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1

Characterize on the Heisenberg group with left invariant Lorentzian metric

Turhan T., Korpinar T.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

423-428

EN

In this paper, we consider the biharmonicity conditions for maps between Riemannian manifolds and we characterize non-geodesic biharmonic curve in Heisenberg group H3 which is endowed with left invariant Lorentzian metric.

2

Variational formulation for incompressible Euler flow / shape-morphing metric and geodesic

Zolesio J. P.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1631-1653

EN

Shape variational formulation for Euler flow has already been considered by the author in (1999a, 2007c). We develop here the control approach considering the convection (or mass transport) as the "state equation" while the speed vector field is the control and we introduce the h-Sobolev curvature which turns to be shape differentiable. The value function defines a new shape metric; we derive existence of geodesic for a p-pseudo metric, verifying the triangle property with a factor 2p-1, for any p > 1. Any geodesic solves the Euler equation for incompressible fluids and, in dimension 3, is not curl free. The classical Euler equation for incompressible fluid (3), coupled with the convection (1) turns to have variational solutions when conditions are imposed on the convected tube ζ while no initial condition has to be imposed on the fluid speed V itself.

3

A note on geodesic and almost geodesic mappings of homogeneous Riemannian manifolds

Formella S.

Opuscula Mathematica

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2005

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Vol. 25, no. 2

181-187

EN

Let M be a differentiable manifold and denote by nabla and nabla~ two linear connections on M. Nabla and nabla~ are said to be geodesically equivalent if and only if they have the same geodesics. A Riemannian manifold (M, g) is a naturally reductive homogeneous manifold if and only if nabla and nabla~ = nabla - T are geodesically equivalent, where T is a homogeneous structure on (M, g) ([7]). In the present paper we prove that if it is possible to map geodesically a homogeneous Riemannian manifold (M, g) onto (M, nabla~), then the map is affine. If a naturally reductive manifold (M, g) admits a nontrivial geodesic mapping onto a Riemannian manifold (formula) then both manifolds are of constant cutvature. We also give some results for almost geodesic mappings (M, g) arr (M, nabla~).

4

Long and short periodic deformations of slender structures caused by temperature and wind

Breuer P., Chmielewski T., Górski P., Konopka E., Tarczyński L.

Zeszyty Naukowe. Budownictwo / Politechnika Opolska

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2003

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z. 47

87-102

EN

Elastic deformations of two high-rise towers caused by wind and sun radiation were evaluated by surveying methods using GPS and levelling. Oscillations by wind mostly occur perpendicular to the wind direction as predicted by the Karman Effect. The frequency of these oscillations appears to be independent of the wind velocity. The influence of sun radiation leads to an elastic deformation in a period of 24 hours in the form of a slim ellipse. The diameter of that ellipse in the East-West direction is twice as large as in the North-South direction. The velocity of displacement is controlled by the intensity of the sun's rays. GPS methods for positioning and levelling for height recording lead to identical deformation results. The geodesic suweying results should be considered in relation to the parameters of the static construction due to Civil Engineering and to the results of physical surveying melhods.