<p> In this thesis we study convexity theorems on the interpolation of linear operators between LP-spaces. An extension of the Riesz-Thorin Theorem to spaces constructed from countably many LP-spaces is given. In addition, results involving analytic families of linear operators between these spaces are obtained. </p> / Thesis / Master of Science (MSc)

Thesis (MSc (Mathematics))--University of Stellenbosch, 2008. / Let T be a non-empty set, A a u-algebra of subsets of T and u : .A -+ Rn a bounded, countably additive measure. A set E E A is called an atom with respect to u if u(E)=/F 0 and, if F E A, FeE, then u(F) = u(E) or u(F) = 0; the measure u is atomic if there exists at least one atom (with respect to u) in A. If no such atom (with respect to u) exists in A, then u is called non-atomic. In 1940 the Russian mathematician A. A. Lyapunov published the Convexity Theorem. According to this theorem the range 'R.{u) of a bounded, finite-dimensional measure u is compact and, in the non-atomic case, convex. Since 1940 much has been published on different aspects of the range of a vector-measure. These aspects range from new and shorter proofs of the Convexity Theorem and the usefulness of it in diverse fields, to research about the geometrical characteristics of the range by using other familiar theorems, like Krein-Milman and Radon-Nikodym. In the survey at hand the Convexity Theorem in itself is studied. Applications in different fields will be looked at as well as pieces about the history of the people and the ideas involved in the development of the theorem.

Les mathématiques tropicales sont une branche des mathématiques correspondant à l'étude d'une algèbre modifiée grâce à la redéfinition de l'addition et de la multiplication. Les mathématiques tropicales sont généralement définies grâce au minimum et à l'addition (algèbre min-plus) mais le terme est parfois utilisé pour désigner l'algèbre max-plus, définie grâce au maximum et à l'addition. Briec et Horvath ont introduit une notion de convexité très proche qui apparait comme un cas limite d’opérateurs utilisés en théorie de l’optimisation par Avriel (1972) et de Ben-Tal (1977). En suivant cette ligne d’investigation, nous allons proposer, dans le domaine de l’économie de la production et de l’optimisation de portefeuille, une certaine classe de modèles économiques à élaborés à partir de ces notions. Pour ce faire, nous introduisons une nouvelle classe de technologie de production permettant de prendre en compte les structures d’homothe´tie-translation dans la mesure de productivité au travers du concept de la Convexité Max-Plus. Ensuite, nous allons établir une relation topologique entre plusieurs classes de modèles convexes généralisés connus. Nous analysons pour cela la limite de Painlevé-Kuratowski des modèles CES-CET et des technologies non paramétriques satisfaisant une hypothèse de rendements d’échelle alpha. On montre que leurs limites topologiques convergent vers les modèles de production B-convexe et Cobb-Douglas. Enfin, nous allons montrer que l'amélioration de l'efficacité technique d’une coalition d’entreprises s'avère compatible avec les technologies de semi-treillis dans un jeu coopératif. Nous introduisons ensuite, le concept d’écart absolu moyen dans la sélection du portefeuille en utilisant le « Shortage Function » qui prend en compte simultanément la réduction des inputs et l’augmentation des outputs comme dans la théorie de la production. Enfin, nous allons étendre le concept de B-convexité et de l’inverse B-convexité en se concentrant sur le calcul des mesures d’efficacité technique dans le graphe. / Tropical algebra is the tropical analogue of linear algebra by redefining the usual operation addition by the maximization operation and the usual addition operation as multiplication. Briec and Horvath introduced a concept of convexity very close to this concept quoted above which appears as one of the limits of use of the theory of optimization by Avriel (1972) and Ben-Tal (1977). Following this line of investigation, we give an overview of contributions involving a semilattice structure of production technologies and an optimization portfolio. To do that, firstly, we propose a framework allowing to consider both semilattice structure and translation homothetic properties in productivity measurement. We introduce the concept of Max-Plus convexity which combine both an upper semilattice structure and an additivity assumption. We establish a topological relation between several classes of known generalized convex models using some basic algebraic convex structures. We analyze the Painlevé-Kuratowski limit of the CES-CET and Alpha-returns to scale models. It is shown that their topological limits yield the B-convex and Cobb-Douglas production models. Moreover, we show that the improvement of technical efficiency is compatible with semilattice technologies in a cooperative game. Then, we introduce a criterion to measure portfolio efficiency based upon the minimization of the maximum absolute deviation and minimum absolute deviation from the expected return using the Shortage function. Finally, we derive simple closed-form expressions to calculate the hyperbolic measure in the case of inverse and B-Convexity that evaluates technical efficiency in the full input-output space.

La Tesis está compuesta por un capítulo introductorio y cuatro capítulosque pasamos a describir.El Capítulo 2 contiene un análisis de las funciones que son posiblementemódulo de convexidad (m.c.) para un espacio de Banach uniformementeconvexo (UC). Se muestra que las funciones m.c. están caracterizadas,salvo equivalencia, por ciertas propiedades clásicas de éstas.En el Capítulo 3, se estudia la noción de m.c. de una función convexadefinida en un espacio de Banach. Éste es el primer trabajo con resultadosgenerales y completos en espacios de Banach. Se muestra que un espacio essuperreflexivo sii admite una función (UC) definida en todo el espacio.En el Capítulo 4 se resuelve un problema establecido por Godefroy yZizler; un espacio de Banach superreflexivo con base de Schauder admiteuna norma (UC) que hace monótona a la base. Se obtienen mejoras deestimaciones de James y Gurari.En el Capítulo 5 el autor estudia la noción del módulo de cuadratura. Éstepermite reconocer la (UC) y la suavidad uniforme. El autor define laversión local, y prueba varias caracterizaciones del comportamientopuntual de la norma. / The thesis consists of one introductory chapter and four chapterscontaining original mathematical results. Let us pass to a briefdescription of the main results.Chapter 2 contains an analysis of the possible modulus of rotundityfunctions (m.r.f) for a given uniformly rotund (UC) Banach space. It isshown that m.r.f. are characterized, up to equivalence, by certainclassical properties of them.In Chapter 3, the notion of m.r. for a convex function defined on a Banachspace is studied. This seems to be the first instance of rather completegeneral results on Banach spaces. It is shown that a Banach space issuperreflexive iff it admits a (UC) function defined on the whole space.In Chapter 4 a problem asked by Godefroy and Zizler is solved; asuperreflexive Banach space with Schauder basis can be renormed by (UC)norm which makes the given basis monotone. An improvement of a result ofGurarii is an immediate corollary.In Chapter 5 the author studies the notion of modulus of squareness. Itallows to recognize (UC) and uniform smoothness. The author succeeds todefine the local version, and proves various characterizations ofpointwise behaviour of the norm.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / In this work, we study some parameters of monophonic convexity in some classes of graphs and we present our results about this subject. We prove that decide if the $m$-interval number is at most 2 and decide if the $m$-percolation time is at most 1 are NP-complete problems even on bipartite graphs. We also prove that the $m$-convexity number is as hard to approximate as the maximum clique problem, which is, $O(n^{1-varepsilon})$-unapproachable in polynomial-time, unless P=NP, for each $varepsilon>0$. Finally, we obtain polynomial time algorithms to compute the $m$-convexity number on hereditary graph classes such that the computation of the clique number is polynomial-time solvable (e.g. perfect graphs and planar graphs). / Neste trabalho, estudamos alguns parÃmetros para a convexidade monofÃnica em algumas classes de grafos e apresentamos nossos resultados acerca do assunto. Provamos que decidir se o nÃmero de $m$-intervalo à no mÃximo 2 e decidir se o tempo de $m$-percolaÃÃo à no mÃximo 1 sÃo problemas NP-completos mesmo em grafos bipartidos. TambÃm provamos que o nÃmero de $m$-convexidade à tÃo difÃcil de aproximar quanto o problema da Clique MÃxima, que Ã, $O(n^{1-varepsilon})$-inaproximÃvel em tempo polinomial, a menos que P=NP, para cada $varepsilon>0$. Finalmente, apresentamos um algoritmo de tempo polinomial para determinar o nÃmero de $m$-convexidade em classes hereditÃrias de grafos onde a computaÃÃo do tamanho da clique mÃxima à em tempo polinomial (como grafos perfeitos e grafos planares).

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2016-12-09T16:43:52Z No. of bitstreams: 2 Dissertação - Julliano Rosa Nascimento - 2016.pdf: 1812313 bytes, checksum: 9bdaa6ddbbe1dd9ce1e9ccdea8016eaf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-13T19:11:50Z (GMT) No. of bitstreams: 2 Dissertação - Julliano Rosa Nascimento - 2016.pdf: 1812313 bytes, checksum: 9bdaa6ddbbe1dd9ce1e9ccdea8016eaf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-12-13T19:11:50Z (GMT). No. of bitstreams: 2 Dissertação - Julliano Rosa Nascimento - 2016.pdf: 1812313 bytes, checksum: 9bdaa6ddbbe1dd9ce1e9ccdea8016eaf (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-11-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we consider the parameter hull number in two graph convexities, the P3- convexity and the geodetic convexity. In the P3-convexity, we present results on the P3- hull number on the Cartesian product, strong product and lexicographic product of graphs. In special, regarding to the Cartesian product, we proved a complexity result, in which we show, given a graph G resulting of a Cartesian product of two graphs and a positive integer k, is NP-complete to decide whether the P3-hull number of G is less than or equal k. We also consider the P3-hull number on complementary prisms GG of connected graphs G and G, in which we show a tighter upper bound than that found in the literature. In the geodetic convexity, we show results of the hull number on complementary prisms GG when G is a tree, when G is a disconnected graph and when G is a cograph. Finally, we also show that in the geodetic convexity, the hull number on the complementary prism GG is unlimited on connected graphs G and G, unlike what happens in the P3-convexity / Nesta dissertação, consideramos o parâmetro número envoltório em duas convexidades em grafos, a convexidade P3 e a convexidade geodética. Na convexidade P3, obtivemos resultados do número envoltório P3 para o produto Cartesiano, produto forte e produto lexicográfico de grafos. Em especial, em relação ao produto Cartesiano, obtivemos um resultado de complexidade, no qual mostramos que, dado um grafo G, resultante de um produto Cartesiano de dois grafos e um inteiro positivo k, é NP-completo decidir se o número envoltório P3 de G é menor ou igual a k. Também consideramos o número envoltório P3 para prismas complementares GG de grafos G e G conexos, em que mostramos um limite superior um pouco mais justo do que o encontrado na literatura. Na convexidade geodética, mostramos resultados do número envoltório para prismas complementares GG quando G é uma árvore, quando G é um grafo desconexo e quando G é um cografo. Por fim, também mostramos que na convexidade geodética o número envoltório do prisma complementar GG pode ser ilimitado para grafos G e G ambos conexos, diferentemente do que ocorre na convexidade P3.

Let K be a compact subset of Cn . The polynomially convex hull of K is defined as The compact set K is said to be polynomially convex if = K. A closed subset is said to be locally polynomially convex at if there exists a closed ball centred at z such that is polynomially convex. The aim of this thesis is to derive easily checkable conditions to detect polynomial convexity in certain classes of sets in This thesis begins with the basic question: Let S1 and S2 be two smooth, totally real surfaces in C2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is locally polynomially convex at the origin? If then it is a folk result that the answer is, “Yes.” We discuss an obstruction to the presumed proof, and use a different approach to provide a proof. When dimR it turns out that the positioning of the complexification of controls the outcome in many situations. In general, however, local polynomial convexity of also depends on the degeneracy of the contact of T0Sj with We establish a result showing this. Next, we consider a generalization of Weinstock’s theorem for more than two totally real planes in C2 . Using a characterization, recently found by Florentino, for simultaneous triangularizability over R of real matrices, we present a sufficient condition for local polynomial convexity at of union of finitely many totally real planes is C2 . The next result is motivated by an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc generated by z and h — where h is a nowhereholomorphic harmonic function on D that is continuous up to ∂D — equals . The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+ R, where R is a nonharmonic perturbation whose Laplacian is “small” in a certain sense. Ideas developed for the latter result, especially the role of plurisubharmonicity, lead us to our final result: a characterization for compact patches of smooth, totallyreal graphs in to be polynomially convex.

In this thesis, the primary powertrain components of a power split hybrid electric vehicle are modeled. In particular, the dynamic model of the energy storage element (i.e., traction battery) is exactly linearized through an input transformation method to take advantage of the proposed optimal control algorithm. A lipschitz continuous and nondecreasing cost function is formulated in order to minimize the net amount of consumed fuel. The globally optimal solution is obtained using a dynamic programming routine that produces the optimal input based on the current state of charge and the future power demand. It is shown that the global optimal control solution can be expressed in closed form for a time invariant and convex incremental cost function utilizing the interval back propagation approach. The global optimality of both time varying and invariant solutions are rigorously proved. The optimal closed form solution is further shown to be applicable to the time varying case provided that the time variations of the incremental cost function are sufficiently small. The real time implementation of this algorithm in Simulink is discussed and a 32.84 % improvement in fuel economy is observed compared to existing rule based methods.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-10-08T19:52:23Z No. of bitstreams: 2 Dissertação - Ronan Gomes de Amorim - 2013.pdf: 1551424 bytes, checksum: 2acf9af7fdc161d745d9a1fcf58ba4b0 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-10-09T11:22:45Z (GMT) No. of bitstreams: 2 Dissertação - Ronan Gomes de Amorim - 2013.pdf: 1551424 bytes, checksum: 2acf9af7fdc161d745d9a1fcf58ba4b0 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-10-09T11:22:45Z (GMT). No. of bitstreams: 2 Dissertação - Ronan Gomes de Amorim - 2013.pdf: 1551424 bytes, checksum: 2acf9af7fdc161d745d9a1fcf58ba4b0 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper presents the main ideas concerning convex sets and functions. Our aim is to deal, didactically, with the main topics concerning convexity, as well as the consequent exploitation of the envolved mathematical concepts. In this sense, we have made a bibliographic revision approaching important theorems, lemmas, corollaries and propositions designed both to first readers and to those who want to work with applications arising from convexity. We hope that this study may constitute an important research source either for students, teachers or researchers who wish to learn more about convex sets. / Neste trabalho, apresentamos as principais ideias concernentes aos conjuntos convexos e às funções convexas. Nosso principal foco é tratar, de forma didática, os principais tópicos envolvidos na convexidade, bem como a consequente exploração dos conceitos matemáticos envolvidos. Nesse sentido, realizamos uma revisão bibliográfica que contemplou teoremas, lemas, corolários e proposições relevantes a um primeiro leitor e a todos que pretendem trabalhar com as aplicações decorrentes da convexidade. Assim, esperamos que este material constitua uma importante fonte de pesquisa a estudantes, professores e pesquisadores que almejem estudar conteúdos relacionados aos conjuntos convexos.

<p>There are several methods avaliable to find areas of interest, but most fail at detecting such areas in cluttered scenes. In this paper two methods will be presented and tested in a qualitative perspective. The first is the darg operator, which is used to detect three dimensional convex or concave objects by calculating the derivative of the argument of the gradient in one direction of four rotated versions. The four versions are thereafter added together in their original orientation. A multi scale version is recommended to avoid the problem that the standard deviation of the Gaussians, combined with the derivatives, controls the scale of the object, which is detected. </p><p>Another feature detected in this paper is rotational symmetries with the help of approximative polynomial expansion. This approach is used in order to minimalize the number and sizes of the filters used for a correlation of a representation of the orientation and filters matching the rotational symmetries of order 0, 1 and 2. With this method a particular type of rotational symmetry can be extracted by using both the order and the orientation of the result. To improve the method’s selectivity a normalized inhibition is applied on the result, which causes a much weaker result in the two other resulting pixel values when one is high. </p><p>Both methods are not enough by themselves to give a definite answer to if the image consists of an area of interest or not, since several other things have these types of features. They can on the other hand give an indication where in the image the feature is found.</p>