ON RICCI RANK OF CARTAN–HADAMARD MANIFOLDS

2002 ◽  
Vol 13 (06) ◽  
pp. 557-578
Author(s):  
DINCER GULER ◽  
FANGYANG ZHENG

In this article, we prove that the maximum rank r of the Ricci tensor of a Cartan–Hadamard manifold Mn satisfies the inequality 2r - 1 ≥ n - s, where n is the dimension and s is the core number, which measures the flatness of Mn. Examples show that this lower bound is sharp.

1972 ◽  
Vol 62 (4) ◽  
pp. 1063-1071 ◽  
Author(s):  
R. D. Adams

Abstract The phases P2KP, P3KP, and P4KP are well recorded from the Novaya Zemlya nuclear explosion of October 14, 1970, with the branch AB at distances of up to 20° beyond the theoretical end point A. This extension is attributed to diffraction around the core-mantle boundary. A slowness dT/dΔ = 4.56±0.02 sec/deg is determined for the AB branch of P4KP, in excellent agreement with recent determinations of the slowness of diffracted P. This slowness implies a velocity of 13.29±0.06 km/sec at the base of the mantle, and confirms recent suggestions of a low-velocity channel above the core-mantle boundary. There is evidence that arrivals recorded before the AB branch of P2KP may lie on two branches, with different slownesses. The ratio of amplitudes of successive orders of multiple inner core reflections gives a lower bound of about 2200 for Q in the outer core.


2016 ◽  
Vol 9 (2) ◽  
Author(s):  
Ilkka Holopainen

AbstractWe construct, by modifying Borbély's example, a 3-dimensional Cartan–Hadamard manifold


2019 ◽  
Vol 9 (2) ◽  
pp. 357-366 ◽  
Author(s):  
Mohammad Dilshad

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.


2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor

An explicit iterative method for solving the variational inequalities on Hadamard manifold is suggested and analyzed using the auxiliary principle technique. The convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. Results can be viewed as refinement and improvement of previously known results.


2005 ◽  
Vol 15 (05n06) ◽  
pp. 799-813
Author(s):  
ROGER C. ALPERIN ◽  
GENNADY A. NOSKOV

We prove that any nonelementary geometrically finite group of isometries of a pinched Hadamard manifold has nonzero algebraic entropy in the sense of M. Gromov. In other words it has uniform exponential growth.


1987 ◽  
Vol 106 ◽  
pp. 79-90
Author(s):  
Takashi Yasuoka

In this paper we shall study hyperbolicity of Hadamard manifolds.In Section 1 we shall define and solve the Dirichlet problem at infinity for Laplacian J, which gives a partial extension of the result of Anderson and Sullivan in Theorem 1 (cf.). In Section 2 we apply the solution of the Dirichlet problem at infinity to a complex analysis on a Kâhler Hadamard manifold whose metric restricted to every geodesic sphere is conformai to that of the standard sphere. It seems that the sphere at infinity of such a manifold admits a CR-structure. In fact we can define a CR-function at infinity on the sphere at infinity. We shall show in Theorem 2 that there exists a holomorphic extension from the sphere at infinity and it coincides with the solution of the Dirichlet problem at infinity, if the Dirichlet problem at infinity is solvable. So we see that such a manifold admits many bounded holomorphic functions. By the similar method we shall show in Theorem 3 that such a manifold is biholomorphic to a strictly pseudoconvex domain in Cn, if the holomorphic sectional curvature Kh(x) is less than −1/(1 + r(x)2), where r(x) is a distance function from a pole. Theorem 3 is a partial answer to a conjecture raised by Green and Wu.


1990 ◽  
Vol 120 ◽  
pp. 181-204 ◽  
Author(s):  
Takashi Shioya

In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.


Author(s):  
Jerome Bertrand ◽  
Kunnath Sandeep

Abstract In this article, we establish estimates on Riesz-type kernels and prove the Adams-type inequality for $W^{k,p}(M)$ functions, where $(M,g)$ is an $n$-dimensional Hadamard manifold with sectional curvature bounded from below and above by a negative constant and $k$ is an integer satisfying $kp=n$.


Author(s):  
Udo Rindelhardt ◽  
Hans-Werner Viehrig ◽  
Joerg Konheiser ◽  
Jan Schuhknecht

Between 1973 and 1990 4 units of the Russian NPP type WWER-440/230 were operated in Greifswald (former GDR). The operation was stopped after the German reunification, because the units did not completely follow western nuclear safety standards. Material probes from the pressure vessels were gained in the frame of the ongoing decommissioning procedure. The investigations of this material started with material from the circumferential core weld of unit 1. This weld was annealed after 13 cycles and operated further for 2 cycles. Additionally, starting with cycle 11, dummy assemblies were inserted to reduce the neutron fluence in the RPV wall. Firstly this paper presents results of the RPV fluence calculations depending on different loading schemes and on the axial weld position based on the Monte Carlo code TRAMO. The results show, that the use of the dummy assemblies reduces the flux by a factor of 2 – 5 depending on the azimuthal position. The fluence increase is reduced to 1/6 at the position of the maximum fluence. The neutron fluence at the different circumferential welds is closely related to their distance to the core. The circumferential core weld (SN0.1.4) received a fluence of 2.4·1019 neutrons/cm2 at the inner surface, it decreases to 0.8·1019 neutrons/cm2 at the outer surface. The neutron fluences at the both other welds are 2 resp. 4 orders of magnitude smaller according to their distances to the core. It should be mentioned that in this cases the fluence gradient can be negative through the wall. The material investigations were done using a trepan from the circumferential core weld. Master Curve and Charpy V-notch testing were applied. Specimens from 7 locations through the thickness of the welding seam were tested. The reference temperature T0 was calculated with the measured fracture toughness values, KJc, at brittle failure of the specimen. Generally the KJc values measured on pre-cracked and side-grooved Charpy size SE(B) specimens of the investigated weld metal follows the course of the Master Curve. The KJc values show a remarkable scatter. In addition the MC SINTAP procedure was applied to determine T0SINTAP of the brittle fraction of the data set. There are remarkable differences between T0 and T0SINTAP indicating macroscopic inhomogeneous weld metal. The highest T0 was about 50°C at a distance of 22 mm from the inner surface of the weld. It is 40 K higher compared with T0 at the inner surface. This is important for the assessment of ductile-to-brittle temperatures measured with sub size Charpy specimens made of weld metal from the inner RPV wall. This material does not represent the most conservative condition. Nevertheless, the Charpy transition temperature TT41J estimated with results of sub size specimens after the recovery annealing was confirmed by the testing of standard Charpy V-notch specimens. The VERLIFE procedure prepared for the integrity assessment of WWER RPV was applied on the measured results. It enables the determination of a reference temperature, RTT0 to index a lower bound fracture toughness curve. This curve agrees with the MC 5% fractile as specified in ASTM E1921-05. The measured KJc values are not enveloped by this lower bound curve. However, the VERLIFE lower bound curve indexed with the SINTAP reference temperature RTT0SINTAP envelops the KJc values. Therefore for a conservative integrity assessment the fracture toughness curve indexed with a RT representing the brittle fraction of a dataset of measured KJc values has to be applied.


2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor

We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.


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