The approach here, mainly based on the nonsmooth dual objects of Mordukhovich, is completely different from that of most of the previous works where generalizations of the alternative theorem of Farkas have been applied.

The results obtained are close to those known in the literature. However, one of the main achievements of this article is that we could also derive necessary optimality conditions for the minmax program of the usual Karush—Kuhn—Tucker type, which seems to be new in this field of study. The author is also grateful to Boris S.

Mordukhovich for inspiring discussion on the upper subdifferential optimality conditions and to Stephan Dempe for insights on bilevel programming. Financial support by the Deutscher Akademischer Austausch Dienst is also acknowledged. Skip to Main Content. Search in: This Journal Anywhere. Advanced search.

Submit an article Journal homepage. Alain B. We call inequality the classic inequality associated to the matrix A with positive entries, and stochastic matrix Q.

Given the classic inequality , we want to compare it to the basic inequality More precisely, we would like to obtain operators V i that satisfy the following: given a matrix A with positive entries and a stochastic matrix Q , there are H i and W i such that inequality becomes inequality We have the following proposition. Such Ruelle operator admits a positive eigenvalue, which gives us an upper bound for the pressure entropy plus a potential associated to the QIFS.

We did not consider the usual space of symbols or a shift operator, as it is assumed in the Ruelle-Perron-Frobenius theory. The references [ 16 ] and [ 21 ] are of fundamental importance in our investigation. A starting point for further investigation could be to study more properties of the QIFS entropy, such as convexity and subadditivity.

In a forthcoming paper we are going to consider relative entropies and quantum conditional expectations. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In.

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## Classics in Applied Mathematics: Generalized Concavity

Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. Correspondence to be sent to: arturoscar. Oxford Academic. Google Scholar. Terra Cunha. Communicated by. Cite Citation.

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Abstract We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Bowen, Y. Sinai and D. It is well-known that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism.

In this work, we investigate a generalization of the classical Thermodynamic Formalism in the sense of Bowen, Sinai and Ruelle for the setting of density matrices.

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We also want to present a concept of pressure for stationary systems acting on density matrices which generalizes the usual one in Ergodic Theory. We call.

## Fonction quasi convexe

For instance, we could choose one of the following:. Note that we can write. We know that every CP mapping which is trace-preserving can be represented in a nonunique way in the Stinespring-Kraus form. We call a unitary trace-preserving CP map a bistochastic map. An example of such a mapping is.

## MSC Classification Codes

We will present a simple example of the kind of problems we are interested here, namely eigenvalues and eigendensity matrices. The eigenvalues are. The following proposition is inspired in [ 18 ].

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We have. By lemma 1 , we have,. Part of the results we present here in this section are variations of results presented in [ 21 ]. Denote by the set of -invariant probability measures on X. Its Schmidt decomposition is. By theorem 2 , we can calculate the entropy of such IFS. In this case, we have. By proposition 5 ,.

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We have by proposition 7 , since h is concave, that is decreasing, and so. Let H be a hermitian operator and V i , linear operators. The inequality follows from [ 21 ], proposition 1. Also by proposition 4 , we have. Nonlinear Programming Classics in Applied Mathematics. Matrix Polynomials Classics in Applied Mathematics. Generalized Concavity Classics in Applied Mathematics Stochastic Processes Classics in Applied Mathematics.

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