Although relativistic physics tend to omit nondynamical "absolute objects" such as a flat metric tensor or a preferred time foliation, there exist interesting questions related to such entities, such as worries about the "flow" of time in special relativity, and the apparent disappearance of time altogether in canonical general relativity. This latter problem is related to the lack of a fixed causal structure with repect to which one might posit "equal-time" commutation relations, for example. In view of these issues, we consider whether including a flat background metric, and perhaps a preferred foliation, is physically worthwhile. We show how a derivation of Einstein's equations from flat spacetime can be generalized to include a preferred foliation, the possible significance of which we discuss, though ultimately we suggest why such a foliation might be present in metaphysics and yet absent from physics. We also derive a new "slightly bimetric" class of theories using the flat spacetime approach. However, such derivations are only formally special relativistic, because they give no heed to the flat metric's causal structure, which the curved effective metric might well violate. After reviewing the history of this problem, we introduce new variables to give a kinematic description of the relation between the two null cones. Then we propose a method to enforce special relativistic causality by using the guage freedom to restrict the configuration space suitably. Consequences for exact solutions, such as the Schwarzschild solution and its 'singularity,' are discussed. Advantages and difficulties regarding adding a mass term to the theory are discussed briefly. / text

Quantum chemistry is plagued by the presence of high-rank quantities, stemming from the N-body nature of the electronic Schrödinger equation. These high-rank quantities present a significant mathematical and computational barrier to the computation of chemical observables, and also drastically complicate the pedagogical understanding of important interactions between particles in a molecular system. The application of physically-motivated rank reduction approaches can help address these to problems. This thesis details recent efforts to apply rank reduction techniques in both of these arenas. With regards to computational tractability, the representation of the 1/r Coulomb repulsion between electrons is a critical stage in the solution of the electronic Schrödinger equation. Typically, this interaction is encapsulated via the order-4 electron repulsion integral (ERI) tensor, which is a major bottleneck in terms of generation, manipulation, and storage. Many rank reduction techniques for the ERI tensor have been proposed to ameliorate this bottleneck, most notably including the order-3 density fitting (DF) and pseudospectral (PS) representations. Here we detail a new and uniquely powerful factorization - tensor hypercontraction (THC). THC decomposes the ERI tensor as a product of five order-2 matrices (the first wholly order-2 compression proposed for the ERI) and offers great flexibility for low-scaling algorithms for the manipulations of the ERI tensor underlying electronic structure theory. THC is shown to be physically-motivated, markedly accurate, and uniquely efficient for some of the most difficult operations encountered in modern quantum chemistry. On the front of chemical understanding of electronic structure theory, we present our recent work in developing robust two-body partitions for ab initio computations of intermolecular interactions. Noncovalent interactions are the critical and delicate forces which govern such important processes as drug-protein docking, enzyme function, crystal packing, and zeolite adsorption. These forces arise as weak residual interactions leftover after the binding of electrons and nuclei into molecule, and, as such, are extremely difficult to accurately quantify or systematically understand. Symmetry-adapted perturbation theory (SAPT) provides an excellent approach to rigorously compute the interaction energy in terms of the physically-motivated components of electrostatics, exchange, induction, and dispersion. For small intermolecular dimers, this breakdown provides great insight into the nature of noncovalent interactions. However, SAPT abstracts away considerable details about the N-body interactions between particles on the two monomers which give rise to the interaction energy components. In the work presented herein, we step back slightly and extract an effective 2-body interaction for each of the N-body SAPT terms, rather than immediately tracing all the way down to the order-0 interaction energy. This effective order-2 representation of the order-N SAPT interaction allows for the robust assignment of interaction energy contributions to pairs of atoms or functional groups (the A-SAPT or F-SAPT partitions), allowing one to discuss the interaction in terms of atom- or functional-group-pairwise interactions. These A-SAPT and F-SAPT partitions can provide deep insight into the origins of complicated noncovalent interactions, e.g., by clearly shedding light on the long-contested question of the nature of the substituent effect in substituted sandwich benzene dimers.

When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfilling this is introduced for the tensor product of three irreps of $su(2)$. This method is then extended to the tensor product of two irreps of $su(3)$. In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over $\mathbb{C}$.

In this dissertation, we investigatethe supersymmetric completion of curvature squared invariants in five and six dimensionsas well as the construction of off-shell Poincar´e supergravities and their matter couplings. We use superconformal calculus in fiveand six dimensions, which are an off- shell formalisms. In fivedimensions,there are twoinequivalentWeyl multiplets: the standard Weyl multiplet and the dilaton Weyl multiplet.The main difference betweenthese twoWeyl multiplets is thatthe dilaton Weyl multipletcontains a graviphoton in its field content whereas the standard Weyl multiplet does not.A supergravity theory based on the standard Weyl multiplet requires coupling to an external vector multiplet. In five dimensions,we construct two new formulations for 2-derivative off-shell Poincar´e supergravity theories and present the internally gauged models. We also construct supersymmetric completions of all curvature squared terms in five dimensional supergravity with eight supercharges.Adopting the dilaton Weyl multiplet, we construct a Weyl squared invariant, the supersymmetric combination of Gauss-Bonnet combination and the Ricci scalar squared invariant as well as all vector multiplets coupled curvature squared invariants. Since the minimal off-shell supersymmetric Riemann tensor squared invariant has been obtained before, both the minimal off-shell and the vector multiplets coupled curvature squared invariants in the dilation Weyl multiplet are complete. We also constructedan off-shell Ricci scalar squared invariant utilizing the standard Weyl multiplet.The supersymmetric Ricci scalar squared in the standard Weyl multiplet is coupled to n number of vector multiplets by construction, and it deforms the very special geometry. We found that in the supersymmetric AdS5 vacuum, the very special geometry defined on the moduli space is modified in a simple way. We study the vacuum solutions with AdS2 × S3 and AdS3 × S2 structures. We also analyze the spectrum around a maximally supersymmetric Minkowski5, and study the magnetic string and electric black hole. Finally, we generalize our procedure for the construction of an off-shell Ricci scalar squared invariant in five dimensions to N = (1, 0), D = 6 supergravity.

In this thesis the application of novel tensor decomposition and tensor representation techniques in highly accurate post Hartree-Fock methods is evaluated. These representation techniques can help to overcome the steep scaling behaviour of high level ab-initio calculations with increasing system size and therefore break the "curse of dimensionality". After a comparison of various tensor formats the application of the "canonical polyadic" format (CP) is described in detail. There, especially the casting of a normal, index based tensor into the CP format (tensor decomposition) and a method for a low rank approximation (rank reduction) of the two-electron integrals in the AO basis are investigated. The decisive quantity for the applicability of the CP format is the scaling of the rank with increasing system and basis set size. The memory requirements and the computational effort for tensor manipulations in the CP format are only linear in the number of dimensions but still depend on the expansion length (rank) of the approximation. Furthermore, the AO-MO transformation and a MP2 algorithm with decomposed tensors in the CP format is evaluated and the scaling with increasing system and basis set size is investigated. Finally, a Coupled-Cluster algorithm based only on low-rank CP representation of the MO integrals is developed. There, especially the successive tensor contraction during the iterative solution of the amplitude equations and the error propagation upon multiple application of the reduction procedure are discussed. In conclusion the overall complexity of a Coupled-Cluster procedure with tensors in CP format is evaluated and some possibilities for improvements of the rank reduction procedure tailored to the needs in electronic structure calculations are shown. / Die vorliegende Arbeit beschäftigt sich mit der Anwendung neuartiger Tensorzerlegungs- und Tensorrepesentationstechniken in hochgenauen post Hartree-Fock Methoden um das hohe Skalierungsverhalten dieser Verfahren mit steigender Systemgröße zu verringern und somit den "Fluch der Dimensionen" zu brechen. Nach einer vergleichenden Betrachtung verschiedener Representationsformate wird auf die Anwendung des "canonical polyadic" Formates (CP) detailliert eingegangen. Dabei stehen zunächst die Umwandlung eines normalen, indexbasierten Tensors in das CP Format (Tensorzerlegung) und eine Methode der Niedrigrang Approximation (Rangreduktion) für Zweielektronenintegrale in der AO Basis im Vordergrund. Die entscheidende Größe für die Anwendbarkeit ist dabei das Skalierungsverhalten das Ranges mit steigender System- und Basissatzgröße, da der Speicheraufwand und die Berechnungskosten für Tensormanipulationen im CP Format zwar nur noch linear von der Anzahl der Dimensionen des Tensors abhängen, allerdings auch mit der Expansionslänge (Rang) skalieren. Im Anschluss wird die AO-MO Transformation und der MP2 Algorithmus mit zerlegten Tensoren im CP Format diskutiert und erneut das Skalierungsverhalten mit steigender System- und Basissatzgröße untersucht. Abschließend wird ein Coupled-Cluster Algorithmus vorgestellt, welcher ausschließlich mit Tensoren in einer Niedrigrang CP Darstellung arbeitet. Dabei wird vor allem auf die sukzessive Tensorkontraktion während der iterativen Bestimmung der Amplituden eingegangen und die Fehlerfortpanzung durch Anwendung des Rangreduktions-Algorithmus analysiert. Abschließend wird die Komplexität des gesamten Verfahrens bewertet und Verbesserungsmöglichkeiten der Reduktionsprozedur aufgezeigt.

Background: To explore normal-appearing white matter (NAWM) in pediatric-onset multiple sclerosis (MS) patients, using diffusion tensor imaging (DTI). DTI study provides measures of WM integrity in adult MS patients. Pediatric MS patients provide a uniquely early window for exploring pathological components of myelin disruption. Methods: DTI data were obtained for 23 pediatric MS patients and 17 healthy children. Images were acquired using GE LX1.5T scanner (DTI parameters: 25 directions, 5mm slice thickness, b=1000s/mm2). Fractional anisotropy (FA) and apparent diffusion co-efficient (ADC) were analyzed in lesions and NAWM throughout corpus callosum (CC) and hemispheres. Results: Altered NAWM integrity in MS patients relative to controls is demonstrated by: reduced FA values (p<0.0001) and elevated ADC values (p<0.05) throughout CC and hemispheres. Conclusions: DTI measures show widespread disruption of WM integrity in children with MS extending beyond visible lesions. These findings implicate diffuse and potentially very early WM degeneration in MS pathobiology.

Background: To explore normal-appearing white matter (NAWM) in pediatric-onset multiple sclerosis (MS) patients, using diffusion tensor imaging (DTI). DTI study provides measures of WM integrity in adult MS patients. Pediatric MS patients provide a uniquely early window for exploring pathological components of myelin disruption. Methods: DTI data were obtained for 23 pediatric MS patients and 17 healthy children. Images were acquired using GE LX1.5T scanner (DTI parameters: 25 directions, 5mm slice thickness, b=1000s/mm2). Fractional anisotropy (FA) and apparent diffusion co-efficient (ADC) were analyzed in lesions and NAWM throughout corpus callosum (CC) and hemispheres. Results: Altered NAWM integrity in MS patients relative to controls is demonstrated by: reduced FA values (p<0.0001) and elevated ADC values (p<0.05) throughout CC and hemispheres. Conclusions: DTI measures show widespread disruption of WM integrity in children with MS extending beyond visible lesions. These findings implicate diffuse and potentially very early WM degeneration in MS pathobiology.

When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfilling this is introduced for the tensor product of three irreps of $su(2)$. This method is then extended to the tensor product of two irreps of $su(3)$. In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over $\mathbb{C}$.

The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its Fourier-Stieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids. Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular G-Hilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)-bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle. The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed. For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra. If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between G-Hilbert bundles and families of Gi-Hilbert bundles. Thanks to this correspondence, the Fourier-Stieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components. The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance. Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a path-connected manifold.

We study the geometry of immanants, which are polynomials on n^2 variables that are defined by irreducible representations of the symmetric group Sn. We compute stabilizers of immanants in most cases by computing Lie algebras of stabilizers of immanants. We also study tensor network states, which are special tensors defined by contractions. We answer a question about tensor network states asked by Grasedyck. Both immanants and tensor network states are related to the Geometric Complexity Theory program, in which one attempts to use representation theory and algebraic geometry to solve an algebraic analogue of the P versus N P problem. We introduce the Geometric Complexity Theory (GCT) program in Section one and we introduce the background for the study of immanants and tensor network states. We also explain the relation between the study of immanants and tensor network states and the GCT program. Mathematical preliminaries for this dissertation are in Section two, including multilinear algebra, representation theory, and complex algebraic geometry. In Section three, we first give a description of immanants as trivial (SL(E) x SL(F )) ><| delta(Sn)-modules contained in the space S^n(E X F ) of polynomials of degree n on the vector space E X F , where E and F are n dimensional complex vectorspaces equipped with fixed bases and the action of Sn on E (resp. F ) is induced by permuting elements in the basis of E (resp. F ). Then we prove that the stabilizer of an immanant for any non-symmetric partition is T (GL(E) x GL(F )) ><| delta(Sn) ><| Z2, where T (GL(E) x GL(F )) is the group of pairs of n x n diagonal matrices with the product of determinants equal to 1, delta(Sn) is the diagonal subgroup of Sn x Sn. We also prove that the identity component of the stabilizer of any immanant is T (GL(E) x GL(F )). In Section four, we prove that the set of tensor network states associated to a triangle is not Zariski closed and we give two reductions of tensor network states from complicated cases to simple cases. In Section five, we calculate the dimension of the tangent space and weight zero subspace of the second osculating space of GL_(n^2) .[perm_n] at the point [perm_n] and determine the Sn x Sn-module structure of this space. We also determine some lines on the hyper-surface determined by the permanent polynomial. In Section six, we give a summary of this dissertation.