The northward movement and collision of the Arabian plate with Eurasia generates compressive stresses and resulting shortening in Iran. Within the Alborz Mountains, North Iran, a complex and not well understood system of strike-slip and thrust faults accomodates a fundamental part of the NNE-SSW oriented shortening. On 28th of May 2004 the Mw 6.3 Baladeh earthquake hit the north-central Alborz Mountains. It is one of the rare and large events in this region in modern time and thus a seldom chance to study earthquake mechanisms and the local ongoing deformation processes. It also demonstrated the high vulnerability of this densily populated region.

This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.

The standard Kaluza-Klein theory is reviewed and its basic equations are rewritten in an anholonomic basis. A five dimensional Yang-Mills type quadratic and cubic curvature gravity model is introduced. By employing the Palatini variational principle, the field equations and the stress-energy tensors of these models are presented. Unification of gravity with electromagnetism is achieved through the Kaluza-Klein reduction mechanism. Reduced curvature invariants,field equations and stress-energy tensors in four dimensional space-time are obtained. The structure of interactions among the gravitational, electromagnetic and massless scalar fields are demonstrated in detail. It is shown that in addition to a set of generalized Maxwell and Yang-Mills type gravity equations the Lorentz force also emerges from this theory. Solutions of the standard Kaluza-Klein theory are explicitly demonstrated to be intrinsically contained in the quadratic model.

Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. The problem of classification of superintegrable systems can be approached by considering associated geometric structures. To this end, we invoke the invariant theory of Killing tensors (ITKT), and the recursive version of the Cartan method of moving frames to derive joint invariants. We are able to intrinsically characterise and interpret the arbitrary parameters appearing in the general form of the Smorodinsky-Winternitz superintegrable potential, where we determine that the more general the geometric structure associated with the SW potential is, the fewer arbitrary parameters it admits. Additionally, we classify the multi-separability of the Tremblay-Turbiner-Winternitz (TTW) system. We provide a proof that only for the case k = +/- 1 does the general TTW system admit orthogonal separation of variables with respect to both Cartesian and polar coordinates. / A study towards the classification of superintegrable systems defined on the Euclidean plane.

In this thesis we study the physical properties of the manifold in general relativity that admits a conformal motion. The results obtained are general as the metric tensor field is not specified. We obtain the Lie derivative along a conformal Killing vector of the kinematical and dynamical quantities for the general energy-momentum tensor of neutral matter. Equations obtained previously are regained as special cases from our results. We also find the Lie derivative of the energy-momentum tensor for the electromagnetic field. In particular we comprehensively study conformal symmetries in the Bianchi I spacetime. The conformal Killing vector equation is integrated to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. These conditions place restrictions on the metric functions. A particular solution is exhibited which demonstrates that these conditions have a nonempty solution set. The solution obtained is a generalisation of the results of Moodley (1991) who considered locally rotationally symmetric spacetimes. The Killing vectors are regained as special cases of the conformal solution. There do not exist any proper special conformal Killing vectors in the Bianchi I spacetime. The homothetic vector is found for a nonvanishing constant conformal factor. We establish that the vacuum Kasner solution is the only Bianchi I spacetime that admits a homothetic vector. Furthermore we isolate a class of vectors from the solution which causes the Bianchi I model to degenerate into a spacetime of higher symmetry. / Thesis (M.Sc.)-University of KwaZulu-Natal, 1992.

In this dissertation we study conformal symmetries in the Stephani universe which is a generalisation of the Robertson-Walker models. The kinematics and dynamics of the Stephani universe are discussed. The conformal Killing vector equation for the Stephani metric is integrated to obtain the general solution subject to integrability conditions that restrict the metric functions. Explicit forms are obtained for the conformal Killing vector as well as the conformal factor . There are three categories of solution. The solution may be categorized in terms of the metric functions k and R. As the case kR - kR = 0 is the most complicated, we provide all the details of the integration procedure. We write the solution in compact vector notation. As the case k = 0 is simple, we only state the solution without any details. In this case we exhibit a conformal Killing vector normal to hypersurfaces t = constant which is an analogue of a vector in the k = 0 Robertson-Walker spacetimes. The above two cases contain the conformal Killing vectors of Robertson-Walker spacetimes. For the last case in - kR = 0, k =I 0 we provide an outline of the integration process. This case gives conformal Killing vectors which do not reduce to those of RobertsonWalker spacetimes. A number of the calculations performed in finding the solution of the conformal Killing vector equation are extremely difficult to analyse by hand. We therefore utilise the symbolic manipulation capabilities of Mathematica (Ver 2.0) (Wolfram 1991) to assist with calculations. / Thesis (M.Sc.)-University of Natal, Durban, 1992.

Material properties of polycrystalline aggregates should manifest the influence of crystallographic texture as defined by the orientation distribution function (ODF). A representation theorem on material tensors of weakly-textured polycrystals was established by Man and Huang (2012), by which a given material tensor can be expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a number of undetermined material parameters. Man and Huang's theorem is based on the classical assumption in texture analysis that ODFs are defined on the rotation group SO(3), which strictly speaking makes it applicable only to polycrystals with (single) crystal symmetry defined by a proper point group. In the present study we consider ODFs defined on the orthogonal group O(3) and extend the representation theorem of Man and Huang to cover pseudotensors and polycrystals with crystal symmetry defined by any improper point group. This extension is important because many materials, including common metals such as aluminum, copper, iron, have their group of crystal symmetry being an improper point group. We present the restrictions on texture coefficients imposed by crystal symmetry for all the 21 improper point groups and we illustrate the extended representation theorem by its application to elasticity.

Principal Components Analysis is a standard tool in data analysis, widely used in data-rich fields such as computer vision, data mining, bioinformatics, and econometrics. For a set of vectors in n dimensions and a natural number k less than n, the method returns a subspace of dimension k whose average squared distance to that set is as small as possible. Besides saving computation by reducing the dimension, projecting to this subspace can often reveal structure that was hidden in high dimension. This thesis considers several novel extensions of PCA, which provably reveals hidden structure where standard PCA fails to do so. First, we consider Robust PCA, which prevents a few points, possibly corrupted by an adversary, from having a large effect on the analysis. When applied to learning noisy logconcave mixture models, the algorithm requires only slightly more separation between component means than is required for the noiseless case. Second, we consider Isotropic PCA, which can go beyond the first two moments in identifying ``interesting' directions in data. The method leads to the first affine-invariant algorithm that can provably learn mixtures of Gaussians in high dimensions, improving significantly on known results. Thirdly, we define the ``Subgraph Parity Tensor' of order r of a graph and reduce the problem of finding planted cliques in random graphs to the problem of finding the top principal component of this tensor.

The principal contribution of this dissertation is a theory of Strongly Orthotropic Continuum Mechanics that is derived entirely from an assertion of geometric strain indeterminacy. Implementable into the finite element method, it can resolve widespread kinematic misrepresentations and offer unique and purportedly exact strain-induced energies by removing the assumptions of strain tensor symmetry. This continuum theory births the proposal of a new class of physical tensors described as the Intrinsic Field Tensors capable of generalising the response of most classical mechanical metrics, a number of specialised formulations and the solutions shown to be kinematically intermediate. A series of numerical examples demonstrate Euclidean objectivity, material frame-indifference, patch test satisfaction, and agreement between the subsequent Material Principal Co-rotation and P??I??C decomposition methods that produce the intermediary stress/strain fields. The encompassing theory has wide applicability owing to its fundamental divergence from conventional mechanics, it offers non-trivial outcomes when applied to even very simple problems and its use of not the Eulerian, Lagrangian but the Intrinsic Frame generates previously unreported results in strongly orthotropic continua.

Thesis (M.S.) -- Mississippi State University. Department of Computer Science and Engineering. / Title from title screen. Includes bibliographical references.