A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional gr...

We investigate the vertical foliation of the standard complex contact structure on Γ \ Sl(2,C), where Γ is a discrete subgroup. We find that, if Γ is nonelementary, the vertical leaves on Γ \Sl(2,C) are holomorphic but not regular. However, if Γ is Kleinian, then Γ \ Sl(2,C) contains an open, dense set on which the vertical leaves are regular, complete and biholomorphic to C∗. If Γ is a uniform...

We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence of a two-dimensional foliation to which the motion is constrained. In particular, we derive a new infinite class of potentials for which the motion is assur...

On this note we prove that a holomorphic foliation of the projective plane with rich, but finite, automorphism group does not have invariant algebraic curves.

Rock anisotropy plays an important role in engineering behavior of rocks. Slates are anisotropic rocks which have long been used for gable roof, floor tiles, borrow materials, and other purposes. The slates studied in this research are calcareous and have a porphyro-lepidoblastic texture. To determine the role of the anisotropy on the tensile strength and fracture pattern, two variables includi...

The tensile strength of rocks plays a noteworthy role in their failure mechanism, and its determination can be beneficial in optimizing the design of the rock structures. Schistose rocks due to their inherent anisotropy in different foliation directions show a diverse strength at each direction. The purpose of this work was to compare and assess the tensile strength of phyllite, which was obtai...