HOUSE_OVERSIGHT_013691.jpg
1.89 MB
Extraction Summary
9
People
2
Organizations
0
Locations
0
Events
1
Relationships
3
Quotes
Document Information
Type:
Academic text / scientific paper page (evidence submission)
File Size:
1.89 MB
Summary
This document is page 191 of a scientific text discussing chaos theory, specifically 'sensitivity to initial conditions' and 'strange attractors' in nonlinear systems. It cites various prominent mathematicians (Ruelle, Kolmogorov, Smale, etc.) and defines dynamical systems and nonlinear transformations. The document bears the Bates stamp 'HOUSE_OVERSIGHT_013691', indicating it was part of a document production to the US House Oversight Committee, likely related to an investigation involving Jeffrey Epstein's connections to the scientific community.
People (9)
| Name | Role | Context |
|---|---|---|
| David Ruelle | Mathematician |
Cited for coining 'sensitivity to initial conditions' (1978) and work on strange attractors.
|
| Takens | Mathematician |
Cited alongside Ruelle (1971).
|
| Rössler | Scientist/Mathematician |
Cited for discovery of a strange attractor via analog computer simulation (1976).
|
| A.N. Kolmogorov | Russian Mathematician |
Cited for work on nonlinear differential equations (1957).
|
| Sinai | Russian Mathematician |
Cited for work on nonlinear differential equations (1959).
|
| V.I. Arnold | Russian Mathematician |
Cited for work on nonlinear differential equations (1968).
|
| Rene Thom | French Mathematician |
Cited for work on nonlinear differential equations (1972).
|
| Steve Smale | U.C. Berkeley Mathematician |
Cited for work on nonlinear differential equations (1967).
|
| Rufus Bowen | Student/Mathematician |
Identified as Steve Smale's student (1975).
|
Organizations (2)
| Name | Type | Context |
|---|---|---|
| U.C. Berkeley |
Affiliation listed for mathematicians Steve Smale and Rufus Bowen.
|
|
| House Oversight Committee |
Implied by the Bates stamp 'HOUSE_OVERSIGHT'.
|
Relationships (1)
Text refers to 'Steve Smale (1967) and his student, Rufus Bowen (1975)'
Key Quotes (3)
"This behavior was called 'sensitivity to initial conditions' by David Ruelle"Source
HOUSE_OVERSIGHT_013691.jpg
Quote #1
"Qualitative and quantitative global similarities were gained while specific solutions were lost in these 'strange attractors' of nonlinear systems."Source
HOUSE_OVERSIGHT_013691.jpg
Quote #2
"The phrase, 'nonlinear transformation' in this context does not imply easily solvable curved functions... but rather allude to expressions containing products, powers and functions"Source
HOUSE_OVERSIGHT_013691.jpg
Quote #3
Full Extracted Text
Complete text extracted from the document (2,304 characters)
as importantly, far apart initial values could be found close together in the limit set.
This behavior was called "sensitivity to initial conditions" by David Ruelle (1978;
Ruelle and Takens, 1971). It is noteworthy, however, that over a range of values of
the parameters, the overall pattern of the orbits of the Lorenz attractor results in
characteristic geometric pictures as well as invariant statistical descriptors.
Qualitative and quantitative global similarities were gained while specific solutions
were lost in these "strange attractors" of nonlinear systems. Analog computer
simulation of a simpler set of equations inspired by nonlinear chemical reaction
kinetics led to the discovery by Rössler (1976) of another early and generic strange
attractor combining sensitivity to initial conditions and characteristic geometries and
measures.
This behavior was called "sensitivity to initial conditions" by David Ruelle (1978;
Ruelle and Takens, 1971). It is noteworthy, however, that over a range of values of
the parameters, the overall pattern of the orbits of the Lorenz attractor results in
characteristic geometric pictures as well as invariant statistical descriptors.
Qualitative and quantitative global similarities were gained while specific solutions
were lost in these "strange attractors" of nonlinear systems. Analog computer
simulation of a simpler set of equations inspired by nonlinear chemical reaction
kinetics led to the discovery by Rössler (1976) of another early and generic strange
attractor combining sensitivity to initial conditions and characteristic geometries and
measures.
It was the Russian mathematicians, A.N. Kolmogorov (1957), Sinai (1959)
and V.I. Arnold (Arnold and Avez, 1968), the French mathematicians, Rene Thom
(1972) and David Ruelle (1978) and the U.C. Berkeley mathematicians, Steve
Smale (1967) and his student, Rufus Bowen (1975), and their associates who
gathered together these and other related computational discoveries and embedded
them in a qualitative theory of nonlinear differential equations, using a variety of
formalisms, including point set and differential topology, geometry, analysis and
ergodic (having an invariant statistical description) measure theory that formally
established the fundamentals for research in nonlinear dynamical systems. Here a
dynamical system refers generally and simply to the components and nonlinear
processes (transformations) that move points (values) in discrete ("map") or
continuous ("differential equation") time around in an appropriately defined space.
The phrase, "nonlinear transformation" in this context does not imply easily solvable
curved functions, such as those representing the sigmoid kinetic or threshold
functions of enzymes and neuronal networks or those that smoothly log transform
the amplitudes of auditory or other sensory modalities in man, but rather allude to
expressions containing products, powers and functions of the computational and/or
experimental variables x_i, such as x_1x_2 , (x_i)^3 or sin(x) .
and V.I. Arnold (Arnold and Avez, 1968), the French mathematicians, Rene Thom
(1972) and David Ruelle (1978) and the U.C. Berkeley mathematicians, Steve
Smale (1967) and his student, Rufus Bowen (1975), and their associates who
gathered together these and other related computational discoveries and embedded
them in a qualitative theory of nonlinear differential equations, using a variety of
formalisms, including point set and differential topology, geometry, analysis and
ergodic (having an invariant statistical description) measure theory that formally
established the fundamentals for research in nonlinear dynamical systems. Here a
dynamical system refers generally and simply to the components and nonlinear
processes (transformations) that move points (values) in discrete ("map") or
continuous ("differential equation") time around in an appropriately defined space.
The phrase, "nonlinear transformation" in this context does not imply easily solvable
curved functions, such as those representing the sigmoid kinetic or threshold
functions of enzymes and neuronal networks or those that smoothly log transform
the amplitudes of auditory or other sensory modalities in man, but rather allude to
expressions containing products, powers and functions of the computational and/or
experimental variables x_i, such as x_1x_2 , (x_i)^3 or sin(x) .
191
HOUSE_OVERSIGHT_013691
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