Monday, January 15, 2018

Hand waving the theory of ratio and window

We are really interested in the stability of loan/deposits, it tends to a point oc convexity.

Call it the ratio, loans/deposit.  Examine delta ratio, its change over some fixed and countable trade sequences going forward.  Our generator is a compact representation of the typical sequences that fill the trade space. Increasing the window size will never add divergence in the bit error. Let us take that as a given, bit error at least does not increase over as window size increases.

So, considering the ratio, as it tends to one the bit bounds gets smaller, the window gets larger. delta ratio, over some fixed window size, goes to zero, as the ratio reaches one, it cannot exceed one. [Note, we expand this condition later and make the bit error have a looser bound].

I should add. The horwitz remainder theorem uses repeating remainder windows to narrow down approximation error. The idea is the same, the generators are derived from the hyperbolic queueing model, via the extension of Shannon to a two sided problem.   The resulting generators being maximum entropy, and compact are also fractional estimators. 

So, the ratio goes from zero to one, and the delta ratio over any fixed window goes to zero. Nothing is concave, there is a maximum ratio times the delta, a point in trade space where the maximum flow occurs from loans to deposits.  That is the maximum ratio variation, or should be, supported by the network.

But all the ratio, price, loan/deposits, yields work equally, there is a pricing sweet spot.   The only condition we assume is that queuing along the generator is stable, which is the sphere packing condition. We are alway filling and emptying, keeping the proper congestion.

Which brings up a very interesting idea that number theory is all about making recursive algorithms queue properly. Prime numbers make great queue structures in their recursive multiply and divide. We really lead into complexity theory, identifying what the bounds are on the complexity of a theory, then derive some good estimates of the strength in its various forces.


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