For p>=2q¡Alet Kp/q be the graph with vertices 0¡A1¡A2¡A¡K¡Ap-1 in which i~j if q<=|i-j|<=p-q. The circular chromatic number Xc(G) of a graph G is the minimum of those p/q for which G admits a homomorphism to Kp/q. The circular clique number Wc(G) of G is the maximum of those p/q for which Kp/q admits a homomorphism to G.. A graph G is circular perfect if for every induced subgraph H of G we have Xc(H)=Wc(H). In this paper,we characterize those rational numbers p/q for which the complement of Kp/q are circular perfect. We also prove that if G(n¡AS) is a circulant graph whose generating set S has cardinality at most 3¡Athen G(n¡AS) is circular perfect.

Thesis (doctoral)--Ruprecht-Karls-Universität zu Heidelberg, 1917. / Cover title. Vita. Includes bibliographical references (p. [63]-68).

Entre 1950 et 1972, Henri Thomas s’engage dans une recherche singulière qui fait l’objet de notre étude et dont il est utile de distinguer deux mouvements.L’énigme qui passionne alors Thomas est celle de l’accès au territoire de l’impossible, entendu comme ce qui outrepasse les limites du possible — irrationnel, illimité, inexpliqué, autres noms du poétique ou du sacré — et seul espace authentique de la poésie.Textuellement, il s’enquiert d’une forme nouvelle capable de dire la recherche de l’impossible, et d’intégrer une dimension épique et poétique. Tous ses récits entre Les Déserteurs (1951) et La Relique (1969) façonnent et affinent une écriture à même de rendre compte de l’aventure à la fois héroïque et poétique des personnages.La forme nouvelle imaginée par Thomas se détourne du récit poétique et des procédés stylistiques qui lui sont liés pour développer une dimension épique portée par la présence d’un héros, qui vient nourrir les questionnements poétiques.D’autre part, il s’agit, grâce à ce récit que Thomas perfectionne pendant vingt ans, de trouver une forme de résolution à la recherche de l’impossible. Durant cette période, l’auteur assume progressivement l’assimilation de l’impossible à une immédiateté qui est aussi la déchirure du sacré. Les références au mythe de Diane, au texte de Klossowski, à la lumière de Hölderlin et à sa quête poétique, et enfin à la présence de la relique, dans les trois derniers récits du cycle étudié, orientent définitivement sa quête vers cette conclusion. Chaque roman l’amène à éclaircir un aspect de sa quête de l’impossible. La quête d’une totalité impensable inscrit résolument Thomas dans une tradition littéraire, de Hölderlin à Mallarmé et à Rimbaud, jusqu’aux poètes du Grand Jeu, et des mystiques à Léon Chestov. Elle l’inclut aussi dans une modernité, une « communauté de l’impossible » qui réunit Artaud, Blanchot, Bataille et Klossowski dans un projet commun, bien que les moyens utilisés pour le mener à bien diffèrent selon les écrivains.Le projet de Thomas mêle donc intimement poétique et narratif, se distinguant de certains mouvements d’avant-garde par sa conservation des éléments traditionnels du roman (personnages, héros, quête…), mais aussi d’une littérature à idées, philosophiques ou politiques, qui l’enfermerait dans le langage du possible.La recherche de l’impossible évolue de pair, chez Thomas, avec la prise de conscience de la nécessité d’un héroïsme qui soit à sa hauteur. Dans ses récits, le dépassement héroïque s’inscrit dans une construction propre à l’épopée, telle que nous l’avons dégagée : un véritable héros, qui répond aux critères du héros romanesque selon Philippe Hamon ou Vincent Jouve.La quête de l’impossible, se résout donc dans la lumière hölderlinienne, lumière philosophique et poétique d’une joie subversive. Ainsi doit se comprendre la quête de réalité parfaite d’Henri Thomas, recherche d’une libération et d’une joie poétique impossibles, qui n’est atteinte que par l’acceptation de son absence. / Between 1950 and 1972, Henri Thomas is engaged in a singular research which is the subject of our study and of which it is useful to distinguish two movements.The enigma that then fascinates Thomas is that of access to the territory of the impossible, understood as what goes beyond the limits of the possible - irrational, unlimited, unexplained, other names of the poetic or the sacred - and the only authentic space of the poetry.Textually, he inquires of a new form capable of saying the search for the impossible, and of integrating an epic and poetic dimension. All his narratives between Les Déserteurs (1951) and La Relique (1969) shape and refine a writing that can account for the heroic and poetic adventure of the characters.The new form imagined by Thomas turns away from the poetic narrative and the stylistic processes connected with it in order to develop an epic dimension borne by the presence of a hero who nourishes poetic questions.On the other hand, it is a question, thanks to this narrative that Thomas perfects during twenty years, to find a form of resolution in search of the impossible. During this period, the author assumes progressively the assimilation of the impossible to an immediacy which is also the tearing of the sacred. References to Diane's myth, to Klossowski's text, to the light of Hölderlin and his poetic quest, and finally to the presence of the relic in the last three narratives of the cycle studied, definitely orientate his quest towards this conclusion. Each novel leads him to clarify an aspect of his quest for the impossible.The quest for an unthinkable total resolutely inscribes Thomas in a literary tradition, from Hölderlin to Mallarmé and Rimbaud, to the poets of the Grand Jeu, and from the mystics to Leon Chestov. It also includes it in a modernity, a "community of the impossible" that unites Artaud, Blanchot, Bataille and Klossowski in a common project, although the means used to carry it out differ according to the writers.Thomas's project is therefore intimately poetic and narrative, distinguished from certain avant-garde movements by its preservation of the traditional elements of the novel (characters, heroes, quest ...), but also from a literature with ideas, philosophical or political , Which would enclose him in the language of the possible.The search for the impossible evolves in tandem, in Thomas, with the realization of the need for heroism that is at his height. In her narratives, the heroic surpassing is part of a construction peculiar to the epic, as we have seen it: a true hero, who meets the criteria of the romantic hero according to Philippe Hamon or Vincent Jouve.The quest for the impossible is resolved in the Hölderlin light, a philosophical and poetic light of subversive joy. Thus must be understood the quest for the perfect reality of Henry Thomas, the search for an impossible liberation and poetic joy, which is attained only by the acceptance of his absence.

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A long standing unanswered question in number theory concerns the existence (or not) of odd perfect numbers. Over time many properties of an odd perfect number have been established and refined. The initial goal of this research was to improve the lower bound on the number of distinct prime factors of an odd perfect number, if one exists, to at least 9. Previous approaches to this problem involved the analysis of a carefully chosen set of special cases with each then being eliminated by a contradiction. This thesis applies an algorithmic, factor chain approach to the problem. The implementation of such an approach as a computer program allows the speed, accuracy and flexibility of modern computer technology to not only assist but even direct the discovery process. In addition to considering odd perfect numbers, several related problems were investigated, concerned with (i) harmonic, (ii) even multiperfect and (iii) odd triperfect numbers. The aim in these cases was to demonstrate the correctness and versatility of the computer code and to fine tune its efficiency while seeking improved properties of these types of numbers. As a result of this work, significant improvements have been made to the understanding of harmonic numbers. The introduction of harmonic seeds, coupled with a straightforward procedure for generating most harmonic numbers below a chosen bound, expands the opportunities for further investigations of harmonic numbers and in particular allowed the determination of all harmonic numbers below 10 to the power 12 and a proof that there are no odd harmonic numbers below 10 to the power 15. When considering even multiperfect numbers, a search procedure was implemented to find the first 10-perfect number as well as several other new ones. As a fresh alternative to the factor chain search, a 0-1 linear programming model was constructed and used to show that all multiperfect numbers divisible by 2 to the power of a, for a being less than or equal to 65, subject to a modest constraint, are known in the literature. Odd triperfect numbers (if they exist) have properties which are similar to, but simpler than, those for odd perfect numbers. An extended test on the possible prime factors of such a number was developed that, with minor differences, applies to both odd triperfect and odd perfect numbers. When applicable, this test allows an earlier determination of a contradiction within a factor chain and so reduces the effort required. It was also shown that an odd triperfect number must be greater than 10 to the power 128. While the goal of proving that an odd perfect number must have at least 9 distinct prime factors was not achieved, due to mainly practical limitations, the algorithmic approach was able to show that for an odd perfect number with 8 distinct prime factors, (i) if it is exactly divisible by 3 to the power of 2a then a = 1, 2, 3, 5, 6 or a is greater than or equal to 31 (ii) if the special component is pi to the power of alpha, pi less than 10 to the 6 and pi to the (alpha+1) less than 10 to the 40, then alpha = 1.

In this thesis, a novel transmission structure is proposed for the Direct Sequence Ultra Wide-Band (DS-UWB) systems. The main purpose of the proposed structure is to eliminate the inter-symbol interference caused by the multi-path environment. In DS-UWB systems, shortening the guard interval is one of the possible ways to achieve higher data rates. However, interference will increase inversely with the length of the guard interval because the signal delay spread caused by the multi-path effect will induce inter-symbol interference. In this thesis, a novel transmission structure that utilizes the autocorrelation properties of the perfect sequence is proposed for interference cancellation in DS-UWB systems. Both computer simulation and mathematical analysis are provided for performance evaluation.

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