Nonexistence for hyperbolic problems on Riemannian manifolds1

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

When the extension property does not hold

A complete extension theorem for linear codes over a module alphabet, equipped with the symmetrized weight composition, is proven. It is also shown that the extension property with respect to a general weight function does not hold for module alphabets, that have a noncyclic socle.

Determination of Weight Functions for Cracks Under Mode-II Loading

For cracks under mode-I loading, it has been demonstrated that a general weight function expression with three unknown parameters can be used to approximate a variety of crack configurations under mode-I loading. For a given crack geometry, the unknown parameters can be determined from reference stress intensity factors (SIFs) together with characteristic properties of the weight functions. It is demonstrated in this paper that a general weight function expression also exists for cracks under mode II loading. The determination of weight functions for cracks in mode II can then also be conducted using reference stress intensity factors (SIFs) together with characteristic properties of the weight functions. This method is used to obtain the mode II weight functions for test specimens including single edge cracked plate, internal center cracked plate and double edge cracked plate. These derived weight functions were further used to calculate the SIFs for the above cracks subjected to several linear and non-linear shear loads and were compared to available SIF solutions.