We propose new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The...

Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions an ellipse $E$ are determined. We prove that, any invariant curve the billiard map on boundary phase space $B^\* E$ ellipse, there exists a sequence whose concentrates curve. use this result to give new proof that ellipses infinitesimally spectrally rigid among $C^\\infty$ domains with symmetries ellipse.

We introduce eigenfunctions of the Reed-Muller transform. Eigenfunctions are functions whose canonical sumof-products expression and PPRM (positive polarity ReedMuller expression) are isomorphic. In the case of symmetric functions, the eigenfunction can be viewed as a function whose reduced truth vector is identical to the reduced ReedMuller spectrum. We show that the number of symmetric (ordin...