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1:57 AM
@LeakyNun Does "hyperbolic" really mean nonzero derivative? I haven't heard that terminology before.
@user76284 I have never heard that either. I would take it with a grain of salt.
it's because the equilibrium point of the corresponding ODE is hyperbolic
but now I think it's the wrong interpretation
1 hour later…
3:10 AM
Can someone suggest a book for very tough multivariable calculus problems?
I'm all ears
1 hour later…
4:15 AM
So I have a question and Im not sure how to really prove it
If we have a convex polygon, and draw diagonals between all non-adjacent sides, we get nC4 intersections, correct? Assuming that no 3 lines intersect at the same point.
I know it takes 4 points to make 2 lines and 2 lines to make an intersection. Hence, nC4 where n = number of sides
But this seems like im missing something
4 hours later…
8:21 AM
Is the choice of a connection equivalent to the choice of a foliation of the total space by the base manifold?
ie : for a total space $E$ with base space $M$ and typical fiber $F$, a foliation of $E$ by $M_f$, $f \in F$
So that the horizontal space are the vectors tangent to the foliation
(with $M_f \cong M$)
8:35 AM
In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a function which is rather irregular, by convolving it with a mollifier the function gets "mollified", that is, its sharp features are smoothed, while still remaining close to the original nonsmooth (generalized) function. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them...
9:08 AM
I really need to grab some text on chaos theory, these things are so fascinating
I do sometimes wonder though. These deterministic chaotic regions were born because the cycles have become so dense that they started to tug each other resulting in these nonlinear behaviour and topological mixing as explained here. What if we can turn these pairwise and higher order cycle interactions off and so they can kept on bifurcate to infinity. Would the resulting pattern look different from chaos?
Is there a name for like
a dual foliation
ie for a manifold $M \cong A \times B$, there's a foliation by $A_b$ and a foliation by $B_a$
9:27 AM
Not sure how relevant these are, as they are two different notions of "dual foilation"
10:09 AM
Is there any way to calculate peak or maxima of a combination of two waves which have different amplitudes and different frequencies?
10:29 AM
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action. == Introduction == A covariant...
You get an n-plane distribution of the total space, but if it were integrable (ie, a foliation) then the connection would be flat and vice versa
@Slereah No, how would you in general choose the complement? And why would it be an integrable foliation? Even if your foliation is Riemannian (essentially, is equipped with a degenerate metric which is degenerate precisely on the foliation) so that you can take a complement, I see no reason this complement should again be a foliation.
See a comment in more detail here: mathoverflow.net/a/62867/40804
Just looking for confirmation: If the joint pdf for $X,Y$ is $f_{X,Y}=2$ and $0<x<y<1$, then the marginal pdf of $X$ is $f_X=\int_x^{1-x}2dy=2-4x$.
Wait, nope, I made a mistake.
This makes more sense: $\int_x^1 2dy=2-2x$
10:45 AM
@MikeMiller Thx
Damn, thought I understood Ehresmann connections for a bit :V
Why does it have to be a foliation (decomposition into submanifolds) to be an intuitive notion?
I suspect whatever intuition you had in that case works just fine more generally
11:16 AM
@MikeMiller I had the notion that as the connection basically connected the points of nearby fibers, it could define a surface of equal "fiber values"
11:26 AM
@Slereah It works for curves, which is how the horizontal distribution gives rise to a parallel transport operator --- if you want to know how a path downstairs is "allowed to move" upstairs, the rule is just that the lift $\widetilde \gamma$ always has its tangent vector lying in the horizontal distribution
Yeah it's always kind of the issue with fiber bundles
I suspect the issue is that you imagine distributions in general (which do sort of "connect points of nearby fibers" in some general sense) give rise to foliations
The examples you can visualize are all one dimensional
Look up pictures of contact structures on R^3
Pretty hard to visualize a sphere bundle on $S^7$ or what have you
11:28 AM
The answer I linked gives fiber bundle examples where the horiz foliation is visualizable but not at all integrable
And yeah, but you sort of stop thinking of them in the same way --- or at least that's the goal
We can only hope
but it's like what's his name says
You never really understand math, you just get used to it
I fucking hate that quote
It's not true at all
You understand it in a different way than maybe you first thought, but you still come to understanding
What a dumbass, that Von Neumann guy, right?
he was the dumbest man to ever live!
but you know how it is
The brain just refuses to believe that the Banach-Tarski paradox makes sense
Because $\mathbb{R}^2$ is just made of fabric
11:32 AM
You'd understand it if you did research in that area for a while, saw how similar ideas come up, how to do different kinds of "doubling processes" like that, etc. I never have so it's magic to me
But when you do the deep dive I really don't think it's just "Meh it's fine now", I think you get to understand why it actually occurs
I just stick with the axiom of choice because apparently the negation of the axiom of choice is even worse
I stick with the axiom of choice because I don't object to nonconstructive existence things, just because I can't see why I can choose an element etc doesn't mean one shouldn't be able to
Apparently with $\neg C$ you can prove that you can split $\mathbb{R}$ into a set of disjoint intervals with a bigger cardinality than $\mathbb{R}$ or some other nonsense
Real numbers were a mistake
Come back Pythagoras
11:37 AM
I might accuse you of neither understanding nor getting used to it
Careful, remember what happened to the guy who kept on about irrationals to the Pythagoreans :V
"Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”"
It's all his fault really
"he perished at sea for his impiety, but he received credit for the discovery"
Publish and perish
Really I get the impression that mathematics get less and less high stake with time
Antiquity you just get killed for saying irrationals exist or drawing circles
12:11 PM
aristoteles didnt get killed for drawing circles
he got killed while drawing circles
Hi there, anyone here that is familiar with sheafs/presheafs of vector fields?
i know at least 1 thing about that
@s.harp that is?
that $\mathfrak{X}$ is a sheaf of modules over $C^\infty$ and if you mod out $\mathfrak m_x \cdot \mathfrak X_x$ (where $\mathfrak m_x$ is the maximal ideal of $C^\infty_x$) from $\mathfrak X_x$ you get $T_xM$
Sounds good, OK, time for helping me here with something?
12:19 PM
Any book suggestions for tough mutlvariable calculus questions?
I'm still waiting :(
I have vector field $J$ $\Bbb R^3\to\Bbb R^3$ that can be expressed in terms of a sum $J=P(d)+D(d)$
@s.harp Archimedes was giving lip about soldiers disturbing his circles
If he hadn't he'd still be alive today
@Slereah oh right, not aristoteles
where the sum $J$ is highly non-trivially independent of the constant vector $d$
Now since $J$ is independent of $d$ for each point in $\Bbb R^3$
one can form a sheaf (I guess) of vector fields one for each point
12:23 PM
Okay. I understand how we can endow $\Bbb{R}^4$ with the structure of an algebra; and with this structure we call $\Bbb{R}^4$ the quaternion algebra. From this we get a group structure on $S^3$, which is the set of all elements in $\Bbb{R}^4 = \langle 1,i,j,k \rangle$ with norm $1$.
My question is, is the norm on $\Bbb{R}^4$ as the quaternion algebra the same as the Euclidean norm?
such that finally one can choose a different d for each r, such that it looks as if J would be independent from a $d(r)$
I would like to have crystal clear mathematical description of this (dirty trick).
You have some kind of a formula that maps a vector $d\in\Bbb R^3$ to a vector field $J$ so that $J$ is linearly independent of $d$ at each point
nothing else you said made sense to me
what are you trying to "get" from this $J$?
A nice algebraic form
for example.
I try to give an example. Gimme some time.
I think that's true. But I don't see how to prove that $S^3$ is isomorphic to $SU(2)$.
isomorphic as groups
@user193319 check out the isomorphism described here: de.wikipedia.org/wiki/SU(2)#Topologie
12:37 PM
@s.harp Thanks, I'll take a look.
@s.harp Lets suppose $$ J_d(x) = x - d - d e^{i\pi}$$ here $J$ is (slighlty) non trivially independent on $d$. We note that $J$ thus is independent in any point $x$ of the specific choice of $d$. Say for some reason I wish to get rid of the $x$, so I could say lets take in each point $x_0$ where we evaluate $J_d(x_0)$ that $d=x_0$ in that way I have formally eliminated the $x$ term. So I have a new expression $J(x_0)=-x_0 e^{i\pi}$.
And that holds for all points $x_0$ in the space.
The point is I have nowhere used $e^{i\pi}=-1$ except implicitely in the notion that $J$ is independent of $d$.
That knowledge could have come from a independent bit of information e.g. something like symmetry considerations or physical considerations.
$J$ is not independent of $d$ at each point, if $x$ is proportional to $d$ then so is $J_d(x)$.
12:54 PM
$d$ originally is a fixed constant (a parameter for the say family of functions $J$).
But as such $J_d(100)=100$ for all $d$, right?
well $100$ is not a point in your space, but for example at the point $x=3d$ $J_d(3d)=3d$ is proportional to $d$ and as such $J_d$ is not independent of $d$ at this point
$100\in\Bbb R$ right?
unless you mean "independent" in the sense that $J_d(x)$ does not depend on $d$ (as opposed to $J_d(x)$ is a vector that is linearly independent to $d$)
yes, but I thought we were looking at $\Bbb R^3$, anyway on $\Bbb R$ any two non-zero numbers are linearly depedendent
I never said anything on "linear dependency" that was your idea from the beginning
I made a baby example for $\Bbb R$
I see, you meant then that the "rule" determining $J_d(x)$ is the same for all $d$
12:58 PM
Hey @Mathein kannst ma schreiben wenn wieder da bist? Ich hätt eine nichtmathematische Frage für dich, falls du mir helfen kannst :P
@s.harp not sure, but its clear that J is not really dependent on $d$?
@s.harp right?
@Rudi_Birnbaum in the sense that $J_d = J_{d'}$ for any $d, d'$ yes
because J_d evaluates using the "secret knowledge" to $x$.
OK fine!
@s.harp I know that this is some kind of wired sheaf thing. But I would like to know exactly how that should be formulated then.
"wired" because $d$ is not a vector space but a constant
I don't see any connection to sheaves, except for the fact that because you have a vector field you may look at at the sub-sheaf of $\mathfrak X$ generated by that specific field (as an $\overline{\Bbb R}$ module, where $\overline{\Bbb R}$ is the constant sheaf with coefficient $\Bbb R$)
I was told its a special kind of sheaf since its not over a vector field but just over a set. I can't understand this info, but maybe you can.
Choosing $d$ is usually called "gauge transformation". And that can be used to make expressions simpler. But this is not a simple gauge transformation since I do not choose $d$ to be $100$ or the like, but I do it in a way that I can set it to be a "function" $d(x)$ more or less.
Instead of using an expression (function, field) I use a continuously indexed family of expressions (functions,fields). And from this I make a continous choice for each $x_0\in\Bbb R^n$
1:12 PM
in the present formulation there is no connection to sheaves that is useful. Maybe your objects $J$ are the solutions to a partial differential equation parametrised by $d$ and it turns out that these solutions are independent of the parameter. If your differential equation is nice enough the solution space will be a sub-sheaf of the vector fields (as an example, local Killing fields are a sheaf), and maybe in the more "grown-up" physics examples these differential equations are interesting ...
... enough that knowing that the solution space is a sheaf is useful (i find it very useful for Killing fields)
@s.harp It kind of makes sense. Its just that there is no differential equation its just the expression for the "solutions" and the external information about the independence on $d$.
and since that expression can be nasty depending on what you want to do with it one tries to "transform" it. So the problem is not to find the solution for a DE but to simplify or adapt the general expression.
But maybe its "nothing but a trick"...?
Hi @ÍgjøgnumMeg
Heile @Rudi :)
Hoffe es geht DIr gut,
kannst schon perfekt deutsch?
Ja siiiia
bin aber noch ned in Heidelberg
noch so 12 täg
bis ich umzühe
Oh aufregend!!
1:24 PM
es kann echt ned schnell genug kommen
Hast schon ne Bude?
ähhh also ich hab mich für eine beworben
hoffentlich krieg is
liegt ca 20km von der Uni entfernt
Sonst Zelt!
also krieg ich die Miles
20 km, das ist schon ein bissl was ...
nimm DIr ein Brompton mit
1:25 PM
Ich nehm das Rennrad mit lol
is aber blöd mit Bus/Zug ...
für Dienstreisen is das echt geil!!
Ja ich tu einfach die ganze Strecke radeln glaub ich
muss eh ein bisschen abziehen
schon so 10kg zugenommen seit Feb.
Semester geht am 1.11. an oder? (mist ..)
1:27 PM
Ehhh glaub am 14. Oktober
Ja ich hab letztens 15 kg abgnommen, und jetzt fress ichs mir gaaaanz langsam wider rauf .. :(
aber da gibts nen Minikurs am 9. Oktober also fahr ich schon am 1. hin
@ÍgjøgnumMeg Ah in AT am 1.10. ..
Ah okey :)
Mist hab 3 VLs ...
im WS ...
1:29 PM
ahaha ich hab mir grad nen stundenplan gebaut
Servus @Rudi_Birnbaum @ÍgjøgnumMeg
Servus @MatheinBoulomenos
@Mathein zeeeas
Bei dir auch alles gut?
Hatte 110 mit 95 schaut man fast wieder menschlich aus ...
also bin grad 110 glaub ich
181cm also schau ich ned sooo schlimm aus aber man spürts halt
1:31 PM
Ich 185 dereinst ...
Ja beim SPort nervts
can i ask one simple chemistry doubt? There is no one active in Chemistry Chats.Please allow me clarify my doubt
Bei mir alles gut :) und bei dir? @Rudi
ja bei rennradler auch weil sies einem immer sofort sagen wenn du a kle mollig bist
1:32 PM
@MathGeek I am a chemist
@MatheinBoulomenos Ja auch, alles ganz locker,
@ÍgjøgnumMeg :-) such Dir andere Freunde ;-)
Nein stimmt schon :-)
hehe naja ich kenn eh keine rennradler aber man sieht's halt im fernsehen/youtube und so :P
Hey @Alessandro :)
@ÍgjøgnumMeg die spinnen sowieseo die Profis, die schauen aus wie 12 jährige wenn man den Kopf nicht sieht und die Mukis ignoriert
hahaha jo echt
so gentechnische Gräuel
how does NH3 solution effect ELECTROLYTIC properties of NaCl?I Know NaCl is good electrolyte in polar solvent only but why can't NaCl bonds be broken in NH3 Solution.@Rudi_Birnbaum
1:35 PM
Ciao @Alessandro come stai? Quando cominciano i lezioni a Bonn?
protonation of NH3 ...
oh!! thanks
NH$_3$ + H$_2$O $\leftrightarrow$ NH$_4^+$ + OH$^-$
Walter White in the house
mind the pKa ...
@ÍgjøgnumMeg genau !
Muss nur noch an der Frisur arbeiten ...
@MathGeek wait
@MathGeek You were asking something different
1:38 PM
also und hoffentich ned an krebs leiden müssen lol
@Mathein kann ich dir irgendwo privat schreiben? :P
kannst mir auch verpiss dich sagen hahaha
@MathGeek You want to know NaCl $\overset{NH_3}{\rightleftharpoons}$, right ?
@ÍgjøgnumMeg äh klar, was ist dir genehm? Email, SMS, Whatsapp, Telegram, Signal, Discord oder Fax hätte ich anzubieten
@Mathein ähm Whatsapp? :)
@ÍgjøgnumMeg okay, ich mail dir meine Nummer
1:42 PM
Cool danke :)
But NaCl does dissolve in liquid ammonia afaik. Maybe not as much as in water ...
@MatheinBoulomenos @ÍgjøgnumMeg Servus Kinder ich muss heim! bye all
Bis bald @Rudi_Birnbaum
@Rudi pfiat di
@Rudi_Birnbaum übrigens auf meinem Blog gibt es gerade eine Einführung in Kategorientheorie falls dich das interessiert :)
@Rudi_Birnbaum please wait
@Rudi_Birnbaum i have one more doubt
can anyone help me in chemistry??
it's very simple question
1:53 PM
@MatheinBoulomenos bene grazie! Il 7 Ottobre (le lezioni*)
2:04 PM
I asked a question on the main site, thought about it and attempted to find an answer, asked two professors, but still cannot find the answer
is $x^n=(1-x)^k$ an example of a moduli space? The $x-$ coordinates of the points in the moduli space would be the solutions to that polynomial equation for some $n,k \in \Bbb N$
and the y-coordinates would be found from plugging the solution back into either side of the equation
thus, the space, would consist of all these points
the points definitely correspond to solutions of algebro-geometric problems, so that checks out
Hi, just wanted to verify something. If A is a set and $b\le |A|$ is a cardinal, then does there necessary exist a subset of A with cardinality $b$?
Seems intuitive but I want to make sure I'm not missing some weird case
2:23 PM
@SirCumference yes
2:35 PM
Assuming choice of course
(back again :-) :51733405 coole Seite
@MathGeek "doubt" is sometimes also called "question", just sayin...
@Rudi_Birnbaum ok!! maam
@AlessandroCodenotti Check your facebook
why everyone here ignores me ??
This isn't a chemistry channel and we're not here to do your homework. What is wrong with you?
2:43 PM
some people are rude :(
Just leave chemistry to the chemistry room and math to the math room :)
@Mike done
@SirCumference With choice, yes, without choice, you can have amorphous sets which are uncountable but all its subsets are either uncountable or finite
chemistry rooms are very inactive
@AlessandroCodenotti Actually, alessandro, are you aware of any choiceless universe models which $b \leq |A|$ implies $b = |A|$?
2:51 PM
don't let this interaction ruin your passion for chemistry
what's an example of a dynamical system in 2 dimensions that cannot be solved using differential equations?
@Secret there is no such thing, $\omega$ and $\omega_1$
3:34 PM
@AlessandroCodenotti actually in the way it's stated is true regardless of AC, what fails is "every uncountable set has a countable subset" or similar statements
@s.harp test
lol, legend says @Balarka and @Akiva managed to ping a future message without edits once
Don't break MSE
ugh missed it
@Secret in order to ping you need a message after the number :P
3:49 PM
:51735430 test
ok I made that number too big, it will be a while before message 51735430 show up
but it seems that number increment in seconds
i think its all messages in the SE network
So if my hypothesis is correct, if someone can write a message at exactly 51735430, it will become a future ping
@Secret test
3:51 PM
@AkivaWeinberger This is hard
Does anyone know which university bought electronic books from american mathematical society(AMS)??
nothing happened
@s.harp that should be it
3:52 PM
@Secret I wonder if I am missing a space...
@Secret maybe if you ping back
:51735458 Test
Off by one!
:51735470 Test
My school only have access to springer, and my friend in JHU says his school doesn't either.
:51735490 Test
:51735495 Test
3:53 PM
:51735501 Test
success, self ping
51735495 is a self reference ping, made at the point the message is made
@Secret Not showing up for me for some reason
Maybe it only checks for messages that were made in the past
so it doesn't know about itself until it's edited
yeah, I cannot seemed to make a future ping
@AkivaWeinberger test
I dunno whatever
hmm, so one can self ping with no edits if they can anticipate their message number and then write the ping that way (which only show up in your computer)
but not future pinging
3:57 PM
@Andrews To answer your question, I don't know which universities buy ebooks from the AMS
@AlessandroCodenotti no you don't need choice
I know I wrote that later
@AkivaWeinberger Thanks anyway.
4:29 PM
oh man, what "field" asin "field theory" in spanish?
i just saw a thesis about classical field theory with a secondary spanish title calling it "la teoria classica de campos"
as in countrysides
or farm fields
The 5/2-twisted Möbius band, allowed to self-intersect, becomes the 1/2-twisted Möbius band
(apparently campos is actually correct )
(counting twists as 360 degree turns)
doesnt that then hold already for the 3/2 twisted thing?
Don't believe it does
'cause a +1-twisted cylinder is the same as a −1-twisted cylinder (if it can self-intersect)
but not a 0-twisted cylinder
so you can only do it mod 720 degree turns
The belt trick
5:01 PM
En álgebra abstracta, un cuerpo (a veces llamado campo como traducción de inglés field) es un sistema algebraico[1]​ en la cual las operaciones llamadas adición y multiplicación se pueden realizar y cumplen las propiedades: asociativa, conmutativa y distributiva de la multiplicación respecto de la adición,[2]​ además de la existencia de inverso aditivo, de inverso multiplicativo y de un elemento neutro para la adición y otro para la multiplicación, los cuales permiten efectuar las operaciones de sustracción y división (excepto la división por cero); estas propiedades ya son familiares de ...
5:44 PM
@LeakyNun thats field in the sense of $\Bbb Q$ etc, not field in the sense of vector field or field theory
in german "Körper"
Anyone here familar with Matlab
I just saw an ad for coffee on Physics.SE. Are there ads on Math.SE as well ? I'm not talking about community ads
@s.harp right we have "Körper" and "Feld" nice :-)
Actually i finished the MatLab Code, but i need help on the "maths" part
Anyone have any idea how to do 6C?
6:13 PM
@amanuel2: If you wrote the code, can't you run the program and get the solution? I mean, the point is just to solve the system of linear equations with $a$ and $b$ as the "right-hand side."
Of course, you can certainly do this by hand, although fractions will be involved.
@TedShifrin hey Ted, long time, how are you ?
and what are you up to these days ?
Salut, @Gabriel. I was just saying the other day I miss our old Parisian days in here :P
The German speakers have taken over :D
I'm doing pretty well, thanks. Missing math teaching. :(
How're you doing?
Ah. I'm afraid our German friend Daniel Fischer is gone for good
I haven't encountered him in person in years. :( Sad. And Pedro is long lost from chat, too, although making great strides in his graduate work.
So you actually met him in person ?
6:19 PM
Oh, Daniel. No, I meant that I've seen occasional answers he's posted on main.
ah ok
I'm finishing my Masters research internship. After that I'll pursue a PhD. I'm planning on going to Rutgers from January to May
Oh, cool. Wow, that happened fast.
do you have news from the other French students that were at the gathering in Paris ?
6:21 PM
So what is your masters on?
machine learning and statistics. The internship was on computational optimal transport
Interesting. Make sure you think about all the statistical stuff geometrically. Semiclassic and I have been discussing how the statisticians hide all the geometry that's in the linear algebra.
Pretty much all the Parisian crowd (including the meme troublemaker) have disappeared, except for one (who's at Polytechnique).
Now I'm blanking on names :(
they both were at Polytechnique
Hypa and Astyx iirc
I never got their last names
Astyx, yes. Duh. Oh, did Hippa end up at Polytechnique, too? I didn't remember that. I wonder what happened to his brother.
Astyx is doing well, ending up more in science/engineering than math. But that's fine by me :)
I hope my neck gets better so I can travel again ...
yes Hypa was definitely at Polytechnique too
6:28 PM
Moi, je ne me rappelle de rien :P
do you still reside in California ?
6:57 PM
@s.harp I just had a look at the wiki article you sent. I don't see how that is an isomorphism from $S^3$ to $SU(2)$. Do those four matrices generate $SU(2)$?
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