607,513 research outputs found
Jet Methods in Time-Dependent Lagrangian Biomechanics
In this paper we propose the time-dependent generalization of an `ordinary'
autonomous human biomechanics, in which total mechanical + biochemical energy
is not conserved. We introduce a general framework for time-dependent
biomechanics in terms of jet manifolds associated to the extended
musculo-skeletal configuration manifold, called the configuration bundle. We
start with an ordinary configuration manifold of human body motion, given as a
set of its all active degrees of freedom (DOF) for a particular movement. This
is a Riemannian manifold with a material metric tensor given by the total
mass-inertia matrix of the human body segments. This is the base manifold for
standard autonomous biomechanics. To make its time-dependent generalization, we
need to extend it with a real time axis. By this extension, using techniques
from fibre bundles, we defined the biomechanical configuration bundle. On the
biomechanical bundle we define vector-fields, differential forms and affine
connections, as well as the associated jet manifolds. Using the formalism of
jet manifolds of velocities and accelerations, we develop the time-dependent
Lagrangian biomechanics. Its underlying geometric evolution is given by the
Ricci flow equation.
Keywords: Human time-dependent biomechanics, configuration bundle, jet
spaces, Ricci flowComment: 13 pages, 3 figure
Functional Classical Mechanics and Rational Numbers
The notion of microscopic state of the system at a given moment of time as a
point in the phase space as well as a notion of trajectory is widely used in
classical mechanics. However, it does not have an immediate physical meaning,
since arbitrary real numbers are unobservable. This notion leads to the known
paradoxes, such as the irreversibility problem. A "functional" formulation of
classical mechanics is suggested. The physical meaning is attached in this
formulation not to an individual trajectory but only to a "beam" of
trajectories, or the distribution function on phase space. The fundamental
equation of the microscopic dynamics in the functional approach is not the
Newton equation but the Liouville equation for the distribution function of the
single particle. The Newton equation in this approach appears as an approximate
equation describing the dynamics of the average values and there are
corrections to the Newton trajectories. We give a construction of probability
density function starting from the directly observable quantities, i.e., the
results of measurements, which are rational numbers.Comment: 8 page
Extrapolation of power series by self-similar factor and root approximants
The problem of extrapolating the series in powers of small variables to the
region of large variables is addressed. Such a problem is typical of quantum
theory and statistical physics. A method of extrapolation is developed based on
self-similar factor and root approximants, suggested earlier by the authors. It
is shown that these approximants and their combinations can effectively
extrapolate power series to the region of large variables, even up to infinity.
Several examples from quantum and statistical mechanics are analysed,
illustrating the approach.Comment: 21 pages, Latex fil
Relations between stochastic and partial differential equations in hilbert spaces
The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation d X (t) = A X (t) d t + B d W (t) and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic t, x h (X (T)) is proved. Interpretation of objects in the equations is given. © 2012 I. V. Melnikova and V. S. Parfenenkova
Resummation Methods for Analyzing Time Series
An approach is suggested for analyzing time series by means of resummation
techniques of theoretical physics. A particular form of such an analysis, based
on the algebraic self-similar renormalization, is developed and illustrated by
several examples from the stock market time series.Comment: Corrections are made to match the published versio
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