We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations.

We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations.

We consider the decomposition of a maximal monotone operator into the sum of an antisymmetric operator and the subdifferential of a proper lower semicontinuous convex function. This is a variant of the well-known decomposition of a matrix into its symmetric and antisymmetric part. We analyze in detail the case when the graph of the operator is a linear subspace. Equivalent conditions of monotonicity are also provided. We obtain several new results on auto-conjugate representations including an explicit formula that is built upon the proximal average of the associated Fitzpatrick function and its Fenchel conjugate. These results are new and they both extend and complement recent work by Penot, Simons and Zălinescu. A nonlinear example shows the importance of the linearity assumption. Finally, we consider the problem of computing the Fitzpatrick function of the sum, generalizing a recent result by Bauschke, Borwein and Wang on matrices to linear relations. / Graduate Studies, College of (Okanagan) / Graduate

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Na Dissertação de Mestrado foi realizado um estudo sobre o comportamento de vigas mistas de aço e concreto biapoiadas sob iteração completa, compostas por perfis classificados como compactos em situação de incêndio. Apesar de ditas biapoiadas, a ideia principal da Dissertação é considerar, na análise em situação de incêndio, a reserva de capacidade existente nas extremidades dessas vigas, usualmente desprezada no dimensionamento à temperatura ambiente, provinda da continuidade da armadura longitudinal negativa presente na laje de concreto, sendo possível dispensar o revestimento contra fogo nesses elementos. Serão abordados, de início, os métodos de dimensionamento ao Estado-Limite Último de vigas mistas à temperatura ambiente, a fim de explicar o comportamento estrutural desse tipo de elemento, dando base para a posterior análise térmica. A análise térmica foi realizada, em uma primeira etapa, por métodos simplificados segundo normas vigentes e tomando hipóteses simplificadoras em que se desprezam esforços indiretos provocados pela dilatação térmica e gradiente térmico, sendo possível aplicar os conceitos no dia a dia de projeto. Numa segunda etapa foi realizada análise termestrutural com modelos numéricos para estudar o comportamento e colapso da viga, sendo os resultados suficientes para validar os valores de TRF encontrados por análise simplificada seguindo métodos normativos. Como conclusão, o ganho de capacidade estrutural da viga, proporcionado pela adoção da ligação mista no apoio, foi suficiente para situações de TRRFs entre 15 min e 21 min, no caso de perfis mais leves, com geometria própria para serem utilizados como vigas de piso de edifícios. Análises mostraram que não é possível justificar a ausência de revestimento contra incêndio para TRRFs de 30 min ou superiores. Tempos inferiores podem ser adotados de acordo com o denominado método do tempo equivalente, limitado a 15 min e são mais usuais para edificações de pequeno porte. / A study will be carried out on the behavior of simply supported, full interaction composite steel and concrete beams composed by compact profiles in fire situation. Despite being designed as simply supported, the main idea is to consider, in the fire situation analysis, the moment resistance capacity reserve on the beam supports, usually neglected during room temperature design, due to the upper longitudinal reinforcement present in the concrete slab, being possible to dispense fireproof coating in these elements. First, room temperature design procedures of composite beams will be approached in order to explain the structural behavior of this type of elements, providing a basis for subsequent thermal analysis. The thermal analysis will be carried out, in a first stage, by simplified methods according to design procedures and adopting simplifying hypotheses in which indirect stresses caused by thermal expansion and thermal gradient are neglected, being possible to apply those concepts in structural design offices. In a second step, thermal stress analyses were performed with aid of numerical models to study the structural behavior and collapse time of the beam. The results were sufficient to validate the fire resistance time values found by the simplified analysis following design methods. As a conclusion, the structural capacity increase of the beam, provided by the adoption of the composite connection at the support in case of lighter steel profiles usually chosen to be used as buildings floor beams, was sufficient for situations of standard fire resistance requirements between 15 min and 21 min. Analyzes have shown that it is not possible to justify the absence of fire resistant coating for standard fire resistance requirements of 30 min or higher. Lower times can be adopted according to the so-called equivalent time method, limited to 15 min, which are more common for small buildings.

Na Dissertação de Mestrado foi realizado um estudo sobre o comportamento de vigas mistas de aço e concreto biapoiadas sob iteração completa, compostas por perfis classificados como compactos em situação de incêndio. Apesar de ditas biapoiadas, a ideia principal da Dissertação é considerar, na análise em situação de incêndio, a reserva de capacidade existente nas extremidades dessas vigas, usualmente desprezada no dimensionamento à temperatura ambiente, provinda da continuidade da armadura longitudinal negativa presente na laje de concreto, sendo possível dispensar o revestimento contra fogo nesses elementos. Serão abordados, de início, os métodos de dimensionamento ao Estado-Limite Último de vigas mistas à temperatura ambiente, a fim de explicar o comportamento estrutural desse tipo de elemento, dando base para a posterior análise térmica. A análise térmica foi realizada, em uma primeira etapa, por métodos simplificados segundo normas vigentes e tomando hipóteses simplificadoras em que se desprezam esforços indiretos provocados pela dilatação térmica e gradiente térmico, sendo possível aplicar os conceitos no dia a dia de projeto. Numa segunda etapa foi realizada análise termestrutural com modelos numéricos para estudar o comportamento e colapso da viga, sendo os resultados suficientes para validar os valores de TRF encontrados por análise simplificada seguindo métodos normativos. Como conclusão, o ganho de capacidade estrutural da viga, proporcionado pela adoção da ligação mista no apoio, foi suficiente para situações de TRRFs entre 15 min e 21 min, no caso de perfis mais leves, com geometria própria para serem utilizados como vigas de piso de edifícios. Análises mostraram que não é possível justificar a ausência de revestimento contra incêndio para TRRFs de 30 min ou superiores. Tempos inferiores podem ser adotados de acordo com o denominado método do tempo equivalente, limitado a 15 min e são mais usuais para edificações de pequeno porte. / A study will be carried out on the behavior of simply supported, full interaction composite steel and concrete beams composed by compact profiles in fire situation. Despite being designed as simply supported, the main idea is to consider, in the fire situation analysis, the moment resistance capacity reserve on the beam supports, usually neglected during room temperature design, due to the upper longitudinal reinforcement present in the concrete slab, being possible to dispense fireproof coating in these elements. First, room temperature design procedures of composite beams will be approached in order to explain the structural behavior of this type of elements, providing a basis for subsequent thermal analysis. The thermal analysis will be carried out, in a first stage, by simplified methods according to design procedures and adopting simplifying hypotheses in which indirect stresses caused by thermal expansion and thermal gradient are neglected, being possible to apply those concepts in structural design offices. In a second step, thermal stress analyses were performed with aid of numerical models to study the structural behavior and collapse time of the beam. The results were sufficient to validate the fire resistance time values found by the simplified analysis following design methods. As a conclusion, the structural capacity increase of the beam, provided by the adoption of the composite connection at the support in case of lighter steel profiles usually chosen to be used as buildings floor beams, was sufficient for situations of standard fire resistance requirements between 15 min and 21 min. Analyzes have shown that it is not possible to justify the absence of fire resistant coating for standard fire resistance requirements of 30 min or higher. Lower times can be adopted according to the so-called equivalent time method, limited to 15 min, which are more common for small buildings.

Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire-one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin-Menshov property and the approximation of Baire- one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides an exposition of topologies (various density topologies as well as the fine topologies in both linear and non-linear potential...