Discrete 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?

https://i.redd.it/thvnxea9lao61.gif

So this is more of just a fun post: I'm curious if anyone had any applications or ideas which used fractional operators . I also wanted to show of my gif.

A bit of background on what's in the GIF and fractional operators in general.

Recall that the 2D discrete Fourier transform, F, is a linear operator on a space of matrices (or d by d arrays). If we apply the Fourier transform 4 times it returns the identity function i.e. F^4 = F(F(F(F))) = I. Note that people have figured out how to let these exponents take non-integer values! This corresponds to fractional Fourier transforms. So for example the half Fourier transform F^(1/2) is something that functions like the square root of the Fourier transform. If we let G = F^(1/2) then we have that G(G)=F, or maybe a bit more concretely, for any matrix/image X, we have that G(G(X)) = F(X). These special exponents behave like regular old exponents in a lot of ways and it has been observed that one can construct F^a for arbitrary real-valued a.

The GIF I've posed takes an image of a pagoda X and applies increasing fractional degrees of Fourier transforms. Specifically the graph shows shows F^a (X) as a goes from 0 to 4.

Links, more on fractional operators

• Fractional fourier transform: https://en.wikipedia.org/wiki/Fractional_Fourier_transform
• It worth nothing that its possible to take the square roots of many linear transforms thereby making it possible to construct fractional versions of these transforms in a straightforward manner. https://en.wikipedia.org/wiki/Square_root_of_a_matrix
• If you think this is cool you should also check out fractional calculus: https://en.wikipedia.org/wiki/Fractional_calculus
  (Remember, the derivative is also a linear operator 🙂....)

Conclusion
I'm curious if anyone has any interesting ideas...

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

I'm having a really hard time with wrapping my head around Predicate Logic. I'm having an especially harder time with Uniqueness.

Need to translate below two statements to Predicate logic using these two functions:

R(x): person x is in this room

O(x,y): person x owns property in state y

(The two statements are independent of each other)

1. There is at least one person in this room who does not own a property in Georgia

My attempt:

∃x(R(x) ∧ ¬O(x,Georgia))

2. There is exactly one person in this room who does not own property in any state. (Cannot use Existential Uniqueness quantifier.)

My attempt:

∃x(R(x) ∧ ∃z∀y(¬O(z,y)→ z=x))

I feel like I'm missing something in part 2.

For the control theory class I'm teaching to my graduate students in mathematics, I've been tracking down literature that will give the appropriate mathematical depth for some of the concepts we are going to be exploring. This is starting with the Laplace transform.

The Laplace transform is so easy to define for functions of exponential type, with the caveat that the integral might not converge for s with small real part, but I honestly, hadn't seen just how many layers there were between there and giving a proper mathematical definition for distributions.

There are the obvious steps that agree with the Fourier transform, like defining a Schwartz like space, then the dual space on that space using a collection of semi-norms. What surprised me was that in the definition of the Laplace transform on distributions, you have to use an approximate method, where you find a sequence of functions in your space that converge to e^{-st} in some fashion.

After flipping back and forth between two textbooks, I put this video together, if anyone is curious. I left some simple exercises and extra reading in the description that I'm giving to my students to help them get the hang of the space.

https://youtu.be/F-xS27UfF30

Cheers! I'm happy for any feedback!

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 heard that someone got 2 for his final mark in this subject

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?

Just watched the 10th episode of season 2 and I'm confused. The ranking is

1. Technique God
2. Dragon God ...

Technique God being Laplace and Dragon God being Orsted. So according to the ranking, it would be fair to say that Laplace > Orsted. But then Man-god just straight up says "oh nah Orsted > Laplace". Like what, the ranking literally says that Laplace is stronger. Someone plz explain

I recently acquired an API 5500 EQ and I must say I'm impressed with the aggressive mids it can produce, however I immediately found the grainy texture in the highs. I'm wondering if this is yin-yang type effect and you can't have the aggressive mids without that grainy texture.

I heard that people swap out the operational amplifiers in the API with ROGUE FIVE™ Discrete Op Amp to remove the grainy texture, and some forums say that it opens up the range, but you lose a little of that aggressive mids.

Does anyone have experience with changing out the operational amplifiers in their API 5500 for ROGUE FIVE™ Discrete Op Amp? or modifying their API Op Amps in general to reduce the grain? What did you learn, observe, or takeaway?

Looking for some anecdotal observations and experiences about altering the sonic signatures of their API 5500 with different Op Amps.

Cheers.

"humanity's greed always gets them in the end"
No truer words have ever been spoken, Laplace...
If you can't tell, this'll be a fun one. I'm gonna get right into it.

Gameplay- I'm a little torn on Laplace's gameplay. On one hand I like how she has magic bolts that be used for insane spread damage and I love how it looks. On the other...her combo, like Maxime's, feels like it could use work, and I am DREADING her secret missions because archer's in this game are not easy to do secret missions as. Also i have no idea what her base Ougi is, I got very lucky and nabbed her summon outfit. So I was using a massive laser beam- very satisfying, if a little limited.

Personality- Oh this woman. At surface level- and what most see- is someone who does fit her "witch" and "devil" monikers perfectly, being a woman who only cares for her appearance and lust. But...this chapter did well to smash that to pieces. I think Bastien is a perfect counter for her, because he can see past her mask (not helped by him seeming to actually lack emotions) and knows she's a good person deep down- aided by the :gem" she was aiming for this chapter appearing to be a way for her to end her "legacy". Which makes me curious if that comment about her being around for decades means she has a curse of some kind that extends her life, making her an actual vampire (though she isn't), or it's something more unsettling. she is the only character with notable canines. There's something hidden in her past and I am intrigued on what the future will reveal.
Also she's just...oh my god every time she interacts with anyone I am letting loose a devil like grin at how she goes to piss off people all the time, and Bastien literally cannot get any of what she is trying to say. She's a damn good character. Shade approves.

Design- CAN SHE GET ANY MORE SEXUAL. Very exposed dress that shows inner boob, her stomach, her dress splits at the end to show off her legs, and heels plus her dress having heart designs? This woman was designed to be sexual and unlike a certain empire spear user, IT WORKS. The entire point of laplace, in any outfit, is to ooze sexuality and lust, because that is the POINT of her character design. Anyone who says she's pure fanservice- Um, duh? that's the POINT, and her personality is in line with that? I love it when a character is fanservice on PURPOSE compared to say...anyone from Tales of Xillia. (Presa from Xillia was the first that came to mind- I have no ide