34,360 research outputs found
A survey on fractional variational calculus
Main results and techniques of the fractional calculus of variations are
surveyed. We consider variational problems containing Caputo derivatives and
study them using both indirect and direct methods. In particular, we provide
necessary optimality conditions of Euler-Lagrange type for the fundamental,
higher-order, and isoperimetric problems, and compute approximated solutions
based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form is in
'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De
Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-201
Calculus of variations with fractional derivatives and fractional integrals
We prove Euler-Lagrange fractional equations and sufficient optimality
conditions for problems of the calculus of variations with functionals
containing both fractional derivatives and fractional integrals in the sense of
Riemann-Liouville.Comment: Accepted (July 6, 2009) for publication in Applied Mathematics
Letter
Overview of Constrained PARAFAC Models
In this paper, we present an overview of constrained PARAFAC models where the
constraints model linear dependencies among columns of the factor matrices of
the tensor decomposition, or alternatively, the pattern of interactions between
different modes of the tensor which are captured by the equivalent core tensor.
Some tensor prerequisites with a particular emphasis on mode combination using
Kronecker products of canonical vectors that makes easier matricization
operations, are first introduced. This Kronecker product based approach is also
formulated in terms of the index notation, which provides an original and
concise formalism for both matricizing tensors and writing tensor models. Then,
after a brief reminder of PARAFAC and Tucker models, two families of
constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models,
are described in a unified framework, for order tensors. New tensor
models, called nested Tucker models and block PARALIND/CONFAC models, are also
introduced. A link between PARATUCK models and constrained PARAFAC models is
then established. Finally, new uniqueness properties of PARATUCK models are
deduced from sufficient conditions for essential uniqueness of their associated
constrained PARAFAC models
Neutral heavy lepton production at next high energy linear colliders
The discovery potential for detecting new heavy Majorana and Dirac neutrinos
at some recently proposed high energy colliders is discussed. These
new particles are suggested by grand unified theories and superstring-inspired
models. For these models the production of a single heavy neutrino is shown to
be more relevant than pair production when comparing cross sections and
neutrino mass ranges.
The process is calculated
including on-shell and off-shell heavy neutrino effects.
We present a detailed study of cross sections and distributions that shows a
clear separation between the signal and standard model contributions, even
after including hadronization effects.Comment: 4 pages including 15 figures, 1 table. RevTex. Accepted in Physical
Review
Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
This chapter presents some numerical methods to solve problems in the
fractional calculus of variations and fractional optimal control. Although
there are plenty of methods available in the literature, we concentrate mainly
on approximating the fractional problem either by discretizing the fractional
term or expanding the fractional derivatives as a series involving integer
order derivatives. The former method, as a subclass of direct methods in the
theory of calculus of variations, uses finite differences, Grunwald-Letnikov
definition in this case, to discretize the fractional term. Any quadrature rule
for integration, regarding the desired accuracy, is then used to discretize the
whole problem including constraints. The final task in this method is to solve
a static optimization problem to reach approximated values of the unknown
functions on some mesh points.
The latter method, however, approximates fractional problems by classical
ones in which only derivatives of integer order are present. Precisely, two
continuous approximations for fractional derivatives by series involving
ordinary derivatives are introduced. Local upper bounds for truncation errors
are provided and, through some test functions, the accuracy of the
approximations are justified. Then we substitute the fractional term in the
original problem with these series and transform the fractional problem to an
ordinary one. Hereafter, we use indirect methods of classical theory, e.g.
Euler-Lagrange equations, to solve the approximated problem. The methods are
mainly developed through some concrete examples which either have obvious
solutions or the solution is computed using the fractional Euler-Lagrange
equation.Comment: This is a preprint of a paper whose final and definite form appeared
in: Chapter V, Fractional Calculus in Analysis, Dynamics and Optimal Control
(Editor: Jacky Cresson), Series: Mathematics Research Developments, Nova
Science Publishers, New York, 2014. (See
http://www.novapublishers.com/catalog/product_info.php?products_id=46851).
Consists of 39 page
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