Non-stationary diffraction problem of a plane oblique pressure wave on the shell in the form of a hyperbolic cylinder taking into account the dissipation effect

The plane non-stationary problem of the dynamics of a thin elastic shell in the form of a hyperbolic cylinder immersed in a liquid under the action of an oblique acoustic pressure wave is considered. To solve this problem, a system of equations is constructed in a related statement. In this case, hydroelasticity problems are reduced to equations of shell dynamics, the damping effect of the liquid (dissipation effect) is taken into account by introducing an integral operator of the convolution type in the time domain. The problem is solved approximately on the basis of the hypothesis of a thin layer taking into account the damping forces in the liquid. The integro-differential equations of shell motion are solved numerically based on the difference discretization of differential operators and the representation of the integral operator by the sum using the trapezoid rule. The kinematic and static parameters of the system are given.

Weingarten spacelike hypersurfaces in a de Sitter space

We study some Weingarten spacelike hypersurfaces in a de Sitter space S1n+1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product Hk(1−coth2 ϱ)× Sn−k(1 − tanh2 ϱ), 1 < k < n − 1 in S1n+1(1), the hyperbolic cylinder H1(1 − coth2 ϱ) × Sn-1(1 − tanh2 ϱ) or spherical cylinder Hn−1(1 − coth2 ϱ) × S1(1 − tanh2 ϱ) in S1n+1 (1)

Complete maximal spacelike surfaces in an anti-de Sitter space {\bf H}^{4}_{2}(c)

In this paper, we prove that if M^2 is a complete maximal spacelike surface of an anti-de Sitter space {\bf H}^{4}_{2}(c) with constant scalar curvature, then S=0, S={-10c\over 11}, S={-4c\over 3} or S=-2c, where S is the squared norm of the second fundamental form of M^{2}. Also(1) S=0 if and only if M^2 is the totally geodesic surface {\bf H}^2(c);(2) S={-4c\over 3} if and only if M^2 is the hyperbolic Veronese surface;(3) S=-2c if and only if M^2 is the hyperbolic cylinder of the totally geodesicsurface {\bf H}^{3}_{1}(c) of {\bf H}^{4}_{2}(c).1991 Mathematics Subject Classifaction 53C40, 53C42.