This thesis concerns inference problems in balanced random effects models with a so-called block circular Toeplitz covariance structure. This class of covariance structures describes the dependency of some specific multivariate two-level data when both compound symmetry and circular symmetry appear simultaneously. We derive two covariance structures under two different invariance restrictions. The obtained covariance structures reflect both circularity and exchangeability present in the data. In particular, estimation in the balanced random effects with block circular covariance matrices is considered. The spectral properties of such patterned covariance matrices are provided. Maximum likelihood estimation is performed through the spectral decomposition of the patterned covariance matrices. Existence of the explicit maximum likelihood estimators is discussed and sufficient conditions for obtaining explicit and unique estimators for the variance-covariance components are derived. Different restricted models are discussed and the corresponding maximum likelihood estimators are presented. This thesis also deals with hypothesis testing of block covariance structures, especially block circular Toeplitz covariance matrices. We consider both so-called external tests and internal tests. In the external tests, various hypotheses about testing block covariance structures, as well as mean structures, are considered, and the internal tests are concerned with testing specific covariance parameters given the block circular Toeplitz structure. Likelihood ratio tests are constructed, and the null distributions of the corresponding test statistics are derived.

In their paper [2] Cuntz, Deninger, and Laca introduced a C*-algebra \mathfrak{T}[R] associated to a number ring R and showed that it was functorial for injective ring homomorphisms and had an interesting KMS-state structure, which they computed directly. Although isomorphic to the Toeplitz algebra of the ax+b-semigroup R⋊R^× of R, their C*-algebra \mathfrak{T}[R] was defined in terms of relations on a generating set of isometries and projections. They showed that a homomorphism φ:\mathfrak{T}[R]→ A is injective if and only if φ is injective on a certain commutative *-subalgebra of \mathfrak{T}[R]. In this thesis we give a direct proof of this result, and go on to show that there is a countable collection of projections which detects injectivity, which allows us to simplify their characterization of faithful representations of \mathfrak{T}[R]. / Graduate / 0405 / jaspar.wiart@gmail.com

Un concepto fundamental en el análisis funcional y sus aplicaciones en física-matemática, es la existencia de conjuntos completos de vectores ortonormales en un espacio de Hilbert. Existen también conjuntos sobrecompletos, que pierden la propiedad de ortogonalidad pero conservan la resolución de la identidad. En particular, los denominados sistemas de estados coherentes son ubicuos en Mecánica Cuántica. Los estados coherentes han sido considerados en el marco de la Mecánica Cuántica por Schrödinger y von Neumann, pero fue mucho más tarde que comenzó el desarrollo sistemático de sus definiciones y del análisis funcional sobre tales bases sobre-completas ([Berezin 1971], [Berezin 1974], [Glauber 1963]). Esta serie de trabajos dio lugar al sistema estándar de estados coherentes en el plano complejo, asociado al grupo de Heisenberg-Weyl como grupo de simetría del espacio de fases de la dinámica clásica de partículas libres. La variable compleja, como parámetro del sistema de estados coherentes, permite describir los vectores del espacio de Hilbert como funciones analíticas enteras en el espacio de Segal-Bargmann. Es precisamente la analiticidad la que permite describir operadores mediante símbolos y permite desarrollar el cálculo simbólico. Posteriores generalizaciones [Perelomov 1986] permitieron de finir y utilizar estados coherentes en variedades más elaboradas, localmente isomorfas a C<SUP>n</SUP>. Diversos conceptos de análisis funcional se generalizan casi trivialmente a estas variedades [Bates 1997, Hurt 1983, Simms 1976] siguiendo el formalismo de estados coherentes del plano. Este formidable aparato analítico inspiró la introducción de estados coherentes en la descripción de sistemas fermiónicos en Mecánica Cuántica. En este caso las variables que parametrizan el sistema de estados coherentes no son puntos en una variedad compleja sino las llamadas variables de Grassmann, elementos nilpotentes, con la propiedad de conjugación pero con un producto anticonmutativo. El eje de esta tesis es la revisión, extensión de propiedades y utilización de estados coherentes en distintos contextos de interés en Mecánica Cuántica y Análisis Funcional.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) Let (G,G+) be a quasi-lattice-ordered group with positive cone G+ Laca and Raeburn have shown that the universal C*-algebra C*(G,G+)introduced by Nica is a crossed product BG+ Xɑ G+ by a semigroup of endomorphisms. Subsequent research centered on totally ordered abelian groups. We generalize the results in [2], [3] and [5] to extend it to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly a-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) XɑG+ and B(G/H)+ XβG+. This leads to one of our main results that B(G/H)+ XβG+ is realized as an induced C*-algebra IndG-H (B(G/H+ Xt(G/H)+). Then we use this result to show the existence of the following short exact sequence of C*-algebras 0-IH+ XɑG+ → BG+ XɑG+ → IndG-H (B(G/H+ Xt(G/H)+) → 0. This leads to show that the ideal IH+ XɑG+ is generated by {iBG+(1-1u):u∊H+} and therefore contained in the commutator ideal CG of the C*-algebra BG+ XɑG+. Moreover, we use our short exact sequence to study the primitive ideals of the C* algebra BG+ XɑG+ which is isomorphic to the Toeplitz albebra T(G) of G.

Research Doctorate - Doctor of Philosophy (PhD) / (**Note: this abstract is a plain text version of the author's abstract, the original of which contains characters and symbols which cannot be accurately represented in this format. The properly formatted abstract can be viewed in the Abstract and Thesis files above.**) Let (G,G+) be a quasi-lattice-ordered group with positive cone G+ Laca and Raeburn have shown that the universal C*-algebra C*(G,G+)introduced by Nica is a crossed product BG+ Xɑ G+ by a semigroup of endomorphisms. Subsequent research centered on totally ordered abelian groups. We generalize the results in [2], [3] and [5] to extend it to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H+ of G+ we introduce a closed ideal IH+ of the C*-algebra BG+. We construct an approximate identity for this ideal and show that IH+ is extendibly a-invariant. It follows that there is an isomorphism between C*-crossed products (BG+/IH+) XɑG+ and B(G/H)+ XβG+. This leads to one of our main results that B(G/H)+ XβG+ is realized as an induced C*-algebra IndG-H (B(G/H+ Xt(G/H)+). Then we use this result to show the existence of the following short exact sequence of C*-algebras 0-IH+ XɑG+ → BG+ XɑG+ → IndG-H (B(G/H+ Xt(G/H)+) → 0. This leads to show that the ideal IH+ XɑG+ is generated by {iBG+(1-1u):u∊H+} and therefore contained in the commutator ideal CG of the C*-algebra BG+ XɑG+. Moreover, we use our short exact sequence to study the primitive ideals of the C* algebra BG+ XɑG+ which is isomorphic to the Toeplitz albebra T(G) of G.

abstract: Both two-way relays (TWR) and full-duplex (FD) radios are spectrally efficient, and their integration shows great potential to further improve the spectral efficiency, which offers a solution to the fifth generation wireless systems. High quality channel state information (CSI) are the key components for the implementation and the performance of the FD TWR system, making channel estimation in FD TWRs crucial. The impact of channel estimation on spectral efficiency in half-duplex multiple-input-multiple-output (MIMO) TWR systems is investigated. The trade-off between training and data energy is proposed. In the case that two sources are symmetric in power and number of antennas, a closed-form for the optimal ratio of data energy to total energy is derived. It can be shown that the achievable rate is a monotonically increasing function of the data length. The asymmetric case is discussed as well. Efficient and accurate training schemes for FD TWRs are essential for profiting from the inherent spectrally efficient structures of both FD and TWRs. A novel one-block training scheme with a maximum likelihood (ML) estimator is proposed to estimate the channels between the nodes and the residual self-interference (RSI) channel simultaneously. Baseline training schemes are also considered to compare with the one-block scheme. The Cramer-Rao bounds (CRBs) of the training schemes are derived and analyzed by using the asymptotic properties of Toeplitz matrices. The benefit of estimating the RSI channel is shown analytically in terms of Fisher information. To obtain fundamental and analytic results of how the RSI affects the spectral efficiency, one-way FD relay systems are studied. Optimal training design and ML channel estimation are proposed to estimate the RSI channel. The CRBs are derived and analyzed in closed-form so that the optimal training sequence can be found via minimizing the CRB. Extensions of the training scheme to frequency-selective channels and multiple relays are also presented. Simultaneously sensing and transmission in an FD cognitive radio system with MIMO is considered. The trade-off between the transmission rate and the detection accuracy is characterized by the sum-rate of the primary and the secondary users. Different beamforming and combining schemes are proposed and compared. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2018

In this work, prestressed elasticity problem as a model of the so-called glacial isostatic adjustment (GIA) process is studied. The model problem is described by a set of partial differential equations (PDE) and discretized with a mixed finite element (FE) formulation. In the presence of prestress the so-constructed system of equations is non-symmetric and indefinite. Moreover, the resulting system of equations is of the saddle point form. We focus on a robust and efficient block lower-triangular preconditioning method, where the lower diagonal block is and approximation of the so-called Schur complement. The Schur complement is approximated by the so-called element-wise Schur complement. The element-wise Schur complement is constructed by assembling exact local Schur complements on the cell elements and distributing the resulting local matrices to the global preconditioner matrix. We analyse the properties of the element-wise Schur complement for the symmetric indefinite system matrix and provide proof of its quality. We show that the spectral radius of the element-wise Schur complement is bounded by the exact Schur complement and that the quality of the approximation is not affected by the domain shape. The diagonal blocks of the lower-triangular preconditioner are combined with inner iterative schemes accelerated by (numerically) optimal and robust algebraic multigrid (AMG) preconditioner. We observe that on distributed memory systems, the top pivot block of the preconditioner is not scaling satisfactorily. The implementation of the methods is further studied using a general profiling tool, designed for clusters. For nonsymmetric matrices we use the theory of Generalized Locally Toeplitz (GLT) matrices and show the spectral behavior of the element-wise Schur complement, compared to the exact Schur complement. Moreover, we use the properties of the GLT matrices to construct a more efficient AMG preconditioner. Numerical experiments show that the so-constructed methods are robust and optimal.

In this thesis, we have dealt primarily with two function algebras. The first one is the space of all holomorphic functions on the unit disc D in the complex plane which are continuous up to the boundary, denoted by A(D). The second one is H1(D), the space of all bounded analytic functions on D. We study results that characterize their maximal ideals. We start with necessary definitions and recall some useful results. In particular, the factorization of Hp functions in terms of Blaschke products, inner and outer functions is stated. Using this factorization, we provide an exposition of a beautiful result, originally by Beurling and rediscovered by Rudin, on the closed ideals of A(D). A maximality theorem by Wermer, which proves that A(D) is itself a maximal closed ideal of H1(D) is proved next. In chapter three, we expand our horizon and look at H1(D) as a dual space to characterize its weak-* closed maximal ideals. In the process we come across the shift operator and a theorem by Beurling, on the shift invariant subspaces of H2(D). We return in our quest to find out more about the maximal ideals of H1(D). The corona theorem states that the maximal ideals of the form Mτ = {ƒ ε H1(D) : ƒ (τ)=0} where τ is in D, are dense in the space of maximal ideals equipped with the Gelfand topology. We describe two approaches to the theorem, one that uses a lemma by Carleson on the existence and special properties of a contour in D. This is followed by a shorter and much more elegant proof by Wolff that uses elementary properties of Hp functions to achieve the same end. We conclude by presenting a proof of the Toeplitz corona theorem.

abstract: Dimensionality reduction methods are examined for large-scale discrete problems, specifically for the solution of three-dimensional geophysics problems: the inversion of gravity and magnetic data. The matrices for the associated forward problems have beneficial structure for each depth layer of the volume domain, under mild assumptions, which facilitates the use of the two dimensional fast Fourier transform for evaluating forward and transpose matrix operations, providing considerable savings in both computational costs and storage requirements. Application of this approach for the magnetic problem is new in the geophysics literature. Further, the approach is extended for padded volume domains. Stabilized inversion is obtained efficiently by applying novel randomization techniques within each update of the iteratively reweighted scheme. For a general rectangular linear system, a randomization technique combined with preconditioning is introduced and investigated. This is shown to provide well-conditioned inversion, stabilized through truncation. Applying this approach, while implementing matrix operations using the two dimensional fast Fourier transform, yields computationally effective inversion, in memory and cost. Validation is provided via synthetic data sets, and the approach is contrasted with the well-known LSRN algorithm when applied to these data sets. The results demonstrate a significant reduction in computational cost with the new algorithm. Further, this new algorithm produces results for inversion of real magnetic data consistent with those provided in literature. Typically, the iteratively reweighted least squares algorithm depends on a standard Tikhonov formulation. Here, this is solved using both a randomized singular value de- composition and the iterative LSQR Krylov algorithm. The results demonstrate that the new algorithm is competitive with these approaches and offers the advantage that no regularization parameter needs to be found at each outer iteration. Given its efficiency, investigating the new algorithm for the joint inversion of these data sets may be fruitful. Initial research on joint inversion using the two dimensional fast Fourier transform has recently been submitted and provides the basis for future work. Several alternative directions for dimensionality reduction are also discussed, including iteratively applying an approximate pseudo-inverse and obtaining an approximate Kronecker product decomposition via randomization for a general matrix. These are also topics for future consideration. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

The dissertation contributes to the further advancement of the theory of various classes of discrete and continuous (integral) convolution operators. The thesis is devoted to the study of sequences of matrices or operators which are built up in special ways from generalized discrete or continuous convolution operators. The generating functions depend on three variables and this leads to considerably more complicated approximation sequences. The aim was to obtain for each case a result analogous to the first Szegö limit theorem providing the first order asymptotic formula for the spectra of regular convolutions.