Nonlinear Systems:
Clue: Gravity bends also. (Lindblom Transformation)
This website represents research for magazine articles not the articles themselves. Lindblom
Volterra:
The Volterra/Wiener representation for nonlinear systems
http://www.maths.dundee.ac.uk/~fdavidso/xcdaa.pdf
*****
"His most famous work was done on integral equations. He began this study in 1884 and in 1896 he published papers on what is now called 'an integral equation of Volterra type'. He continued to study functional analysis applications to integral equations producing a large number of papers on composition and permutable functions."
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Volterra.html
"We introduce a generalized version of the Volterra-Wiener functional expansion and show its fruitfulness in the treatment of nonlinear stochastic processes.
As an application we derive in a simple way the statistical properties of the single-mode laser radiation both in the steady state and in the transient regime.
Besides deriving already known results with good accuracy, we give for the first time the transient correlation function for the photon number."
©1976 The American Physical Society
http://prola.aps.org/abstract/PRA/v14/i1/p383_1
A Logical Extension:
Abstract:
"Many processes display nonlinear behavior if they are driven over a large operating range.
If linear controllers cannot yield satisfactory control performance, nonlinear control techniques have to be employed.
This requires the knowledge of nonlinear process s.
This paper presents an overview about process architectures originating from the fields of neural networks and fuzzy systems, based on which nonlinear -based controllers can be designed.
Three commonly used -based control approaches are described.
Depending on the controller design approach and later controller implementation, different demands on the architecture arise.
These demands concern the exploitation of the linear control techniques, the incorporation of prior process knowledge and the fulfillment of hardware requirements.
These issues will be discussed and nonlinear ing and control of an industrial- scale heat exchanger based on neuro-fuzzy network will be presented as an illustrative example."
Keywords:
Nonlinear System,Local Architecture,Modeling,Control,Predictive Control,Internal Control,Heat Exchanger
http://w3.rt.e-technik.tu-darmstadt.de/~fink/FinkEtal2003.en.html
Local Linear Neuro-Fuzzy s
http://www.iee.org/Publish/Books/Control/index.cfm?book=CE%20070
http://elindblom.bravehost.com
"In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions.
Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity).
Moreover, it satisfies a more technical completeness requirement which ensures that limits exist when expected, which facilitates various definitions from calculus.
Hilbert spaces provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis.
Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics."
http://en.wikipedia.org/wiki/Hilbert_space
In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior.
It occurs when a system interacts with its environment in such a way that different portions of its wavefunction can no longer interfere with each other.
In the many-worlds interpretation of quantum mechanics, decoherence is responsible for the appearance of wavefunction collapse.
http://en.wikipedia.org/wiki/Decoherence
Superposition and entanglement
Decoherence occurs when a system loses phase coherence between different portions of its quantum mechanical state.
It then no longer exhibits quantum interference between those portions (as might be seen in a double-slit experiment).
Decoherence is caused by interactions with a second system which may be thought of as either "the environment" or as "a measuring device".
In the latter view, the interactions may be considered to be quantum measurements.
As a result of an interaction, the wave functions of the system and the measuring device become entangled with each other.
Decoherence happens when different portions of the system's wavefunction become entangled in different ways with the measuring device.
For two portions of the entangled system's state to interfere, the original system and the measuring device must both evolve into the same state.
If the measuring device has many degrees of freedom, it is very unlikely for this to happen.
As a consequence, the system behaves as a classical statistical ensemble of the different portions rather than as a single coherent quantum superposition of them.
From the perspective of the measuring device, in each member of the ensemble the system appears to have collapsed onto a state with precise values for the measured attributes.
http://en.wikipedia.org/wiki/Decoherence#Superposition_and_entanglement
superposition and entanglement
http://en.wikipedia.org/wiki/Quantum_computer
bravenet.com