The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.

The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisors: Aloysius K. Mok and Martin D.F. Wong. Vita. Includes bibliographical references. Available also from UMI company.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 140-146). Available also in a digital version from Dissertation Abstracts.

D.Phil. / Domination in graphs is now well studied in graph theory and the literature on this subject has been surveyed and detailed in the two books by Haynes, Hedetniemi, and Slater [45, 46]. In this thesis, we continue the study of domination, by adding to the theory; improving a number of known bounds and solving two previously published conjectures. With the exception of the introduction, each chapter in this thesis corresponds to a single paper already published or submitted as a journal article. Despite the seeming disparity in the content of some of these articles, there are two overarching goals achieved in this thesis. The rst is an attempt to partition the vertex set of a graph into two sets, each holding a speci c domination-type property. The second is simply to improve known bounds for various domination parameters. In particular, an edge weighting function is presented which has been useful in providing some of these bounds. Although the research began as two separate areas of focus, there has been a fair degree of overlap and a number of the results contained in this thesis bridge the gap quite pleasingly. Specially, Chapter 11 uses the edge weighting function to prove a bound on one of the sets in our most fundamental partitions, while the improvement on a known bound presented in Chapter 7 was inspired by considering the possible existence of another partition. This latter proof relies implicitly on the `almost' existence of such a partition.

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Ce mémoire de maîtrise en composition musicale présente cinq oeuvres composées entre l'automne 1988 et le printemps 1992. L'oeuvre pour basson ou saxophone ténor et percussions Duo concertant est conçue sous forme de dialogue et demande une fine complicité de la part des Interprètes. Dorléac pour bols, cuivres, piano et percussions, intègre à la fois une forme rondo et une forme d'arche dont le point culminant se situe aux deux tiers de l'oeuvre. Relevés no.7. pour quatuor à cordes, est Inspiré des sept parties d'un poème de Guillevic Etier. L'oeuvre Séquences voilées pour quatuor à cordes et orchestre se présente sous la forme de succession de séquences reliées entre elles par un voile harmonique. Double éclat, pour hautbois d'amour/cor anglais, piano/ célesta, orchestre à cordes et percussions, exploite à la limite les possibilités du hautbois d'amour et du cor anglais (soliste). / Québec Université Laval, Bibliothèque 2014

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: p. 213-217. / Ph.D.

Reproduction de : Thèse de doctorat : Sûreté du logiciel et calcul à haute performance : Toulouse, INPT : 2007. / Titre provenant de l'écran-titre. Bibliogr. 193 réf.

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