scholarly journals THE GROWTH RATE OF A JACOBI VECTOR FIELD ON A HADAMARD MANIFOLD WHOSE GEODESIC FLOW IS OF ANOSOV TYPE

2000 ◽  
Vol 54 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Tetsuo UKAI
Keyword(s):  
Growth Rate ◽  
Vector Field ◽  
Geodesic Flow ◽  
Hadamard Manifold ◽  
Jacobi Vector

TOPOLOGICAL ENTROPY VERSUS GEODESIC ENTROPY

1994 ◽  
Vol 05 (02) ◽  
pp. 213-218 ◽  
Author(s):  
GABRIEL P. PATERNAIN ◽  
MIGUEL PATERNAIN
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Exponential Growth ◽  
Geodesic Flow ◽  
Compact Riemannian Manifold ◽  
Betti Numbers ◽  
Exponential Growth Rate ◽  
Sphere Bundle ◽  
Space Of Paths ◽  
Simply Connected

Using Yomdin's Theorem [8], we show that for a compact Riemannian manifold M, the geodesic entropy — defined as the exponential growth rate of the average number of geodesic segments between two points — is ≤ the topological entropy of the geodesic flow of M. We also show that if M is simply connected and N ⊂ M is a compact simply connected submanifold, then the exponential growth rate of the sequence given by the Betti numbers of the space of paths starting in N and ending in a fixed point of M, is bounded above by the topological entropy of the geodesic flow on the normal sphere bundle of N.

scholarly journals Solving Yosida inclusion problem in Hadamard manifold

2019 ◽  
Vol 9 (2) ◽  
pp. 357-366 ◽  
Author(s):  
Mohammad Dilshad
Keyword(s):  
Approximate Solution ◽  
Vector Field ◽  
Variational Inequalities ◽  
Hadamard Manifold ◽  
Inclusion Problem ◽  
Approximation Operator ◽  
Hadamard Manifolds ◽  
Yosida Approximation ◽  
Type Algorithm ◽  
Monotone Vector Field

Abstract We consider a Yosida inclusion problem in the setting of Hadamard manifolds. We study Korpelevich-type algorithm for computing the approximate solution of Yosida inclusion problem. The resolvent and Yosida approximation operator of a monotone vector field and their properties are used to prove that the sequence generated by the proposed algorithm converges to the solution of Yosida inclusion problem. An application to our problem and algorithm is presented to solve variational inequalities in Hadamard manifolds.

scholarly journals Exceptional minimal sets of C1+α-group actions on the circle

1991 ◽  
Vol 11 (3) ◽  
pp. 455-467 ◽  
Author(s):  
S. Hurder
Keyword(s):  
Growth Rate ◽  
Fixed Point ◽  
Topological Entropy ◽  
Geodesic Flow ◽  
Closing Lemma ◽  
Finitely Generated ◽  
Exceptional Set ◽  
Minimal Set ◽  
Finitely Generated Group ◽  
Positive Topological Entropy

AbstractWe prove two extensions of Sacksteder's Theorem for the action A: Γ × S1 → S1 of a finitely-generated group Γ on the circle by C1+α-diffeomorphisms. If the action A has an exceptional minimal set K with a gap endpoint of exponential orbit growth rate, or if the action A on K has positive topological entropy, then the exceptional set K is hyperbolic. That is, A has a linearly contracting fixed-point in K. A key point of the paper is to prove a foliation closing lemma using the foliation geodesic flow technique.

scholarly journals A Statistical Cohomogeneity One Metric on the Upper Plane with Constant Negative Curvature

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Limei Cao ◽  
Didong Li ◽  
Erchuan Zhang ◽  
Zhenning Zhang ◽  
Huafei Sun
Keyword(s):  
Vector Field ◽  
Hyperbolic Space ◽  
Negative Curvature ◽  
Statistical Manifold ◽  
Constant Negative Curvature ◽  
Cohomogeneity One ◽  
Geometrical Structures ◽  
Connected Manifold ◽  
Jacobi Vector ◽  
Simply Connected

we analyze the geometrical structures of statistical manifoldSconsisting of all the wrapped Cauchy distributions. We prove thatSis a simply connected manifold with constant negative curvatureK=-2. However, it is not isometric to the hyperbolic space becauseSis noncomplete. In fact,Sis approved to be a cohomogeneity one manifold. Finally, we use several tricks to get the geodesics and explore the divergence performance of them by investigating the Jacobi vector field.

The threshold energy for atom displacement in fcc metals

1990 ◽  
Vol 48 (4) ◽  
pp. 530-531
Author(s):  
Wilfried Sigle ◽  
Matthias Hohenstein ◽  
Alfred Seeger
Keyword(s):  
Growth Rate ◽  
Elevated Temperatures ◽  
Threshold Energy ◽  
Dislocation Loops ◽  
Absolute Minimum ◽  
Voltage Electron ◽  
Atom Displacement ◽  
Electron Microscopes ◽  
Fcc Metal

Prolonged electron irradiation of metals at elevated temperatures usually leads to the formation of large interstitial-type dislocation loops. The growth rate of the loops is proportional to the total cross-section for atom displacement,which is implicitly connected with the threshold energy for atom displacement, Ed . Thus, by measuring the growth rate as a function of the electron energy and the orientation of the specimen with respect to the electron beam, the anisotropy of Ed can be determined rather precisely. We have performed such experiments in situ in high-voltage electron microscopes on Ag and Au at 473K as a function of the orientation and on Au as a function of temperature at several fixed orientations.Whereas in Ag minima of Ed are found close to <100>,<110>, and <210> (13-18eV), (Fig.1) atom displacement in Au requires least energy along <100>(15-19eV) (Fig.2). Au is thus the first fcc metal in which the absolute minimum of the threshold energy has been established not to lie in or close to the <110> direction.

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

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