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1

Convergence results for a class of pramarts and superpramarts in Banach spaces

Castaing C., Salvadori A.

Commentationes Mathematicae

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2013

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Vol. 53, No. 2

223--253

2

Relative functional entropy in convex analysis

Raissouli M.

Journal of Applied Analysis

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2011

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Vol. 17, nr 2

231-239

EN

The purpose of this work is to extend the relative entropy S(A|B) = A 1/2 log(A -1/2 BA -1/2)A 1/2 from positive operators to convex functionals. Our functional approach implies immediately, in a fast way, some simplifications and improvements for that of positive operators already discussed in the literature.

3

A short proof of the separable reduction theorem

Penot J. P.

Demonstratio Mathematica

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2010

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Vol. 43, nr 3

653-663

EN

We present a simple proof of the separable reduction theorem, a crucial result of nonsmooth analysis which allows to extend to Asplund spaces the results known for separable spaces dealing with Fréchet subdifferentials. It relies on elementary results in convex analysis and avoids certain technicalities.

4

Lagrange principle and necessary conditions

Tikhomirov V.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1589-1605

EN

Necessary conditions of extremum (from the times of Fermat and Lagrange till our times) for extremal problems where smoothness is interlaced with convexity, and some type of regularity takes place, correspond to a unique general principle, which is due to Lagrange. This report is devoted to the Lagrange principle in the theory of optimization.

5

Axiomatic formulation of thermodynamics ideal gas laws

Grzesikiewicz W., Wakulicz A.

Journal of KONES

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2006

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Vol. 13, No. 2

103-110

EN

The relationships between physical quantities determining thermodynamic state of an ideal gas are analysed in the paper. It will be proved that such relationships can be obtained based on convex analysis of the formula describing specific energy of the gas. This formula is expressed as a functional with specific volume and specific entropy being its constrains. The resulting constitutive equations of ideal gas, determining pressure and temperature as a function of entropy and specific volume, are given. It will be proved that the equation of the ideal gas formulated by Clapeyron, can be easy obtain from the constitutive equations, eliminating the variable describing entropy. Moreover, it will be shown that the functional of specific energy of ideal gas is convex. Because of this mathematical property of the functional, Legendre transform is used in order to determine three conjugated functionals, i.e. enthalpy, free enthalpy and free energy. The method of description of thermodynamic relationships to be introduced herein, differs from well-known classic handbook’s presentations. Moreover, it permits a better understanding of the structure of thermodynamics equations.

6

Extremum principles for nonpotential and initial-value problems

Telega J. J.

Archives of Mechanics

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2002

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Vol. 54, nr 5-6

663-690

EN

The aim of this paper is to derive extremum and saddle-point principles for a class of nonpotential and initial-value problems. The procedure used is based on an extension of the procedure primarily used by Brezis and Ekeland [7, 8] to classical parabolic equations. In essence, this approach exploits some fundamental notions of convex analysis.

7

Minimal pairs in classes of frustums

Kaczmarek S.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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1999

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[Z] 39

75-86

EN

In this paper we investigate a special type of convex sets called general frustums. We give methods that make possible to go from pairs of frustums to smaller equivalent pairs, not necessarily minimal. We present criteria of minimality for pairs of frustums. We find minimal pairs equivalent, to pairs of frustums created by proper parallel cutting of frustum.