Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews-McCoy-Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power $p>1$ loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.Comment: 18 pages. We included an example for the loss of convexity and pinching. In the third version we dropped the concavity assumption on F. Comments are welcom

A Nonlinear Baroclinic Wave-Mean Oscillation with Multiple Normal Mode Instabilities

An unstable, nonlinear baroclinic wave-mean oscillation is found in a strongly supercritical quasigeostrophic f-plane numerical channel model with 3840 Fourier components. The growth of linear disturbances to this time-periodic oscillation is analyzed by computing time-dependent normal modes (Floquet vectors). Two different Newton–Picard methods are used to compute the unstable solution, the first based on direct computation of a large set of Floquet vectors, and the second based on an efficient iterative solver. Three different growing normal modes are found, which modify the wave structure of the wave-mean oscillation in two essentially different ways. The dynamics of the instabilities are qualitatively similar to the baroclinic dynamics of the wave-mean oscillation. The results provide an example of time-dependent normal mode instability of a strongly nonlinear time-dependent baroclinic flow

The structure of quantum Lie algebras for the classical series B_l, C_l and D_l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at $q=1$. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint ---> adjoint. We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B_l, C_l and D_l. In the quantum case also the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.Comment: 25 pages, amslatex, eepic. Final version for publication in J. Phys. A. Minor misprints in eqs. 5.11 and 5.12 correcte

Theory and computation of covariant Lyapunov vectors

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects

We study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactification with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One type are "magnetic bions" which carry net magnetic charge and induce a mass gap for gauge fluctuations. Another type are "neutral bions" which are magnetically neutral, and their understanding requires a generalization of multi-instanton techniques in quantum mechanics - which we refer to as the Bogomolny-Zinn-Justin (BZJ) prescription - to compactified field theory. The BZJ prescription applied to bion-anti-bion topological molecules predicts a singularity on the positive real axis of the Borel plane (i.e., a divergence from summing large orders in peturbation theory) which is of order N times closer to the origin than the leading 4-d BPST instanton-anti-instanton singularity, where N is the rank of the gauge group. The position of the bion--anti-bion singularity is thus qualitatively similar to that of the 4-d IR renormalon singularity, and we conjecture that they are continuously related as the compactification radius is changed. By making use of transseries and Ecalle's resurgence theory we argue that a non-perturbative continuum definition of a class of field theories which admit semi-classical expansions may be possible.Comment: 112 pages, 7 figures; v2: typos corrected, discussion of supersymmetric models added at the end of section 8.1, reference adde

Circulating levels of dickkopf-1, osteoprotegerin and sclerostin are higher in old compared with young men and women and positively associated with whole-body bone mineral density in older adults

Summary: Bone mineral density declines with increasing older age. We examined the levels of circulating factors known to regulate bone metabolism in healthy young and older adults. The circulating levels of dickkopf-1, osteocalcin, osteoprotegerin and sclerostin were positively associated with WBMD in older adults, despite the average WBMD being lower and circulating dickkopf-1, osteoprotegerin and sclerostin being higher in old than young. Purpose: To investigate the relationship between whole-body bone mineral density (WBMD) and levels of circulating factors with known roles in bone remodelling during 'healthy' ageing. Methods: WBMD and fasting plasma concentrations of dickkopf-1, fibroblast growth factor-23, osteocalcin, osteoprotegerin, osteopontin and sclerostin were measured in 272 older subjects (69 to 81 years; 52% female) and 171 younger subjects (18-30 years; 53% female). Results: WBMD was lower in old than young. Circulating osteocalcin was lower in old compared with young, while dickkopf-1, osteoprotegerin and sclerostin were higher in old compared with young. These circulating factors were each positively associated with WBMD in the older adults and the relationships remained after adjustment for covariates (r-values ranging from 0.174 to 0.254, all p<0.01). In multivariate regression, the body mass index, circulating sclerostin and whole-body lean mass together accounted for 13.8% of the variation with WBMD in the older adults. In young adults, dickkopf-1 and body mass index together accounted for 7.7% of variation in WBMD. Conclusion: Circulating levels of dickkopf-1, osteocalcin, osteoprotegerin and sclerostin are positively associated with WBMD in community-dwelling older adults, despite the average WBMD being lower and circulating dickkopf-1, osteoprotegerin and sclerostin being higher in old than young

Integration with the human genome of peptide sequences obtained by high-throughput mass spectrometry

A crucial aim upon the completion of the human genome is the verification and functional annotation of all predicted genes and their protein products. Here we describe the mapping of peptides derived from accurate interpretations of protein tandem mass spectrometry (MS) data to eukaryotic genomes and the generation of an expandable resource for integration of data from many diverse proteomics experiments. Furthermore, we demonstrate that peptide identifications obtained from high-throughput proteomics can be integrated on a large scale with the human genome. This resource could serve as an expandable repository for MS-derived proteome information

Links between cardiovascular disease and osteoporosis in postmenopausal women: serum lipids or atherosclerosis per se?

INTRODUCTION AND HYPOTHESIS: Epidemiological observations suggest links between osteoporosis and risk of acute cardiovascular events and vice versa. Whether the two clinical conditions are linked by common pathogenic factors or atherosclerosis per se remains incompletely understood. We investigated whether serum lipids and polymorphism in the ApoE gene modifying serum lipids could be a biological linkage. METHODS: This was an observational study including 1176 elderly women 60–85 years old. Women were genotyped for epsilon (ɛ) allelic variants of the ApoE gene, and data concerning serum lipids (total cholesterol, triglycerides, HDL-C, LDL-C, apoA1, ApoB, Lp(a)), hip and spine BMD, aorta calcification (AC), radiographic vertebral fracture and self-reported wrist and hip fractures, cardiovascular events together with a wide array of demographic and lifestyle characteristics were collected. RESULTS: Presence of the ApoE ɛ4 allele had a significant impact on serum lipid profile, yet no association with spine/hip BMD or AC could be established. In multiple regression models, apoA1 was a significant independent contributor to the variation in AC. However, none of the lipid components were independent contributors to the variation in spine or hip BMD. When comparing the women with or without vertebral fractures, serum triglycerides showed significant differences. This finding was however not applicable to hip or wrist fractures. After adjustment for age, severe AC score (≥6) and/or manifest cardiovascular disease increased the risk of hip but not vertebral or wrist fractures. CONCLUSION: The contribution of serum lipids to the modulators of BMD does not seem to be direct but rather indirect via promotion of atherosclerosis, which in turn can affect bone metabolism locally, especially when skeletal sites supplied by end-arteries are concerned. Further studies are needed to explore the genetic or environmental risk factors underlying the association of low triglyceride levels to vertebral fractures

A new approach to comprehensively evaluate the morphological properties of the human femoral head : example of application to osteoarthritic joint

Osteoarthritis affects the morphological properties of the femoral head. The goal of this study was to develop a method to elucidate whether these changes are localised to discrete regions, or if the reported trends in microstructural changes may be identified throughout the subchondral bone of the human femoral head. Whole femoral heads extracted from osteoarthritic (n = 5) and healthy controls (n = 5) underwent microCT imaging 39 μm voxel size. The subchondral bone plate was virtually isolated to evaluate the plate thickness and plate porosity. The trabecular bone region was divided into 37 volumes of interest spatially distributed in the femoral head, and bone morphometric properties were determined in each region. The study showed how the developed approach can be used to study the heterogeneous properties of the human femoral head affected by a disease such as osteoarthritis. As example, in the superior femoral head osteoarthritic specimens exhibited a more heterogeneous micro-architecture, with trends towards thicker cortical bone plate, higher trabecular connectivity density, higher trabecular bone density and thicker structures, something that could only be observed with the newly developed approach. Bone cysts were mostly confined to the postero-lateral quadrants extending from the subchondral region into the mid trabecular region. Nevertheless, in order to generalise these findings, a larger sample size should be analysed in the future. This novel method allowed a comprehensive evaluation of the heterogeneous micro-architectural properties of the human femoral head, highlighting effects of OA in the superior subchondral cortical and trabecular bone. Further investigations on different stages of OA would be needed to identify early changes in the bone