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March 27

2 Questions related to 'Grids' (from a math perspective)

Hi,

In reviewing some stuff I did for a undergraduate project (in computer graphics) a number of years ago, I found myself asking 2 question..

(i)Firstly I found myself wondering what a regular 'grid' was from an abstract math perspective.

Ideally, I'd also like eventually to have some note on Wikiversity about how certain types of grid are developed from an abstract perspective http://en.wikiversity.org/wiki/User:ShakespeareFan00/Grid_definitions

Any ideas on where to start?

My inital rather cursory scanning around suggest topics like Lattice Graph and something to do with a Cartesian product applied to graphs would be relevant to rectilineal 'girds' at any rate..

However there are other types of grpah with regularised arrangements that could also be considerd grids, (like for example those defined in polar rather than cartersian space.)

(ii) I noted that the academic overseeing my project had provided me with a 'distance' preserving transform called a 'slew' transform. (http://en.wikiversity.org/wiki/User:ShakespeareFan00/Slew_transform) I'd be very appreciative if someone here was able to provide an independent citation confirming the correctness (and or existence of 'slew' transforms), alterantivly, if someone has an explantion of how a slew transform can be developed from first principles I'd be most appreciative.

Sfan00 IMG (talk) 16:17, 27 March 2011 (UTC)[reply]

It sounds like Lattice (group) might be a good place to start. Perhaps tessellation would also be of interest. For spaces other than Euclidean ones, the most obvious plan would be first to decide on a group of allowed transformations/motions of the space (such as for example the group of isometries, or the transformations of the complex plane defined by multiplication with a constant) and then look for lattice (discrete subgroup) of that. –Henning Makholm (talk) 16:29, 27 March 2011 (UTC)[reply]

Thanks, not that I am able to understand some of what you are saying... Sfan00 IMG (talk) 17:57, 27 March 2011 (UTC)[reply]

Laplace Transforms

I am supposed to use Laplace transforms to solve $y''-3y'+2y=\delta (t)$ . On doing so, I arrive at ${\mathcal {L}}[y]={\frac {5}{s-2}}-{\frac {4}{s-1}}$ and, after applying inverse Laplace transforms, we have $y=5e^{2t}-4e^{t}$ but when I sub this back into the original differential equation, I have $0=\delta (t)$ ; so close and yet so far. What have I done wrong / do I need to do to achieve the correct answer? Thanks. asyndeton talk 17:46, 27 March 2011 (UTC)[reply]

I get

L[y]=(s-2)^{-1}-(s-1)^{-1}

, so one solution is

y=(e^{2t}-e^{t})H(t)

where H is the Heaviside function. Sławomir Biały (talk) 19:02, 27 March 2011 (UTC)[reply]

Just realised I used the boundary conditions for another question. y(0)=y'(0)=0 was an excellent guess on your behalf! And thanks for the Heaviside bit, it's exactly what I needed. asyndeton talk 19:20, 27 March 2011 (UTC)[reply]

Solution curves

One of my homework problems is to show that the solution curves of dy/dx = -y(2x^3-y^3)/(x(2y^3-x^3)) are of the form x^3 + y^3 = 3Cxy. Differentiating with respect to x, I get 3x^3 + 3y^2y' = 3(xy'+y); solving for y' yields y'=(y-x^2)/(y^2-x). I'm pretty sure this isn't equal to the original expression- am I going about this wrong? 129.219.58.12 (talk) 19:43, 27 March 2011 (UTC)[reply]

When you differentiated the expression

x^{3}+y^{3}=3cxy

you lost the c in the derivative. Try again; I think you'll probably be able to get it now. asyndeton talk 20:32, 27 March 2011 (UTC)[reply]

Fourier Transform

I have to show that $e^{-a|t|}={\frac {a}{\pi }}\int \limits _{-\infty }^{\infty }\!{\frac {e^{iwt}}{w^{2}+a^{2}}}dw$ using the Fourier inversion formula. I'm not quite sure what to do; do I simply integrate the right hand side? If so, could someone provide me with a hint on where to start? I'm not quite sure. Thanks. asyndeton talk 20:38, 27 March 2011 (UTC)[reply]

The right-hand side can be calculated by the Cauchy integral formula, if it is done carefully. But I think what is intended is instead to show that the Fourier transform of the left-hand side is equal to

a/\pi (w^{2}+a^{2})

, which amounts to showing

{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{-iwt}e^{-a|t|}\,dt={\frac {a}{\pi (w^{2}+a^{2})}}.

--Sławomir Biały (talk) 20:45, 27 March 2011 (UTC)[reply]