Tesis sobre el tema "Graph projection"

The goal of information visualization is to transform abstract data into efficient visual format which help humans extract useful information from the data collected. Two distinctive types of data are of particular interest to the information visualization community: one is multidimensional data in which each data point/entity is described by various attributes. The other is relational data which represent the interaction between the entities. These two types of data are common in the real world but difficult for people to understand. Many visualization approaches have been developed to help people gain insight in to such data sets. This thesis focuses on visualizing the multivariate networks which contains the aforementioned two types of data. The word “multivariate" has two meanings: l) The entities have multiple attributes. 2) The same set of entities can have different relationships. Visualizations should help viewers to compare attributes of different entities, recognize entities’ roles in various relationships and detect correlation between attributes and relationships. We present a two-phase approach to visualize the multivariate networks. In the first phase, vertices with similar attribute values and short graph distance are mapped to proximate regions on the representation. In the second phase. the layout generated in the first phase is adjusted to separate overlapping vertices. reduce vertex—edge overlaps and edge crossings. A user study is carried out to evaluate the effectiveness of this hybrid approach. Results are compared against the commonly used glyph-based technique. The user study shows that the hybrid approach helps users to find correlations between the vertices’ attribute and relationship. In addition, the users find the hybrid visualization facilitates comparing different relationships between the same set of entities. Finally, the hybrid approach is applied to visualize four international trading networks.

Cette thèse étudie le problème de la protection de la vie privée dans le LinkedOpen Data (ou « LOD », en français « web des données ouvertes » ou encore « donnéesliées ouvertes »). Ce travail se situe à l’intersection d’une longue série de travaux sur laconfidentialité des données et le LOD. Notre objectif est d’étudier l’impact des aspectssémantiques sur la publication des données et sur les fuites éventuelles d’information.Nous considérons RDF comme le format de représentation du LOD et la confidentialitédifférentielle (DP) comme le principal critère de protection de la vie privée. La DP a étéinitialement conçue pour définir la confidentialité dans le domaine des bases de donnéesrelationnelle. Elle est basée sur une quantification de la difficulté pour un attaquantd’identifier, en observant le résultat d’un algorithme, quelle base de données parmis unvoisinage a été utilisée pour le produire.Les objectifs de cette thèse sont au nombre de quatre: O1) améliorer la protectiondes données LOD. En particulier, proposer une approache permettant de construire desméchanismes DP utilisables sur RDF ; O2) étudier comment les définitions des voisinagessur les bases de données relationnelles en présence de contraintes de clés étrangères (FK) peuvent être traduites en RDF : O3) proposer de nouvelles définitions de voisinages sur des bases de données relationnelles équivalente à des notions existantes de voisinage sur les graphes (avec une sémantique précise) et O4) proposer un formalisme facilitant laconception et l’implémentation de mécanismes d’anonymisation de données RDF.Concernant O1, nous proposons une nouvelle approche basée sur la projection degraphes pour adapter le concept de DP à RDF. Pour O2, nous déterminons le modèlede protection qui correspond à la traduction de modèles déjà existants pour des basesde données relationnelles sous contraintes FK. Pour O3, nous introduisons le conceptde restrict deletion neighborhood (voisinage d’effacement limité) équivalent envoisinage de type "typed-node" (noeud typé). Nous proposons également une relaxation de la définition permettant de traduite les voisinages "typed-outedge" (arc sortanttypé). Pour O4, nous proposons un langage de transformation de graphes basé sur leconcept de réécriture de graphes, qui sert de fondation pour construire divers mécanismes d’anonymisation sur des graphes attribués.L’ensemble de nos contributions théoriques ont été implémentées par des prototypes"preuve de concept" et ont été évalués sur des jeux de données réels, afin de montrerl’applicabilité de nos travaux à des cas d’usage réels
This thesis studies the problem of privacy in linked open data (LOD). Thiswork is at the intersection of long lines of work on data privacy and linked open data.Our goal is to study how the presence of semantics impacts the publication of data andpossible data leaks. We consider RDF as the format to represent LOD and DifferentialPrivacy (DP) as the main privacy concept. DP was initially conceived to define privacyin the relational database (RDB) domain and is based on a quantification of the difficultyfor an attacker observing an output to identify which database among a neighborhoodis used to produce it.The objective of this thesis is four-fold: O1) to improve the privacy of LOD. Inparticular, to propose an approach to construct usable DP-mechanisms on RDF; O2) tostudy how neighborhood definitions over RDB in the presence of foreign key (FK) constraints translate to RDF; O3) to propose new neighborhood definitions over relationaldatabase translating into existing graph concepts to ease the design of DP mechanisms;and O4) to support the implementation of sanitization mechanisms for RDF graphs witha rigorous formal foundation.For O1, we propose a novel approach based on graph projection to adapt DP toRDF. For O2, we determine the privacy model resulting from the translation of popularprivacy model over RDB with FK constraints to RDF. For O3, we propose the restrictdeletion neighborhood over RDB with FK constraints whose translation to the RDFgraph world is equivalent to typed-node neighborhood. Moreover, we propose a looserdefinition translating to typed-outedge neighborhood. For O4, we propose a graphtransformation language based on graph rewriting to serve as a basis for constructingvarious sanitization mechanisms on attributed graphs.We support all our theoretical contributions with proof-of-concept prototypes thatimplement our proposals and are evaluated on real datasets to show the applicability ofour work

Avec la croissance importante du besoin d'analyser une grande masse de données structurées tels que les composés chimiques, les structures de protéines ou même les réseaux sociaux, la fouille de sous-graphes fréquents est devenue un défi réel en matière de fouille de données. Ceci est étroitement lié à leur nombre exponentiel ainsi qu'à la NP-complétude du problème d'isomorphisme d'un sous-graphe général. Face à cette complexité, et pour gérer cette taille importante de l'espace de recherche, les méthodes classiques de fouille de graphes ont exploré des heuristiques de recherche basées sur le support, le langage de description des exemples (limitation aux chemins, aux arbres, etc.) ou des hypothèses (recherche de sous-arborescence communes, de chemins communs, etc.). Dans le cadre de cette thèse, nous nous basons sur une méthode d'appariement de graphes issue du domaine de la programmation par contraintes, nommée AC-projection, qui a le mérite d'avoir une complexité polynomiale. Nous introduisons des approches de fouille de graphes permettant d'améliorer les approches existantes pour ce problème. En particulier, nous proposons deux algorithmes, FGMAC et AC-miner, permettant de rechercher les sous-graphes fréquents à partir d'une base de graphes. Ces deux algorithmes profitent, différemment, des propriétés fortes intéressantes de l'AC-projection. En effet, l'algorithme FGMAC adopte un parcours en largeur de l'espace de recherche et exploite l'approche par niveau introduite dans Apriori, tandis que l'algorithme AC-miner parcourt l'espace en profondeur par augmentation de motifs, assurant ainsi une meilleure mise à l'échelle pour les grands graphes. Ces deux approches permettent l'extraction d'un type particulier de graphes, il s'agit de celui des sous-graphes AC-réduits fréquents. Dans un premier temps, nous prouvons, théoriquement, que l'espace de recherche de ces sous-graphes est moins important que celui des sous-graphes fréquents à un isomorphisme près. Ensuite, nous menons une série d'expérimentations permettant de prouver que les algorithmes FGMAC et AC-miner sont plus efficients que ceux de l'état de l'art. Au même temps, nous prouvons que les sous-graphes AC-réduits fréquents, en dépit de leur nombre sensiblement réduit, ont le même pouvoir discriminant que les sous-graphes fréquents à un isomorphisme près. Cette étude est menée en se basant sur une évaluation expérimentale de la qualité des sous-graphes AC-réduits fréquents dans un processus de classification supervisée de graphes
With the important growth of requirements to analyze large amount of structured data such as chemical compounds, proteins structures, social networks, to cite but a few, graph mining has become an attractive track and a real challenge in the data mining field. Because of the NP-Completeness of subgraph isomorphism test as well as the huge search space, frequent subgraph miners are exponential in runtime and/or memory use. In order to alleviate the complexity issue, existing subgraph miners have explored techniques based on the minimal support threshold, the description language of the examples (only supporting paths, trees, etc.) or hypothesis (search for shared trees or common paths, etc.). In this thesis, we are using a new projection operator, named AC-projection, which exhibits nice complexity properties as opposed to the graph isomorphism operator. This operator comes from the constraints programming field and has the advantage of a polynomial complexity. We propose two frequent subgraph mining algorithms based on the latter operator. The first one, named FGMAC, follows a breadth-first order to find frequent subgraphs and takes advantage of the well-known Apriori levelwise strategy. The second is a pattern-growth approach that follows a depth-first search space exploration strategy and uses powerful pruning techniques in order to considerably reduce this search space. These two approaches extract a set of particular subgraphs named AC-reduced frequent subgraphs. As a first step, we have studied the search space for discovering such frequent subgraphs and proved that this one is smaller than the search space of frequent isomorphic subgraphs. Then, we carried out experiments in order to prove that FGMAC and AC-miner are more efficient than the state-of-the-art algorithms. In the same time, we have studied the relevance of frequent AC-reduced subgraphs, which are much fewer than isomorphic ones, on classification and we conclude that we can achieve an important performance gain without or with non-significant loss of discovered pattern's quality

Cette thèse aborde le problème du manque de performance des outils exploitant des représentationsà base de graphes en reconnaissance des formes. Nous proposons de contribuer aux nouvellesméthodes proposant de tirer partie, à la fois, de la richesse des méthodes structurelles et de la rapidité des méthodes de reconnaissance de formes statistiques. Deux principales contributions sontprésentées dans ce manuscrit. La première correspond à la proposition d'une nouvelle méthode deprojection explicite de graphes procédant par analyse multi-facettes des graphes. Cette méthodeeffectue une caractérisation des graphes suivant différents niveaux qui correspondent, selon nous,aux point-clés des représentations à base de graphes. Il s'agit de capturer l'information portéepar un graphe au niveau global, au niveau structure et au niveau local ou élémentaire. Ces informationscapturées sont encapsulés dans un vecteur de caractéristiques numériques employantdes histogrammes flous. La méthode proposée utilise, de plus, un mécanisme d'apprentissage nonsupervisée pour adapter automatiquement ses paramètres en fonction de la base de graphes àtraiter sans nécessité de phase d'apprentissage préalable. La deuxième contribution correspondà la mise en place d'une architecture pour l'indexation de masses de graphes afin de permettre,par la suite, la recherche de sous-graphes présents dans cette base. Cette architecture utilise laméthode précédente de projection explicite de graphes appliquée sur toutes les cliques d'ordre 2pouvant être extraites des graphes présents dans la base à indexer afin de pouvoir les classifier.Cette classification permet de constituer l'index qui sert de base à la description des graphes etdonc à leur indexation en ne nécessitant aucune base d'apprentissage pré-étiquetées. La méthodeproposée est applicable à de nombreux domaines, apportant la souplesse d'un système de requêtepar l'exemple et la granularité des techniques d'extraction ciblée (focused retrieval)
This thesis addresses the problem of lack of efficient computational tools for graph based structural pattern recognition approaches and proposes to exploit computational strength of statistical pattern recognition. It has two fold contributions. The first contribution is a new method of explicit graph embedding. The proposed graph embedding method exploits multilevel analysis of graph for extracting graph level information, structural level information and elementary level information from graphs. It embeds this information into a numeric feature vector. The method employs fuzzy overlapping trapezoidal intervals for addressing the noise sensitivity of graph representations and for minimizing the information loss while mapping from continuous graph space to discrete vector space. The method has unsupervised learning abilities and is capable of automatically adapting its parameters to underlying graph dataset. The second contribution is a framework for automatic indexing of graph repositories for graph retrieval and subgraph spotting. This framework exploits explicit graph embedding for representing the cliques of order 2 by numeric feature vectors, together with classification and clustering tools for automatically indexing a graph repository. It does not require a labeled learning set and can be easily deployed to a range of application domains, offering ease of query by example (QBE) and granularity of focused retrieval

La cryo-microscopie électronique est une technique tomographique permettant de reconstituer la structure 3D d’un objet complexe en biologie à partir d’un jeu d’acquisitions. Ces images de l’objet complexe sont appelées les projections et sont acquises sous orientations inconnues. Un des avantages de la cryo-microscopie électronique est l’obtention d’un modèle 3D de très haute résolution de l’objet dans un état naturel. La procédure de reconstruction comporte plusieurs étapes telles que l’alignement, la classification des projections, l’estimation de leurs orientations et le raffinement des projections. Lors de ces étapes, la distance entre deux projections est fréquemment mesurée. Le travail réalisé au cours de cette thèse s’organise autour de la recherche théorique d’une distance entre des projections non-orientées avec comme objectif l’amélioration de la procédure de reconstruction tomographique en cryo-microscopie électronique. La contribution de ce travail de thèse est une méthode permettant d’estimer la différence angulaire entre deux projections dans les cas 2D et 3D. Notre méthode est basée sur la construction d’un graphe de voisinage dont les sommets sont les projections, dont les arêtes relient des projections voisines et sont pondérées par une approximation locale de la différence angulaire. Le calcul de ces poids repose sur les propriétés des moments de projection. Notre méthode est testée sur des images simulées de différentes résolutions et de différents niveaux du bruit. La comparaison avec des autres méthodes d’estimation de la différence angulaire est aussi réalisée
Cryo-electron microscopy is a tomographic technique allowing to reconstruct a 3D model of complex structure in biology from a set of acquired images. These images are known as the tomographic projections and are taken at unknown directions. The advantage of the cryo-electron microscopy is the 3D reconstruction at very high resolution. The reconstruction procedure consists of many steps such as projection alignment, projection classification, orientation estimation and projection refinement. During these steps, the distance between two projections is frequently measured. The work in this thesis aims at studying the distances mesured between two unknown-direction projections with the objective of improving the reconstruction result in the cryo-electron microscopy. The contribution of this thesis is the developement of a method for estimating the angular difference between two projections in 2D and 3D. Our method is based on the construction of a neighborhood graph whose vertices are the projections, whose edges link the projection neighbors and are weighted by a local approximation of the angular difference. The calculation of the weights relies on the projection moment properties. The proposed method has been tested on simulated images with different resolutions and at different noise levels. The comparison with others estimation methods of angular difference has been realised

We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.

Social media services and applications enable billions of users to share an unprecedented amount of social information, which is further augmented by location and collocation information from mobile phones, and can be aggregated to provide an accurate digital representation of the social world. This dissertation argues that extracted social knowledge from this wealth of information can be embedded in the design of novel distributed, socially-aware applications and services, consequently improving system response time, availability and resilience to attacks, and reducing system overhead. To support this thesis, two research avenues are explored. First, this dissertation presents Prometheus, a socially-aware peer-to-peer service that collects social information from multiple sources, maintains it in a decentralized fashion on user-contributed nodes, and exposes it to applications through an interface that implements non-trivial social inferences. The system's socially-aware design leads to multiple system improvements: 1) it increases service availability by allowing users to manage their social information via socially-trusted peers, 2) it improves social inference performance and reduces message overhead by exploiting naturally-formed social groups, and 3) it reduces the opportunity of attackers to influence application requests. These performance improvements are assessed via simulations and a prototype deployment on a local cluster and on a worldwide testbed (PlanetLab) under emulated application workloads. Second, this dissertation defines the projection graph, the result of decentralizing a social graph onto a peer-to-peer system such as Prometheus, and studies the system's network properties and how they can be used to design more efficient socially-aware distributed applications and services. In particular: 1) it analytically formulates the relation between centrality metrics such as degree centrality, node betweenness centrality, and edge betweenness centrality in the social graph and in the emerging projection graph, 2) it experimentally demonstrates on real networks that for small groups of users mapped on peers, there is high association of social and projection graph properties, 3) it shows how these properties of the (dynamic) projection graph can be accurately inferred from the properties of the (slower changing) social graph, and 4) it demonstrates with two search application scenarios the usability of the projection graph in designing social search applications and unstructured P2P overlays. These research results lead to the formulation of lessons applicable to the design of socially-aware applications and distributed systems for improved application performance such as social search, data dissemination, data placement and caching, as well as for reduced system communication overhead and increased system resilience to attacks.

Dans cette thèse, nous etudions deux principaux problèmes de la théorie des graphes: problème d’homomorphisme des graphes planaires (signés) et problème de cycle hamiltonien.Comme une extension du théorème des quatre couleurs, il est conjecturé([80], [41]) que chaque graphe signé cohérent planaire de déséquilibré-maille d+1(d>1) admet un homomorphisme à cube projective signé SPC(d) de dimension d. La question suivant étalés naturelle:Est-ce que SPC(d) une borne optimale de déséquilibré-maille d+1 pour tous les graphes signés cohérente planaire de déséquilibré-maille d+1?Au Chapitre 2, nous prouvons que: si (B,Ω) est un graphe signé cohérente dedéséquilibré-maille d qui borne la classe des graphes signés cohérents planaires de déséquilibré-maille d+1, puis |B| ≥2^{d−1}. Notre résultat montre que si la conjecture ci-dessus est vérifiée, alors le SPC(d) est une borne optimale à la fois en terme du nombre des sommets et du nombre de arêtes.Lorsque d=2k, le problème est équivalent aux problème des graphes:est-ce que PC(2k) une borne optimale de impair-maille 2k+1 pour P_{2k+1} (tous les graphes planaires de impair-maille au moins 2k+1)? Notez que les graphes K_4-mineur libres sont les graphes planaires, est PC(2k) aussi une borne optimale de impair-maille 2k+1 pour tous les graphes K_4-mineur libres de impair-maille 2k+1? La réponse est négative, dans[6], est donné une famille de graphes d’ordre O(k^2) que borne les graphes K_4-mineur libres de impair-maille 2k+1. Est-ce que la borne optimale? Au Chapitre 3, nous prouvons que: si B est un graphe de impair-maille 2k+1 qui borne tous les graphes K_4-mineur libres de impair-maille 2k+1, alors |B|≥(k+1)(k+2)/2. La conjonction de nos résultat et le résultat dans [6] montre que l’ordre O(k^2) est optimal. En outre, si PC(2k) borne P_{2k+1}, PC(2k) borne également P_{2r+1}(r>k).Cependant, dans ce cas, nous croyons qu’un sous-graphe propre de P(2k) serait suffisant à borner P_{2r+1}, alors quel est le sous-graphe optimal de PC2k) qui borne P_{2r+1}? Le premier cas non résolu est k=3 et r= 5. Dans ce cas, Naserasr [81] a conjecturé que le graphe Coxeter borne P_{11}. Au Chapitre 4, nous vérifions cette conjecture pour P_{17}.Au Chapitres 5, 6, nous étudions les problèmes du cycle hamiltonien. Dirac amontré en 1952 que chaque graphe d’ordre n est hamiltonien si tout sommet a un degré au moins n/2. Depuis, de nombreux résultats généralisant le théorème de Dirac sur les degré ont été obtenus. Une approche consiste à construire un cycle hamiltonien à partir d'un ensemble de sommets en contrôlant leur position sur le cycle. Dans cette thèse, nous considérons deux conjectures connexes. La première est la conjecture d'Enomoto: si G est un graphe d’ordre n≥3 et δ(G)≥n/2+1, pour toute paire de sommets x,y dans G, il y a un cycle hamiltonien C de G tel que dist_C(x,y)=n/2.Notez que l’ ́etat de degre de la conjecture de Enomoto est forte. Motivé par cette conjecture, il a prouvé, dans [32], qu’une paire de sommets peut être posé des distances pas plus de n/6 sur un cycle hamiltonien. Dans [33], les cas δ(G)≥(n+k)/2 sont considérés, il a prouvé qu’une paire de sommets à une distance entre 2 à k peut être posé sur un cycle hamiltonien. En outre, Faudree et Li ont proposé une conjecture plus générale: si G est un graphe d’ordre n≥3 et δ(G)≥n/2+1, pour toute paire de sommets x,y dans G et tout entier 2≤k≤n/2, il existe un cycle hamiltonien C de G tel que dist_C(x,y)=k. Utilisant de Regularity Lemma et Blow-up Lemma, au chapitre 5, nous donnons une preuve de la conjeture d'Enomoto conjecture pour les graphes suffisamment grand, et dans le chapitre 6, nous donnons une preuve de la conjecture de Faudree et Li pour les graphe suffisamment grand
In this thesis, we study two main problems in graph theory: homomorphism problem of planar (signed) graphs and Hamiltonian cycle problem.As an extension of the Four-Color Theorem, it is conjectured ([80],[41]) that every planar consistent signed graph of unbalanced-girth d+1(d>1) admits a homomorphism to signed projective cube SPC(d) of dimension d. It is naturally asked that:Is SPC(d) an optimal bound of unbalanced-girth d+1 for all planar consistent signed graphs of unbalanced-girth d+1?In Chapter 2, we prove that: if (B,Ω) is a consistent signed graph of unbalanced-girth d which bounds the class of consistent signed planar graphs of unbalanced-girth d, then |B|≥2^{d-1}. Furthermore,if no subgraph of (B,Ω) bounds the same class, δ(B)≥d, and therefore,|E(B)|≥d·2^{d-2}.Our result shows that if the conjecture above holds, then the SPC(d) is an optimal bound both in terms of number of vertices and number of edges.When d=2k, the problem is equivalent to the homomorphisms of graphs: isPC(2k) an optimal bound of odd-girth 2k+1 for P_{2k+1}(the class of all planar graphs of odd-girth at least 2k+1)? Note that K_4-minor free graphs are planar graphs, is PC(2k) also an optimal bound of odd-girth 2k+1 for all K_4-minor free graphs of odd-girth 2k+1 ? The answer is negative, in [6], a family of graphs of order O(k^2) bounding the K_4-minor free graphs of odd-girth 2k+1 were given. Is this an optimal bound? In Chapter 3, we prove that: if B is a graph of odd-girth 2k+1 which bounds all the K_4-minor free graphs of odd-girth 2k+1,then |B|≥(k+1)(k+2)/2. Our result together with the result in [6] shows that order O(k^2) is optimal.Furthermore, if PC(2k) bounds P_{2k+1},then PC(2k) also bounds P_{2r+1}(r>k). However, in this case we believe that a proper subgraph of PC(2k) would suffice to bound P_{2r+1}, then what’s the optimal subgraph of PC(2k) that bounds P_{2r+1}? The first case of this problem which is not studied is k=3 and r=5. For this case, Naserasr [81] conjectured that the Coxeter graph bounds P_{11} . Supporting this conjecture, in Chapter 4, we prove that the Coxeter graph bounds P_{17}.In Chapter 5,6, we study the Hamiltonian cycle problems. Dirac showed in 1952that every graph of order n is Hamiltonian if any vertex is of degree at least n/2. This result started a new approach to develop sufficient conditions on degrees for a graph to be Hamiltonian. Many results have been obtained in generalization of Dirac’s theorem. In the results to strengthen Dirac’s theorem, there is an interesting research area: to control the placement of a set of vertices on a Hamiltonian cycle such that thesevertices have some certain distances among them on the Hamiltonian cycle.In this thesis, we consider two related conjectures, one is given by Enomoto: if G is a graph of order n≥3, and δ(G)≥n/2+1, then for any pair of vertices x, y in G, there is a Hamiltonian cycle C of G such that dist_C(x, y)=n/2. Motivated by this conjecture, it is proved,in [32],that a pair of vertices are located at distances no more than n/6 on a Hamiltonian cycle. In [33], the cases δ(G) ≥(n+k)/2 are considered, it is proved that a pair of vertices can be located at any given distance from 2 to k on a Hamiltonian cycle. Moreover, Faudree and Li proposed a more general conjecture: if G is a graph of order n≥3, and δ(G)≥n/2+1, then for any pair of vertices x, y in G andany integer 2≤k≤n/2, there is a Hamiltonian cycle C of G such that dist_C(x, y) = k. Using Regularity Lemma and Blow-up Lemma, in Chapter 5, we give a proof ofEnomoto’s conjecture for graphs of sufficiently large order, and in Chapter 6, we give a proof of Faudree and Li’s conjecture for graphs of sufficiently large order

We will discuss three ways to bound the chromatic number on a Cayley graph. 1. If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. 2. We will prove a general statement that all vertex-transitive maximal triangle-free graphs on n vertices with valency greater than n/3 are 3-colourable. Since Cayley graphs are vertex-transitive, the bound of general graphs also applies to Cayley graphs. 3. Since Cayley graphs for abelian groups arise from vector spaces, we can view the connection set as a set of points in a projective geometry. We will give a characterization of all large complete caps, from which we derive that all maximal triangle-free cubelike graphs on 2ⁿ vertices and valency greater than 2ⁿ/4 are either bipartite or 4-colourable.

The property that a graph has an embedding in the projective plane is closed under taking minors. Thus by the well known Graph Minor theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L, but Archdeacon's proof never appeared in any refereed journal. In this thesis we develop a modern approach and technique for finding the list L, independent of previous work. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected. In this thesis we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. For proving our main result, we use a technique which can be considered as a variation and generalization of the method that Robertson, Seymour and Thomas used for non-planar extension of planar graphs. Using this technique, besides our main result, we also classify the set of minor minimal obstructions for a-, ac-, abc-planarity for rooted graphs. (A rooted graph (G,a,b,c) is a-planar if there exists a split of the vertex a to a' and a' in G such that the new graph G' obtained by the split has an embedding in a disk such that the vertices a', b, a', c are on the boundary of the disk in the order listed. We define b- and c-planarity analogously. We say that the rooted graph (G,a,b,c) is ab-planar if it is a-planar or b-planar, and we define abc-planarity analogously.)

Projeções Multidimensionais são úteis para gerar visualizações adequadas para apoiar a análise exploratória de uma grande variedade de dados complexos e de alta dimensionalidade. Tarefas de análise visual de dados têm se beneficiado de projeções para explorar dados textuais, de imagens, de sensores, entre outros. Porém, diferentes técnicas de projeção e diferentes parametrizações de uma mesma técnica produzem resultados distintos para um mesmo conjunto de dados, pois as técnicas adotam estratégias distintas para representar os dados originais em um espaço cuja dimensionalidade permite sua visualização. Atualmente, ainda há poucos recursos para avaliar a qualidade dessas projeções e, em geral, as soluções existentes avaliam propriedades específicas, demandando grande esforço do analista para uma avaliação mais abrangente. Neste trabalho, introduzimos um arcabouço para computar medidas de avaliação de projeções com enfoque em análise de vizinhanças e de agrupamentos. Para elaborar este arcabouço, foi realizado um estudo de percepção para entender melhor como os usuários observam as projeções e foi conduzida uma investigação de representações dos dados capazes de favorecer a identificação de vizinhanças e de agrupamentos. Os padrões identificados no estudo de percepção auxiliaram a validar a representação dos dados, em que foi proposto um modelo de grafo, chamado de Extended Minimum Spanning Tree (EMST), capaz de capturar características condizentes com as observações dos participantes no estudo. O grafo EMST também foi validado por meio de dois estudos comparativos de identificação de vizinhanças e de agrupamentos. Com base neste arcabouço foram propostas cinco medidas de qualidade de projeções multidimensionais, duas delas para avaliar características relacionadas à separação visual das classes, e outras três para avaliar a preservação de propriedades do espaço original, especificamente a separação das classes, as vizinhanças e os agrupamentos. As medidas são ilustradas por meio de sua aplicação a conjuntos de dados artificiais, favorecendo a sua interpretação, bem como a conjuntos de dados reais, evidenciando a sua potencial utilidade em cenários reais. Também é apresentada uma comparação das novas medidas de preservação de vizinhanças com medidas similares descritas na literatura, permitindo identificar diferenças e semelhanças entre elas.
Multidimensional projections are valuable tools to generate visualizations that support exploratory analysis of a wide variety of complex high-dimensional data. Many examples are found in the literature of visual data analysis tasks that employ projections to explore, for instance, text, image, network and sensor data. Nonetheless, dierent projection techniques applied to a particular data set, or even alternative parameterizations of a single technique, can produce very distinct outcomes, as techniques adopt different strategies to reduce data dimensionality. Few resources are available to support assessing projection quality and, in general, existing solutions focus on specific properties. Thus, a broader assessment typically requires considerable human effort. In this work we introduce a framework to compute projection evaluation measures that focus on neighborhoods and clusters. To elaborate this framework we conducted (i) an experimental study to better understand how users perceive projections and (ii) an investigation of possible data representations capable of favoring the identification of neighborhoods and clusters. The observations resulting from the experimental study have been considered to propose and validate a novel graph data model, called Extended Minimum Spanning Tree (EMST), which captures data properties shown to be consistent with the observations by the participants in the study. The EMST graph has been validated by means of two comparative studies conducted to identify neighborhoods and clusters in multidimensional data. Under this framework, five novel measures of projection quality are introduced, two of them to assess properties related to the visual separation of classes, and three to assess the preservation of data properties in the original space, in particular the preservation of class separation, the preservation of neighborhoods and the preservation of groups. The quality measures have been applied to projections of synthetic data sets, favoring their understanding and interpretation, and also to projections of real data sets, illustrating their potential applicability in real scenarios. The newly introduced neighborhood preservation measures are also compared with existing methods in order to illustrate their differences and similarities.

This thesis was originally motivated by considering vector space analogues of problems in extremal set theory, but our main results concern colouring a graph that is intimately related to these vector space analogues. The vertices of the q-Kneser graph are the k-dimensional subspaces of a vector space of dimension v over F_q, and two k-subspaces are adjacent if they have trivial intersection. The new results in this thesis involve colouring the q-Kneser graph when k=2. There are two cases. When k=2 and v=4, the chromatic number is q²+q. If k=2 and v>4, the chromatic number is (q^(v-1)-1)/(q-1). In both cases, we characterise the minimal colourings. We develop some theory for colouring the q-Kneser graph in general.

Root systems are a fundamental concept in the theory of Lie algebra. In this thesis, we will use two different kind of graphs to represent the group generated by reflections acting on the elements of the root system. The root systems we are interested in are those of type A2, B2 and G2. After drawing the graphs, we will study the algebraic groups corresponding to those root systems. We will use three different techniques to give a geometric description of the homogeneous spaces G/P where G is the algebraic group corresponding to the root system and P is one of its parabolic subgroup. Finally, we will make a link between the graphs and the multiplication of basis elements in the Chow group CH(G/P).

L’objet de cette thèse est de proposer des approches nouvelles permettant l’utilisation d’algorithmes d’apprentissage automatique travaillant usuellement des données tabulaires aux graphes. Un graphe est une structure de donnée composée de nœuds reliés entre eux par des liens. Cette structure peut être représentée sous la forme d’une matrice, où chaque connexion entre de nœuds est représentée par une valeur non nulle, permettant une manipulation des données plus facile. Néanmoins, par leurs différences structurelles, la transposition d’un algorithme exploitant des données tabulaires aux graphes ne donne pas les résultats escomptés. Deux caractéristiques rendent cette adaptation difficile : la faible connectivité des nœuds ainsi que la distribution en loi de puissance du degré des nœuds. Ces caractéristiques conduisent toutes les deux à des matrices creuses pauvres en information tout en nécessitant beaucoup de mémoire de stockage. Dans ces travaux, nous proposons plusieurs manières de prendre en compte ces différences pour deux types de graphes particuliers. Dans la première partie, nous nous intéressons aux graphes de citations et à leur représentation dans l’optique de la veille technologique, tandis que la seconde partie s’adresse aux graphes bipartites utilisés principalement par les systèmes de recommandation. Ces adaptations permettent la réalisation de taches usuelles en apprentissage automatique, telle que le partitionnement et la visualisation des données. Pour le cas des graphes bipartites, des algorithmes spécifiques de co-partitionnement sont proposés pour la segmentation conjointe des deux parties. La troisième partie prend un revers différent. La méthode développée exploite le graphe des k plus proches voisins construit à partir des données tabulaires afin de corriger des erreurs de classifications. Les différentes méthodes développées utilisent diverses approches pour emmagasiner plus d’information dans un vecteur par rapport à l’encodage binaire habituel, permettant de travailler les graphes avec des algorithmes usuel d’apprentissage automatique
This thesis proposes new approaches to process graph using machine learning algorithms designed for tabular data. A graph is a data structure made of nodes linked to each others by edges. This structure can be represented under a matrix form where the connection between two nodes is represented by a non-zero value, simplifying the manipulation of the data. Nonetheless, the transposition of an algorithm adapted to tabular data to graphs would not give the expected results because of the structural differences. Two characteristics make the transposition difficult: the low nodes’ connectivity and the power-law distribution of nodes’ degree. These two characteristics both lead to sparse matrices with low information content while requiring a large memory. In this work, we propose several methods that consider these two graph’s specificities. In the first part, we focus on citation graphs which belong to the directed acyclic graph category and can be exploited for technical watch, while the second part is dedicated to bipartite graphs mainly use by recommender systems. These adaptations permit the achievement of usual machine learning tasks, such as clustering and data visualization. Specific co-clustering algorithms were designed to segment jointly each side of a bipartite graph and identify groups of similar nodes. The third part approaches graphs from a different perspective. The developed approach exploits the k nearest neighbours graph built from the tabular data to help correcting classification errors. These different methods use diverse methods to embed more information in a vector compared to the usual binary encoding, allowing to process graphs with usual machine learning algorithm

We apply the semidefinite programming method to obtain a new upper bound on the cardinality of codes made of subspaces of a linear vector space over a finite field. Such codes are of interest in network coding.Next, with the same method, we prove an upper bound on the cardinality of sets avoiding one distance in the Johnson space, which is essentially Schrijver semidefinite program. This bound is used to improve existing results on the measurable chromatic number of the Euclidean space.We build a new hierarchy of semidefinite programs whose optimal values give upper bounds on the independence number of a graph. This hierarchy is based on matrices arising from simplicial complexes. We show some properties that our hierarchy shares with other classical ones. As an example, we show its application to the problem of determining the independence number of Paley graphs.

Cette thèse a pour première vocation d'être un état de l'art sur le calcul quantique, sinon exhaustif, simple d'accès (chapitres 1, 2 et 3). La partie originale de cet essai consiste en deux approches mathématiques du calcul quantique concernant quelques systèmes quantiques : la première est de nature algébrique et fait intervenir des structures particulières : les corps et les anneaux de Galois (chapitre 4), la deuxième fait appel à la géométrie dite projective (chapitre 5). Cette étude a été motivée par le théorème de Kochen et Specker et par les travaux de Peres et Mermin qui en ont découlé

Cette thèse a pour première vocation d’être un état de l’art sur le calcul quantique, sinon exhaustif, simple d’accès (chapitres 1, 2 et 3). La partie originale de cet essai consiste en deux approches mathématiques du calcul quantique concernant quelques systèmes quantiques : la première est de nature algébrique et fait intervenir des structures particulières : les corps et les anneaux de Galois (chapitre 4), la deuxième fait appel à la géométrie dite projective (chapitre 5). Cette étude a été motivée par le théorème de Kochen et Specker et par les travaux de Peres et Mermin qui en ont découlé
The first vocation of this thesis would be a state of the art on the field of quantum computation, if not exhaustive, simple access (chapters 1, 2 and 3). The original (interesting) part of this treatise consists of two mathematical approaches of quantum computation concerning some quantum systems : the first one is an algebraic nature and utilizes some particular structures : Galois fields and rings (chapter 4), the second one calls to a peculiar geometry, known as projective one (chapter 5). These two approaches were motivated by the theorem of Kochen and Specker and by work of Peres and Mermin which rose from it

Le polyèdre associé à un problème d'optimisation combinatoire est l'enveloppe convexe des (vecteurs d'incidence des) solutions réalisables de ce problème. De nombreux problèmes d'optimisation combinatoire se formulent comme une maximisation de fonctions linéaires sur les polyèdres qui leurs sont associés. La description du polyèdre par un système d'inéquations linéaires est intimement liée à la résolution du problème correspondant, par le biais de la programmation linéaire. Afin de déterminer un tel système, une approche classique consiste à décomposer le problème en sous-problèmes tels que les polyèdres associés soient connus ; une composition ultérieure de ces derniers conduit à une description du polyèdre associé au problème considéré. L'objet principal de cette thèse est l'étude de la composition des polyèdres. Dans un premier temps, une approche de composition, basée sur la programmation dynamique et les méthodes de projection polyédrale, est étudiée et des résultats généraux sont proposés, permettant ainsi d'unifier des recherches existantes dans ce domaine. Cette approche est, ensuite, appliquée à la composition de polyèdres associés au problème du voyageur de commerce. En seconde partie, considérant le problème du stable, des opérations sur les graphes (composition par identification de sous-graphes de deux graphes donnés, adjonction d'une nouvelle arête) sont traitées. Des résultats polyédraux sont donc donnés, et des conséquences concernant la perfection et la h-perfection des graphes sont montrés

This thesis deals with issues automatic composing panoramic photos from individual photos. Gradually examines the various steps of algorithms and methods used in them, which are used in creating panoramas. It also focuses on the design of the own system based on methods discussed to construct panoramas. This system is implemented using OpenCV library and it is created also a graphical interface using a Qt library. Finally, are in this thesis evaluated outcomes of this designed and implemented system on available datasets.

The main goal of this thesis is a contribution to the classification of all incidence geometries

of rank two on which some group PSL(2,q), q a prime power, acts flag-transitively.

Actually we require that the action be RWPRI (residually weakly primitive) and (2T)1

(doubly transitive on every residue of rank one). In fact our definition of RWPRI requires

the geometry to be firm (each residue of rank one has at least two elements) and RC

(residually connected).

The main goal is achieved in this thesis.

It is stated in our "Main Theorem". The proof of this theorem requires more than 60pages.

Quite surprisingly, our proof in the direction of the main goal uses essentially the classification

of all subgroups of PSL(2,q), a famous result provided in Dickson’s book "Linear groups: With an exposition of the Galois field theory", section 260, in which the group is called Linear Fractional Group LF(n, pn).

Our proof requires to work with all ordered pairs of subgroups up to conjugacy.

The restrictions such as RWPRI and (2T)1 allow for a complete analysis.

The geometries obtained in our "Main Theorem" are bipartite graphs; and also locally 2-arc-transitive

graphs in the sense of Giudici, Li and Cheryl Praeger. These graphs are interesting in their own right because of

the numerous connections they have with other fields of mathematics.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

Cette recherche avait pour objectif principal d’examiner les relations entre l’estime de soi des jeunes enfants et leur adaptation scolaire. Notre intention était également de comparer les points de vue des enfants et de leur perception de soi. 119 élèves de 65 et de CE2 ainsi que leurs parents et leurs enseignants ont participé à cette étude. L’estime de soi a été appréhendée à l’aide d’un outil informatique permettant l’exploration de la perception de soi et de l’école (EPSE), inspiré du Self Perception Profile (Harter, 1982). Une épreuve projective graphique a permis d’explorer les capacités représentatives des enfants de 5 et 8 ans au sujet de leur vécu scolaire. Les points de vue des enseignants et des parents ont été respectivement recueillis à partir de questionnaires renseignés pour chacun des enfants de l’étude. Les analyses des résultats ont montré que les enfants sont capables, dès S ans, d’estimer leur valeur en tant que personne et en tant qu’élève. Les enfants de CE2 proposent néanmoins une évaluation plus objective. L’épreuve projective graphique a révélé que tous les enfants sont capables de représenter des lieux et des activités en lien avec l’école. Les enfants de 5 ans produisent des dessins moins élaborés et plus centrés sur des activités ludiques que ceux de 8 ans, qui représentent des scènes évoquant plus souvent le travail scolaire. Il est apparu que les enfants qui expriment au moins une fois à I’EPSE des perceptions négatives d’eux-mêmes ou de l’école produisent des dessins de moins bonne qualité que ceux qui se perçoivent de manière positive. L’étude des réponses des adultes aux questionnaires montre que leurs points de vue influencent celui des enfants et que les enseignants sont globalement plus sévères que les parents pour apprécier les compétences des élèves. L’ensemble des résultats recueillis tend à confirmer l’existence de liens entre l’estime de soi des enfants et leur adaptation scolaire. Ils corroborent l’idée selon laquelle l'estime de soi des enfants se construit à partir de leurs ressentis dans leur milieu familial et scolaire et en fonction des feedbacks et des attentes des éducateurs de référence (parents et enseignants)
This research was primarily to review the relationship between self-esteem of young children and their school adjustment. Our intention was also to compare the views of children, teachers and parents about children’s academic skills and their self-perception. 119 pupils of kindergarten and 3rd grade and their parents and teachers participated in this study. Self-esteem was apprehended with the aid of a computer tool for exploring the perception of self and school (EPSE), inspired from the Self Perception Profile (Harter, 1982). A projective test chart has explored the capacity representing children of 5 and 8 years about their school experiences. The views of teachers and parents were respectively collected from questionnaires completed for each child in the study. The analysis results showed that children are able, from 5 years to estimate their value as a person and as a student. Children CE2 nevertheless offer a more objective assessment. The projective test chart showed that all children capable of representing places and activities associated with the school. 5 years children produce pictures less developed and more focused on leisure activities than 8 years represent scenes evoking mostly school work. It appeared that children who express at least once in EPSE negative perceptions of themselves or about school produce designs of lower quality than those who perceive themselves in positive ways. The study of adult responses to questionnaires showed that their views influence those of children and teachers are generally more severe than parents to assess students’ skills. All the results obtained tend to confirm the existence of links between self-esteem of children and their school adjustment. They corroborate the idea that children‘s self-esteem is constructed from their feelings within their family and school and depending on feedback and expectations of their teachers reference (parents and teachers)

With the growing number of XML documents on theWeb it becomes essential to effectively organise these XML documents in order to retrieve useful information from them. A possible solution is to apply clustering on the XML documents to discover knowledge that promotes effective data management, information retrieval and query processing. However, many issues arise in discovering knowledge from these types of semi-structured documents due to their heterogeneity and structural irregularity. Most of the existing research on clustering techniques focuses only on one feature of the XML documents, this being either their structure or their content due to scalability and complexity problems. The knowledge gained in the form of clusters based on the structure or the content is not suitable for reallife datasets. It therefore becomes essential to include both the structure and content of XML documents in order to improve the accuracy and meaning of the clustering solution. However, the inclusion of both these kinds of information in the clustering process results in a huge overhead for the underlying clustering algorithm because of the high dimensionality of the data. The overall objective of this thesis is to address these issues by: (1) proposing methods to utilise frequent pattern mining techniques to reduce the dimension; (2) developing models to effectively combine the structure and content of XML documents; and (3) utilising the proposed models in clustering. This research first determines the structural similarity in the form of frequent subtrees and then uses these frequent subtrees to represent the constrained content of the XML documents in order to determine the content similarity. A clustering framework with two types of models, implicit and explicit, is developed. The implicit model uses a Vector Space Model (VSM) to combine the structure and the content information. The explicit model uses a higher order model, namely a 3- order Tensor Space Model (TSM), to explicitly combine the structure and the content information. This thesis also proposes a novel incremental technique to decompose largesized tensor models to utilise the decomposed solution for clustering the XML documents. The proposed framework and its components were extensively evaluated on several real-life datasets exhibiting extreme characteristics to understand the usefulness of the proposed framework in real-life situations. Additionally, this research evaluates the outcome of the clustering process on the collection selection problem in the information retrieval on the Wikipedia dataset. The experimental results demonstrate that the proposed frequent pattern mining and clustering methods outperform the related state-of-the-art approaches. In particular, the proposed framework of utilising frequent structures for constraining the content shows an improvement in accuracy over content-only and structure-only clustering results. The scalability evaluation experiments conducted on large scaled datasets clearly show the strengths of the proposed methods over state-of-the-art methods. In particular, this thesis work contributes to effectively combining the structure and the content of XML documents for clustering, in order to improve the accuracy of the clustering solution. In addition, it also contributes by addressing the research gaps in frequent pattern mining to generate efficient and concise frequent subtrees with various node relationships that could be used in clustering.

碩士
靜宜大學
資訊管理學系研究所
99
In the research fields of the sequential pattern mining, many proposed algorithms made efforts on improving the mining efficiency as well as customized the mining algorithms for specific application domains. Contiguous item sequential pattern mining is a novel technique to extract single-item sequential patterns where each pair of adjacent elements in the patterns is connected in the original sequences. The contiguous item sequential patterns can be used widely in many popular data mining research fields such as the biological data mining, movement pattern mining, and web usage mining. In this study, we propose a new algorithm termed TPGP(Two-Phase Graph Projection). In the beginning, TPGP scans the sequence database once and connected the information which between entries in the sequences is saved in the projected map. By traversing the projected map, we can find the supersets of contiguous single item sequential patterns. Then, the algorithm constructs a tree structure based on the sequences in the supersets found in the first stage and traverses the tree to discover all of the contiguous single item sequential patterns. We conducted a series of experiments on the synthetic datasets generated by the IBM data generator. The Up-Down tree algorithm is compared with the proposed TPGP algorithm. The experimental results show that TPGP outperforms the UDtree method in both CPU and memory usages.

Bachelor Honours - Bachelor of Mathematics (Honours)
The c*-algebra C*(E) of a discrete graph E is generated by a family of orthogonal projections and partial isometries. If a discrete group G acts on E, this action induces an action of G on C*(E). In [5], Kumijan and Pask showed that if E is locally finite and the action G on C*E is free, then the C*-algebra C*(GE) of the quotient graph is Morita equivalent to the crossed product C*(E) Xα G. This result has a striking similarity with a theorem of Green [3, Theorem 14], which implies that, if X is a locally compact space and G is a locally compact group which acts freely and properly on X, then C₀(GX) (the C*-algebra of continuous functions ∫:GX → ℂ such that for all ∈>0, the set {z ∈ GX : |∫(z)|≥∈} is compact) is Morita equivalent to the crossed product C₀(X) Xα G.

Lau, Siu Cheong.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.
Includes bibliographical references (leaves 51-52).
Abstracts in English and Chinese.
Chapter 1 --- In search of minimal cycles --- p.9
Chapter 1.1 --- What do we mean by cycles? --- p.9
Chapter 1.2 --- Integral currents --- p.10
Chapter 1.3 --- Calibration theory --- p.13
Chapter 2 --- Motivation from the Hodge Conjecture --- p.17
Chapter 2.1 --- Hodge theory on Riemannian manifolds --- p.17
Chapter 2.2 --- Hodge decomposition in Kahler manifolds --- p.19
Chapter 2.3 --- The Hodge conjecture --- p.22
Chapter 3 --- Variation of cycles in symmetric orbit --- p.26
Chapter 3.1 --- Variational formulae --- p.26
Chapter 3.2 --- Stability of cycles in Sm and CPn --- p.29
Chapter 3.3 --- Symmetric orbit in Euclidean space --- p.31
Chapter 3.4 --- Projective spaces in simple Jordan algebra --- p.39
Chapter 3.4.1 --- Introduction to simple Jordan algebra --- p.39
Chapter 3.4.2 --- Projective spaces as symmetric orbits --- p.41
Chapter 3.4.3 --- Computation of second fundamental form --- p.43
Chapter 3.4.4 --- The main theorem --- p.45
Chapter 3.5 --- Future directions --- p.49
Bibliography --- p.51

The main results of this dissertation show evidence supporting the Successive Surface Scaffolding Conjecture. This is a new conjecture that, if true, guarantees the existence of all the wye-delta-order minimal obstructions of a surface S as subgraphs of the irreducible triangulations of the surface S with a crosscap added. A new data structure, i.e. an augmented rotation system, is presented and used to create an exponential-time algorithm for embedding graphs in any surface with a constant-time check of the change in genus when inserting an edge. A depiction is a new formal definition for representing an embedding graphically, and it is shown that more than one depiction can be given for nonplanar embeddings, and that sometimes two depictions for the same embedding can be drastically different from each other. An algorithm for finding the essential cycles of an embedding is given, and is used to confirm for the projective-plane obstructions, a theorem that shows any embedding of an obstruction must have every edge in an essential cycle. Obstructions of a general surface S that are minor-minimal and not double-wye-delta-minimal are shown to each have an embedding on the surface S with a crosscap added. Finally, open questions for further research are presented.
Graduate