Path integrals and finite dimensional filters

1995 ◽  
pp. 209-229 ◽  
Author(s):  
S. J. Maybank
Keyword(s):  
Path Integrals ◽  
Finite Dimensional

scholarly journals Ordering, Symbols and Finite-Dimensional Approximations of Path Integrals

1994 ◽  
Vol 92 (3) ◽  
pp. 669-685 ◽  
Author(s):  
T. Kashiwa ◽  
S. Sakoda ◽  
S. V. Zenkin
Keyword(s):  
Path Integrals ◽  
Finite Dimensional

scholarly journals PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE

2007 ◽  
Vol 19 (09) ◽  
pp. 967-1044 ◽  
Author(s):  
ADRIAN P. C. LIM
Keyword(s):  
Compact Manifold ◽  
Compact Riemannian Manifold ◽  
Path Integrals ◽  
Energy Functional ◽  
Normalization Constant ◽  
Finite Dimensional ◽  
Riemannian Volume ◽  
Volume Measure ◽  
Constant E ◽  
Geodesic Paths

A typical path integral on a manifold, M is an informal expression of the form [Formula: see text] where H(M) is a Hilbert manifold of paths with energy E(σ) < ∞, f is a real-valued function on H(M), [Formula: see text] is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret [Formula: see text] as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, [Formula: see text] of [0, τ], let [Formula: see text] be the finite dimensional Riemannian submanifold of H(M) consisting of piecewise geodesic paths adapted to [Formula: see text]. Under certain curvature restrictions on M, it is shown that [Formula: see text] where [Formula: see text] is a "normalization" constant, E : H(M) → [0,∞) is the energy functional, [Formula: see text] is the Riemannian volume measure on [Formula: see text], ν is Wiener measure on continuous paths in M, and ρ is a certain density determined by the curvature tensor of M.

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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