87 research outputs found
Fractional variational calculus for nondifferentiable functions
We prove necessary optimality conditions, in the class of continuous
functions, for variational problems defined with Jumarie's modified
Riemann-Liouville derivative. The fractional basic problem of the calculus of
variations with free boundary conditions is considered, as well as problems
with isoperimetric and holonomic constraints.Comment: Submitted 13-Aug-2010; revised 24-Nov-2010; accepted 28-March-2011;
for publication in Computers and Mathematics with Application
Anomalous g-Factors for Charged Leptons in a Fractional Coarse-Grained Approach
In this work, we investigate aspects of the electron, muon and tau
gyromagnetic ratios (g-factor) in a fractional coarse-grained scenario, by
adopting a Modified Riemann-Liouville (MRL) fractional calculus. We point out
the possibility of mapping the experimental values of the specie's g-factors
into a theoretical parameter which accounts for fractionality, without
computing higher-order QED calculations. We wish to understand whether the
value of (g-2) may be traced back to a fractionality of space-time.The
justification for the difference between the experimental and the theoretical
value g=2 stemming from the Dirac equation is given in the terms of the
complexity of the interactions of the charged leptons, considered as
pseudo-particles and "dressed" by the interactions and the medium. Stepwise, we
build up a fractional Dirac equation from the fractional Weyl equation that, on
the other hand, was formulated exclusively in terms of the helicity operator.
From the fractional angular momentum algebra, in a coarse-grained scenario, we
work out the eigenvalues of the spin operator. Based on the standard
electromagnetic current, as an analogy case, we write down a fractional
Lagrangian density, with the electromagnetic field minimally coupled to the
particular charged lepton. We then study a fractional gauge-like invariance
symmetry, formulate the covariant fractional derivative and propose the spinor
field transformation. Finally, by taking the non-relativistic regime of the
fractional Dirac equation, the fractional Pauli equation is obtained and, from
that, an explicit expression for the fractional g-factor comes out that is
compared with the experimental CODATA value. Our claim is that the different
lepton species must probe space-time by experiencing different fractionalities,
once the latter may be associated to the effective interactions of the
different families with the medium.Comment: 15 page
A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String
We introduce a fractional theory of the calculus of variations for multiple
integrals. Our approach uses the recent notions of Riemann-Liouville fractional
derivatives and integrals in the sense of Jumarie. Main results provide
fractional versions of the theorems of Green and Gauss, fractional
Euler-Lagrange equations, and fractional natural boundary conditions. As an
application we discuss the fractional equation of motion of a vibrating string.Comment: Accepted for publication in the Journal of Mathematical Physics
(14/January/2010
On the Zitterbewegung Transient Regime in a Coarse-Grained Space-Time
In the present contribution, by studying a fractional version of Dirac's
equation for the electron, we show that the phenomenon of Zitterbewegung in a
coarse-grained medium exhibits a transient oscillatory behavior, rather than a
purely oscillatory regime, as it occurs in the integer case, . Our
result suggests that, in such systems, the Zitterbewegung-type term related to
a trembling motion of a quasiparticle is tamed by its complex interactions with
other particles and the medium. This can justify the difficulties in the
observation of this interesting phenomenon. The possibility that the
Zitterbewegung be accompanied by a damping factor supports the viewpoint of
particle substructures in Quantum Mechanics.Comment: 11 pages, 1 figure. This paper has been published in J. Adv. Phys. 7
(2015) 144
Non-asymptotic fractional order differentiators via an algebraic parametric method
Recently, Mboup, Join and Fliess [27], [28] introduced non-asymptotic integer
order differentiators by using an algebraic parametric estimation method [7],
[8]. In this paper, in order to obtain non-asymptotic fractional order
differentiators we apply this algebraic parametric method to truncated
expansions of fractional Taylor series based on the Jumarie's modified
Riemann-Liouville derivative [14]. Exact and simple formulae for these
differentiators are given where a sliding integration window of a noisy signal
involving Jacobi polynomials is used without complex mathematical deduction.
The efficiency and the stability with respect to corrupting noises of the
proposed fractional order differentiators are shown in numerical simulations
No Violation of the Leibniz Rule. No Fractional Derivative
We demonstrate that a violation of the Leibniz rule is a characteristic
property of derivatives of non-integer orders. We prove that all fractional
derivatives D^a, which satisfy the Leibniz rule D^(fg)=(D^a f) g + f (D^a g),
should have the integer order a=1, i.e. fractional derivatives of non-integer
orders cannot satisfy the Leibniz rule.Comment: 6 page
Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations
Recently, fractional differential equations have been investigated via the
famous variational iteration method. However, all the previous works avoid the
term of fractional derivative and handle them as a restricted variation. In
order to overcome such shortcomings, a fractional variational iteration method
is proposed. The Lagrange multipliers can be identified explicitly based on
fractional variational theory.Comment: 12 pages, 1 figure
A Robust Variable Step Size Fractional Least Mean Square (RVSS-FLMS) Algorithm
In this paper, we propose an adaptive framework for the variable step size of
the fractional least mean square (FLMS) algorithm. The proposed algorithm named
the robust variable step size-FLMS (RVSS-FLMS), dynamically updates the step
size of the FLMS to achieve high convergence rate with low steady state error.
For the evaluation purpose, the problem of system identification is considered.
The experiments clearly show that the proposed approach achieves better
convergence rate compared to the FLMS and adaptive step-size modified FLMS
(AMFLMS).Comment: 15 pages, 3 figures, 13th IEEE Colloquium on Signal Processing & its
Applications (CSPA 2017
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