[EM] Exact spatial model probabilities?

[Forest Simmons]

To paraphrase Dickens ... they were the good old days ... and the bad old
days!

Daniel's idea reminds me of Robert  B-J's comment about Heaviside and
Dirac  functions. Dirac unit impulse functions are approximated by
Gaussians with infinitesimal variance in the Theory of Distributions. And
every Probability distribution is a convolution of itself with a Dirac
delta ... which is useful because Laplace transforms turn convolution
products into algebraic products.

Electrical engineers are used to approximating all kinds of input signals
with sums of standard functions ... impulse, step, ramp, sinusoids,
Gaussians, that have well known Laplace and Fourier Transforms.

How useful it is to be able to go back and forth between the time and
frequency domains! Even in quantum mechanics ... the more compact the
support of a wave function, the more spread out its Fourier transform, and
vice-versa. That's the wave mechanical basis of the uncertainty principle.

A convenient way to get the mean, variance, and higher moments of a
probability distribution (think voter distribution) is by finding the
Taylor/MacLauren coefficients of the Laplace or Fourier transforms of the
distribution functions.

[Fourier and Laplace transforms differ by a 90 degree rotation in the
complex frequency domain.So what I say for one also goes for the other
without needing to mention it every time.]

My wife needs the phone ... more later..

W

El mar., 25 de ene. de 2022 9:40 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 25.01.2022 06:06, Forest Simmons wrote:
> > Thanks Ted and Daniel. Very interesting!
> >
> > In the early 70's we did our minuteman missile simulations on a room
> > size mainframe IBM 360/65 computer with FORTRAN code, double precision
> > arithmetic ... punched cards interface.... and all. We got one
> > turnaround per night.... night because the Top Secret runs had to be
> > totally isolated from the daytime use of the computer. In 1974 our group
> > got ahold of a couple of the mini-computers that were just coming out
> > ... TTY "ticker tape" interface at first then (unreliable, but more
> > convenient) floppy discs. Very slow, single precision, but interactive
> > BASIC for the spine of the simulation. We employed pseudo-double
> > precision for the numerical integration, and modified BASIC so we could
> > call on assembled bottle-neck subroutines, etc.
>
> It's things like these that makes me think that current computers are
> capable of vastly more than they're currently being used for. Computers
> with less than 1M of RAM could be used to calculate missile
> trajectories, run industrial process control, etc. We now have 16G or more.
>
> Of course, I know that part of the reason is that programmer time is now
> the most scarce resource. The programs that are developed now (mostly
> user-facing stuff) are much slower than they need to be in part because
> it would take too much time and effort to optimize down to the bare metal.
>
> -km
>
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