This thesis is devoted to investigation of some properties of the permanent function over the set Omega_n of n-by-n doubly stochastic matrices. It contains some basic properties as well as some partial progress on Foregger's conjecture. CONJECTURE[Foregger] For every n\in N, there exists k=k(n)>1 such that, for every matrix A\in Omega_n, per(A^k)<=per(A). In this thesis the author proves the following result. THEOREM For every c>0, n\in N, for all sufficiently large k=k(n,c), for all A\in\Omega_n which minimum nonzero entry exceeds c, per(A^k)<=per(A). This theorem implies that for every A\in\Omega_n, there exists k=k(n,A)>1 such that per(A^k)<=per(A).

The Raman spectrum of polycrystalline or matrix-isolated S₂N₂ shows three bands attributable to its Raman active fundamentals, including two in close proximity; the possibility of Fermi resonance is discounted. The infrared spectrum.of polycrystalline S₂N₂ shows five bands, including three attributable to the infrared active fundamentals, while the others are associated with some intermediate species in the polymerisation of S₂N₂. The vibrational spectra of matrix-isolated S₄N₄ are consistent with previous observations in the solid state and in solution, also with the established cage structure of the molecule. The stretching force constants of S₂N₂ and S₄N₄, lower than those predicted on the basis of observations on acyclic S-N molecules, are correlated with the strain in the molecules and their associated thermodynamic instability. The interaction force constants indicate delocalised π-bonding, apparently more extensive in S₂N₂. Substantial cross-ring S-S bonding is evident in S₄N₄; S-S interactions in S₂N₂ are apparently non-bonded and repulsive in nature. The infrared spectrum of matrix-isolated Cr0C1₃ contains bands attributable to the fundamentals of this molecule, along with several indicating the presence of Cr0₂C1₂ and possibly other related molecules. The Raman spectrum shows just three strong bands, all below 250 cm^-1, assumed to arise from the deformation fundamentals of Cr0C1₃; the form of the spectrum is attributed to absorption or fluorescence. The force constants derived for Cr0C1₃ correspond closely to their counterparts in V0C1₃ and Cr0₂C1₂, suggesting similar force fields in the three molecules. The infrared spectrum of the volatile products of the reaction between PC1₃ and NaN₃ indicates the presence of several molecules, possibly including C1₂PN₃ and oligomers of C1₂ P = N, although no definite conclusions are drawn. Spectroscopic evidence also suggests that the reaction between (CH₃)₂PC1 and NaN₃ yields (CH₃)₂PN₃ as a major product, although observations such as the effect of ultraviolet photolysis remain unexplained.

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The use of fibre-reinforced polymer matrix composite materials is growing at a faster rate than GDP in many countries. An improved understanding of their processing and mechanical behaviour would extend the potential applications of these materials. For unidirectional composites, it is predicted that localised absence of fibres is related to longitudinal compression failure. The use of woven reinforcements permits more effective manufacture than for unidirectional fibres. It has been demonstrated experimentally that compression strengths of woven composites are reduced when fibres are clustered. Summerscales predicted that clustering of fibres would increase the permeability of the reinforcement and hence expedite the processing of these materials. Commercial fabrics are available which employ this concept using flow-enhancing bound tows. The net effect of clustering fibres is to enhance processability whilst reducing the mechanical properties. The effects reported above were qualitative correlations. Gross differences in the appearance of laminate sections are apparent for different weave styles. For the quantification of subtle changes in fabric architecture, the use of automated image analysis is essential. Griffm used Voronoi tessellation to measure the microstructures of composites made using flow-enhancing tows. The data was presented as histograms with no single parameter to quantify microstructure. This thesis describes the use of automated image analysis for the measurement of the microstructures of woven fibre-reinforced composites, and pioneers the use of fractal dimensions as a single parameter for their quantification. It further considers the process-property- structure relationships for commercial and experimental fabric reinforcements in an attempt to resolve the processing versus properties dilemma. A new flow-enhancement concept has been developed which has a reduced impact on laminate mechanical properties.

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Shipment consolidation is a logistics strategy whereby many small shipments are combined into a few larger loads. The economies of scale achieved by shipment consolidation help in reducing the transportation costs and improving the utilization of logistics resources. The fundamental questions about shipment consolidation are i) to how large a size should the consolidated loads be allowed to accumulate? And ii) when is the best time to dispatch such loads? The answers to these questions lie in the set of decision rules known as shipment consolidation policies. A number of studies have been done in an attempt to find the optimal consolidation policy. However, these studies are restricted to only a few types of consolidation policies and are constrained by the input parameters, mainly the order arrival process and the order weight distribution. Some results on the optimal policy parameters have been obtained, but they are limited to a couple of specific types of policies. No comprehensive method has yet been developed which allows the evaluation of different types of consolidation policies in general, and permits a comparison of their performance levels. Our goal in this thesis is to develop such a method and use it to evaluate a variety of instances of shipment consolidation problem and policies. In order to achieve that goal, we will venture to use matrix-analytic methods to model and solve the shipment consolidation problem. The main advantage of applying such methods is that they can help us create a more versatile and accurate model while keeping the difficulties of computational procedures in check. More specifically, we employ a discrete batch Markovian arrival process (BMAP) to model the weight-arrival process, and for some special cases, we use phase-type (PH) distributions to represent order weights. Then we model a dispatch policy by a discrete monotonic function, and construct a discrete time Markov chain for the shipment consolidation process. Borrowing an idea from matrix-analytic methods, we develop an efficient algorithm for computing the steady state distribution of the Markov chain and various performance measures such as i) the mean accumulated weight per load, ii) the average dispatch interval and iii) the average delay per order. Lastly, after specifying the cost structures, we will compute the expected long-run cost per unit time for both the private carriage and common carriage cases.

In this thesis we revisit two classical definitions of angle in an inner product space: real-part angle and Hermitian angle. Special attention is paid to Krein’s inequality and its analogue. Some applications are given, leading to a simple proof of a basic lemma for a trace inequality of unitary matrices and also its extension. A brief survey on recent results of angles between subspaces is presented. This naturally brings us to the world of majorization. After introducing the notion of majorization, we present some classical as well as recent results on eigenvalue majorization. Several new norm inequalities are derived by making use of a powerful decomposition lemma for positive semidefinite matrices. We also consider coneigenvalue majorization. Some discussion on the possible generalization of the majorization bounds for Ritz values is presented. We then turn to a basic notion in convex analysis, the Legendre-Fenchel conjugate. The convexity of a function is important in finding the explicit expression of the transform for certain functions. A sufficient convexity condition is given for the product of positive definite quadratic forms. When the number of quadratic forms is two, the condition is also necessary. The condition is in terms of the condition number of the underlying matrices. The key lemma in our derivation is found to have some connection with the generalized Wielandt inequality. A new inequality between angles in inner product spaces is formulated and proved. This leads directly to a concise statement and proof of the generalized Wielandt inequality, including a simple description of all cases of equality. As a consequence, several recent results in matrix analysis and inner product spaces are improved.