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1

Variational characterizations for eigenfunctions of analytic self-adjoint operator functions

Katsouleas G., Maroulas J.

Opuscula Mathematica

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2013

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Vol. 33, no. 2

307--321

EN

In this paper we consider Rellich's diagonalization theorem for analytic self-adjoint operator functions and investigate variational principles for their eigenfunctions and interlacing statements. As an application, we present a characterization for the eigenvalues of hyperbolic operator polynomials.

2

Variational principles for set-valued mappings with applications to multiobjective optimization

Bao T. Q., Mordukhovich B. S.

Control and Cybernetics

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2007

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Vol. 36, no 3

531-562

EN

This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which-being certainly of independent interest-are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang, formulated via an appropriate subdifferential notion for set-valued mappings with values in partially ordered spaces. Based on these variational principles and corresponding tools of generalized differentiation, we derive new conditions of the coercivity and Palais-Smale types ensuring the existence of optimal solutions to set-valued optimization problems with noncompact feasible sets in infinite dimensions and then obtain necessary optimality and suboptimality conditions for nonsmooth multiobjective optimization problems with general constraints, which are new in both finite-dimensional and infinite-dimensional settings.

3

Thermodynamic aspects of variational principles for fluids with heat flow

Sieniutycz S.

Archives of Thermodynamics

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2005

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Vol. 26, no. 2

45-71

EN

Processing some known results of nonequilibrium statisctical mechanics we focus on nonequilibrium corrections Δs to entropy s of the fluid in terms of the nonequilibrium density distribution function, f. To evaluate corrections Δe to the energy e or kinetic potential L we apply a relationship that links energy and entropy representations of thermodynamics. We also evaluate coefficients of wave model of heat conduction, such as: relaxation time, propagation speed and thermal inertia. With corrections to L we formulate a quadratic Lagrangian and a variational principle of Hamilton's (least action) type for a fluid with heat flux, or other random-type effect, in the field or Eulerian representation of fluid motion. We discuss canonical and generalized conservation laws and show that satisfaction of the second law of thermodynamics under the constraint of canonical conservation laws.

4

Extended Ginzburg-Landau theory of magnetically active anisotropic superconductors

Rogula D.

Journal of Technical Physics

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2004

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Vol. 45, no 3

243-251

EN

A formulation of thermodynamical theory of magnetically active, anisotropic materials admitting coexistence of the superconductivity and magnetic order is proposed. The theory is based on the Ginzburg-Landau approach extended to multi-component order parameters and the states of thermodynamic quasi-equilibrium far from the superconducting phase transition. The field equations are derived under assumption of the U(1) gauge invariance. The questions of exact anisotropic similarity transformations as well as approximate anisotropic scaling are discussed.

5

Steady interphase heat transfer in nonlinear media - a variational principle

Sieniutycz S.

Archives of Thermodynamics

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2002

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Vol. 23, no 3

79-90

EN

A Format-like principle of minimum time is formulated to predict the change in direction of the heat flux within each phase and through the interface between two immiscible media differring with value of the specific thermal resistance. The basis for the principle are single-phase extremum properties of the entropy production and a related Onsagerian framework for heat transfer in the entropy representation. It is shown that a thermal ray spanned between two given points takes the shape that assures that its relatively large part resides in the region of lower thermal resistance (a 'rarer' region of the medium). In other words, the energy flux bends into the direction which ensures its shape corresponding with the longest residence time of energy in regions of larger conductivity. This property makes one possible to predict shapes of thermal rays or paths of the heat flow and to develop the corresponding Hamilton-Jacobi formalism that describes moton of thermal wavefronts.

6

Termodynamika nierównowagowa a inżynieria procesowa

Sieniutycz S.

Inżynieria Chemiczna i Procesowa

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2001

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T. 22, z. 3A

111--124

PL

Przedstawiono zarys teorii i zastosowań termodynamiki nierównowagowej, istotnych dla techniki procesowej. Naświetlono znaczenie dyscypliny, związek z inżynierią procesową, główne reprezentacje i kierunki teorii oraz zasadnicze wyniki. Rozważono zastosowania w układach fizykochemicznych i biologicznych: optymalnie sterowane operacje jednostkowe, procesy w maszynach cieplnych, nieliniowy ruch ciepła, zjawiska relaksacyjne i samo-propagujące fronty reakcyjno-dyfuzyjne. Podkreślono własność ekstremalnego zachowanie się układu termodynamicznego w obecności ograniczeń, która prowadzi do równań kinetycznych, praw zachowania i optymalnych parametrów (zasady wariacyjne i optymalizacja). Omówiono także perspektywy dyscypliny i wyzwania na przyszłość.

EN

The paper outlines the theory and applications of nonequilibrium thermodynamics essential in chemical engineering. We focus on the significance of the discipline, its link with the process engineering, main representations and directions of the theory, and basic results. Applications in physiochemical and biological systems are considered which involve: optimally controlled unit operations, processes in thermal machines, nonlinear heat transfer, relaxation phenomena and self-propagating reaction-diffusion fronts. We stress the property of extremum behaviour of the thermodynamic system in presence of constraints, which leads to kinetic equations, conservation laws and optimal parameters (variational principles and optimization). Discussed are also perspectives of the discipline and its future challenges.

7

On the chemical potential/electronegativity equalization in density functional theory

Nalewajski R.F.

Polish Journal of Chemistry

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1998

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Vol. 72, nr 7S

1763-1778

EN

General variational principles of the Kohn-Sham (KS) density functional theory are interpreted as the corresponding chemical potential/electronegativity equalization equations. The unconstrained (ground-state) and constrained (excited) electron configuratons are examined for both the system global description and for the case of its partitioning into mutually closed subsystems, e.g., reactants. The chemical potential discontinuity for the integer numbers of electrons at zero temperature is stressed, and the KS orbital description of he charge transfer (CT) between reactants is discussed. Using the appropriate ensemble formulation of the KS theory thein situ chemical potential/electronegativity difference, the driving 'force' behind the inter-reactant CT, is linked to the relevant KS frontier eigenvalues of polarized reactants.