163 research outputs found
Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presence of Coarse-Graining in Time and Space
One considers the motion of nonlinear systems close to their equilibrium positions in the presence of coarse-graining in time on the one hand, and coarse-graining in time on the other hand. By considering a coarse-grained time as a time in which the increment is not dt but rather (dt)c > dt, one is led to introduce a modeling in terms of fractional derivative with respect to time; and likewise for coarse-graining with respect to the space variable x. After a few prerequisites on fractional calculus via modified Riemann-Liouville derivative, one examines in a detailed way the solutions of fractional linear differential equations in this framework, and then one uses this result in the linearization of nonlinear systems close to their equilibrium positions
Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense
Emergence of order in selection-mutation dynamics
We characterize the time evolution of a d-dimensional probability
distribution by the value of its final entropy. If it is near the
maximally-possible value we call the evolution mixing, if it is near zero we
say it is purifying. The evolution is determined by the simplest non-linear
equation and contains a d times d matrix as input. Since we are not interested
in a particular evolution but in the general features of evolutions of this
type, we take the matrix elements as uniformly-distributed random numbers
between zero and some specified upper bound. Computer simulations show how the
final entropies are distributed over this field of random numbers. The result
is that the distribution crowds at the maximum entropy, if the upper bound is
unity. If we restrict the dynamical matrices to certain regions in matrix
space, for instance to diagonal or triangular matrices, then the entropy
distribution is maximal near zero, and the dynamics typically becomes
purifying.Comment: 8 pages, 8 figure
Entropic trade-off relations for quantum operations
Spectral properties of an arbitrary matrix can be characterized by the
entropy of its rescaled singular values. Any quantum operation can be described
by the associated dynamical matrix or by the corresponding superoperator. The
entropy of the dynamical matrix describes the degree of decoherence introduced
by the map, while the entropy of the superoperator characterizes the a priori
knowledge of the receiver of the outcome of a quantum channel Phi. We prove
that for any map acting on a N--dimensional quantum system the sum of both
entropies is not smaller than ln N. For any bistochastic map this lower bound
reads 2 ln N. We investigate also the corresponding R\'enyi entropies,
providing an upper bound for their sum and analyze entanglement of the
bi-partite quantum state associated with the channel.Comment: 10 pages, 4 figure
Scale relativity and fractal space-time: theory and applications
In the first part of this contribution, we review the development of the
theory of scale relativity and its geometric framework constructed in terms of
a fractal and nondifferentiable continuous space-time. This theory leads (i) to
a generalization of possible physically relevant fractal laws, written as
partial differential equation acting in the space of scales, and (ii) to a new
geometric foundation of quantum mechanics and gauge field theories and their
possible generalisations. In the second part, we discuss some examples of
application of the theory to various sciences, in particular in cases when the
theoretical predictions have been validated by new or updated observational and
experimental data. This includes predictions in physics and cosmology (value of
the QCD coupling and of the cosmological constant), to astrophysics and
gravitational structure formation (distances of extrasolar planets to their
stars, of Kuiper belt objects, value of solar and solar-like star cycles), to
sciences of life (log-periodic law for species punctuated evolution, human
development and society evolution), to Earth sciences (log-periodic
deceleration of the rate of California earthquakes and of Sichuan earthquake
replicas, critical law for the arctic sea ice extent) and tentative
applications to system biology.Comment: 63 pages, 14 figures. In : First International Conference on the
Evolution and Development of the Universe,8th - 9th October 2008, Paris,
Franc
Fractional conservation laws in optimal control theory
Using the recent formulation of Noether's theorem for the problems of the
calculus of variations with fractional derivatives, the Lagrange multiplier
technique, and the fractional Euler-Lagrange equations, we prove a Noether-like
theorem to the more general context of the fractional optimal control. As a
corollary, it follows that in the fractional case the autonomous Hamiltonian
does not define anymore a conservation law. Instead, it is proved that the
fractional conservation law adds to the Hamiltonian a new term which depends on
the fractional-order of differentiation, the generalized momentum, and the
fractional derivative of the state variable.Comment: The original publication is available at http://www.springerlink.com
Nonlinear Dynamic
Measuring Information Transfer
An information theoretic measure is derived that quantifies the statistical
coherence between systems evolving in time. The standard time delayed mutual
information fails to distinguish information that is actually exchanged from
shared information due to common history and input signals. In our new
approach, these influences are excluded by appropriate conditioning of
transition probabilities. The resulting transfer entropy is able to distinguish
driving and responding elements and to detect asymmetry in the coupling of
subsystems.Comment: 4 pages, 4 Figures, Revte
Cadmium accumulation and interactions with zinc, copper, and manganese, analysed by ICP-MS in a long-term Caco-2 TC7 cell model
The influence of long-term exposure to cadmium (Cd) on essential minerals was investigated using a Caco-2
TC7 cells and a multi-analytical tool: microwave digestion and inductively coupled plasma mass spectrometry.
Intracellular levels, effects on cadmium accumulation, distribution, and reference concentration
ranges of the following elements were determined: Na, Mg, Ca, Cr, Fe, Mn, Co, Ni, Cu, Zn, Mo, and Cd.
Results showed that Caco-2 TC7 cells incubated long-term with cadmium concentrations ranging from 0 to
10 lmol Cd/l for 5 weeks exhibited a significant increase in cadmium accumulation. Furthermore, this
accumulation was more marked in cells exposed long-term to cadmium compared with controls, and that
this exposure resulted in a significant accumulation of copper and zinc but not of the other elements
measured. Interactions of Cd with three elements: zinc, copper, and manganese were particularly studied.
Exposed to 30 lmol/l of the element, manganese showed the highest inhibition and copper the lowest on
cadmium intracellular accumulation but Zn, Cu, and Mn behave differently in terms of their mutual
competition with Cd. Indeed, increasing cadmium in the culture medium resulted in a gradual and significant
increase in the accumulation of zinc. There was a significant decrease in manganese from 5 lmol
Cd/l exposure, and no variation was observed with copper.
Abbreviation: AAS – Atomic absorption spectrometry; CRM– Certified reference material; PBS – Phosphate
buffered saline without calcium and magnesium; DMEM – Dubelcco’s modified Eagle’s medium
Stochastic evolution equations driven by Liouville fractional Brownian motion
Let H be a Hilbert space and E a Banach space. We set up a theory of
stochastic integration of L(H,E)-valued functions with respect to H-cylindrical
Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in
the interval (0,1). For Hurst parameters in (0,1/2) we show that a function
F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical
Liouville fBm if and only if it is stochastically integrable with respect to an
H-cylindrical fBm with the same Hurst parameter. As an application we show that
second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by
space-time noise which is white in space and Liouville fractional in time with
Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous
both and space.Comment: To appear in Czech. Math.
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