4D function basis for physical application

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4D function basis for physical application

Postby acoustician » Thu Jan 16, 2020 12:39 pm

Hello!
I've stumbled a couple of times on this forum while doing some research for my scientific project, which will most likely by the subject of my PhD thesis. Long story short - I want to develop a 4D representation for acoustical directivity by modelling time or frequency as 4th physical dimension and I need some 4D basis functions. As far as I know, the only ones used in engineering are hyperspherical harmonics, which I'm certainly going to use, but I was wondering whether or not I can find something else. Since I don't really need the extra dimension to be angular (time or frequency seem to be more of a linear dimension), I was thinking about some basis defined in spherindrical coordinates. This seems to be the only place in the internet that actually knows or cares about spherindrical coordinate system, so here's my question to you - do you know of any spherindrical basis?
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Re: 4D function basis for physical application

Postby wendy » Thu Jan 16, 2020 2:21 pm

Welcome to the forum.

The greater attention here is paid to geometric 4d, ie E4 and its variants. Space-time is different in that it is a stack of flick-cards, which form an animation of history using D_0 as its time-axis.

None the same, spherindrical coordinates are essentially sphere + line, in much the same way as cylinderical is circle + line. It would give, for example, \( t, r, \theta, \phi\), in much the same as adding \(t \) to ordinary spherical coordinates.

The most common coordinate system used here is an oblique 4-linear system, used largely to set polytopes. But feel free to ask, we have many different experts here.
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Re: 4D function basis for physical application

Postby acoustician » Thu Jan 16, 2020 5:42 pm

The whole point of my model is that time/frequency is considered a spatial dimension. So I actually want to parametrize a 4D geometrical shape, if you will. I'm also fully aware of what a sperindrical coordinate system is, just don't know about any 4D basis functions defined within this system. I know it's very specific and scientific question, but I have a feeling that there might be some mathematics freaks in here that could potentially know something more about it.
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Re: 4D function basis for physical application

Postby wendy » Fri Jan 17, 2020 12:19 pm

You could ask your question. We deal with all sorts of dimensions here. So i imagine someone would help.
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Re: 4D function basis for physical application

Postby student91 » Thu Jan 30, 2020 1:23 am

In order to have a countable (i.e. not big) basis of functions for your space, you need to have some kind of periodicity (you have to be able to reduce to compact support).

e.g. if you do a Fourier series, because the system is periodic you obtain a nice sum, whereas a non-periodic Function is treated with a Fourier transform resulting in it being written as an integral (I mean you need an integral to get the original function back).

So I am not sure what your question precisely is, normal 4D-space will not allow for a countable basis of functions.

For the sake of argument, let us assume your function has sperindrical coordinates (r,phi,psi,z), where (r,phi,psi) are spherical coordinates and (z) is your 4th direction. Then phi and psi already have compact support (they are bounded by some multiple of pi), but r and z are not in any clear way compact. Hence, if you do not know anything about the distribution over r resp. z, you will not be able to write your function on a countable basis.
Nevertheless, you can still just do your Fourier things on these axes separately. e.g. do your spherical harmonics on (phi,psi) for every r and z, and then you can possibly also do a Laplace transform in the r direction and a Fourier transform in the z direction. Is that what you are looking for?
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