$. And if $n$ is not a prime power, e.g. distinct primes $p$ and $q$ divide $n$, then you can take the multiples of $p$ and $q$ to give maximal subgroups
so $n$ is a prime power
kek
Thought the better of that attempt at a counter
But yeah algebra and number theory are the ends of all human endeavor
clearly you have not done any actual mathematics
come to the church of geometry
21:30
Lol I don't like geometry
I did geometry once and it was awful
@BalarkaSen ἈΓΕΩΜΈΤΡΗΤΟΣ ΜΗΔΕῚΣ ΕἸΣΊΤΩ
sucks to be you then lol
If algebra isn't math, it's better than math
Maybe algebraic geometry is alright
21:31
As long as it's arithmetic stuff
i can settle on algebraic geometry
Complex Algebraic is illegal
Algebraic geometry gave me PTSD
@Daminark Forster explains Grothendieck's theory of fundamental groups using pictures
No algebraist can do that
Integrals on Riemann surfaces ftw:)
21:32
Sorry about that lol
@TastyRomeo lol you spoiled the answer
it's an exercise Tobias gave me
oh
oops
Though given how little I know of elliptic function theory...
21:33
@Semiclassical yeah good stuff
periods on Riemann surfaces give embeddings to moduli spaces
cool shit
give me a non-abelian group with order prime power
B(2,3)... nope
D8 or Q8, better
@BalarkaSen SNIPED
I am going to link you horrible stuff now
21:34
horrible stuff that will leave you scarred forever
Pictures are bad anyway @Balarka
@BalarkaSen I'm ready
Are you sure?
actually I've never watched 2G1C so I don't know how far things can scar me...
[because people say that it can be quite scar-ry]
never watch 2G1C
21:36
@BalarkaSen I would never, lol
those are actually scary and disgusting
@BalarkaSen come on, what is it
@TastyRomeo what did you not like about alg geo? I know nothing of it yet
lmao
how do you know so many things
Well, I was just traumatised by the professor who taught me alg geo
or rather, tried to teach it to me
21:38
tried to teach me it
(we used Hartshorne, btw :^) )
tried it to teach me
tried to teach it me
it tried to teach me
me, it tried to teach
to teach me it, tried
I did end up scoring a 19/20 by mindlessly proving stuff and applying theorems and lemmas
But I never felt like I understood it
21:39
@Daminark Hence, Hartshorne is bad for health
mir es lehren versuchte
Actually a grad student I know in number theory told me about this book by Qing Liu with more of an Arithmetic focus
@BalarkaSen the bit of that which I knew about once upon a time was
essayait me l'enseigner
tentaba me lo ensener
Suppose you've got a family of Riemann surfaces, parametrized by one complex number
Eg y^2-x(x-1)(x-w)=0
21:42
@LeakyNun Did you listen to the song
@BalarkaSen yes
So long as w != 0,1,infinity the resulting surface is smooth
Good. Now try this
I unironically like that song
@BalarkaSen try this
Now, if you change w a little then the topology doesn't change; it's always a torus
21:44
@TastyRomeo do you need choice to choose an element from $G \setminus M$?
Suppose you pick one of the cycles on this torus and compute the period of dx/y
@BalarkaSen lol wtf lighting a cigarette with the gas
@LeakyNun not quite my tempo
I like the LOBBY beats in the background
And then consider it as a function of w.
@BalarkaSen I've always considered it a talent to be able to pronounce bh / dh / gh
21:47
yeah thats pretty common in Indian dialect
I have an idea of dh because it's in Arabic (and it's the way I say th half the time)
@Daminark no, the Arabic one is nowhere close
dh is actual aspirated "d", not some voiced "th"
But how does bh work?
@Daminark same
Oh wait hmm
21:47
that's why I consider it to be a talent
I mean I don't even have a reference on how it's supposed to sound
@Daminark say "d" like in English "dent", but aspirated
like with a puff of air, just like when you say "teeth" instead of "steep" (compare the two "t" there)
@BalarkaSen and not get confused with ph / th / kh
come on, a four-way distinction in Hindi
English has border-line three-way (which is why it's so hard to describe)
@Daminark the same manner and place of articulation are these four consonants: bh / b / ph / p
the difference between voiced and aspirated (2x2 = V4)
Shite

« first day (2605 days earlier)      last day (2714 days later) »