Rob, (01)
I like holograms, and I agree with the following point: (02)
> To anyone with an interest I recommend studying why a
> holographic memory should be able to store more data.
> How each element in a hologram contributes to the signal.
> Why in a hologram each piece gives you the whole picture,
> but adding pieces gives you more resolution. (03)
There are also many related mathematical techniques that
transform digital data into continuous forms. The Thom-
Wildgen methods, for example, are very different from
holograms, but they also relate digital data to continuous
mathematical methods. I believe each of these methods
captures one view of the elephant, but putting them all
together to reconstruct the elephant requires more work. (04)
> The trade-off being that the information of each element
> becomes somewhat indefinite, even contradictory. (05)
The phrase "somewhat indefinite" is appropriate. But I would
not use the word 'contradictory' because that is a misuse of
a technical term from one field that is more misleading
than helpful when applied to a very different field. (06)
If you would like more references, I suggest: (07)
http://www.cs.cmu.edu/~lblum/PAPERS/TuringMeetsNewton.pdf
Computing over the reals. Where Turing meets Newton. (08)
Excerpts below. (09)
However, there is still much more to be said about these issues.
And metaphors that misuse technical terms in either discrete or
continuous mathematics are more confusing than helpful. (010)
John
_____________________________________________________________________ (011)
Computing over the Reals: Where Turing Meets Newton (012)
Lenore Blum
Computer Science Department
Carnegie Mellon University (013)
The classical (Turing) theory of computation has been extraordinarily
successful in providing the foundations and framework for theoretical
computer science. Yet its dependence on 0's and 1's is fundamentally
inadequate for providing such a foundation for modern scientific
computation where most algorithms --with origins in Newton, Euler,
Gauss, et. al. -- are real number algorithms. (014)
In 1989, Mike Shub, Steve Smale and I introduced a theory of
computation and complexity over an arbitrary ring or field R.
If R is Z2 = ({0, 1}, +, ?), the classical computer science theory
is recovered. If R is the field of real numbers, Newton's algorithm,
the paradigm algorithm of numerical analysis, fits naturally into
our model of computation... (015)
In this paper, I discuss these results and indicate how basic notions
from numerical analysis such as condition, round-off and approximation
are being introduced into complexity theory, bringing together ideas
germinating from the real calculus of Newton and the discrete
computation of computer science. The canonical reference for this
material is the book, _Complexity and Real Computation_ ... (016)
More fundamental differences arise with the distinct underlying spaces,
the mathematics employed and problems tackled by each tradition. In
numerical analysis and scientific computation, algorithms are generally
defined are over the reals or complex numbers and the relevant
mathematics is that of the continuum. On the other hand, 0s and 1s
are the basic bits of the theory of computation of computer science
and the mathematics employed is generally discrete. The problems of
numerical analysts tend to come from the classical tradition of
equation solving and the calculus. Those of the computer scientist
tend to have more recent combinatorial origins. The highly developed
theory of computation and complexity theory of computer science in
general is unnatural for analyzing problems arising in numerical
analysis, yet no comparable formal theory has emanated from the
latter.... (017)
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