1,751 research outputs found
Life-Space Foam: a Medium for Motivational and Cognitive Dynamics
General stochastic dynamics, developed in a framework of Feynman path
integrals, have been applied to Lewinian field--theoretic psychodynamics,
resulting in the development of a new concept of life--space foam (LSF) as a
natural medium for motivational and cognitive psychodynamics. According to LSF
formalisms, the classic Lewinian life space can be macroscopically represented
as a smooth manifold with steady force-fields and behavioral paths, while at
the microscopic level it is more realistically represented as a collection of
wildly fluctuating force-fields, (loco)motion paths and local geometries (and
topologies with holes). A set of least-action principles is used to model the
smoothness of global, macro-level LSF paths, fields and geometry. To model the
corresponding local, micro-level LSF structures, an adaptive path integral is
used, defining a multi-phase and multi-path (multi-field and multi-geometry)
transition process from intention to goal-driven action. Application examples
of this new approach include (but are not limited to) information processing,
motivational fatigue, learning, memory and decision-making.Comment: 25 pages, 2 figures, elsar
Tsallis entropy composition and the Heisenberg group
We present an embedding of the Tsallis entropy into the 3-dimensional
Heisenberg group, in order to understand the meaning of generalized
independence as encoded in the Tsallis entropy composition property. We infer
that the Tsallis entropy composition induces fractal properties on the
underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we
justify why the underlying configuration/phase space of systems described by
the Tsallis entropy has polynomial growth for both discrete and Riemannian
cases. We provide a geometric framework that elucidates Abe's formula for the
Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian
spaces.Comment: 26 pages, No figures, LaTeX2e. To be published in Int. J. Geom.
Methods Mod. Physic
Comparison and Rigidity Theorems in Semi-Riemannian Geometry
The comparison theory for the Riccati equation satisfied by the shape
operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds
of arbitrary index, using one-sided bounds on the Riemann tensor which in the
Riemannian case correspond to one-sided bounds on the sectional curvatures.
Starting from 2-dimensional rigidity results and using an inductive technique,
a new class of gap-type rigidity theorems is proved for semi-Riemannian
manifolds of arbitrary index, generalizing those first given by Gromov and
Greene-Wu. As applications we prove rigidity results for semi-Riemannian
manifolds with simply connected ends of constant curvature.Comment: 46 pages, amsart, to appear in Comm. Anal. Geo
Embedding Riemannian Manifolds by the Heat Kernel of the Connection Laplacian
Given a class of closed Riemannian manifolds with prescribed geometric
conditions, we introduce an embedding of the manifolds into based on
the heat kernel of the Connection Laplacian associated with the Levi-Civita
connection on the tangent bundle. As a result, we can construct a distance in
this class which leads to a pre-compactness theorem on the class under
consideration
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