Let G = {g1, ¡K,gn} be a finite abelian group, and let LG = [gij ] be the Latin square defined by gij = gi + gj. Denote by k(G) the largest number of mutually orthogonal system containing LG. In 1948, Paige showed that if the Sylow 2-subgroup of G is not cyclic, then LG has a transversal. In this paper, we give an constructive proof for this theorem and give some upper bound and lower bound for the number k(G).

The Banach-Stone Theorem (respectly, Kadison Theorem) says that two abelian (respectively, general) C*-algebras are isomorphic as C*-algebras (respectively, JB*-algebras) if and only if they are isomorphic as Banach spaces. We are interested in using different structures to determine C*-algebras. Here, we would like to study the disjointness structures of C*-algebras and investigate if it suffices to determine C*-algebras. There are at least four versions of disjointness structures: zero product, range orthogonality, domain orthogonality and doubly orthogonality. In this thesis, we first study these disjointness structures in the case of standard operator algebras. Then we extend these results to general C*-algebras, namely, C*-algebras with continuous trace.

International Telemetering Conference Proceedings / October 25-28, 1999 / Riviera Hotel and Convention Center, Las Vegas, Nevada / The mathematics basis that can form a telemetry system is orthogonal functions. Three kinds of orthogonal functions are used up to now. First of them is sine and cosine function family. The second one is block pulse function family. The third one is Walsh function family. Their corresponding telemetry systems are FDM, TDM and SDM (CDM). Later we introduced an orthogonal function which is called Bridge function. The corresponding system is named telemetry system based on Bridge function. Now a new kind of orthogonal function, Generalized Bridge function, has been found. It can be applied to practical multiplex of information transmission. In this paper the author provides the design concept, block diagram, operational principle and technical realization of telemetry system based on Generalized Bridge function.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Group Representation Theory is the mathematical language best suited to describing the symmetry properties of a structure, and a structural analysis can utilises Group Representation Theory to provide the most efficient and systematic method of exploiting the full symmetry properties of any symmetric structure. Group Representation Theory methods currently exist for the Stiffness Niethod of structural analysis, where the stiffness matrix of a structure is block-diagonalised into a number of independent submatrices, each of which relates applied loads and displacements with a particular type of symmetry. This dissertation extends the application of Group Representation Theory to the equilibrium and compatibility matrices which are commonly used in the Force Method of structural analysis. Group Representation Theory is used to find symmetry-adapted coordinate systems for both the external vector space which is suitable for representing the loads applied to a structure, and the internal vector space wh",t-k is-suitable for representing the internal forces. Using these symmetry-adapted coordinate systems the equilibrium matrix is block-diagonalised into a number of independent submatrix blocks, thus decomposing the analysis into a number of subproblems which require less computational effort. Each independent equilibrium submatrix block relates applied loads and internal forces with particular symmetry properties, and hence any states of self-stress or inextensional mechanisms in one of these equilibrium submatrix blocks will necessarily have ~rresponding symmetry properties. Thus, a symmetry analysis provides valuable insight into the behaviour of symmetric structures by helping to identify and classif:)'. any states of self-stress .or inextensional mechanisms present in a structure. In certain cases it is also possible for a symmetry analysis to identify when a structure contains a :ijnite rather than infinitesimal mechanism. To do this a symmetry analysis must b~ carried out using the symmetry properties of the inextensional mechanism of interest. If the analysis shows that any states of self-stress which exist in the structure have "lesser" symmetry properties, then the states of self-stress exist independently from the mechanism and cannot prevent its finite motion.

The results presented in this dissertation refer to the geometry of Minkowski spaces, i.e., of real finite-dimensional Banach spaces. First we study geometric properties of radial projections of bisectors in Minkowski spaces, especially the relation between the geometric structure of radial projections and Birkhoff orthogonality. As an application of our results it is shown that for any Minkowski space there exists a number, which plays somehow the role that $\sqrt2$ plays in Euclidean space. This number is referred to as the critical number of any Minkowski space. Lower and upper bounds on the critical number are given, and the cases when these bounds are attained are characterized. Moreover, with the help of the properties of bisectors we show that a linear map from a normed linear space $X$ to another normed linear space $Y$ preserves isosceles orthogonality if and only if it is a scalar multiple of a linear isometry. Further on, we examine the two tangent segments from any exterior point to the unit circle, the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, the distances from the normalization of the sum of two unit vectors to those two vectors, and the extension of the notions of orthocentric systems and orthocenters in Euclidean plane into Minkowski spaces. Also we prove theorems referring to chords of Minkowski circles and balls which are either concurrent or parallel. All these discussions yield many interesting characterizations of the Euclidean spaces among all (strictly convex) Minkowski spaces. In the final chapter we investigate the relation between the length of a closed curve and the length of its midpoint curve as well as the length of its image under the so-called halving pair transformation. We show that the image curve under the halving pair transformation is convex provided the original curve is convex. Moreover, we obtain several inequalities to show the relation between the halving distance and other quantities well known in convex geometry. It is known that the lower bound for the geometric dilation of rectifiable simple closed curves in the Euclidean plane is $\pi/2$, which can be attained only by circles. We extend this result to Minkowski planes by proving that the lower bound for the geometric dilation of rectifiable simple closed curves in a Minkowski plane $X$ is analogously a quarter of the circumference of the unit circle $S_X$ of $X$, but can also be attained by curves that are not Minkowskian circles. In addition we show that the lower bound is attained only by Minkowskian circles if the respective norm is strictly convex. Also we give a sufficient condition for the geometric dilation of a closed convex curve to be larger than a quarter of the perimeter of the unit circle.

Master of Science / Department of Statistics / Kun Chen / The singular value decomposition (SVD) is a commonly used matrix factorization technique in statistics, and it is very e ective in revealing many low-dimensional structures in a noisy data matrix or a coe cient matrix of a statistical model. In particular, it is often desirable to obtain a sparse SVD, i.e., only a few singular values are nonzero and their corresponding left and right singular vectors are also sparse. However, in several existing methods for sparse SVD estimation, the exact orthogonality among the singular vectors are often sacri ced due to the di culty in incorporating the non-convex orthogonality constraint in sparse estimation. Imposing orthogonality in addition to sparsity, albeit di cult, can be critical in restricting and guiding the search of the sparsity pattern and facilitating model interpretation. Combining the ideas of penalized regression and Bregman iterative methods, we propose two methods that strive to achieve the dual goal of sparse and orthogonal SVD estimation, in the general framework of high dimensional multivariate regression. We set up simulation studies to demonstrate the e cacy of the proposed methods.

Dynamics of a beam-based micro-electromechanical resonator is investigated theoretically and experimentally. The resonant structure comprises a micro-beam and two electrostatic comb-drives, one for exciting the vibration, and the other for detecting the response. Two identical resonators of this type can form a double-ended tuning fork. An analytical linear model of these resonators is developed by assuming the beam to obey the thin beam theory subjected to an axial force. The comb-drives are initially treated as a point mass. The point mass is free to be placed anywhere along the beam span. The exact natural frequencies and mode shapes of vibration are obtained. Further, the mass is considered to have rotary inertia. The influence of the rotary inertia on the natural frequencies and mode shapes of vibration are investigated. Subsequently, the model of a beam with a guided mass is studied to determine the upper limit of the natural frequencies of the resonator. The advantage of this model over the previous ones is in providing detailed insight into the dynamics of the resonator, particularly when the comb-drives are placed at locations other than the mid-point of the beam. It has been shown that the mode shapes of vibration of these resonators are not orthogonal to each other under its classic definition. The orthogonality condition of the mode shapes of the beam-lumped mass system is introduced, and used for studying the forced vibration response. The nonlinear vibration of the system due to stretching is considered for the case of free vibration and the primary resonance. The nonlinear model is used to investigate the effect of damping on the resonator response. The interaction of the electrostatic governing equations and the mechanical model is studied. This model is employed for designing the experiment circuits for testing fabricated resonators. The fabrication processes used are explained, and the design parameters of each resonator are provided. The experimental results are reported, and used to find the axial force and stress of the resonant beams. The model and results of this dissertation can be used in the design of beam-based micromachined resonators for different applications.

Dynamics of a beam-based micro-electromechanical resonator is investigated theoretically and experimentally. The resonant structure comprises a micro-beam and two electrostatic comb-drives, one for exciting the vibration, and the other for detecting the response. Two identical resonators of this type can form a double-ended tuning fork. An analytical linear model of these resonators is developed by assuming the beam to obey the thin beam theory subjected to an axial force. The comb-drives are initially treated as a point mass. The point mass is free to be placed anywhere along the beam span. The exact natural frequencies and mode shapes of vibration are obtained. Further, the mass is considered to have rotary inertia. The influence of the rotary inertia on the natural frequencies and mode shapes of vibration are investigated. Subsequently, the model of a beam with a guided mass is studied to determine the upper limit of the natural frequencies of the resonator. The advantage of this model over the previous ones is in providing detailed insight into the dynamics of the resonator, particularly when the comb-drives are placed at locations other than the mid-point of the beam. It has been shown that the mode shapes of vibration of these resonators are not orthogonal to each other under its classic definition. The orthogonality condition of the mode shapes of the beam-lumped mass system is introduced, and used for studying the forced vibration response. The nonlinear vibration of the system due to stretching is considered for the case of free vibration and the primary resonance. The nonlinear model is used to investigate the effect of damping on the resonator response. The interaction of the electrostatic governing equations and the mechanical model is studied. This model is employed for designing the experiment circuits for testing fabricated resonators. The fabrication processes used are explained, and the design parameters of each resonator are provided. The experimental results are reported, and used to find the axial force and stress of the resonant beams. The model and results of this dissertation can be used in the design of beam-based micromachined resonators for different applications.