The results of a systematic 35Cl, 81Br, and 127I SSNMR spectroscopic study of a series of halogen-substituted anilinium halide salts are presented. Solid-state NMR of these nuclides, bromine-/81 and iodine-127 in particular, is not well established. Twenty-one compounds thought to exhibit halogen bonding were prepared based on modified literature procedures, and two crystal structures were solved. Experiments show that collection of SSNMR spectra of the anions is feasible, though ultrahigh magnetic fields (21.1 T) and variable offset data acquisition were found to be essential. Electric field gradient and chemical shift tensors are measured experimentally for all 21 compounds, significantly expanding the body of data for the quadrupolar halogen nuclei. Quadrupolar coupling constants for chlorine-35 ranged from 2.12 to 6.04 MHz, for bromine-81 from 12.3 to 45.3 MHz, and for iodine-127 from 57.50 to 152.50 MHz. Gauge-including projector-augmented wave density functional theory (GIPAW-DFT) calculations were used to provide insight as to how the NMR parameters vary with local environment and long-range crystal packing. Overall, calculations reproduced the experimental trends in quadrupolar coupling constants and chemical shift tensor span (Ω) but failed to provide quantitative agreement within experimental error. Experimental and computational data were analyzed in order to provide insight into how halogen bonding influences NMR parameters. Several trends were elucidated from this study, including an inverse correlation between Ω and the length of the shortest halogen-halide contact (d). In selected bromine compounds, for example, Ω (81Br) was measured to increase from 120 to 240 ppm as d decreased from 3.838 to 3.443 Å. In summary, this study has demonstrated the feasibility and utility of quadrupolar halogen SSNMR, and that these techniques may prove useful in characterizing halogen bonding interactions in solids.

In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.

The results of a systematic 35Cl, 81Br, and 127I SSNMR spectroscopic study of a series of halogen-substituted anilinium halide salts are presented. Solid-state NMR of these nuclides, bromine-/81 and iodine-127 in particular, is not well established. Twenty-one compounds thought to exhibit halogen bonding were prepared based on modified literature procedures, and two crystal structures were solved. Experiments show that collection of SSNMR spectra of the anions is feasible, though ultrahigh magnetic fields (21.1 T) and variable offset data acquisition were found to be essential. Electric field gradient and chemical shift tensors are measured experimentally for all 21 compounds, significantly expanding the body of data for the quadrupolar halogen nuclei. Quadrupolar coupling constants for chlorine-35 ranged from 2.12 to 6.04 MHz, for bromine-81 from 12.3 to 45.3 MHz, and for iodine-127 from 57.50 to 152.50 MHz. Gauge-including projector-augmented wave density functional theory (GIPAW-DFT) calculations were used to provide insight as to how the NMR parameters vary with local environment and long-range crystal packing. Overall, calculations reproduced the experimental trends in quadrupolar coupling constants and chemical shift tensor span (Ω) but failed to provide quantitative agreement within experimental error. Experimental and computational data were analyzed in order to provide insight into how halogen bonding influences NMR parameters. Several trends were elucidated from this study, including an inverse correlation between Ω and the length of the shortest halogen-halide contact (d). In selected bromine compounds, for example, Ω (81Br) was measured to increase from 120 to 240 ppm as d decreased from 3.838 to 3.443 Å. In summary, this study has demonstrated the feasibility and utility of quadrupolar halogen SSNMR, and that these techniques may prove useful in characterizing halogen bonding interactions in solids.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Em comunicaÃÃes cooperativas, dois ou mais terminais de transmissÃo sÃo combinados para aumentar a diversidade e/ou a potencia dos sinais que chegam a um determinado receptor. Portanto, mesmo que os dispositivos nÃo disponham de mais de uma antena, ou que entÃo haja uma grande perda por propagaÃÃo entre dois pontos comunicantes, os diversos elementos transmissores podem atuar como um arranjo virtual de antenas, obtendo-se assim vantagens dos sistemas de mÃltiplas antenas (MIMO), sobretudo o aumento da capacidade de transmissÃo. Recentemente, a chamada analise tensorial tem se mostrado uma abordagem eficiente entÃo para a estimaÃÃo de canais em sistemas com diversidade cooperativa. Contudo, nos poucos trabalhos dedicados a essa tarefa, a utilizaÃÃo da decomposiÃÃo tensorial PARAFAC para a modelagem dos sinais recebidos nÃo possibilitou o desenvolvimento de tÃcnicas de estimaÃÃo conjunta de canais e sÃmbolos. Com a idÃia de se evitar o uso de sequencias de treinamento, que limita a eficiÃncia espectral da transmissÃo por dedicar uma parte da largura de banda apenas para a tarefa de estimaÃÃo dos canais, o objetivo desta tese à prover novas estratÃgias de comunicaÃÃo, em termos de sistemas de transmissÃo e receptores semi-cegos, baseados em tensores adaptados a sistemas cooperativos MIMO unidirecionais de dois saltos. Dois sistemas de transmissÃo sÃo propostos utilizando uma codificaÃÃo espaÃo-temporal do tipo Khatri-Rao na fonte e duas estrategias de processamento Amplify-and-Forward (AF) no relay. Para estes sistemas, nomeados PT2-AF e NP-AF, os sinais recebidos no chamado nà de destino satisfazem os modelos tensoriais do tipo PARATUCK2 e Nested PARAFAC. Explorando as propriedades de unicidade destes modelos tensoriais estabelecidas nesta tese, vÃrios receptores semi-cegos sÃo derivados. Alguns destes receptores sÃo do tipo ALS, enquanto outros sÃo de soluÃÃes baseadas na factorizaÃÃo de produtos de Khatri-Rao. Resultados de simulaÃÃo sÃo apresentados para ilustrar os desempenhos dos receptores propostos em comparaÃÃo a alguns estimadores supervisionados. / In cooperative communication systems, two or more transmitting terminals are combined to increase the diversity and/or the power of the signals arriving at a particular receiver. Therefore, even if the devices do not have more than one antenna, or if a significant propaga- tion loss is present between the two communicating nodes, the various transmitting elements can act as a virtual antenna array, thus obtaining the benefits of the multiple antenna (MIMO) systems, especially the increase in the capacity. Recently, tensor decompositions have been introduced as an efficient approach for channel estimation in cooperative com- munication systems. However, among the few works devoted to this task, the utilization of the PARAFAC tensor decomposition for modeling the received signals did not allow the development of techniques for joint symbol and channel estimation. Aiming to avoid the use of pilot sequences, which limits the overall spectral efficiency by dedicating a portion of the bandwidth only for the channel estimation task, the objective of this thesis is to provide new tensor-based strategies, including transmission systems and semi-blind receivers, for one-way two-hop MIMO relaying systems. Based on a Khatri-Rao space-time coding at the source and two different Amplify-and-Forward (AF) relaying strategies, two transmission schemes are proposed. For these systems, named PT2-AF and NP-AF, the received signals at the destination node follow respectively a PARATUCK2 and a nested PARAFAC tensor model. Exploiting uniqueness properties of these tensor models which are established in the thesis, several semi-blind receivers are derived. Some of these receivers are of iterative form us- ing an ALS algorithm, whereas some other ones have closed-form solutions associated with Khatri-Rao factorizations. Some simulation results are finally presented to illustrate the per- formance of the proposed receivers which are compared to some state-of-the-art supervised techniques.

The results of a systematic 35Cl, 81Br, and 127I SSNMR spectroscopic study of a series of halogen-substituted anilinium halide salts are presented. Solid-state NMR of these nuclides, bromine-/81 and iodine-127 in particular, is not well established. Twenty-one compounds thought to exhibit halogen bonding were prepared based on modified literature procedures, and two crystal structures were solved. Experiments show that collection of SSNMR spectra of the anions is feasible, though ultrahigh magnetic fields (21.1 T) and variable offset data acquisition were found to be essential. Electric field gradient and chemical shift tensors are measured experimentally for all 21 compounds, significantly expanding the body of data for the quadrupolar halogen nuclei. Quadrupolar coupling constants for chlorine-35 ranged from 2.12 to 6.04 MHz, for bromine-81 from 12.3 to 45.3 MHz, and for iodine-127 from 57.50 to 152.50 MHz. Gauge-including projector-augmented wave density functional theory (GIPAW-DFT) calculations were used to provide insight as to how the NMR parameters vary with local environment and long-range crystal packing. Overall, calculations reproduced the experimental trends in quadrupolar coupling constants and chemical shift tensor span (Ω) but failed to provide quantitative agreement within experimental error. Experimental and computational data were analyzed in order to provide insight into how halogen bonding influences NMR parameters. Several trends were elucidated from this study, including an inverse correlation between Ω and the length of the shortest halogen-halide contact (d). In selected bromine compounds, for example, Ω (81Br) was measured to increase from 120 to 240 ppm as d decreased from 3.838 to 3.443 Å. In summary, this study has demonstrated the feasibility and utility of quadrupolar halogen SSNMR, and that these techniques may prove useful in characterizing halogen bonding interactions in solids.

The representer theorem from the reproducing kernel Hilbert space theory is the origin of many kernel-based machine learning and signal modelling techniques that are popular today. Most kernel functions used in practical applications behave in a homogeneous manner across the domain of the signal of interest, and they are called stationary kernels. One open problem in the literature is the specification of a non-stationary kernel that is computationally tractable. Some recent works solve large-scale optimization problems to obtain such kernels, and they often suffer from non-identifiability issues in their optimization problem formulation. Many practical problems can benefit from using application-specific prior knowledge on the signal of interest. For example, if one can adequately encode the prior assumption that edge contours are smooth, one does not need to learn a finite-dimensional dictionary from a database of sampled image patches that each contains a circular object in order to up-convert images that contain circular edges. In the first portion of this thesis, we present a novel method for constructing non-stationary kernels that incorporates prior knowledge. A theorem is presented that ensures the result of this construction yields a symmetric and positive-definite kernel function. This construction does not require one to solve any non-identifiable optimization problems. It does require one to manually design some portions of the kernel while deferring the specification of the remaining portions to when an observation of the signal is available. In this sense, the resultant kernel is adaptive to the data observed. We give two examples of this construction technique via the grayscale image up-conversion task where we chose to incorporate the prior assumption that edge contours are smooth. Both examples use a novel local analysis algorithm that summarizes the p-most dominant directions for a given grayscale image patch. The non-stationary properties of these two types of kernels are empirically demonstrated on the Kodak image database that is popular within the image processing research community. Tensors and tensor decomposition methods are gaining popularity in the signal processing and machine learning literature, and most of the recently proposed tensor decomposition methods are based on the tensor power and alternating least-squares algorithms, which were both originally devised over a decade ago. The algebraic approach for the canonical polyadic (CP) symmetric tensor decomposition problem is an exception. This approach exploits the bijective relationship between symmetric tensors and homogeneous polynomials. The solution of a CP symmetric tensor decomposition problem is a set of p rank-one tensors, where p is fixed. In this thesis, we refer to such a set of tensors as a rank-one decomposition with cardinality p. Existing works show that the CP symmetric tensor decomposition problem is non-unique in the general case, so there is no bijective mapping between a rank-one decomposition and a symmetric tensor. However, a proposition in this thesis shows that a particular space of rank-one decompositions, SE, is isomorphic to a space of moment matrices that are called quasi-Hankel matrices in the literature. Optimization over Riemannian manifolds is an area of optimization literature that is also gaining popularity within the signal processing and machine learning community. Under some settings, one can formulate optimization problems over differentiable manifolds where each point is an equivalence class. Such manifolds are called quotient manifolds. This type of formulation can reduce or eliminate some of the sources of non-identifiability issues for certain optimization problems. An example is the learning of a basis for a subspace by formulating the solution space as a type of quotient manifold called the Grassmann manifold, while the conventional formulation is to optimize over a space of full column rank matrices. The second portion of this thesis is about the development of a general-purpose numerical optimization framework over SE. A general-purpose numerical optimizer can solve different approximations or regularized versions of the CP decomposition problem, and they can be applied to tensor-related applications that do not use a tensor decomposition formulation. The proposed optimizer uses many concepts from the Riemannian optimization literature. We present a novel formulation of SE as an embedded differentiable submanifold of the space of real-valued matrices with full column rank, and as a quotient manifold. Riemannian manifold structures and tangent space projectors are derived as well. The CP symmetric tensor decomposition problem is used to empirically demonstrate that the proposed scheme is indeed a numerical optimization framework over SE. Future investigations will concentrate on extending the proposed optimization framework to handle decompositions that correspond to non-symmetric tensors.

Orientador: Diego Sebastian Ledesma / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T22:36:25Z (GMT). No. of bitstreams: 1 Paula_PedroManfrimMagalhaesde_M.pdf: 1130226 bytes, checksum: bbd8d375ddf7846ed2eafe024103e682 (MD5) Previous issue date: 2015 / Resumo: Este trabalho apresenta um estudo sobre variedades Riemannianas que possuem um tensor de Bakry-Émery-Ricci com limitações. Inicialmente abordamos tanto aspectos da geometria Riemanniana tradicional como métricas e geodésicas, quanto aspectos mais avançados como as fórmulas de Bochner, Weitzenböck e o teorema de Hodge. Em seguida discutimos a convergência de Gromov-Hausdorff e suas propriedades, além de serem apresentados alguns teoremas como os de Kasue e Fukaya. Por fim estudamos as propriedades topológicas e geométricas de variedades com limitação no tensor de Bakry-Émery-Ricci e o comportamento de tais limitações com respeito à submersões e à convergência de Gromov-Hausdorff / Abstract: This work presents a study about Riemannian manifolds having a Bakry-Émery-Ricci tensor with bounds. Initially we approached both the traditional aspects of Riemannian geometry like metrics and geodesics, as more advanced aspects like the Bochner, Weitzenböck formulas and the Hodge's theorem. Then we discussed the Gromov-Hausdorff convergence and its properties, in addition to showing some theorems as those from Kasue and Fukaya. Lastly we studied the topological and geometric properties of manifolds with bounds on the Bakry-Émery-Ricci tensor and the behavior of these bounds with respect to submersions and the Gromov-Hausdorff convergence / Mestrado / Matematica / Mestre em Matemática

The tensor theory is a branch of Multilinear Algebra that describes the relationship between sets of algebraic objects related to a vector space. Tensor theory together with tensor analysis is usually known to be tensor calculus. This thesis presents a formal category treatment on tensor notation, tensor calculus, and differential manifold. The focus lies mainly on acquiring and understanding the basic concepts of tensors and the operations over them. It looks at how tensor is adapted to differential geometry and continuum mechanics. In particular, it focuses more attention on the application parts of mechanics such as; configuration and deformation, tensor deformation, continuum kinematics, Gauss, and Stokes' theorem with their applications. Finally, it discusses the concept of surface forces and stress vector.

Optical flow is the apparent motion of brightness patterns within an image scene. Algorithms used to calculate the optical flow for a sequence of images are useful in a variety of applications, including motion detection and obstacle avoidance. Typical optical flow algorithms are computationally intense and run slowly when implemented in software, which is problematic since many potential applications of the algorithm require real-time calculation in order to be useful. To increase performance of the calculation, optical flow has recently been implemented on FPGA and GPU platforms. These devices are able to process optical flow in real-time, but are generally less accurate than software solutions. For this thesis, two different optical flow algorithms have been implemented to run on a GPU using NVIDIA's CUDA SDK. Previous FPGA implementations of the algorithms exist and are used to make a comparison between the FPGA and GPU devices for the optical flow calculation. The first algorithm calculates optical flow using 3D gradient tensors and is able to process 640x480 images at about 238 frames per second with an average angular error of 12.1 degrees when run on a GeForce 8800 GTX GPU. The second algorithm uses increased smoothing and a ridge regression calculation to produce a more accurate result. It reduces the average angular error by about 2.3x, but the additional computational complexity of the algorithm also reduces the frame rate by about 1.5x. Overall, the GPU outperforms the FPGA in frame rate and accuracy, but requires much more power and is not as flexible. The most significant advantage of the GPU is the reduced design time and effort needed to implement the algorithms, with the FPGA designs requiring 10x to 12x the effort.

The aim of this thesis is to investigate the tensor decomposition of structured tensors related to SL(n)-irreducible representations. Structured tensors are multilinear objects satisfying specific symmetry relations and their decompositions are of great interest in the applications. In this thesis we look for the decompositions of tensors belonging to irreducible representations of SL(n) into sum of elementary objects associated to points of SL(n)-rational hoogeneous varieties. This family includes Veronese varieties (symmetric tensors), Grassmann varieties (skew-symmetric tensors), and flag varieties. A classic tool to study the decomposition of symmetric tensors is the apolarity theory, which dates back to Sylvester. An analogous skew-symmetric apolarity theory for skew-symmetric tensors have been developed only few years ago. In this thesis we describe a global apolarity theory called Schur apolarity theory, which is suitable for tensors belonging to any irreducible representation of SL(n). Examples, properties and applications of such apolarity are studied with details and original results both in algebra and geoemtry are provided.