46,522 research outputs found
Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems
In this paper, a new class of generalized of nonconvex multitime multiobjective variational problems is considered. We prove the sufficient optimality conditions for efficiency and proper efficiency in the considered multitime multiobjective variational problems with univex functionals. Further, for such vector variational problems, various duality results in the sense of Mond-Weir and in the sense of Wolfe are established under univexity. The results established in the paper extend and generalize results existing in the literature for such vector variational problems
Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
The approach to the theory of many-particle interacting systems from a
unified standpoint, based on the variational principle for free energy is
reviewed. A systematic discussion is given of the approximate free energies of
complex statistical systems. The analysis is centered around the variational
principle of N. N. Bogoliubov for free energy in the context of its
applications to various problems of statistical mechanics and condensed matter
physics. The review presents a terse discussion of selected works carried out
over the past few decades on the theory of many-particle interacting systems in
terms of the variational inequalities. It is the purpose of this paper to
discuss some of the general principles which form the mathematical background
to this approach, and to establish a connection of the variational technique
with other methods, such as the method of the mean (or self-consistent) field
in the many-body problem, in which the effect of all the other particles on any
given particle is approximated by a single averaged effect, thus reducing a
many-body problem to a single-body problem. The method is illustrated by
applying it to various systems of many-particle interacting systems, such as
Ising and Heisenberg models, superconducting and superfluid systems, strongly
correlated systems, etc. It seems likely that these technical advances in the
many-body problem will be useful in suggesting new methods for treating and
understanding many-particle interacting systems. This work proposes a new,
general and pedagogical presentation, intended both for those who are
interested in basic aspects, and for those who are interested in concrete
applications.Comment: 60 pages, Refs.25
A Minty variational principle for set optimization
Extremal problems are studied involving an objective function with values in
(order) complete lattices of sets generated by so called set relations.
Contrary to the popular paradigm in vector optimization, the solution concept
for such problems, introduced by F. Heyde and A. L\"ohne, comprises the
attainment of the infimum as well as a minimality property. The main result is
a Minty type variational inequality for set optimization problems which
provides a sufficient optimality condition under lower semicontinuity
assumptions and a necessary condition under appropriate generalized convexity
assumptions. The variational inequality is based on a new Dini directional
derivative for set-valued functions which is defined in terms of a "lattice
difference quotient": A residual operation in a lattice of sets replaces the
inverse addition in linear spaces. Relationships to families of scalar problems
are pointed out and used for proofs: The appearance of improper scalarizations
poses a major difficulty which is dealt with by extending known scalar results
such as Diewert's theorem to improper functions
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
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