305,039 research outputs found
A Generalized Fractional Calculus of Variations
We study incommensurate fractional variational problems in terms of a
generalized fractional integral with Lagrangians depending on classical
derivatives and generalized fractional integrals and derivatives. We obtain
necessary optimality conditions for the basic and isoperimetric problems,
transversality conditions for free boundary value problems, and a generalized
Noether type theorem.Comment: This is a preprint of a paper whose final and definitive form will
appear in Control and Cybernetics. Paper submitted 01-Oct-2012; revised
25-March-2013; accepted for publication 17-April-201
Classification of Generalized Symmetries for the Vacuum Einstein Equations
A generalized symmetry of a system of differential equations is an
infinitesimal transformation depending locally upon the fields and their
derivatives which carries solutions to solutions. We classify all generalized
symmetries of the vacuum Einstein equations in four spacetime dimensions. To
begin, we analyze symmetries that can be built from the metric, curvature, and
covariant derivatives of the curvature to any order; these are called natural
symmetries and are globally defined on any spacetime manifold. We next classify
first-order generalized symmetries, that is, symmetries that depend on the
metric and its first derivatives. Finally, using results from the
classification of natural symmetries, we reduce the classification of all
higher-order generalized symmetries to the first-order case. In each case we
find that the generalized symmetries are infinitesimal generalized
diffeomorphisms and constant metric scalings. There are no non-trivial
conservation laws associated with these symmetries. A novel feature of our
analysis is the use of a fundamental set of spinorial coordinates on the
infinite jet space of Ricci-flat metrics, which are derived from Penrose's
``exact set of fields'' for the vacuum equations.Comment: 57 pages, plain Te
Generalized Fermi-Dirac Functions and Derivatives: Properties and Evaluation
The generalized Fermi-Dirac functions and their derivatives are important in
evaluating the thermodynamic quantities of partially degenerate electrons in
hot dense stellar plasmas. New recursion relations of the generalized
Fermi-Dirac functions have been found. An effective numerical method to
evaluate the derivatives of the generalized Fermi-Dirac functions up to third
order with respect to both degeneracy and temperature is then proposed,
following Aparicio. A Fortran program based on this method, together with a
sample test case, is provided. Accuracy and domain of reliability of some
other, popularly used analytic approximations of the generalized Fermi-Dirac
functions for extreme conditions are investigated and compared with our
results.Comment: accepted for publication in Comp. Phys. Com
Hilfer-Prabhakar Derivatives and Some Applications
We present a generalization of Hilfer derivatives in which Riemann--Liouville
integrals are replaced by more general Prabhakar integrals. We analyze and
discuss its properties. Further, we show some applications of these generalized
Hilfer-Prabhakar derivatives in classical equations of mathematical physics,
like the heat and the free electron laser equations, and in
difference-differential equations governing the dynamics of generalized renewal
stochastic processes
Gauging Higher Derivatives
The usual prescription for constructing gauge-invariant Lagrangian is
generalized to the case where a Lagrangian contains second derivatives of
fields as well as first derivatives. Symmetric tensor fields in addition to the
usual vector fields are introduced as gauge fields. Covariant derivatives and
gauge-field strengths are determined.Comment: 12 pages, LaTex
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