Mathematics

user76284

1:57 AM

@LeakyNun Does "hyperbolic" really mean nonzero derivative? I haven't heard that terminology before.

Semiclassical

Context?

Mike Miller

@user76284 I have never heard that either. I would take it with a grain of salt.

Leaky Nun

it's because the equilibrium point of the corresponding ODE is hyperbolic

but now I think it's the wrong interpretation

AbhigyanC

3:10 AM

Can someone suggest a book for very tough multivariable calculus problems?

I'm all ears

krauser126

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

Slereah

8:21 AM

Hey

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 $M_f \cong M$)

Secret

8:35 AM

Mollifier

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...

Secret

9:08 AM

Bifurcation spectrogram

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Tangent bifurcations leading to periodic orbits

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Hopf Bifurcation Diagram with Vector Field (with audio)

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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?

Slereah

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 $B_a$

Secret

9:27 AM

arxiv.org/abs/1808.07352, arxiv.org/abs/1509.02071

Not sure how relevant these are, as they are two different notions of "dual foilation"

Ajay Mishra

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?

Mike Miller

10:29 AM

Ehresmann connection

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

Rithaniel

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: $\int_x^1 2dy=2-2x$

Slereah

10:45 AM

@MikeMiller Thx

Damn, thought I understood Ehresmann connections for a bit :V

Mike Miller

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

Slereah

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"

Mike Miller

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

Slereah

Yeah it's always kind of the issue with fiber bundles

Mike Miller

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

Slereah

The examples you can visualize are all one dimensional

Mike Miller

Look up pictures of contact structures on R^3

Slereah

Pretty hard to visualize a sphere bundle on $S^7$ or what have you

Mike Miller

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

Slereah

We can only hope

but it's like what's his name says

You never really understand math, you just get used to it

Mike Miller

I fucking hate that quote

Slereah

heh

Mike Miller

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?

Slereah

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

Mike Miller

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

Slereah

I just stick with the axiom of choice because apparently the negation of the axiom of choice is even worse

Mike Miller

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

Slereah

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

Mike Miller

Whatever

Slereah

Real numbers were a mistake

Come back Pythagoras

Mike Miller

11:37 AM

lol

I might accuse you of neither understanding nor getting used to it

;)

Slereah

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

s.harp

12:11 PM

aristoteles didnt get killed for drawing circles

he got killed while drawing circles

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$Mathematics$ Mathematics