Early-transition-metal alkylidene chemistry. Synthesis of tantalum silyl alklidene complexes and reactions of tantalum alkylidene complexes with silanes

This dissertation describes research conducted on the chemistry of tantalum silyl complexes, reactions of alkylidene complexes with phenylsilanes, kinetic and mechanistic studies of the reaction of a tantalum alkylidene complex with phenylsilanes, and related organometallic chemistry. An overview and summary of the Ph.D. research project is given in Chapter 1. Chapter 2 describes the synthesis and characterization of the thermally unstable tantalum silyl alkylidene complexes (Me3ECH2)2Ta[=CHEMe3]SiPh2But (E = C, 1; Si, 2). Complexes 1 and 2 were found to decompose by elimination of HSiPh2But to unidentified products and (Me3SiCH2)4Ta2(µ-CSiMe3)2, respectively. Reaction of 1 or 2 with PMe3 leads to elimination of HSiPh2But and the formation of bis(phosphine)bis(alkylidene) complexes (Me3ECH2)Ta(PMe3)2[=CHEMe3]2 (E = C, 3; Si, 4). Chapter 3 presents the reactions of the tantalum alkylidene complexes (Me3SiCH2)3 Ta(PMe3)=CHSiMe3 (7), 4, and (Me3SiCH2)Ta(PMe3)2[=CHCMe3]2 (8) with phenylsilanes PhRSiH2 (R = Ph, Me) and (PhSiH2)2CH2 to yield novel bis(silyl)-substituted alkylidene complexes as well as the first reported 1, 1\u27metalla-3-silacyclobutadiene and 1, 1 \u27-metalla-3,5-disilacyclohexadiene complexes through the preferential reaction of the silanes with the alkylidene ligands of 4, 7, and 8. Chapter 4 describes the mechanistic investigation of the formation of the bis(silyl)-substituted alkylidene complex (Me3SiCH2)3Ta=C(SiMe3)SiPhMeH (9a) from the reaction of 7 with PhMeSiH2. Kinetic studies were found to be consistent with a dissociative mechanism in which a PMe3 molecule dissociated from the tantalum center in 7, forming an open coordination site on the metal for the reaction with PhMeSiH2. Deuterium labeling studies were consistent with a mechanism involving hydride and silyl intermediates. The final chapter discusses attempted syntheses of dianionic bis(silyl)lithium and -potassium compounds as potential ligands for the synthesis of poly- or persilyl early-transition-metal complexes. Two previously unreported bis(silyl)methane and -ethane complexes [(Me3Si)3Si]2CH2 (18) and (Me3Si)3SiCH2CH2Si(SiMe3)3 (19) were synthesized as possible ligand precursors

Interaction of sine-Gordon kinks with defects: The two-bounce resonance

A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.Comment: 26 pages, 17 figure

Semi-indefinite-inner-product and generalized Minkowski spaces

In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi- and indefinite-inner-products and define the semi-indefinite-inner-product and the corresponding structure, the semi-indefinite-inner-product space. We give a generalized concept of Minkowski space embedded in a semi-indefinite-inner-product space using the concept of a new product, that contains the classical cases as special ones. In the second part of this paper we investigate the real, finite dimensional generalized Minkowski space and its sphere of radius $i$. We prove that it can be regarded as a so-called Minkowski-Finsler space and if it is homogeneous one with respect to linear isometries, then the Minkowski-Finsler distance its points can be determined by the Minkowski-product

Hard loss of stability in Painlev\'e-2 equation

A special asymptotic solution of the Painlev\'e-2 equation with small parameter is studied. This solution has a critical point $t_*$ corresponding to a bifurcation phenomenon. When $t<t_*$ the constructed solution varies slowly and when $t>t_*$ the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution, using a sequence of scaling and matching procedures

Operational Research in Education

Operational Research (OR) techniques have been applied, from the early stages of the discipline, to a wide variety of issues in education. At the government level, these include questions of what resources should be allocated to education as a whole and how these should be divided amongst the individual sectors of education and the institutions within the sectors. Another pertinent issue concerns the efficient operation of institutions, how to measure it, and whether resource allocation can be used to incentivise efficiency savings. Local governments, as well as being concerned with issues of resource allocation, may also need to make decisions regarding, for example, the creation and location of new institutions or closure of existing ones, as well as the day-to-day logistics of getting pupils to schools. Issues of concern for managers within schools and colleges include allocating the budgets, scheduling lessons and the assignment of students to courses. This survey provides an overview of the diverse problems faced by government, managers and consumers of education, and the OR techniques which have typically been applied in an effort to improve operations and provide solutions

Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations