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GR Glossary

http://archive.ncsa.uiuc.edu/Cyberia/NumRel/glossary.html


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Space and Time
 
In 1917, just a year after the theory of general relativity was published, Einstein suggested that the theory might be applied on a cosmic scale to describe the space and time of the Universe as a whole. According to general relativity, the overall distribution and movement of the contents of the Universe (i.e. the distribution of energy and momentum, and the flow of momentum from place to place) should determine the large scale properties of space and time.

In particular, it should determine whether space is infinite or not, and whether the Universe, which is currently expanding, will go on expanding forever, or whether it will eventually recollapse. The idea of applying general relativity on such a large scale is a bold one, but it has been very fruitful. In particular it has spawned the subject of relativistic cosmology, which is largely concerned with the study of mathematical models of the Universe based on simple assumptions about the contents of the Universe and their behaviour.


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GR

Courtesy: StevenA

In order to create much of anything useful, you need 3 components in the mix. These could be described in many ways but fundamentally they can be equated to:

1) A set of elements (this could be seen as a linear function or value)
2) A non-linear process (a method of combining these into something new or differentiating between already existing members)
3) ... and time always appears to come along for the ride. Time can encompass a lot but fundamentally I don't believe it needs to do anything more than provide a sequence/chronology or ordering to the two steps above.

We could describe the first requirement in various ways as memory, storage, space or dimensions.

Time implies a preferred vantage point to observations (this could be similar to selecting a current set of inputs and outputs or a differential comparison between past and present).

Both those are generally rather easy to work with mathematically, but the complexity arises from the 2nd component - non-linear processes.

We can use almost any non-linearity and create any other non-linearity. There can be some problems with efficiency in mapping from one form of non-linearity to another, but if we restrict ourselves to working with discrete systems only and avoid infinite and continuous fields, most any non-linearity will work. We fundamentally just need an efficient way to map one set of data in 2 orthogonal ways to perform operations with properties equivalent to addition and multiplication, for example, but that's not even necessary. A binary threshold comparison in a discrete system could be enough to substitute for many multiplicative properties, though having an operation providing more than binary information would generally be better.

But, for example, let's take the two operations of addition and squaring and see what can be done with these alone:

x*y=((x+y)^2-(x-y)^2)/4

We could iterate integer powers of a value in, I believe, log^2 time by using a recursive structure. The same could be done for fractional powers by using a binary search, or feedback.

We could generate various functions over time, per some spectrum, by non-linear hetrodyning of a signal and adding or substracting various components to it.

We've got Euclidean and relativistic spaces with c^2=a^2+b^2...

Apparently even the prime numbers can be rather directly related to a choatic bifurcation (x(t)=1-u*x(t-1)^2):

http://arxiv.org/abs/math.DS/0601517

You could even generate a fourier transform via. just additions and squaring alone (I won't bother to detail it, but it follows along the same lines). The spectrum of an atom could be see in terms of the ratio between sums and differences of squares (possibly a nice physical computational engine):

wavelength=k(1/a^2-1/b^2)=k*(b^2-a^2)/(a^2+b^2)

You can even make all this simpler and consider a fine stream of physical probabilities applied to a binary logical operation. For example an AND function applied to two independent (or orthogonal/relatively prime frequency component of a single source) calculates probability(x)*probability(y) (the 2 input NAND function is general purpose and could be seen as calculating 1-a*b, from which an adequate approximation of any other function should be possible to generate with enough recursion). If applied to a single source, you can utilize orthogonal relationships in the frequency spectrum by placing "tap points" of this filter at relatively prime locations in the stream (so you'd have an uneven spiral of information cycling through a single processing point at relatively prime locations in the sequence). That may not appear immediately very efficient in terms of accuracy for a result over time, but when you consider the dynamic response, it has a high bandwidth and if you're "beaming" a signal over space, the "memory" or storage costs almost nothing.

Anyway, my intent wasn't really to describe the power of typical addition and multiplication but instead to encourage development of an pair of alternate orthogonal relationships that could provide a method to reduce the fundamental complexity of calculating these (though of course having an efficient physical implementation is the ultimate goal). This of course relates to a ton of mathematical fields that are all related upon the same thing - an efficient manner in which to process non-linear relationships in more than a single dimension.

For example, the function:

x={x: x<=1 ; 1+(x-1)/2 : x>1} is a very simple operation to perform on a serial bitstream for a value in binary (it's just a conditional bit shift). But this provides a single non-linear point and a recursive structure of this could be more useful (... which if done a little different could get you characteristics related to phi/golden mean ... hmmm), though not as easy to utilize as, for example, a logarithm.

Gray coding (and not just in binary) has some very symettrical features that might prove useful as well.

Anyway, if there's a way to reproduce a fractal structure with very efficient operations (I keep thinking binomial expansion and the sierpinski triangle could be a good starting point as the structure is discrete, has well defined characteristics, including gaussian/statistical/Euclidean and exponential features, and you can find many other exponential structures embedded within it - gaussians also appear to be able to get you a free conversion between time and frequency domains), then it could be just a matter of finding the chain of ways to remap it into other domains.

Well, that's about it. There's a challenge that would revolutionize most all of science, mathematics and physics and could likely provide a much deeper insight into "how it all works", if someone can find a better system of mathematics that possesses these features.








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--Last edited by saucer on 2007-08-10 20:17:16 --

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Minkowskian


http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec7.html



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