A graph is called star extremal if its fractional chromatic number is equal to its circular chromatic number. Given integers n,k,k' such that 1<=k<=k'<=n/2,the circulant graph G(n,S_k,k') has vertex set [n]={0,1,2,...,n-1} in which i~j if k<=|i-j|<=k' or n-k'<= |i-j|<=n-k. It was known that for n=q(k+k')+r,where 0<=r <k+k', if k'>=5/4k,then G(n,S_k,k') is star extremal. In the thesis, we prove that if k'>=7/6k and q>=4, then G(n,S_k,k') is star extremal.

The search for max-cliques in graphs is a well established NP-complete problem in graph theory and algorithm design, with many algorithms designed to make use of internal structures of specific types of graphs. We study the extension of the problem of searching for max-cliques in graphs to hypergraphs with constant edge size k, and adapt existing algorithms for graphs to work in k-hypergraphs. In particular, we are interested in the generalization of circulant graphs to circulant k-hypergraphs, and provide a definition of this type of hypergraph. We design and implement a new algorithm to perform max-clique searches on circulant k-hypergraphs. This algorithm combines ideas from a Russian doll algorithm for max-cliques in graphs (Ostergard 2002) with an algorithm based on necklaces for a class of circulant k-hypergraphs (Tzanakis, Moura, Stevens and Panario 2016). We examine the performance of our new algorithm against a set of adapted algorithms (backtracking and Russian doll search for general k-hypergraphs, and necklace-based search for circulant k-hypergraphs) in a set of benchmarking experiments across various densities and edge sizes. This study reveals that the new algorithm outperforms the others when edge density of the hypergraph is high, and that the pure necklace-based algorithm is best in the case of low densities. Finally, we use our new algorithm to perform an exhaustive search on circulant 4-hypergraphs constructed from linear feedback shift register sequences on finite fields of order q that yields covering arrays. The search is completed for 2 <= q <= 5 which solves the open case of q=5 left by Tzanakis et al.

An interesting type of distributed system is a collection of identical subsystems that interact in a distinct pattern. A notable example is a ring, more commonly referred to as a circulant system. It is well known that control problems for circulant systems can be simplified by exploiting their common connection with the shift operator. Based on an examination of the algebraic properties underlying this connection, we identify a broader class of systems that share common base transformations. We call it the class of linear patterned systems. Members with meaningful physical interpretations include symmetric circulant systems, triangular Toeplitz systems and certain hierarchical systems. A geometric approach is employed to study the basic control properties of patterned systems, including controllability, observability and decomposition. Controller synthesis for several stabilization problems is then considered, and we show that a patterned solution to the problems exists if a general solution exists.

An interesting type of distributed system is a collection of identical subsystems that interact in a distinct pattern. A notable example is a ring, more commonly referred to as a circulant system. It is well known that control problems for circulant systems can be simplified by exploiting their common connection with the shift operator. Based on an examination of the algebraic properties underlying this connection, we identify a broader class of systems that share common base transformations. We call it the class of linear patterned systems. Members with meaningful physical interpretations include symmetric circulant systems, triangular Toeplitz systems and certain hierarchical systems. A geometric approach is employed to study the basic control properties of patterned systems, including controllability, observability and decomposition. Controller synthesis for several stabilization problems is then considered, and we show that a patterned solution to the problems exists if a general solution exists.

We determine necessary and sufficient conditions for a Cayley digraph of the cyclic group of order n to have the property that any other Cayley digraph of a cyclic group of order n is isomorphic to the first if and only if an isomorphism between the two digraphs is a group automorphism of the cyclic group of order n.

No description available.

The purpose of this dissertation is to provide a theoretical model to design RF coils using multiconductor transmission line (MTL) structures for MRI applications. In this research, an MTL structure is represented as a multiport network using its port admittance matrix. Resonant conditions and closed-form solutions for different port resonant modes are calculated by solving the eigenvalue problem of port admittance matrix using block matrix algebra. A mathematical proof to show that the solution of the characteristic equation of the port admittance matrix is equivalent to solving the source side input impedance is presented. The proof is derived by writing the transmission chain parameter matrix of an MTL structure, and mathematically manipulating the chain parameter matrix to produce a solution to the characteristic equation of the port admittance matrix. A port admittance matrix can be formulated to take one of the forms depending on the type of MTL structure: a circulant matrix, or a circulant block matrix (CB), or a block circulant circulant block matrix (BCCB). A circulant matrix can be diagonalized by a simple Fourier matrix, and a BCCB matrix can be diagonalized by using matrices formed from Kronecker products of Fourier matrices. For a CB matrix, instead of diagonalizing to compute the eigenvalues, a powerful technique called “reduced dimension method� can be used. In the reduced dimension method, the eigenvalues of a circulant block matrix are computed as a set of the eigenvalues of matrices of reduced dimension. The required reduced dimension matrices are created using a combination of the polynomial representor of a circulant matrix and a permutation matrix. A detailed mathematical formulation of the reduced dimension method is presented in this thesis. With the application of the reduced dimension method for a 2n+1 MTL structure, the computation of eigenvalues for a 4n X 4n port admittance matrix is simplified to the computation of eigenvalues of 2n matrices of size 2 X 2. In addition to reduced computations, the model also facilitates analytical formulations for coil resonant conditions. To demonstrate the effectiveness of the proposed methods (2n port model and reduced dimension method), a two-step approach was adopted. First, a standard published RF coil was analyzed using the proposed models. The obtained resonant conditions are then compared with the published values and are verified by full-wave numerical simulations. Second, two new dual tuned coils, a surface coil design using the 2n port model, and a volume coil design using the reduced dimensions method are proposed, constructed, and bench tested. Their validation was carried out by employing 3D EM simulations as well as undertaking MR imaging on clinical scanners. Imaging experiments were conducted on phantoms, and the investigations indicate that the RF coils achieve good performance characteristics and a high signal-to-noise ratio in the regions of interest.

L’analyse de l’ADN circulant dans le contexte de la tumorogenèse et comme outil diagnostic Le projet de thèse ici décrit est fondé sur la découverte remarquable qu'une quantité importante d’ADN circule dans le sang de patients atteints de cancer [1-5]. Le développement d'une technologie basée sur la détection de l'ADN circulant représente une avancée scientifique et médicale pour le diagnostic et le suivi dans la prise en charge thérapeutique des patients atteints de cancer. Malgré de nombreuses études menées au cours des dix dernières années [4,5] sur l'ADN circulant, les origines de la libération de l'ADN circulant dans les liquides biologiques sont hypothétiques et sa structure n'est pas élucidée. Ces données ne valident pas jusqu'à présent l'ADN circulant en tant que biomarqueur. Pour cette raison, les objectifs du groupe dirigé par Alain Thierry sont axés sur l'élucidation des formes structurelles de l'ADN circulant. Ainsi, en utilisant des souris nude xénogreffées avec des lignées cellulaires tumorales humaines de cancer colorectal ainsi que des échantillons sanguins cliniques provenant de patients atteints de cancer colorectal, l'équipe a montré que la concentration en ADN circulant était corrélée positivement avec la taille de la tumeur [6, 7] et ces résultats se sont révélés optimaux pour des tailles inférieures à 100 pb. Une discrimination significative entre les individus sains et les patients du cancer a été observée grâce à l'analyse de la fragmentation de l'ADN circulant. L'originalité de ces découvertes a donné naissance à la technologie Intplex récemment breveté par le CNRS [8]. L'objectif de la thèse est de valider la quantification et la fragmentation de l'ADN circulant comme un outil de diagnostic et de suivi de la maladie dans la prise en charge du cancer en analysant de près les facteurs qui peuvent influencer la quantification et la fragmentation de l'ADN circulant. Grâce au modèle animal développé par l’équipe et l’étroite collaboration avec les centres anti-cancéreux, différents paramètres seront analysés. Une partie de la thèse se concentrera sur la comparaison et la standardisation des résultats en fonction de nombreux facteurs spécifiques à la tumeur, comme son type, sa progression, sa différenciation et sa localisation tissulaire. Le travail de thèse portera également sur l'influence de facteurs individuels pouvant affecter la quantité et la fragmentation de l'ADN circulant: âge, sexe, antécédents médicaux, états physiologiques spécifiques, situations physiopathologiques. L'influence du traitement sera également explorée. Des études seront menées afin de standardiser l'analyse biologique: influence du rythme circadien, prise de nourriture .... Techniquement, la variation analytique et l'influence des facteurs pré-analytiques seront déterminées afin d’établir un guide de bonnes pratiques analytiques pour éliminer tout artefact susceptible d’affecter la quantité et l'intégrité de l'ADN circulant dans les échantillons. Ces deux paramètres seront testés dans une évaluation clinique prospective multicentrique sur une cohorte de 450 patients atteints de cancer colorectal. Ce travail garantit un impact considérable dans la littérature et dans la pratique clinique comme test non invasif de diagnostic et de suivi et comme un outil pour améliorer les connaissances de base sur l'ADN circulant et le cancer. / Analysis of circulating DNA circulating in the context of tumorigenesis and as a diagnostic tool The thesis project described here is based on the remarkable discovery that a significant amount of DNA circulates in blood of cancer patients [1-5]. The development of a technology based on the detection of circulating DNA represents a scientific and medical breakthrough for diagnosis and follow up in therapeutic care of cancer patients. Despite numerous studies conducted over the last decade [4,5] on circulating DNA, origins of release of circulating DNA in biological fluids are hypothetical and its structure is unclear. These data do not validate so far circulating DNA as a biomarker. For this reason, objectives of the group led by Alain Thierry are focused on elucidating structural forms of circulating DNA. Thus, using nude mice xenografted with human tumor cell lines of colorectal cancer as clinical samples from colorectal cancer patients, the team showed that the concentration was positively correlated with tumor size [6 , 7] and these results were optimal for sizes below 100 bp. A significative discrimination between healthy individuals and cancer patients was found by the analysis of circulating DNA fragmentation. The originality of these discoveries gave rise to the Intplex technology recently patented by the CNRS [8]. The aim of the thesis is to validate quantification and fragmentation of circulating DNA as a diagnostic and follow-up test in the management of cancer by closely analyzing the factors that may influence quantification and fragmentation of circulating DNA. Thanks to mouse model developed by the team and close collaboration with clinical cancer centers, different parameters will be analyzed. One part of the thesis will be focused on comparison and standardization of the different results depending on many factors specific to the tumor, such as its type, its progression, its differentiation and its tissue localization. The thesis work will also focus on the influence of individual factors that may affect the quantity and the fragmentation of circulating DNA: age, sex, medical history, specific physiological states, pathophysiological situations. The influence of treatment will also be explored. Studies will be undertaken in order to standardize the biological analysis: influence of circadian rhythm, food intake.... Technically, the analytical variation and the influence of pre-analytical factors will be determined to establish a good practice guide to eliminate any artifacts altering the amount and integrity of circulating DNA in the samples. These two parameters will be tested in a prospective multicentric clinical evaluation on a cohort including 450 patients with colorectal cancer. This work warrants a significant impact in the literature and in cancer clinical practice as a non invasive diagnostic and follow-up test and as a tool to improve the basic knowledge on circulating DNA and cancer.

We consider three problems in this dissertation, all under the unifying theme of random matrix products. The first and second problems are concerned with weak convergence in stochastic matrices and circulant matrices, respectively, and the third is concerned with the numerical calculation of the Lyapunov exponent associated with some random Fibonacci sequences. Stochastic matrices are nonnegative matrices whose row sums are all equal to 1. They are most commonly encountered as transition matrices of Markov chains. Circulant matrices, on the other hand, are matrices where each row after the first is just the previous row cyclically shifted to the right by one position. Like stochastic matrices, circulant matrices are ubiquitous in the literature.In the first problem, we study the weak convergence of the convolution sequence mu to the n, where mu is a probability measure with support S sub mu inside the space S of d by d stochastic matrices, d greater than or equal to 3. Note that mu to the n is precisely the distribution of the product X sub 1 times X sub 2 times and so on times X sub n of the mu distributed independent random variables X sub 1, X sub 2, and so on, X sub n taking values in S. In [CR] Santanu Chakraborty and B.V. Rao introduced a cyclicity condition on S sub mu and showed that this condition is necessary and sufficient for mu to the n to not converge weakly when d is equal to 3 and the minimal rank r of the matrices in the closed semigroup S generated by S sub mu is 2. Here, we extend this result to any d bigger than 3. Moreover, we show that when the minimal rank r is not 2, this result does not always hold.The second problem is an investigation of weak convergence in another direction, namely the case when the probability measure mu's support S sub mu consists of d by d circulant matrices, d greater than or equal to 3, which are not necessarily nonnegative. The resulting semigroup S generated by S sub mu now lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence mu to the n to analyze the problem. Our approach is based on the work of Mukherjea and his collaborators, who in [LM] and [DM] presented a method based on a bookkeeping of the possible structure of the compact kernel K of S.The third problem considered in this dissertation is the numerical determination of Lyapunov exponents of some random Fibonacci sequences, which are stochastic versions of the classical Fibonacci sequence f sub (n plus 1) equals f sub n plus f sub (n minus 1), n greater than or equal to 1, and f sub 0 equal f sub 1 equals 1, obtained by randomizing one or both signs on the right side of the defining equation and or adding a "growth parameter." These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent. Following techniques presented by Embree and Trefethen in their numerical paper [ET], we study the behavior of the Lyapunov exponents as a function of the probability p of choosing plus in the sign randomization.

Le but de cette thèse est de caractériser le comportement de la phase solide marquant le régime turbulent dans la zone dilué d’une colonne à lit fluidisé circulant et définir sa limite supérieure qui marque la transition au régime de fluidisation rapide. Une série d’expériences ont été réalisé dans ce sens surtout pour déterminer les vitesses des particules et leurs écarts types en utilisant la technique laser doppler vélocimétrie. L’étude est divisée en deux parties principales, la première a consisté à déterminer le comportement moyen de la phase solide et la structure macroscopique du lit ainsi que son comportement fluctuant à partir de l’analyse du mouvement axial et transversal des particules avec et sans séparation des particules suivant leurs sens de mouvement (ascendante/ descendante, dirigée vers le centre de la colonne / la paroi). La deuxième partie a consisté d’étudier la transition entre le régime turbulent et le régime de fluidisation rapide à partir de l’analyse de l’évolution des grandeurs moyennes et fluctuantes du lit localement au centre et en paroi (surtout les vitesses des particules, leurs écarts-types, et le taux de particules en circulation). Sur la base de ces observations nous avons défini un nouveau régime intermédiaire entre le régime turbulent et le régime de fluidisation rapide que nous avons appelé régime "Pré-fluidisation rapide" qui partage beaucoup de caractéristiques avec le régime de fluidisation rapide mais sans variation sensible sur le taux de solide récupéré en sortie. / The aim of this thesis is to characterize the solid phase behavior marking the turbulent fluidization regime in the dilute zone of a circulating fluidized bed riser and set its upper limit, which marks the transition to the fast fluidization regime. A series of experiments were conducted in this direction, especially to determine the particle velocities and their standard deviations using laser Doppler velocimetry technique. The study is divided into two main parts, the first was to determine the average solid phase behavior and the macroscopic structure of the bed in this region and its fluctuating behavior from the analysis of axial and transversal particle motions with and without separation of particles according to their movement direction (up / down, directed toward the center of the column / wall). The second part consists of studying the transition between the turbulent and fast fluidization regime from the analysis of the evolution of mean and fluctuating flow quantities locally in the center riser and in the wall (especially the particle velocity, their standard deviations, and the particle flow rate). Based on the results observations, we have identified a new intermediate regime between the turbulent and the fast fluidization regime that we called "Pre-fast fluidization" regime which shares many characteristics with the fast fluidization regime, but without any significant change in the solid flow rate collected on the column exit.