The Singularity Problem for Space-Times with Torsion

The problem of a rigorous theory of singularities in space-times with torsion is addressed. We define geodesics as curves whose tangent vector moves by parallel transport. This is different from what other authors have done, because their definition of geodesics only involves the Christoffel connection, though studying theories with torsion. We propose a preliminary definition of singularities which is based on timelike or null geodesic incompleteness, even though for theories with torsion the paths of particles are not geodesics. The study of the geodesic equation for cosmological models with torsion shows that the definition has a physical relevance. It can also be motivated, as done in the literature, remarking that the causal structure of a space-time with torsion does not get changed with respect to general relativity. We then prove how to extend Hawking's singularity theorem without causality assumptions to the space-time of the ECSK theory. This is achieved studying the generalized Raychaudhuri equation in the ECSK theory, the conditions for the existence of conjugate points and properties of maximal timelike geodesics. Hawking's theorem can be generalized, provided the torsion tensor obeys some conditions. Thus our result can also be interpreted as a no-singularity theorem if these additional conditions are not satisfied. In other words, it turns out that the occurrence of singularities in closed cosmological models based on the ECSK theory is less generic than in general relativity. Our work is to be compared with previous papers in the literature. There are some relevant differences, because we rely on a different definition of geodesics, we keep the field equations of the ECSK theory in their original form rather than casting them in a form similar to general relativity with a modified energy momentum tensor,Comment: 17 pages, plain-tex, published in Nuovo Cimento B, volume 105, pages 75-90, year 199

Comparison of the Utility and Validity of Three Scoring Tools to Measure Skin Involvement in Patients With Juvenile Dermatomyositis

OBJECTIVE: To compare the abbreviated Cutaneous Assessment Tool (CAT), Disease Activity Score (DAS), and Myositis Intention to Treat Activity Index (MITAX) and correlate them with the physician's 10-cm skin visual analog scale (VAS) in order to define which tool best assesses skin disease in patients with juvenile dermatomyositis. METHODS: A total of 71 patients recruited to the UK Juvenile Dermatomyositis Cohort and Biomarker Study were included and assessed for skin disease using the CAT, DAS, MITAX, and skin VAS. The Childhood Myositis Assessment Scale (CMAS), manual muscle testing of 8 groups (MMT8), muscle enzymes, inflammatory markers, and physician's global VAS were recorded. Relationships were evaluated using Spearman's correlations and predictors with linear regression. Interrater reliability was assessed using intraclass correlation coefficients. RESULTS: All 3 tools showed correlation with the physician's global VAS and skin VAS, with DAS skin showing the strongest correlation with skin VAS. DAS skin and CAT activity were inversely correlated with CMAS and MMT8, but these correlations were moderate. No correlations were found between the skin tools and inflammatory markers or muscle enzymes. DAS skin and CAT were the quickest to complete (mean ± SD 0.68 ± 0.1 minutes and 0.63 ± 0.1 minutes, respectively). CONCLUSION: The 3 skin tools were quick and easy to use. The DAS skin correlated best with the skin VAS. The addition of CAT in a bivariate model containing the physician's global VAS was a statistically significant estimator of skin VAS score. We propose that there is scope for a new skin tool to be devised and tested, which takes into account the strengths of the 3 existing tools

On the supersymmetries of anti de Sitter vacua

We present details of a geometric method to associate a Lie superalgebra with a large class of bosonic supergravity vacua of the type AdS x X, corresponding to elementary branes in M-theory and type II string theory.Comment: 16 page

A natural Finsler--Laplace operator

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representations and computations of spectral data.Comment: 25 pages, v2: minor modifications, changed the introductio

Nonstandard Drinfeld-Sokolov reduction

Subject to some conditions, the input data for the Drinfeld-Sokolov construction of KdV type hierarchies is a quadruplet (\A,\Lambda, d_1, d_0), where the $d_i$ are $\Z$-gradations of a loop algebra \A and \Lambda\in \A is a semisimple element of nonzero $d_1$-grade. A new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented. The proposal relies on splitting the $d_1$-grade zero part of \A into a vector space direct sum of two subalgebras. This permits one to interpret certain Gelfand-Dickey type systems associated with a nonstandard splitting of the algebra of pseudo-differential operators in the Drinfeld-Sokolov framework.Comment: 19 pages, LaTeX fil

Quantum Mechanics of Yano tensors: Dirac equation in curved spacetime

In spacetimes admitting Yano tensors the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to operators that in the classical limit commute with the Dirac operator and generate conserved quantities. We show that the result is preserved in the full quantum theory, that is, Yano symmetries are not anomalous. This was known for Yano tensors of rank two, but our main result is to show that it extends to Yano tensors of arbitrary rank. We also describe the conformal Yano equation and show that is invariant under Hodge duality. There is a natural relationship between Yano tensors and supergravity theories. As the simplest possible example, we show that when the spacetime admits a Killing spinor then this generates Yano and conformal Yano tensors. As an application, we construct Yano tensors on maximally symmetric spaces: they are spanned by tensor products of Killing vectors.Comment: 1+32 pages, no figures. Accepted for publication on Classical and Quantum Gravity. New title and abstract. Some material has been moved to the Appendix. Concrete formulas for Yano tensors on some special holonomy manifolds have been provided. Some corrections included, bibliography enlarge

D-Matter

We study the properties and phenomenology of particle-like states originating from D-branes whose spatial dimensions are all compactified. They are non-perturbative states in string theory and we refer to them as D-matter. In contrast to other non-perturbative objects such as 't Hooft-Polyakov monopoles, D-matter states could have perturbative couplings among themselves and with ordinary matter. The lightest D-particle (LDP) could be stable because it is the lightest state carrying certain (integer or discrete) quantum numbers. Depending on the string scale, they could be cold dark matter candidates with properties similar to that of wimps or wimpzillas. The spectrum of excited states of D-matter exhibits an interesting pattern which could be distinguished from that of Kaluza-Klein modes, winding states, and string resonances. We speculate about possible signatures of D-matter from ultra-high energy cosmic rays and colliders.Comment: 25 pages, 5 figures, references adde

On "Dotsenko-Fateev" representation of the toric conformal blocks

We demonstrate that the recent ansatz of arXiv:1009.5553, inspired by the original remark due to R.Dijkgraaf and C.Vafa, reproduces the toric conformal blocks in the same sense that the spherical blocks are given by the integral representation of arXiv:1001.0563 with a peculiar choice of open integration contours for screening insertions. In other words, we provide some evidence that the toric conformal blocks are reproduced by appropriate beta-ensembles not only in the large-N limit, but also at finite N. The check is explicitly performed at the first two levels for the 1-point toric functions. Generalizations to higher genera are briefly discussed.Comment: 10 page

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fr\'echet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fr\'echet minimizer and the asymptotic distribution of the sample Fr\'echet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds