No desenvolvimento de projetos de iluminação de ambientes, um dos objetivos que se destaca é a busca pelo conforto visual, que emprega metodologias de resolução conhecidas, como o Método dos Lumens e o Método Ponto a Ponto. A luz visível está contida no espectro da radiação térmica e, portanto, o fluxo luminoso pode ser relacionado ao fluxo de radiação térmica. Determinar as posições e as competências das fontes de luz necessárias na superfície de projeto ganha importância quando o comportamento, em termos de uniformidade ou de fluxo radiante, é especificado. O presente trabalho visa a estabelecer diferentes valores de fluxo em duas regiões distintas da superfície do projeto. Por meio do posicionamento das fontes de luz, é estabelecido um fluxo maior na região denominada principal e um fluxo menor na região denominada secundária. A modelagem matemática da radiação térmica (Método das Radiosidades) é aplicada ao projeto de iluminação, considerando-se as características da visão humana e o comportamento das fontes de luz. Na modelagem, é considerada uma cavidade retangular tridimensional com superfícies cinza e com condição de parede fria, na qual o poder emissivo das paredes é nulo. As fontes de luz são representadas por unidades de malha no teto. A relação de equações é resolvida por metodologia inversa, usando o algoritmo de Otimização Extrema Generalizada (GEO). Este algoritmo é classificado como um método de otimização estocástica de busca global para a resolução de sistemas considerados inicialmente mal condicionados. A posição e a potência das fontes luminosas são determinadas pela resolução do sistema de equações, de forma a proporcionar um fluxo de radiação duas vezes maior na região principal em relação à região secundária. A função objetivo do processo consiste em minimizar a diferença entre o fluxo desejado e os valores de fluxo de radiação incidente nas duas regiões da superfície de projeto. Em virtude das características de simetria do problema, a relação é estabelecida para apenas um quarto da cavidade. Assim, por exemplo, aplicar a metodologia com 9 fontes de luz a um quarto da região resulta em 36 fontes de luz em toda a cavidade. Os resultados mostram que é possível encontrar um arranjo de fontes de luz preestabelecendo-se duas condições de potência. / In the development of environmental illumination projects, one of the main goals to be achieved is the visual comfort, which is usually done by known methodologies, like the Lumens Method and the Point by Point Method. Since the visible light is contained in the spectrum of thermal radiation, the luminous flux can be related to the thermal radiation flux. The determination of the position and power of the light sources required by the design surface gains an higher importance whenever a behavior is specified, should it be in terms of uniformity or in therms of radiant flux. In this work, we describe a method that allows the establishment of different flux values in two distinct regions of the design surface, which are referred by the names main region and secondary region. Through the spatial arrangement of the light sources, the method sets a more intense flux in the main region and a less intense one in the secondary region. The mathematical model of thermal radiation, known as Radiosity Method, is applied to the illumination design, along with the characteristics of the human vision and the behavior of light sources. In this model, a rectangular three-dimensional cavity is considered. It has gray surfaces and exhibits the conditions of a cold wall, in which the emissivity power of the walls is null. The light sources are represented by a mesh unit in the ceiling. The system of equations is solved by inversemethodology, using the Generalized Extremal Optimization (GEO) algorithm. This algoritm is classified as being a stochastic optimization method of global search to solve systems that are initially considered ill-conditioned. By solving this system, the position and power of light sources can be determined, and this is done in such a way that the flux radiation in the main region is twice more intense then the one in the secondary region. The target function of the whole process is to minimize the difference between the desired flux and the incident flux radiation values for each one of the two design surface regions. We further explore the problem symmetry, solving the equation system for only a quarter of the cavity. This way, if the methodology is applied with nine light sources into a quarter of the region, the entire cavity will behave as if it has 36 light sources. Our results show that, given two prescribed conditions of power, it is possible to find an arrangement of light sources.

The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states. / KMBT_363 / Adobe Acrobat 9.54 Paper Capture Plug-in

The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.

Cette thèse traite de problèmes liés à la théorie des limitesd'objets combinatoires, une récente théorie qui a permis de tisserdes liens entre différents domaines tels que la combinatoire,l'analyse, la géométrie ou la théorie de la probabilité.Cette thèse applique des méthode venant de cette théorie à des problèmesde combinatoire extrémale.Dans un premier chapitre, je développe une théorie des limites d'objetsappelés emph{types d'ordre}, un objets qui encode des configurationsd'ensembles de points du plan. Le type d'ordre d'un ensemble de pointssuffit à caractériser de nombreuses propriétés essentielles de cet ensemblede point comme, par exemple, son enveloppe convexe.Je montre qu'une limite de type d'ordre peut être représentée par un objetanalogue à un graphon à valeurs O ou 1.Je fais ensuite le lien entre limites de type d'ordre et la distributionnaturelle de limite de type d'ordre obtenue par l’échantillonnage de pointsdu plan suivant une certaine probabilité.De cette manière, toute probabilité sur le plan engendre une limite de typed'ordre. Je montre d'une part que cette correspondance n'est pas surjective-c'est à dire qu'il existe des limites de type d'ordre ne venant pas de probabilitédu plan- et j'étudie d'autre part son injectivité.Je montre que si le support d'une mesure de probabilité est assez gros, par exemple siil contient une boule ouvert, alors la limite que cette mesure engendre suffit à caractériser cette mesure à une transformation projective près.Un second chapitre traite de test de propriété.Un testeur de propriété est un algorithme aléatoire permettant de séparerles objets ayant une certaine propriété des objet à distance au moins εde l'avoir, au sens de la distance d'édition.Ce domaine donne des algorithmes extrêmement rapides, et en particulierdes algorithmes dont la complexité ne dépends pas de la taille de l'entréemais seulement du paramètre de précision ε.Un résultat fondamental de cet domaine pour les graphes montré par Alonet Shapira est le suivant : toute classe de graphe héréditaire possède un teltesteur.Cette thèse contribue à la question suivante :Quelles classes de graphes possède un testeur dont la complexité est unpolynôme en 1/ε ?Je montre qu'en particulier la classe des graphes d'intervales possède un teltesteur.La théorie des algèbres de drapeaux est un outil étroitement lié aux limites degraphes denses qui donne une méthode pour démontrer des bornes sur certainsparamètres combinatoires à l'aide d'un ordinateur.Dans un troisième chapitre, je présente un programme écrit durant ma thèsequi implémente cette méthode.Ce programme fonctionne comme une bibliothèque pour calculer dans les algèbresde drapeaux, manipuler des inégalités sur les drapeaux ou encoder des problèmesd'optimisations par une instance de programme semi-défini positif qui peutensuite être résolu par un solveur externe.Ce programme est en particulier utilisé pour obtenir un nouvelle borne pour le cas triangulaire de la conjecture de Caccetta-Häggkvist. / This thesis is focused on problems related to the theory of combinatorial limits.This theory opened links between different fields such asanalysis, combinatorics, geometry and probability theory.In this thesis, we apply ideas coming from this framework toproblems in extremal combinatorics.In a first chapter we develop a theory of limits for emph{order types},a geometrical object that encodes configuration of a set of points in theplane by the mean of the orientations of their triangles.The order type of a point set suffices to determine many of its properties,such as for instance the boundary of its convex hull.We show that the limit of a converging sequence of order typescan be represented by random-free object analogous to a graphon.Further, we link this notion to the natural distributions of order typesarising from the sampling of random points from some probability measureof the plane.We observe that in this mean, every probability measure gives rise to a limitof order types.We show that this map from probability measure on the plane to limit oforder type is not surjective.Concerning its injectivity,we prove that if a measure has large enough support, for instance if its supportcontains an open ball, the limit of order types the measure generatessuffices to essentially determine this measure.A second chapter is focused on property testing.A tester is a randomized algorithm for distinguishing between objects satisfyinga property from those that are at some distance at least εfrom having itby means of the edition distance.This gives very efficient algorithms, and in particular algorithms whosecomplexity does not depend on the size of the input but only on the parameter ε.For graphs, it has been shown by Alon and Shapira that every hereditary propertyhas such a tester.We contribute to the following question :which classes of graphs have a one-sided property tester with a number of queries that is a polynomial in 1/ε ?We give a proof that the class of interval graphs has such a tester.The theory of flag algebras is a framework introduced by Razborovclosely related to dense limit of graphs, that gives a way to systematicallyderive bounds for parameters in extremal combinatorics.In a third chapter we present a program developed during my Phd.that implements this method.This program works as a library that can compute flag algebras,manipulate inequalities on densities and encode the optimization of some parameterin a semi-definite positive instance that can be given to a dedicated solverto obtain a bound on this parameter.This program is in particular used to obtain a new bound forthe triangle case of the Caccetta-Häggkvist conjecture.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22974号 / 理博第4651号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 山口 孝男, 教授 藤原 耕二, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Cette thèse a pour thème la coloration de diverses classes de graphes épars. Shearer montra en 1983 [She83] que le ratio d'indépendance des graphes sans triangle de degré maximal d est au moins (1-o(1))ln d/d, et 13 ans plus tard Johansson [Joh96] démontra que le nombre chromatique de ces graphes est au plus O(d/ln d) quand d tend vers l'infini. Ce dernier résultat fut récemment amélioré par Molloy [Mol19], qui montra que la borne (1+o(1))d/ln d est valide quand d tend vers l'infini.Tandis que le résultat de Molloy s'exprime à l'aide d'un paramètre global, le degré maximal du graphe, nous montrons qu'il est possible de l'étendre à la coloration locale. Il s'agit de la coloration par liste, où la taille de la liste associée à chaque sommet ne dépend que de son degré. Avec une méthode différente se basant sur les propriétés de la distribution hard-core sur les ensembles indépendants d'un graphe, nous obtenons un résultat similaire pour la coloration fractionnaire locale, avec des hypothèses plus faibles. Nous démontrons également un résultat concernant la coloration fractionnaire locale des graphes où chaque sommet est contenu dans un nombre borné de triangles, et une borne principalement optimale sur le taux d'occupation — la taille moyenne des ensembles indépendants — de ces graphes. Nous considérons également les graphes de maille 7, et prouvons des résultats similaires qui améliorent les bornes précédemment connues quand le degré maximal du graphe est au plus 10^7. Finalement, pour les graphes d-réguliers où d vaut 3, 4, ou 5, de maille g variant entre 6 et 12, nous démontrons de nouvelles bornes inférieures sur le ratio d'indépendance.Le Chapitre 2 est dédié à la coloration à distance t d'un graphe, qui généralise la notion de coloration forte des arêtes. Nous cherchons à étendre le théorème de Johansson à la coloration à distance t, par l'exclusion de certains cycles. Le résultat de Johansson s'obtient par exclusion des triangles, ou des cycles de taille k pour n'importe quelle valeur de k. Nous montrons que l'exclusion des cycles de taille 2k, pour n'importe quel k>t, a un effet similaire sur le nombre chromatique à distance t, et sur l'indice chromatique à distance t+1. En outre, quand t est impair, une conclusion similaire peut se faire pour le nombre chromatique à distance t par l'exclusion des cycles de d'une taille impaire fixée valant au moins 3t. Nous étudions l'optimalité de ces résultats à l'aide de constructions de nature combinatoire, algébrique, et probabiliste.Dans le Chapitre 3, nous nous intéressons à la densité bipartie induite des graphes sans triangle, un paramètre relaxant celui de la coloration fractionnaire. Motivés par une conjecture de Esperet, Kang, et Thomassé [EKT19], qui prétend que la densité bipartie induite de graphes sans triangle de degré moyen d est au moins de l'ordre de ln d, nous démontrons cette conjecture quand d est suffisamment grand en termes du nombre de sommets n, à savoir d est au moins de l'ordre de (n ln n)^(1/2). Ce résultat ne pourrait être amélioré que par une valeur de l'ordre de ln n, ce que nous montrons à l'aide d'une construction reposant sur le processus sans triangle. Nos travaux se ramènent à un problème intéressant, celui de déterminer le nombre chromatique fractionnaire maximal d'un graphe épars à n sommets. Nous prouvons des bornes supérieures non triviales pour les graphes sans triangle, et pour les graphes dont chaque sommet appartient à un nombre borné de triangles.Cette thèse est reliée aux nombres de Ramsey. À ce jour, le meilleur encadrement connu sur R(3,t) nous est donné par le résultat de Shearer, et par une analyse récente du processus sans triangle [BoKe13+,FGM13+], ce qui donne(1-o(1)) t²/(4 ln t) < R(3,t) < (1+o(1)) t²/ln t. (1)Beaucoup de nos résultats ne pourraient être améliorés à moins d'améliorer par la même occasion (1), ce qui constituerait une révolution dans la théorie de Ramsey quantitative. / This thesis focuses on generalisations of the colouring problem in various classes of sparse graphs.Triangle-free graphs of maximum degree d are known to have independence ratio at least (1-o(1))ln d/d by a result of Shearer [She83], and chromatic number at most O(d/ln d) by a result of Johansson [Joh96], as d grows to infinity. This was recently improved by Molloy, who showed that the chromatic number of triangle-free graphs of maximum degree d is at most (1+o(1))d/ln d as d grows to infinity.While Molloy's result is expressed with a global parameter, the maximum degree of the graph, we first show that it is possible to extend it to local colourings. Those are list colourings where the size of the list associated to a given vertex depends only on the degree of that vertex. With a different method relying on the properties of the hard-core distribution on the independent sets of a graph, we obtain a similar result for local fractional colourings, with weaker assumptions. We also provide an analogous result concerning local fractional colourings of graphs where each vertex is contained in a bounded number of triangles, and a sharp bound for the occupancy fraction — the average size of an independent set — of those graphs. In another direction, we also consider graphs of girth 7, and prove related results which improve on the previously known bounds when the maximum degree does not exceed 10^7. Finally, for d-regular graphs with d in the set {3,4,5}, of girth g varying between 6 and 12, we provide new lower bounds on the independence ratio.The second chapter is dedicated to distance colourings of graphs, a generalisation of strong edge-colourings. Extending the theme of the first chapter, we investigate minimal sparsity conditions in order to obtain Johansson-like results for distance colourings. While Johansson's result follows from the exclusion of triangles — or actually of cycles of any fixed length — we show that excluding cycles of length 2k, provided that k>t, has a similar effect for the distance-t chromatic number and the distance-(t+1) chromatic index. When t is odd, the same holds for the distance-t chromatic number by excluding cycles of fixed odd length at least 3t. We investigate the asymptotic sharpness of our results with constructions of combinatorial, algebraic, and probabilistic natures.In the third chapter, we are interested in the bipartite induced density of triangle-free graphs, a parameter which conceptually lies between the independence ratio and the fractional chromatic number. Motivated by a conjecture of Esperet, Kang, and Thomassé [EKT19], which states that the bipartite induced density of a triangle-free graph of average degree d should be at least of the order of ln d, we prove that the conjecture holds for when d is large enough in terms of the number of vertices n, namely d is at least of the order of (n ln n)^(1/2). Our result is shown to be sharp up to term of the order of ln n, with a construction relying on the triangle-free process. Our work on the bipartite induced density raises an interesting related problem, which aims at determining the maximum possible fractional chromatic number of sparse graph where the only known parameter is the number of vertices. We prove non trivial upper bounds for triangle-free graphs, and graphs where each vertex belongs to a bounded number of triangles.All the content of this thesis is a collection of specialisations of the off-diagonal Ramsey theory. To this date, the best-known bounds on the off-diagonal Ramsey number R(3,t) come from the aforementioned result of Shearer for the upper-bound, and a recent analysis of the triangle-free process [BoKe13+,FGM13+] for the lower bound, giving(1-o(1)) t²/(4 ln t) < R(3,t) < (1+o(1)) t²/ln t. (1)Many of our results are best possible barring an improvement of (1), which would be a breakthrough in off-diagonal Ramsey theory.

Das Konzept des Minimum-Energie-Pfades (MEP) dient als grundlegendes Modell, um den Ablauf chemischer Reaktionen zu verstehen. Es basiert auf der Theorie des Übergangszustandes, derzufolge Ausgangsstoffe und Reaktionsprodukte energetisch stabile Zustände bilden, die durch eine Potentialbarriere voneinander getrennt sind. Beim Übergang der Reaktanten in die Produkte gilt es, diese Barriere zu ¨uberwinden.

The need to understand the structure of complex networks increases as both their complexity and the dependency of human society on them grows. Network centralities help to recognize the key elements of these networks. Betweenness centrality is a network centrality measure based on shortest paths. More precisely, the contribution of a pair of vertices u, v to a vertex w ̸= u, v is the fraction of the shortest uv-paths which lead through w. Betweenness centrality is then given by the sum of contributions of all pairs of vertices u, v ̸= w to w. In this work, we have summarized known results regarding both exact values and bounds on betweenness. Additionally, we have improved an existing bound and obtained more exact formulation for r-regular graphs. We have made two major contributions about betweenness uniform graphs, whose vertices have uniform betweenness value. The first is that all betweenness uniform graphs of order n with maximal degree n − k have diameter at most k, by which we have solved a conjecture posed in the literature. The second major result is that betweenness uniform graphs nonisomorphic to a cycle that are either vertex- or edge-transitive are 3-connected, by which we have partially solved another conjecture. 1

Union-free families of subsets of [n] = {1, . . ., n} have been studied in Frankl and Füredi (Eur J Combin 5:127-131, 1984). In this paper, we provide a complete characterization of maximal symmetric difference-free families of subsets of [n].

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19259号 / 理博第4114号 / 新制||理||1592(附属図書館) / 32261 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 入谷 寛, 教授 加藤 毅, 教授 吉川 謙一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM