Laplace operator Puns

This may sound pretty odd to do but just wanted to say it can be done!!!! (Also you can solve ODES with nonzero IC's with Laplace if you "shift" into a new coordinate system).

Suppose ∆ is given explicitly on M in some coordinates. How might I retrieve g in the same coordinates?

I am working on a problem involving the Navier-Stokes Momentum equation in cylindrical coordinates (r,theta,z). There is a term "\del^ 2{u_\theta}" but I am not entirely sure what this means. I thought \del^2 was the Laplace operator but I've only seen it involving a vector, not a component of a vector. How would I expand this? From the solution to the problem, that term converts to "1/rd/dr(r*du_theta/dr)". How was this derived?

Let a be in the spectrum of ∆. Will ∆-a have a Green's function/fundamental solution? What conditions on the manifold are necessary?

Irrc, if ∆ is a constant coefficient differential operator, then the answer is yes.

(speaking of which... Wikipedia claiming that a constant coefficient differential operator is the most important case is just stupid.)

I'm serous, it comes to my attention that we can keep a "sub whatever" for pages and pages but something as simple as ∇^2 is often written as ∆, which may be very confusing sometimes since the differentiation is also written this way. So, why?

Does the s variable make sense to anyone in this way? Can it even be described like this? Laplace transforms were always a black box to me, I think this could be the key for me to understand them better.

If someone has an easier way to understand it, I would like to hear it. Thanks!

Suppose [; U \subset \mathbb{R}^n ;] is bounded and open, with smooth boundary. Then for [; u \in C_c^\infty (U) ;] we have [; \int_U |\Delta u| = \sum_{i,j=1}^n \int_U |\partial_i \partial_j u| ;] .

I can see how [; \le ;] is true by the triangle inequality, but can't really see why it should be an equality. Any hints are appreciated!

So the Laplace operator is if I understand right the divergence of the gradiënt of a function.

Now imagine the function: f(x,y)=x^2 + y^2. Then div grad f = f''(x) + f''(y) = 2 + 2 = 4

Now if we take the point f(3,2) we get 3^2+2^2 = 13.

If we calculate the average of the neighboring points, i.e. f(3,1;2), f(2,9;2), f(3;2,1), f(3;1,9) we get:

1/4 (9.61+4+8,41+4+9+4,41+9+3,61)=52,04/4=13.01

I don't see the value of the Laplace operator in connection with the average of the neighboring points? What am I doing wrong.

My mission is to numerically understand the fact that a Laplace operator is finding the average

Links to better explain what I mean: http://physics.stackexchange.com/questions/20714/laplace-operators-interpretation

https://www.quora.com/What-is-an-intuitive-explanation-of-the-Laplace-operator-or-Laplacian-operator

1. u(x,y) = 3x^(2)y -y^(3). Show that u_xx + u_yy = 0. (∆u = 0).

2. u(x,y) = ln(x^2 + y^(2)). Show that ∆u = 0.

3. Derive the Laplace operator for polar coordinates (using other words). Basically show that the bottom is true for u(x,y) when [;x = r\cos(\theta), y = r\sin(\theta);].

I am not planning on using this operator or any of these techniques on the test or quizzes, I am just wondering what these problems say about this operator and any interesting properties that one should know about it.

I know the the laplace-Beltrami is the more general form of the laplace and it can work for surfaces as well but I'm confused because they are both defined on the wiki as

\Delta = div grad f(x)

So I'm wondering how they are different?

I am quite confused about how to solve this problem:

Let U c R^n be a domain, g a riemannian metric field on that domain. f a smooth function on U. The laplace beltrami operator Lg = 1/det(g)sum(d_j g(sqrt(det(g))*g^(ij)*d_i)) where d is the partial derivative (d_i gmeans we take the partial derivative of g with respect to the ith coordinate xi) and we sum over all i and j.

Show: Lg(f) = g^(ij)*del_i(d_j f)

Where with del_i we mean the covariant derivative on a vector field, if vl is a covector field, then del_i vl = d_i vl - vj G^j _(il) where G are the Christoffel symbols.

I have no idea about where to start here. I thought about somehow calculating the derivative of det(g), but I can't see how to go about that? I googled and found that you could use: det(g)=exp(tr(log(g))) or something like that, however, then we always end up getting a trace in there, and as far as I know the trace doesn't just disappear?

Anybody got any idea?

I need to prove that it is impossible that an eigenfunction (except the first one because it is constant and therefore don't have any nodal lines) of the Laplace operator on the unit disc with Neumann boundary condition only have nodal lines that do not touch the boundary of the disc.

I know the proof for the second eigenfunction, but that's all.

In fact, given the general form of the eigenfunctions on the disc, I can prove that assertion, but I need a more elegant proof that would perhaps reduce the general case to the case of the second eigenfunction or make use of the following inequality :

\mu_{k+1} <= \lambda_k for all k

where \mu and \lambda are respectively eigenvalues of the Laplace operator on the disc with Neumann (resp. Dirichlet) boundary condition.

Any help?

Hello everyone! And happy new year!

This spring I am teaching a graduate course on Control Theory in the mathematics department at my university, and this is going to be a great learning experience for me. I got my PhD almost a decade ago studying functional analysis and operator theory, but then I went on to do two postdocs focusing on nonlinear controls.

I've always felt that I missed out on learning linear controls, and so I'm using this class to really dive into the subject. I have about 10 books I'm pulling from, where I am trying to strike the balance between advanced mathematical material and some more boots on the ground (for a mathematician) control theory.

It's surprisingly difficult to find a textbook that covers both the mathematics and the control theory well. Sontag's text does a decent job, but some of the topics I want to cover (like H infinity control) aren't in there. However, Doyle, Francis, and Tannenbaum's textbook covers H infinity controls, but only mention the essential mathematics in passing.

And none seem to really go deep enough to give a rigorous definition of the Laplace transform on Distributions (like the delta function)! Yamamoto's textbook From Vector Space to Function Spaces does a half way decent job, but then pushes the important proofs off into references. So I have a whole library I'm using to teach a single class.

H infinity control theory is a great little space to explore the interconnection between some operator theory and controls. It rests on the mathematical framework of Nevanlinna Pick interpolation, which concerns operators over reproducing Kernel Hilbert Spaces (specifically the Hardy space of the half plane). But I'll also go into PID controllers, cover Nyquist's stability theorem, and other fundamental concepts from controls.

This video here is my introduction to the course, and I'm currently editing the lecture on the definition of the Laplace transform for distributions. It should be a lot of fun!

Let me know if you have any pointers, references, or advice. I'm happy to learn as much as I can :)

I just finished it and I kind of regrettably rushed it, because I knew how many volumes was left that I kind of kept reading for the sake of finally getting to the conclusion. I wished I was reading without knowing how many volumes was left so I wasn't so focused on how it's going to end.

My only complaints was the final battle with Gisu was a little anti-climatic, we knew how powerful Orsted was, and the only time he seemed touchable was when he first fought Rudeus and after that he was like Saitama. It felt like the chance of victory for Hitogami, at least for Rudeus's story felt so impossible, and yes they kept hammering on the fact that Orsted would end up using too much mana and couldn't have enough to fight Laplace and Hitogami, but it felt like such an arbitrary consequence where we didn't even know the exact amount that would had made it so the loop would be a complete loss, and history has proven that things that were thought impossible from his previous loop didn't hold true, and Hitogami's defeat is so far in the future, it wouldn't had even been a plot hole had he even used up every ounce of mana in the final fight.

I also liked how Rudeus, even though he's shown to be an OP MC, wasn't all that powerful and still needed a lot of help to win against top tier characters in the verse. There was at times how frustratingly weak he was at the end, but it's probably because I'm use to the MC being the strongest person and just solo carrying the entire show, at a certain point.

It was left with a lot of room to expand on and it gave a satisfying enough ending where we can conclude Rudeus story and move on with Lara's without any second thoughts. I just wished we at least could verify a little more on Nanahoshi end because I am sure a lot of people came up with this theory that they were brought into this world to defeat Hiogami when it was revealed who Nanahoshi was during the Magic Academy arc. Even though we went through A LOT of source material, we only scrapped the surface of it all, for what was essentially just the beginning of a larger story.

Going to be so excited for this to all get animated, but it's going to take like 8 years at minimum or some shit to conclude given how the anime industry operates... Unless they get enough budget to have it running weekly like a Shounen which is unlikely.

Some time ago I came across a fantastic talk on the moonshine conjecture called can't you just feel the moonshine? and I thought it was just brilliant; since I'm writing my undergrad thesis I really want something that will catch the eye of the commission, do you have any suggestions? The topic is discrete differential geometry, in particular the use of the Laplace-Beltrami operator in computer graphics and mesh processing. I kinda liked

>A spectre is haunting geometry processing—the spectrum of the laplacian

but it's maybe a bit too political hehe. Ideas?

Lately I found an interest in solving PDEs numerically. I've done some simulations on rectangular grids(heat equation, wave equation with location dependent velocity etc), and I'd like to do something more sophisticated. My end goal is to perform numerical simulations on curved surfaces, but simple things first.

Flat domain

Given I've got a flat domain discretized into a bunch of triangles with a value associated at each point. I would like to calculate the spatial derivatives at point P given the values at P's immediate neighbors. I figured I could calculate a linear function that interpolates values at vertices belonging to each neighboring triangle, calculate the derivatives based on each interpolant and then do a weighted average. However I'm not sure whether the weight should be with respect to the surface area S, or the edge length l?

Abstracting the ambient space

Is it possible/sensible to express the gradient/laplace operator not in terms of derivatives with respect to the axes of ambient space, but in terms of some sort local barycentric coordinates spanned on the neighboring vertices? Would the differential equation change in any way? E.g would the equation u_{t} = ∇^(2)u remain the same, or would some additional factors show up?

Curved domains

What considerations would I need to take if the domain weren't flat? For example if the domain would approximate the surface of a sphere or a torus? I feel like there's a need to account for the curvature of the surface in the differential operators. This is why I thought about expressing the gradient not with respect to ambient space but with respect to the local space defined by the simplicial complex(should it be a good idea).

All of this is for fun/learning, so one can assume everything is sufficiently nice, smooth and not an edge case.

Preamble

I wrote this a while back like since like 2013 lost it and found it in an old harddrive sometime early this year and waited till Resurrections came out and to think about it before putting this up.
This write up doesn't reference any of that film's narrative, also I'm ignoring a lot of the Matrix Online

Yeah it's a bit fanfictiony, giving a reread but I still like it enough to put it up here.

Quick review of The Matrix Resurrections
I liked it, it felt very personal and auteur-driven (and a bit of catering of Baudrillard's critiques)

A lot of decent ideas and very interesting subtext but the establishment and worldbuilding was lacking, I would've loved to see an extended version of this as a limited series a la Watchmen.

I wanted more of the crew of the Mnemosyne TBH

I thought I would've missed Fishburne but I more missed Weaving if anything.

This is probably due to covid but everything felt too small, IO felt too small, the new matrix felt too small, action was okay but too small scale and uninspired for my liking.
The core conceit involving Neo and Trinity is what sticks and mostly works in my opinion.

Format and conceptualization

In my personal opinion the ideal format, is an 8-10 episode miniseries a la HBO's Watchmen.

This is the main reason a lot of legacy franchises suffer so much because it's solid ideas that in no way can be fit into 2-21/2 hr movie. (cough terminator)

The main protagonist is Sati, with her growing up in the fallout of the end of the War and the main themes would be justice, grievance, forgiveness and reconciliation in a post-colonialist context and the idea of deriving purpose from that.

A lot of focus would be on the Oracle and The Architect and their full history.
Spoon Kid and The Kid would be important bit players as well

Plot.

So it's about fifteen or so years after the end of Revolutions, Sati operates as a liaison between machines and humans, slow efforts at reconciliation by slowly waking humans up and freeing them and letting machines free to occupy the world

The first step is to solve the scorched sky problem and then revitalise earth's ecosystem by a type of nanotech engineering cloud that would create synthetic biology and would kickstart real biology to flourish from the genetic data in the matrix.
She's in a relationship (polypan) with spoon kid and The Kid and she spends a lot of time with human

However there are machines that don't agree wi

Do your worst!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

I'm surprised it hasn't decade.

They’re on standbi

What fundamental properties causes Laplace Transform to be chosen for transfer function theory