213 research outputs found
A General Backwards Calculus of Variations via Duality
We prove Euler-Lagrange and natural boundary necessary optimality conditions
for problems of the calculus of variations which are given by a composition of
nabla integrals on an arbitrary time scale. As an application, we get
optimality conditions for the product and the quotient of nabla variational
functionals.Comment: Submitted to Optimization Letters 03-June-2010; revised 01-July-2010;
accepted for publication 08-July-201
Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales
We obtain Euler-Lagrange and transversality optimality conditions for
higher-order infinite horizon variational problems on a time scale. The new
necessary optimality conditions improve the classical results both in the
continuous and discrete settings: our results seem new and interesting even in
the particular cases when the time scale is the set of real numbers or the set
of integers.Comment: This is a preprint of a paper whose final and definite form will
appear in Journal of Optimization Theory and Applications (JOTA). Paper
submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for
publication 15-April-201
Transversality Conditions for Infinite Horizon Variational Problems on Time Scales
We consider problems of the calculus of variations on unbounded time scales.
We prove the validity of the Euler-Lagrange equation on time scales for
infinite horizon problems, and a new transversality condition.Comment: Submitted 6-October-2009; Accepted 19-March-2010 in revised form; for
publication in "Optimization Letters"
Direct and Inverse Variational Problems on Time Scales: A Survey
We deal with direct and inverse problems of the calculus of variations on
arbitrary time scales. Firstly, using the Euler-Lagrange equation and the
strengthened Legendre condition, we give a general form for a variational
functional to attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be
published in the Springer Volume 'Modeling, Dynamics, Optimization and
Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer
Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted,
after a revision, 19/Jan/201
Towards the Formalization of Fractional Calculus in Higher-Order Logic
Fractional calculus is a generalization of classical theories of integration
and differentiation to arbitrary order (i.e., real or complex numbers). In the
last two decades, this new mathematical modeling approach has been widely used
to analyze a wide class of physical systems in various fields of science and
engineering. In this paper, we describe an ongoing project which aims at
formalizing the basic theories of fractional calculus in the HOL Light theorem
prover. Mainly, we present the motivation and application of such formalization
efforts, a roadmap to achieve our goals, current status of the project and
future milestones.Comment: 9 page
Improved Bounds on Quantum Learning Algorithms
In this article we give several new results on the complexity of algorithms
that learn Boolean functions from quantum queries and quantum examples.
Hunziker et al. conjectured that for any class C of Boolean functions, the
number of quantum black-box queries which are required to exactly identify an
unknown function from C is ,
where is a combinatorial parameter of the class C. We
essentially resolve this conjecture in the affirmative by giving a quantum
algorithm that, for any class C, identifies any unknown function from C using
quantum black-box
queries.
We consider a range of natural problems intermediate between the exact
learning problem (in which the learner must obtain all bits of information
about the black-box function) and the usual problem of computing a predicate
(in which the learner must obtain only one bit of information about the
black-box function). We give positive and negative results on when the quantum
and classical query complexities of these intermediate problems are
polynomially related to each other.
Finally, we improve the known lower bounds on the number of quantum examples
(as opposed to quantum black-box queries) required for -PAC
learning any concept class of Vapnik-Chervonenkis dimension d over the domain
from to . This new lower bound comes
closer to matching known upper bounds for classical PAC learning.Comment: Minor corrections. 18 pages. To appear in Quantum Information
Processing. Requires: algorithm.sty, algorithmic.sty to buil
Euler-Lagrange equations for composition functionals in calculus of variations on time scales
In this paper we consider the problem of the calculus of variations for a
functional which is the composition of a certain scalar function with the
delta integral of a vector valued field , i.e., of the form
. Euler-Lagrange
equations, natural boundary conditions for such problems as well as a necessary
optimality condition for isoperimetric problems, on a general time scale, are
given. A number of corollaries are obtained, and several examples illustrating
the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems
(DCDS-B); revised 10-March-2010; accepted 04-July-201
Universality in Systems with Power-Law Memory and Fractional Dynamics
There are a few different ways to extend regular nonlinear dynamical systems
by introducing power-law memory or considering fractional
differential/difference equations instead of integer ones. This extension
allows the introduction of families of nonlinear dynamical systems converging
to regular systems in the case of an integer power-law memory or an integer
order of derivatives/differences. The examples considered in this review
include the logistic family of maps (converging in the case of the first order
difference to the regular logistic map), the universal family of maps, and the
standard family of maps (the latter two converging, in the case of the second
difference, to the regular universal and standard maps). Correspondingly, the
phenomenon of transition to chaos through a period doubling cascade of
bifurcations in regular nonlinear systems, known as "universality", can be
extended to fractional maps, which are maps with power-/asymptotically
power-law memory. The new features of universality, including cascades of
bifurcations on single trajectories, which appear in fractional (with memory)
nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201
The Hahn Quantum Variational Calculus
We introduce the Hahn quantum variational calculus. Necessary and sufficient
optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange
problems, are studied. We also show the validity of Leitmann's direct method
for the Hahn quantum variational calculus, and give explicit solutions to some
concrete problems. To illustrate the results, we provide several examples and
discuss a quantum version of the well known Ramsey model of economics.Comment: Submitted: 3/March/2010; 4th revision: 9/June/2010; accepted:
18/June/2010; for publication in Journal of Optimization Theory and
Application
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