Saturday, January 05, 2008

Infinite Probes in Optimization

Infinite Probes in Optimization .
Andre Willers
5 Jan 2008

Definitions:
n! = n*(n-1)*(n-2)*(n-3)*…*1

The Euclidean Constant e .
e = (1/1! + 1/2! + 1/3!+ 1/4!… )
=2.178…

I will make it as simple as possible using costs and temporal sequences as examples .

Assume that the elements of a system are identifiable and repetitive .
The rules of Arithmetic can then be applied .

Take any such system . By definition it can then be subdivided indefinitely . Each subdivision of r elements can have r! possible permutations (or states) .

Let there be a CostReserve to cover unknown costs in the failure of any element in a group .

Key Assumption
Let at least one element of each r be Aristotelian False (ie a failure cost item)

The Total cost of failure will be the sum of the cost of failures of the elements.
(ReserveCost) *( (1/1! + 1/2! + 1/3!+ 1/4!… ))= (TotalCost)

Your Reserve times the sum of individual failures = total
This defines the boundary .

This is the optimization equation .

This is equal to
ReserveCost / TotalCost = 1/e ~ 0.37

This means that you should not consider an endeavour where the rate of return is less than 37% . This is the long-term break-even point from a probabilistic viewpoint .

This equation is independent of space or time .

Any society (any at all ) that does not deliver 37%+ returns gets zapped by the chaotic downturns .

Since vested interests always decrease rates of returns , empires rise and fall .

The probe can be used in any infinite series , but care has to be taken to define the terms .

An Interesting corollary :

The alert reader would have noticed that the above is true for when there is always at least one false item per group .
What if there is at least one true item per group ? This is approached as follows:

In trading systems , individual items can be identified . The cost of failure in an individual item is then = 1/ (r!/r) = 1/ (r-1)!
The summation is then
(ReserveCost/Item)*(1*1/1! + 2*1/2! + 3*1/3!+ 4*1/4!… ))= (TotalCost)
(ReserveCost/Item)*(1 + 1 + 1/2!+ 1/3!… ))= (TotalCost)
(ReserveCost/Item) / (TotalCost) = 1/(1+e) ~ 0.27

As can be seen , trading systems will survive the collapse of empires . There is a whole 10% difference favouring traders . Recovery systems will then also favor traders .

This is the underlying reason why Stalinistic Communism failed and Chinese went trader . The VOC and East India Co . are good examples .

Ten percent is just too much .

Ho-ho-ho !
The average between 0.37 and 0.27 is roughly 0.33 . A third .Can you see where this is relevant in any infinite situation where uncertainty rules?

Non-Failure Ratio
A non-failure ratio of (1- 1/3) ~2/3 summated over infinity = (2/3) / (1-2/3) =2 .

This means that any system that can be described or broken up into smaller groups is strongly ordered . Successful subgroups survive by a factor of 2 . This is built into the mathematics of the Universe . Life is one such sub-group , so are particles ,etc . Negative entropy is the rule . Life is everywhere . Positive entropy only holds in the most primitive and earliest systems .

Beautiful!

Quantum and non-quantum systems are the same .

Singularity Points.
27% and 37% are singularity points , as are (100-27)=63% and 53% . SocioEconomic systems destabilize at these points . Regardless whether they are human , alien , japanese , chinese or any other sort of ese .
Any society where more than a third of productive enterprise is spent without engendering new wealth creation is doomed .

Tax rates of more than 27% reduce trade . Tax rates between 27% and 37% will let it limp along . Tax rates over 37% is a disaster waiting to happen . It is that simple , and no amount of pontificating will change the realities of economics .

The tax rate anchored between 0% and 27% will follow a catenary . The optimal will be halfway (12.5%) . This would be the optimal VAT rate . Note that deviations from this (up or down) leads to social unrest . The unrest is not linear , but catenary . In other words , it gets rapidly worse as it approaches the anchor points of 0 or 27 % . This is true of any civilization , human or alien . It is inherent in the mathematics .

These are absolute rates , not relative .

The RepoRate .
The tax on new money . If the reporate plus profits stay away from the singularity points (27% , 37%) there is no problem.

What about undefined realities ? Refer to
http://andreswhy.blogspot.com “Topos:
Note in the definitions above , that as groups are divided by n , and n->infinity , the system approaches undefined “thingies” . Hence , it can be used as a probe into undefined realms .

Marriage:
An interesting corollary is optimizing pairings of two . The theory above says that random pairing will lead to a failure rate of 37% . Observation seems to bear this out .

To optimize : Note and keep track the best features of the first 37/100 daters , then seriously consider the first anybody after that who equals or better the standards of the first 37

This result has been independently reached in games theory

Note that your body is independently doing this sort of infinite analysis . Hence speed dating , cocktail parties , mingling , etc .

A good algorithm is to speed date at least 50 persons . Do not do this if you cannot handle rejection , since this concentrates years of normal rejection into a few weeks .

A therapeutic mechanism? Sects use this whipsaw mechanism of rejection (by the outside world) and warm acceptance by the sect to create mini-societies .
Hence cold-canvassing . Rejection binds them closer . Any converts are a bonus . Most real converts are made by on-on-one contact .

But that is also the definition of most successful families .

If you are stupid and ugly , and can handle speed dating , you are a better man than I am , Human .

Virtual speed dating .
Analogous to cures for fear of heights , fear of confined spaces , etc . Considerable success has been achieved in reprogramming the amygdala for these and related phobias using virtual reality .

A Virtual Rejection website will make money .
That’s OK if somebody learns from it .
Another way for actresses or avatars to make money .
“I was rejected 1000 times by Lara”
“Balls rejected by Lara TombRaider as not dusty enough.”
The mind boggles .

Andre