We consider two one-dimensional harmonic chains coupled along their length via linear springs. Casting the elastic wave equation for this system in a Dirac-like form reveals a directional representation. The elastic band structure, in a spectral representation, is constituted of two branches corresponding to symmetric and antisymmetric modes. In the directional representation, the antisymmetric states of the elastic waves possess a plane wave orbital part and a 4x1 spinor part. Two of the components of the spinor part of the wave function relate to the amplitude of the forward component of waves propagating in both chains. The other two components relate to the amplitude of the backward component of waves. The 4x1 spinorial state of the two coupled chains is supported by the tensor product Hilbert space of two identical subsystems composed of a non-interacting chain with linear springs coupled to a rigid substrate. The 4x1 spinor of the coupled system is shown to be in general not separable into the tensor product of the two 2x1 spinors of the uncoupled subsystems in the directional representation. (C) 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Glioblastoma is a highly malignant tumor which mostly recurs locally around the resected contrast enhancement. However, it is difficult to identify tumor invasiveness pre-surgically, especially in non-enhancing areas. Thus, the aim of this thesis was to utilize multimodal MR technique to identify and characterize the peritumoral progression zone that eventually leads to tumor progression. Patients with newly diagnosed cerebral glioblastoma were included consecutively from our cohort between 2010 and2014. The presurgical MRI sequences included volumetric T1-weighted with contrast, FLAIR, T2-weighted, diffusion-weighted imaging, diffusion tensor and perfusion MR imaging. Postsurgical and follow-up MRI included structural and ADC images. Image deformation, caused by disease nature and surgical procedure, renders routine coregistration methods inadequate for MRIs comparison between different time points. Therefore, a two-staged non-linear semi-automatic coregistration method was developed from the modification of the linear FLIRT and non-linear FNIRT functions in FMRIB’s Software Library (FSL). Utilising the above mentioned coregistration method, a volumetric study was conducted to analyse the extent of resection based on different MR techniques, including T1 weighted with contrast, FLAIR and DTI measures of isotropy (DTI-p) and anisotropy (DTI-q). The results showed that patients can have a better clinical outcome with a larger resection of the abnormal DTI q areas. Further study of the imaging characteristics of abnormal peritumoural DTI-q areas, using MRS and DCS-MRI, showed a higher Choline/NAA ratio (p = 0.035), especially higher Choline (p = 0.022), in these areas when compared to normal DTI-q areas. This was indicative of tumour activity in the peritumoural abnormal DTI-q areas. The peritumoural progression areas were found to have distinct imaging characteristics. In these progression areas, compared to non-progression areas within a 10 mm border around the contrast enhancing lesion, there was higher signal intensity in FLAIR (p = 0.02), and T1C (p < 0.001), and there were lower intensity in ADC (p = 0.029) and DTI-p (p < 0.001). Further applying radiomics features showed that 35 first order features and 77 second order features were significantly different between progression and non-progression areas. By using supervised convolutional neural network, there was an overall accuracy of 92.4% in the training set (n = 37) and 78.5% in the validation set (n=14). In summary, multimodal MR imaging, particularly diffusion tensor imaging, can demonstrate distinct characteristics in areas of potential progression on preoperative MRI, which can be considered potential targets for treatment. Further application of radiomics and machine learning can be potentially useful when identifying the tumor invasive margin before the surgery.

Made available in DSpace on 2015-04-22T22:16:09Z (GMT). No. of bitstreams: 1 Dissertacao Final Amauri.pdf: 2958074 bytes, checksum: de9a6605623a316acb109cd1a0c50ad2 (MD5) Previous issue date: 2009-02-20 / FAPEAM - Fundação de Amparo à Pesquisa do Estado do Amazonas / Urich Köthe proposes improvements in the calculation of the structure tensor, that will be presented in this work. First, the tax of sampling is altered, therefore the standard method for calculation of that tensor violates the Shannon s sampling theorem; at second, using a non-linear filter to calculate the entrances of the structure tensor, the integration of information coming from out of the edge is minimized. / Ullrich Köthe propõe melhorias no cálculo do tensor de estrutura, que serão apresentadas neste trabalho. Primeira, a taxa de amostragem é alterada, pois o método padrão para cálculo desse tensor viola o teorema de amostragem de Shannon-Whittaker; segunda, usando um filtro não-linear para calcular as entradas do tensor de estrutura, a integração de informação oriunda de fora da aresta é minimizada.

Dans cette thèse, nous explorons de nouvelles méthodes d’analyse d’images pour la détection précoce des changements métaboliques cérébraux causés par la maladie d’Alzheimer. Nous introduisons deux apports méthodologiques que nous appliquons à un ensemble de données réelles. Le premier est basé sur l’apprentissage automatique afin de créer une carte des informations pertinentes pour la classification d'un ensemble d’images. Pour cela nous échantillonnons des blocs de Voxels selon un algorithme de Monte-Carlo. La mise en œuvre d’une classification basée sur ces patchs 3d a pour conséquence la réduction significative du volume de patchs à traiter et l’extraction de caractéristiques dont l’importance est statistiquement quantifiable. Cette méthode s’applique à différentes caractéristiques et est adaptée à des types d’images variés. La résolution des cartes produites par cette méthode peut être affinée à volonté et leur contenu informatif est cohérent avec des résultats antérieurs obtenus dans la littérature. Le second apport méthodologique porte sur la conception d’un nouvel algorithme de décomposition de tenseur d’ordre important, adapté à notre application. Cet algorithme permet de réduire considérablement la consommation de mémoire et donc en évite la surcharge. Il autorise la décomposition rapide de tenseurs, y compris ceux de dimensions très déséquilibrées. Nous appliquons cet algorithme en tant que méthode d’extraction de caractéristiques dans une situation où le clinicien doit diagnostiquer des stades précoces de la maladie d'Alzheimer en utilisant la TEP-FDG seule. Les taux de classification obtenus sont souvent au-dessus des niveaux de l’état de l’art. / In this thesis we want to explore novel image analysis methods for the early detection of metabolic changes in the human brain caused by Alzheimer's disease (AD). We will present two methodological contributions and present their application to a real life data set. We present a machine learning based method to create a map of local distribution of classification relevant information in an image set. The presented method can be applied using different image characteristics which makes it possible to adapt the method to many kinds of images. The maps generated by this method are very localized and fully consistent with prior findings based on Voxel wise statistics. Further we preset an algorithm to draw a sample of patches according to a distribution presented by means of a map. Implementing a patch based classification procedure using the presented algorithm for data reduction we were able to significantly reduce the amount of patches that has to be analyzed in order to obtain good classification results. We present a novel non-negative tensor factorization (NTF) algorithm for the decomposition of large higher order tensors. This algorithm considerably reduces memory consumption and avoids memory overhead. This allows the fast decomposition even of tensors with very unbalanced dimensions. We apply this algorithm as feature extraction method in a computer-aided diagnosis (CAD) scheme, designed to recognize early-stage ad and mild cognitive impairment (MCI) using fluorodeoxyglucose (FDG) positron emission tomography (PET) scans only. We achieve state of the art classification rates.

This dissertation explores aspects of the representation theory for tensor algebras, which are non-selfadjoint operator algebras Muhly and Solel introduced in 1998, by developing a cohomology theory for completely bounded Hilbert modules. Similar theories have been developed for Banach modules by Johnson in 1970, for operator modules by Paulsen in 1997, and for Hilbert modules over the disc algebra by Carlson and Clark in 1995. The framework presented here was motivated by a desire to further understand the completely bounded representation theory for tensor algebras on Hilbert spaces. The focal point of this thesis is the first Ext group, Ext1, which is defined as equivalence classes of short exact sequences of completely bounded Hilbert modules. Alternate descriptions of this group are presented. For general operator algebras, Ext1 can be realized as the collection completely bounded derivations equivalent up to an inner derivation. When the operator algebra is a tensor algebra, Ext1 can be described as a quotient space of intertwining operators, a description analogous to a result of Ferguson in 1996 in the case of the classical disc algebra. A theorem of Sz.-Nagy and Foias from 1967, concerning contractions in triangular form, is applied to analyze derivations that are off-diagonal corners of completely contractive representations. It is proved that, in some cases, this analysis determines when all derivations must be inner or suggests ways to construct non-inner derivations. In the third chapter, a characterization is given of completely bounded representations of a tensor algebra in terms of similarities of contractive intertwiners. Also proven is that for a Csup*;-correspondence X over a Csup*;-algebra A and the Toeplitz algebra T(X), Mn(T(X))= T(Mn(X)). The analogous statement for tensor algebras is deduced as a corollary. In the final chapter, a brief survey of non-abelian category theory is provided. Extensions of completely bounded Hilbert modules over operator algebras are defined. Theorems asserting the projectivity of isometric modules and injectivity of coisometric modules by Carlson, Clark, Foias, and Williams in 1995 are generalized to the noncommutative setting of tensor algebras using commutant lifting. A result of Popesecu in 1996 for noncommutative disc algebras is also covered in the general framework of this thesis.

Characterizing a manifold up to isometry is a challenging task. A manifold is a topological space. One may equip a manifold with a metric, and generally speaking, this metric determines how the manifold “looks". An example of this would be the unit sphere in R3. While we typically envision the standard metric on this sphere to give it its familiar shape, one could define a different metric on this set of points, distorting distances within this set to make it seem perhaps more ellipsoidal, something not isometric to the standard round sphere. In an effort to distinguish manifolds up to isometry, we wish to compute meaningful invariants. For example, the Riemann curvature tensor and its surrogates are examples of invariants one could construct. Since these objects are generally too complicated to compare and are not real valued, we construct scalar invariants from these objects instead. This thesis will explore these invariants and exhibit a special family of manifolds that are not flat on which all of these invariants vanish. We will go on to properly define, and gives examples of, manifolds, metrics, tangent vector fields, and connections. We will show how to compute the Christoffel symbols that define the Levi-Civita connection, how to compute curvature, and how to raise and lower indices so that we can produce scalar invariants. In order to construct the curvature operator and curvature tensor, we use the miracle of pseudo-Riemannian geometry, i.e., the Levi-Civita connection, the unique torsion free and metric compatible connection on a manifold. Finally, we examine Generalized Plane Wave Manifolds, and show that all scalar invariants of Weyl type on these manifolds vanish, despite the fact that many of these manifolds are not flat.

The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.

Thesis (Ph. D. in Physics)--Vanderbilt University, Dec. 2007. / Title from title screen. Includes bibliographical references.

<p>The pumping performance of the heart is affected by the mechanical properties of the muscle fibre part of the cardiac wall, the myocardium. The myocardium has a complex structure, where muscle fibres have different orientations at different locations, and during the cardiac cycle, the myocardium undergoes large elastic deformations. Hence, myocardial strain pattern is complex. In this thesis work, a computation method for myocardial strain and a detailed map of myocardial transmural strain during the cardiac cycle are found by the use of surgically implanted metallic markers and beads. The strain is characterized in a local cardiac coordinate system. Thereafter, non-invasive phase contrast magnetic resonance imaging (PC-MRI) is used to compare strain at different myocardial regions. The difference in resolution between marker data and PC-MRI data is elucidated and some of the problems associated with the low resolution of PC-MRI are given.</p>

<p>The heart is a complex three-dimensional structure with mechanical properties that are inhomogeneous, non-linear, time-variant and anisotropic. These properties affect major physiological factors within the heart, such as the pumping performance of the ventricles, the oxygen demand in the tissue and the distribution of coronary blood flow.</p><p>During the cardiac cycle the heart muscle tissue is deformed as a consequence of the active contraction of the muscle fibers and their relaxation respectively. A mapping of this deformation would give increased understanding of the mechanical properties of the heart. The deformation induces strain and stress in the tissue which are both mechanical properties and can be described with a mathematical tensor object.</p><p>The aim of this master's thesis is to develop a visualization tool for the strain tensor objects that can aid a user to see and/or understand various differences between different hearts and spatial and temporal differences within the same heart. Preferably should the tool be general enough for use with different types of data.</p>