509 research outputs found

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

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    We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form FpF^{-p}, where p>1p>1 and FF is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature F=HF=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>1p>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

    Unstable periodic orbits in a chaotic meandering jet flow

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    We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and atmospheric currents. A method to detect and locate the unstable periodic orbits and classify them by the origin and bifurcations is developed. We consider in detail period-1 and period-4 orbits playing an important role in chaotic advection. We introduce five classes of period-4 orbits: western and eastern ballistic ones, whose origin is associated with ballistic resonances of the fourth order, rotational ones, associated with rotational resonances of the second and fourth orders, and rotational-ballistic ones associated with a rotational-ballistic resonance. It is a new kind of nonlinear resonances that may occur in chaotic flow with jets and/or circulation cells. Varying the perturbation amplitude, we track out the origin and bifurcations of the orbits for each class

    Kinematic studies of transport across an island wake, with application to the Canary islands

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    Transport from nutrient-rich coastal upwellings is a key factor influencing biological activity in surrounding waters and even in the open ocean. The rich upwelling in the North-Western African coast is known to interact strongly with the wake of the Canary islands, giving rise to filaments and other mesoscale structures of increased productivity. Motivated by this scenario, we introduce a simplified two-dimensional kinematic flow describing the wake of an island in a stream, and study the conditions under which there is a net transport of substances across the wake. For small vorticity values in the wake, it acts as a barrier, but there is a transition when increasing vorticity so that for values appropriate to the Canary area, it entrains fluid and enhances cross-wake transport.Comment: 28 pages, 13 figure

    The homotopy type of the loops on (n1)(n-1)-connected (2n+1)(2n+1)-manifolds

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    For n2n\geq 2 we compute the homotopy groups of (n1)(n-1)-connected closed manifolds of dimension (2n+1)(2n+1). Away from the finite set of primes dividing the order of the torsion subgroup in homology, the pp-local homotopy groups of MM are determined by the rank of the free Abelian part of the homology. Moreover, we show that these pp-local homotopy groups can be expressed as a direct sum of pp-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math., Birkhauser/Springer, 2018. arXiv admin note: text overlap with arXiv:1510.0519

    Linking and causality in globally hyperbolic spacetimes

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    The linking number lklk is defined if link components are zero homologous. Our affine linking invariant alkalk generalizes lklk to the case of linked submanifolds with arbitrary homology classes. We apply alkalk to the study of causality in Lorentz manifolds. Let MmM^m be a spacelike Cauchy surface in a globally hyperbolic spacetime (Xm+1,g)(X^{m+1}, g). The spherical cotangent bundle STMST^*M is identified with the space NN of all null geodesics in (X,g).(X,g). Hence the set of null geodesics passing through a point xXx\in X gives an embedded (m1)(m-1)-sphere SxS_x in N=STMN=ST^*M called the sky of x.x. Low observed that if the link (Sx,Sy)(S_x, S_y) is nontrivial, then x,yXx,y\in X are causally related. This motivated the problem (communicated by Penrose) on the Arnold's 1998 problem list to apply link theory to the study of causality. The spheres SxS_x are isotopic to fibers of (STM)2m1Mm.(ST^*M)^{2m-1}\to M^m. They are nonzero homologous and lk(Sx,Sy)lk(S_x,S_y) is undefined when MM is closed, while alk(Sx,Sy)alk(S_x, S_y) is well defined. Moreover, alk(Sx,Sy)Zalk(S_x, S_y)\in Z if MM is not an odd-dimensional rational homology sphere. We give a formula for the increment of \alk under passages through Arnold dangerous tangencies. If (X,g)(X,g) is such that alkalk takes values in Z\Z and gg is conformal to gg' having all the timelike sectional curvatures nonnegative, then x,yXx, y\in X are causally related if and only if alk(Sx,Sy)0alk(S_x,S_y)\neq 0. We show that x,yx,y in nonrefocussing (X,g)(X, g) are causally unrelated iff (Sx,Sy)(S_x, S_y) can be deformed to a pair of Sm1S^{m-1}-fibers of STMMST^*M\to M by an isotopy through skies. Low showed that if (\ss, g) is refocussing, then MM is compact. We show that the universal cover of MM is also compact.Comment: We added: Theorem 11.5 saying that a Cauchy surface in a refocussing space time has finite pi_1; changed Theorem 7.5 to be in terms of conformal classes of Lorentz metrics and did a few more changes. 45 pages, 3 figures. A part of the paper (several results of sections 4,5,6,9,10) is an extension and development of our work math.GT/0207219 in the context of Lorentzian geometry. The results of sections 7,8,11,12 and Appendix B are ne

    Target Space Duality between Simple Compact Lie Groups and Lie Algebras under the Hamiltonian Formalism: I. Remnants of Duality at the Classical Level

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    It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie group GG with a bi-invariant metric and a generating function Γ\Gamma suggested in the physics literature, we follow the above line of thought and work out the canonical transformation Φ\Phi generated by Γ\Gamma together with an \Ad-invariant metric and a B-field on the associated Lie algebra g\frak g of GG so that GG and g\frak g form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation Φ\Phi including a careful analysis of its domain and image. The geometry of the T-dual structure on g\frak g is lightly touched.Comment: Two references and related comments added, also some typos corrected. LaTeX and epsf.tex, 36 pages, 4 EPS figures included in a uuencoded fil

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

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    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=1q=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

    Normal Cones and Thompson Metric

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    The aim of this paper is to study the basic properties of the Thompson metric dTd_T in the general case of a real linear space XX ordered by a cone KK. We show that dTd_T has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of dTd_T and some results concerning the topology of dTd_T, including a brief study of the dTd_T-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for KK. One considers various completeness properties and one studies the relations between them. Since dTd_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 dTd_T and order-unit (semi)norms u|\cdot|_u are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although dTd_T and u|\cdot|_u are only topological (and not metrical) equivalent on KuK_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

    Application of the National Osteoporosis Foundation Guidelines to postmenopausal women and men: the Framingham Osteoporosis Study.

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    Summary We applied the 2008 National Osteoporosis Foundation (NOF) Guidelines to Framingham Osteoporosis Study participants and found nearly one half of Caucasian postmenopausal women and one sixth of men aged 50 years and older would be recommended for osteoporosis treatment. Given the high proportion of persons recommended for treatment, NOF Guidelines may need to be re-evaluated with respect to budget impact. Introduction Little is known about the public health impact of the NOF Guidelines. Therefore, we determined the proportion of US Caucasians recommended for treatment of osteoporosis according to NOF Guidelines (2003 and 2008). Methods One thousand nine hundred and forty-six postmenopausal women and 1,681 men aged ≥50 years from the Framingham Study with information on bone mineral density (1987–2001) were included. Information on clinical predictors was used to estimate the 10-year probability of hip and major osteoporotic fracture by FRAX® (version 3.0). Results Overall proportion of women meeting treatment criterion was less when the 2008 NOF Guidelines were applied (41.1%) compared with 2003 Guidelines (47.8%). The proportion of women aged 75 years increased slightly (78.3% in 2003, 86.0% in 2008). Seventeen percent of men aged ≥50 years met treatment criterion (2.5% aged 50–64 years, 49.8% aged >75 years). Conclusions Nearly one half of Caucasian postmenopausal women and one sixth of men aged 50 years and older would be recommended for osteoporosis treatment according to 2008 NOF Guidelines. Given the high proportion of persons recommended for treatment, NOF Guidelines may need to be re-evaluated with respect to budget impact
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