京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21328号 / 理博第4424号 / 新制||理||1635(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 並河 良典, 教授 上 正明, 教授 森脇 淳 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let \text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$. A {\em Leonard system} on $V$ is a sequence \[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\] where $\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$; $\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$ are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and nonzero when $\vert i-j \vert = 1$. % Leonard systems arise in connection with orthogonal polynomials, representations of many nice algebras, and the study of some highly regular combinatorial objects. We shall use the construction of Leonard pairs of classical type from finite-dimensional modules of $\mathit{sl}_2$ and the construction of Leonard pairs of basic type from finite-dimensional modules of $U_q(\mathit{sl}_2)$. Suppose $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ is a Leonard system. For $0 \leq i \leq d$, let \[ U_i = (E^*_0V+E^*_1V+\cdots + E^*_iV)\cap (E_iV+E_{i+1}V+\cdots + E_dV). \] Then $U_0$, $U_1$, \ldots, $U_d$ is the {\em split decomposition of $V$ for $\Phi$}. % The split decomposition of $V$ for $\Phi$ gives rise to canonical matrix representations of $\A$ and $\B$ in terms of useful parameters for the Leonard system. %These canonical matrix representations for $\A$, $\B$ are respectively lower bidiagonal and upper bidiagonal. In this thesis, we consider when certain Leonard systems share a split decomposition. We say that Leonard systems $\Phi:=(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d)$ and $\hat{\Phi}:=(\hat{\A} ;\hat{\B}; \lbrace \hat{E}_i\rbrace_{i=0}^d; \lbrace \hat{E^*}_i\rbrace_{i=0}^d)$ are {\em friends} when $\A = \hat{\A}$ and $\Phi$, $\hat{\Phi}$ have the same split decomposition. % We obtain Leonard systems which share a split decomposition by constructing them from closely related module structures for either $\mathit{sl}_2$ or $U_q(\mathit{sl}_2)$ on $V$. We then describe friends by a parametric classification. In this manner we describe all pairs of friends of classical and basic types. In particular, friendship is not entirely a property of isomorphism classes.

International Telemetering Conference Proceedings / October 22-25, 2001 / Riviera Hotel and Convention Center, Las Vegas, Nevada / In current practice sensor data is digitized and input into computers, displays, and recorders. To try to reduce the volume of digitized data, our original hypothesis was that by selecting a subset of digital values from an over-sampled signal, we could improve signal identification and improve perhaps Nyquist performance. Our investigations did not lead to significant improvements but did clarify our thinking regarding the usage of digitized data.

In 1985, Moffat-Takaoka (MT) algorithm was developed to solve the all pairs shortest path (APSP) problem. This algorithm manages to get time complexity of O(n² log n) expected time when the end-point independent model of probabilistic assumption is used. However, the use of a critical point introduced in this algorithm has made the implementation of this algorithm quite complicated and the running time of this algorithm is difficult to analyze. Therefore, this study introduces a new deterministic algorithm for the APSP that provides an alternative to the existing MT algorithm. The major advantages of this approach compared to the MT algorithm are its simplicity, intuitive appeal and ease of analysis. Moreover, the algorithm was shown to be efficient as the expected running time is the same O(n² log n). Performance of a good algorithm depends on the data structure used to speed up the operations needed by the algorithm such as insert, delete-min and decrease-key operations. In this study, two new data structures have been implemented, namely quaternary and dimensional heaps. In the experiment carried out, the quaternary heap that employed similar concept with the trinomial heap with a special insertion cache function performed better than the trinomial heap when the number of n vertices was small. Likewise, the dimensional heap data structure executed the decrease-key operation efficiently by maintaining the thinnest structure possible through the use of thin and thick edges, far surpassing the existing binary, Fibonacci and 2-3 heaps data structures when a special acyclic graph was used. Taken together all these promising findings, a new improved algorithm running on a good data structure can be implemented to enhance the computing accuracy and speed of todays computing machines.

This research focused on the synthesis of novel ogranoboron reagents in efforts to perform a variety of synthetic transformations, and additionally, the development of new methodologies to generate drug-like scaffolds. Initially, three novel tripod ligands were synthesized, and two were effectively chelated to boron to provide the desired organoborates. Such organoborates were employed in nucleophilic additions where they were found to be ineffective, whereas some activity was observed in Suzuki-Miyaura cross-coupling reactions. An additional project on organoboron compounds was conducted and focused on the development of organoboron frustrated Lewis pairs (FLPs) to facilitate the storage and transfer of hydrogen, nucleophilic addition reactions, and Claisen rearrangements. A new method for synthesizing a pyrrolidine diol unit was accomplished, and this intermediate was utilized to synthesize two FLPs. The reactivity of the FLPs with small molecules was assessed, and the pyrrolidine diol unit was subsequently evaluated for its ability to undergo a multicomponent reaction (MCR) to yield compounds possessing beneficial biological activity. Further research in this area was conducted, and a 5-aminoimidazole scaffold was synthesized employing a new MCR which is more efficient than previously reported methodologies. 5-Aminoimidazoles are frequently found in compounds which possess desirable biological activity, and this novel method was employed to generate a library of eleven 5-aminoimidazoles. Additionally, two post condensation modification reactions were developed. During initial studies, a side product was observed which was identified as a dihydrotriazine, which is another biologically appealing chemotype. Therefore, an enhanced method of synthesizing this product was developed, and a library of eleven dihydrotriazines was produced. In summary, novel organoboron reagents were synthesized, and their activity was evaluated. The pyrrolidine diol utilized to synthesize FLPs was applied towards an MCR. Furthermore, a novel MCR was developed for the synthesis of 5-aminoimidazoles, and an enhanced protocol for the synthesis of dihydrotriazines was found.

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Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Barley, during the past three decades, has been used considerably for the study of linkage groups and character inheritance. The commercial varieties are diploids, having seven pairs of chromosomes. Six linkage groups have been established. According to studies on interchange by Kramer et al. (1954), two linkage groups, formerly designated as III and VII, may be separate arms of the same chromosome. They are now generally considered as one linkage group.