Principal Fiber Research Articles

Gender dimorphism of white matter integrity assessed by diffusion tensor magnetic resonance imaging in abstinent alcoholic men and women

Background Alcoholism is a debilitating disorder associated with widespread cognitive and neurological abnormalities. However, there is limited scientific literature evaluating gender-specific similarities and differences in microstructural white matter pathology associated with alcoholism. In our prior work, we used diffusion tensor magnetic resonance imaging to examine the integrity of white matter fiber tracts in the brains of abstinent alcoholic (ALC) men compared with nonalcoholic (NC) men. We found that ALC men had decreased fractional anisotropy (FA) within white matter fiber tracts connecting to frontal and limbic networks, primarily of the right hemisphere (Harris et al., 2008). In the current project, we sought to confirm our prior findings in abstinent ALC men and, additionally, examine whether different white matter abnormalities were present in abstinent ALC women. Methods As determined by our manual inspection, 60-direction high-quality diffusion tensor imaging images were acquired from 30 abstinent (at least 4 weeks) ALC participants (21 women) and 25 NC controls (17 women). Tract-based spatial statistics tools included in FSL 5.0 were used to analyze a tensor model that yielded regional FA values for each participant. To examine the effects of gender, we built a 2 X 2 ANOVA design with three planned comparisons of primary interest: ALC group-by-gender interaction, ALC women versus NC women, and ALC men versus NC men. Results We observed FA deficits in ALC men relative to NC men, with a similar effect size and variability as observed in our prior study. In contrast, ALC women displayed strikingly greater FA values compared to NC women in widespread white matter regions, including most principal long-association fiber tracts. Also, they had greater FA for local white matter architecture in the dorsolateral and ventral prefrontal regions, as well as the sublenticular extended amygdala. When controlling for multiple comparisons, the higher FA observed in ALC women remained significant. For many regions, groupby-gender interaction effects were observed. However, likely due to the small sample sizes for men, the interaction effects did not survive threshold-free cluster enhancement, the correction procedure for multiple comparisons used in these analyses. Conclusions These results suggest antithetical gender abnormalities in white matter tracts of ALC brains. Whereas abstinent ALC men displayed deficits consistent with our prior study, our new findings for abstinent ALC women demonstrated increased FA values. These distinct patterns of white matter abnormalities point toward a differential underlying neural basis for gender-specific propensity and/or sequelae to long-term alcoholism, and suggest implications for further investigation of possible gender-specific approaches to prevention and treatment.

Geometric Arbitrage and Spectral Theory

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the no-free-lunch-with vanishing-risk condition if it is only if 0 is in the spectrum.

Norm Continuous Unitary Representations of Lie Algebras of Smooth Sections

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Frechet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γ_c(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γ_c(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C^∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗A, where A is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)_0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SU_n(A)_0 with A a complete involutive commutative continuous inverse algebra.

On some flat connection associated with locally symmetric surface

Abstract For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P(M,G). Associated with - satisfying some particular conditions - local basis of TM local connection form of such a connection is an R(G)-valued 1-form build from the dual basis ω1, ω2 and from the local connection form ω of ▽. The structural equations of (M,∇) are equivalent to the condition dΩ-Ω∧Ω=0. This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.

Classical higgs fields

We consider a classical gauge theory on a principal fiber bundle P → X in the case where its structure group G is reduced to a subgroup H in the presence of classical Higgs fields described by global sections of the quotient fiber bundle P/H → X. We show that matter fields with the exact symmetry group H in such a theory are described by sections of the composition fiber bundle Y → P/H → X, where Y → P/H is the fiber bundle with the structure group H, and the Lagrangian of these sections is factored by virtue of the vertical covariant differential determined by a connection on the fiber bundle Y → P/H.

Principal strain-induced fiber orientation evolution in the Csf/Mg composites with a large deformation

In this article, the fiber orientation evolution in the Csf/Mg composites with a large deformation is investigated. The principal strain-induced fiber orientation evolution mechanism, which states that the fiber orientation evolution in the Csf/Mg composites with a large deformation is determined by the principal strains, is proposed. The fiber orientation distribution factors [Formula: see text] taking the form of the principal strains are proposed to quantitatively characterize the fiber orientation distribution in the Csf/Mg composites. The fiber orientation factors [Formula: see text] predicted by the finite element simulation based on the principal strain-induced fiber orientation evolution mechanism are compared against those [Formula: see text] measured from the micrographs of the extrusion experiments of Csf/Mg composites. The results demonstrate that the principal strain-induced fiber orientation evolution mechanism is valid and convenient to predict the fiber orientation evolution in the Csf/Mg composites with a large deformation. In the extruded Csf/Mg composites, the fibers are reoriented toward the direction of the maximum principal strain and deviated from the direction of the minimum principal strain.

On globally non-trivial almost-commutative manifolds

Within the framework of Connes’ noncommutative geometry, we define and study globally non-trivial (or topologically non-trivial) almost-commutative manifolds. In particular, we focus on those almost-commutative manifolds that lead to a description of a (classical) gauge theory on the underlying base manifold. Such an almost-commutative manifold is described in terms of a “principal module,” which we build from a principal fibre bundle and a finite spectral triple. We also define the purely algebraic notion of “gauge modules,” and show that this yields a proper subclass of the principal modules. We describe how a principal module leads to the description of a gauge theory, and we provide two basic yet illustrative examples.

Geometric phases of water waves

Recently, Banner et al. (Phys. Rev. Lett., 112 (2014) 114502) highlighted a new fundamental property of open ocean wave groups, the so-called crest slowdown. For linear narrow-band waves, this is related to the geometric and dynamical phase velocities Ud and Ug associated with the parallel transport through the principal fiber bundle of the wave dynamics with U(1) symmetry. The theoretical predictions are shown to be in fair agreement with ocean field observations, from which the average crest speed with and .

Algebraic aspects of gauge theories

Gauge theories are primary tools in modern elementary particle physics. The generally recognized mathematical foundations of these theories are in differential geometry, namely, in the theory of connections in a principal fiber bundle. We propose another approach to the mathematical description of gauge theories based on a combination of algebraic and geometric methods.

Automatic clustering and population analysis of white matter tracts using maximum density paths

We introduce a framework for population analysis of white matter tracts based on diffusion-weighted images of the brain. The framework enables extraction of fibers from high angular resolution diffusion images (HARDI); clustering of the fibers based partly on prior knowledge from an atlas; representation of the fiber bundles compactly using a path following points of highest density (maximum density path; MDP); and registration of these paths together using geodesic curve matching to find local correspondences across a population. We demonstrate our method on 4-Tesla HARDI scans from 565 young adults to compute localized statistics across 50 white matter tracts based on fractional anisotropy (FA). Experimental results show increased sensitivity in the determination of genetic influences on principal fiber tracts compared to the tract-based spatial statistics (TBSS) method. Our results show that the MDP representation reveals important parts of the white matter structure and considerably reduces the dimensionality over comparable fiber matching approaches.

Geometric uncertainty relation for mixed quantum states

In this paper we use symplectic reduction in an Uhlmann bundle to construct a principal fiber bundle over a general space of unitarily equivalent mixed quantum states. The bundle, which generalizes the Hopf bundle for pure states, gives in a canonical way rise to a Riemannian metric and a symplectic structure on the base space. With these we derive a geometric uncertainty relation for observables acting on quantum systems in mixed states. We also give a geometric proof of the classical Robertson-Schrödinger uncertainty relation, and we compare the two. They turn out not to be equivalent, because of the multiple dimensions of the gauge group for general mixed states. We give examples of observables for which the geometric relation provides a stronger estimate than that of Robertson and Schrödinger, and vice versa.

Geometry of quantum evolution for mixed quantum states

The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert–Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.

A Characterization of Invariant Connections

Given a principal fibre bundle with structure group $S$, and a fibre transitive Lie group $G$ of automorphisms thereon, Wang's theorem identifies the invariant connections with certain linear maps $\psi\colon \mathfrak{g}\rightarrow \mathfrak{s}$. In the present paper, we prove an extension of this theorem which applies to the general situation where $G$ acts non-transitively on the base manifold. We consider several special cases of the general theorem, including the result of Harnad, Shnider and Vinet which applies to the situation where $G$ admits only one orbit type. Along the way, we give applications to loop quantum gravity.

Geometry of matrix product states: Metric, parallel transport, and curvature

We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e., the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a Kähler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.

Wong’s equations in Yang-Mills theory

AbstractWong’s equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong’s equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.

Deformations of generalized holomorphic structures

A deformation theory of generalized holomorphic structures in the setting of (generalized) principal fibre bundles is developed. It allows the underlying generalized complex structure to vary together with the generalized holomorphic structure. We study the related differential graded Lie algebra, which controls the deformation problem via the Maurer–Cartan equation. As examples, we check the content of the Maurer–Cartan equation in detail in the special cases where the underlying generalized complex structure is symplectic or complex. A deformation theorem, together with some non-obstructed examples, is also included.

From SU(2) Gauge Theory to Qubits on the Fuzzy Sphere

We consider a classical pure SU(2) gauge theory, and make an ansatz, which separates the space-temporal degrees of freedom from the internal ones. This ansatz is gauge-invariant but not Lorentz invariant. In a limit case of the ansatz, obtained through a contraction map, and corresponding to a vacuum solution, the SU(2) gauge field reduces to an operator, which is the product of the generator of a global U(1) group times a Pauli matrix. We give a geometrical interpretation of the ansatz and of the contraction map in the framework of principal fiber bundles. Then, we identify the internal degrees of freedom of the gauge field with the non-commutative coordinates of the fuzzy sphere in the fundamental representation and obtain a one qubit state.

THE YANG–MILLS FUNCTIONAL AND BOGOMOLOV INEQUALITY FOR ARBITRARY PRINCIPAL BUNDLES OVER KÄHLER MANIFOLDS

We study the Yang–Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kähler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang–Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin–Kobayashi correspondence we prove that every (poly-)stable principal Kℂ-bundle has to satisfy this generalized Bogomolov type inequality.

Almost Product Structure in Differentiable Principal Fibre Bundle over Lorentzian Almost Paracontact Manifold

Differentiable principal fibre bundles have been defined and studied by Kobayashi and Nomizu [3] and others. In this paper, I have studied an almost product structure in the manifold P of a differentiable principal fibre bundle (P, M , G , P p ). Keywords. Differentiable manifold, Fibre bundle, left invariant , Almost product structure.

Collagen fiber alignment and maximum principal strain in the glenohumeral capsule predict location of failure during uniaxial extension

The glenohumeral joint is frequently dislocated resulting in injury to the glenohumeral capsule. Repair techniques that focus on restoring the capsule after dislocation to re-establish its stabilizing function could benefit from predictions of the location of failure in this continuous sheet of tissue with a random collagen fiber alignment in the unloaded state. Therefore, the objective of this study was to determine the collagen fiber alignment and maximum principal strain in all regions of the capsule during uniaxial extension to failure and to determine whether these parameters could predict the location of tissue failure. Collagen fiber alignment, quantified using a small-angle light-scattering device, and maximum principal strain in the capsule were determined at 5% increments of elongation until tissue failure. A contingency table analyzed with Fischer's exact test demonstrated that peak collagen fiber alignment, represented by the normalized orientation index (p < 0.001) and maximum principal strain (p < 0.001), is significant in predicting location of failure. The direct correlation between the maximum principal strain and collagen fiber alignment measured prior to failure to the location of tissue failure suggests these parameters can be used as a predictive tool to help locate the areas of the glenohumeral capsule that are susceptible to failure. In the future, changes in collagen fiber alignment following injury could be used to develop a constitutive model for injured capsular tissue.