PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying (Fv2-a2)(Fv2+a2)(Fv2 - b2)(Fv2 + b2) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2n-dimensional Lagrange manifold E is defined and the F(±a2; ±b2)-structure on the vertical tangent space TV (E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on T(E). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r-parallel, r¯ anti half parallel when r = r¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

Fundamental Vector Fields

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.

Determining the Embedded Pile Length for Large-Diameter Monopiles

The use of the monopile support structure farther offshore requires a large amount of construction steel. Designing an efficient foundation can significantly reduce the amount of steel needed and thus the total costs. This paper evaluates the applicability of current foundation design criteria for large-diameter monopiles. Emphasis will be on the vertical tangent criterion as suggested by Germanischer Lloyd and the “zero-toe-kick” criterion for determining the required embedded pile length under static loading conditions. The lateral behavior of a total of 40 different design cases of monopile support structures in water depths ranging from 15 to 35 m has been studied. The soil conditions ranged from loose to very dense sand, which is typical for the North Sea. It has been concluded that the vertical tangent and the “zero-toe-kick” criteria leads to overly conservative embedded pile lengths. A preliminary design approach is presented, which is based on the knowledge that shortening the embedded pile length will decrease the natural frequency of the support structure. The results from this preliminary design approach study have been compared with the current monopile design practice, and it was concluded that the embedded monopile length can be reduced while achieving both lateral stability and maintaining small values of deflection at mudline and the pile toe.

A link between the shape of the austenite–martensite interface and the behaviour of the surface energy

Let Ω ⊂ R2 denote a bounded Lipschitz domain and consider some portion Γ0 of ∂Ω representing the austenite–twinned-martensite interface which is not assumed to be a straight segment. We prove that for an elastic energy density ϖ: R2 → [0 ∞) such that ϖ(0, ±1) = 0. Here, W(Ω) consists of all functions u from the Sobolev class W1, ∞(Ω) such that |uy| = 1 almost everywhere on Ω together with u = 0 on Γ0. We will first show that, for Γ0 having a vertical tangent, one cannot always expect a finite surface energy, i.e. in the above problem, the condition in general cannot be included. This generalizes a result of [12] where Γ0is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary ∂Ω, that is, one can even take Γ0 = ∂Ω. We also show that the existence or non-existence of minimizers depends on the shape of the austenite–twinned-martensite interface Γ0.

REFRACTIVE INDEX OF He4: LIQUID

Changes in the phase refractive index n with temperature have been measured between 1.6 and 4.2° K at λ = 5462.27 Å, for liquid He4 at its saturated vapor pressure, using a metal optical cryostat and a Jamin interferometer. A novel adaptation of the Jamin interferometer has been made so that, in addition, an absolute determination of the group refractive index, nG, could be made using white light of "effective wavelength" 5595 ± 40 Å. When the dispersion correction is made, the phase index for the Hg green line at T55E = 3.700° K is n = 1.026124 ± 0.000035. The relative measurements have been adjusted to this value. The more than 200 experimental points show a random scatter of less than 5 × 10−6 in index. Using Kerr's density data the polarizability is thus (N0α) = (0.12454 ± 0.00021) cm3 mole−1 at λ = 5462.27 Å, for liquid He4 at 3.7 °K. Within experimental error (N0α) is found to be independent of temperature. Thus the refractive index data may be considered as a measurement of the liquid density and coefficient of expansion.The region near the λ-point is of special interest. The expansion coefficient determined from the refractive index, βn, may be represented within experimental error by 103βnI = +41.5 + 14.5 log|T−Tλ| for T > Tλ, from about 0.1° above Tλ to within 0.01° of Tλ; and by 103βnII = −1.5 + 14.5 log |T−Tλ| for T < Tλ, from about 0.1° below Tλ to within 0.002° of Tλ. This implies that the density–temperature curve has both a vertical tangent and a point of inflection at the λ-point; and that the maximum in density occurs about 0.001° above the λ-point.