The Conscience of the Colony ... An apparent attribute called intellect.

[Drifter]

Poincare` Recurrence Theorum:

Poincaré recurrence theorem
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In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics.
The theorem is named after Henri Poincaré, who published it in 1890.
Contents

• 1 Precise formulation
• 2 Discussion of proof
• 3 Formal statement of the theorem
  
  • 3.1 Theorem 1
  • 3.2 Theorem 2
• 4 Quantum mechanical version
• 5 See also
• 6 References

[edit] Precise formulation

Any dynamical system defined by an ordinary differential equation determines a flow map f t mapping phase space on itself. The system is said to be volume-preserving if the volume of a set in phase space is invariant under the flow. For instance, all Hamiltonian systems are volume-preserving because of Liouville's theorem. The theorem is then: If a flow preserves volume and has only bounded orbits, then for each open set there exist orbits that intersect the set infinitely often.[1]
As an example, the deterministic baker's map exhibits Poincaré recurrence which can be demonstrated in a particularly dramatic fashion when acting on 2D images. A given image, when sliced and squashed hundreds of times, turns into a snow of apparent "random noise". However, when the process is repeated thousands of times, the image reappears, although at times marred with greater or lesser bits of noise.



http://en.wikipedia.org/wiki/Poincar...rrence_theorem

[Drifter]

Ergodic theory

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Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics.
A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important examples are ergodic theorems of Birkhoff and von Neumann. For the special class of ergodic systems, the time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger properties, such as mixing and equidistribution, have also been extensively studied.
The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.
The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).

[Drifter]

Harmonic analysis

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Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called "harmonics" (in physics), hence the name "harmonic analysis," but the name "harmonic" in this context is generalized beyond its original meaning of integer frequency multiples. In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience. The classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley-Wiener theorem is an example of this. The Paley-Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported. This is a very elementary form of an uncertainty principle in a harmonic analysis setting. See also Convergence of Fourier series.
Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.

[Drifter]

What is Spanda?
http://www.amazon.com/Spanda-Karikas.../dp/8120808215

Bhava Spandana:
“Bhava Spandana is like making a person jump and look beyond the wall. He sees beyond his limitation experientially. Once he sees that, he knows that he must go over the wall one day to see what is on the other side.” -SADHGURU
http://www.ishafoundation.org/Isha-Y...a-spandana.isa

[Drifter]

Nicloa Tesla 'spanks' a Pulsar: http://www.scam.com/showthread.php?p=1085338

[Drifter]

around 38 minutes into this documentary an interesting encounter Tesla has, he says best explains the source of his engenius ideations and theorums. http://www.youtube.com/watch?v=3M_RT9UX3ZY

[mkirkpatrick]

Hey Greg! Can you hear me in them thar mountains? Do you have a computer there? If so then please use it! Waiting for your reply mate.



regards michael.

[labelwench]

An interesting article that you might enjoy reading....

Several of the nearly 200 scholars nominated what are arguably the two most powerful scientific theories ever developed. "Darwin's natural selection wins hands down," argues Dawkins, emeritus professor at Oxford University.

"Never in the field of human comprehension were so many facts explained by assuming so few," he says of the theory that encompasses everything about life, based on the idea of natural selection operating on random genetic mutations.

Einstein's theory of relativity, which explains gravity as the curvature of space, also gets a few nods.

To neuroscientist Robert Sapolsky of Stanford University, the most beautiful idea is emergence, in which complex phenomena almost magically come into being from extremely simple components.

For instance, a human being arises from a few thousand genes. The intelligence of an ant colony - labor specialization, intricate underground nests - emerges from the seemingly senseless behavior of thousands of individual ants.

"Critically, there's no blueprint or central source of command," says Sapolsky. Each individual ant has a simple algorithm for interacting with the environment, "and out of this emerges a highly efficient colony."

Among other tricks, the colony has solved the notorious Traveling Salesman problem, or the challenge of stopping at a long list of destinations by the shortest route possible.

From the Baltimore Sun, Jan.16th, 2012
http://www.baltimoresun.com/business...,4109791.story

[Graybeard]

Its the simplicity that is truly amazing.

No one ever thought, while pondering the imponderables, life the Universe and why is it all .... that its underpinning, it generating ability ... could have such simple algos driving the complexity.

The theory of natural Selection will eventually encompass the universe as it is not limited to just 'life' ... but may be applied to any situation.

A side note: The intelligence or information of the colony, (as opposed to the intelligence or information possessed by an individual member) does not actually exist ... but is presupposed and disposed to become apparent only by the actions of those who can observe from a distance and objectively. It is derivable information available only to those who can derive.

It is ephemeral .... non-existent ... this subtly is the reason that theories such as Intelligent Design become acceptable to many when a deeper observation would show clearly that simplicity has created an illusion-trap that some of the greatest minds have fallen into. Such is the side effects of, and the burden of, carrying an ego.

"Critically, there's no blueprint or central source of command," says Sapolsky. Each individual ant has a simple algorithm for interacting with the environment, "and out of this emerges a highly efficient colony."

great post

cool bananas ... greg