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The classical Schottky problem is concerned with characterization of Jacobian varieties of compact Riemann surfaces among all abelian varieties, or the identification of the Jacobian locus J (M g) in the moduli space A g of principally polarized abelian varieties as an algebraic subvariety. By viewing A g as a noncompact metric space coming from its structure as a locally symmetric space and J (M g) as a metric subspace, we compare the subspace metric d and the induced length metric ℓ on J (M g) . Consequently, we clarify the nature of the metric distortion of the subspace J (M g) and hence settle a problem posed by Farb (2006) on the metric distortion of J (M g) inside A g in a certain sense (see Theorem 1.5 and Corollary 1.6). [ABSTRACT FROM AUTHOR]

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the wellposedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. [ABSTRACT FROM AUTHOR]

This paper presents analysis on two 3D mesh to 2D map strategies applied to unwrap images of rock tunnels and facilitate visualization of large datasets. First, we examined mesh parameterization algorithms which are used in computer graphics to convert a 3D mesh model to a 2D representation. We found that while these methods were automatic and could provide 2D maps with minimal metric distortion (ie: conservation of lengths in 3D when mapped to 2D), they exhibited twisted shapes and were not intuitive to interpret. Second, we proposed two novel approaches, combining mesh deformation algorithms, which are used in computer animation to reshape a 3D mesh to resemble a 3D plane, and projection onto a 2D plane. We found that while these methods required user interaction and introduced a greater amount of metric distortion, their outputs were fairly intuitive to interpret. To compare the relative merits of mesh parameterization and mesh deformation and projection, the different strategies are applied to a 8.2 m wide by 41 m long by 6.7 m high subsection of a mining tunnel. The metric distortion produced was calculated and their respective output 2D maps are presented and discussed. [ABSTRACT FROM AUTHOR]

Abstract: The synthesis of chloro-vanadato-apatites M 5(VO4)3Cl (M = Ca, Sr, Ba) in a two step solid state procedure is reported. IR spectra show a considerable shift of the frequencies attributed to the tetrahedral group to lower energies compared with the phosphato-apatites. The structure of these compounds is determined by powder diffraction data and refined using Rietveld methods. An elaborate analysis of the data shows that the chloride anions are to be found at different positions along a channel along with varying probability. They are fairly well localised in the case of the Sr and the Ba compound. In the Ca compound they are spread over a larger region along c. A refinement according to a new concept of “metric distortion” gives a good fit of the data and indicates that the delocalisation of electron density seen in a Difference-Fourier plot is best described by a static picture of disorder. [Copyright &y& Elsevier]

Abstract: During the past decade, several computational approaches have been proposed for the task of mapping highly convoluted surfaces of the human brain to simpler geometric objects such as a sphere or a topological disc. We report the results of a quantitative comparison of FreeSurfer, CirclePack, and LSCM with respect to measurements of geometric distortion and computational speed. Our results indicate that FreeSurfer performs best with respect to a global measurement of metric distortion, whereas LSCM performs best with respect to angular distortion and best in all but one case with a local measurement of metric distortion. FreeSurfer provides more homogeneous distribution of metric distortion across the whole cortex than CirclePack and LSCM. LSCM is the most computationally efficient algorithm for generating spherical maps, while CirclePack is extremely fast for generating planar maps from patches. [Copyright &y& Elsevier]

In this article, we propose an approach to integrate cortical morphology measures for improving the discrimination of individuals with and without very mild Alzheimer's disease (AD). FreeSurfer was applied to scans collected from 83 participants with very mild AD and 124 cognitively normal individuals. We generated cortex thickness, white matter convexity (aka "sulcal depth"), and white matter surface metric distortion measures on a normalized surface atlas in this first study to integrate high resolution gray matter thickness and white matter surface geometric measures in identifying very mild AD. Principal component analysis was applied to each individual structural measure to generate eigenvectors. Discrimination power based on individual and combined measures are compared, based on stepwise logistic regression and 10-fold cross-validation. Global AD likelihood index and surface-based likelihood maps were also generated. Our results show complementary patterns on the cortical surface between thickness, which reflects gray matter atrophy, convexity, which reflects white matter sulcal depth changes and metric distortion, which reflects white matter surface area changes. The classifier integrating all 3 types of surface measures significantly improved classification performance compared with classification based on single measures. The principal component analysis-based approach provides a framework for achieving high discrimination power by integrating high-dimensional data, and this method could be very powerful in future studies for early diagnosis of diseases that are known to be associated with abnormal gyral and sulcal patterns., (Copyright © 2015 Elsevier Inc. All rights reserved.)