Inside every eukaryotic cell is the nucleus, organelles, and the surrounding cytoplasm, which typically accounts for 50% of the cell volume. The cytoplasm is a complex mixture of water, protein, and a dynamic polymer network. Cells use cytoplasmic streaming to transmit chemical signals, to distribute nutrients, and to generate forces involved in locomotion. In this talk we present two different models related to cytoplasmic streaming in amoeboid cells. In the first part of the talk, we present a computational model to describe the dynamics of blebbing, which occurs when the cytoskeleton detaches from the cell membrane, resulting in the pressure-driven flow of cytosol towards the area of detachment and the local expansion of the cell membrane. The model is used to explore the relative roles in bleb dynamics of cytoplasmic viscosity, permeability of the cytoskeleton, and elasticity of the membrane and cytoskeleton. In the second part of the talk we examine how flow-induced instabilities of cytoplasm are related to the structural organization of the giant amoeboid cell Physarum polycephalum. We use a multiphase flow model that treats both the cytosol and cytoskeleton as fluids each with its own material properties and internal forces, and we discuss instabilities of the sol/gel mixture that produce flow channels within the gel. We analyze a reduced model and offer a new and general explanation for how fluid flow is involved in cytoskeletal reorganization.

While the superconvergence phenomenon is well understood for the h-version finite element method, the relevant study for the p-version finite element method and the spectral method is lacking.

In this work, superconvergence properties for some high-order orthogonal polynomial interpolations are studied. The results are twofold: When interpolating function values, we identify those points where the first and second derivatives of the interpolant converge faster; When interpolating the first derivative,we locate those points where the function value of the interpolant superconverges. For both cases we consider various Chebyshev polynomials, but for the latter case, we also include the counterpart Legendre polynomials.

The talk will review some elementary, but amusing, results concerning
surface spectra for discrete Laplacians on half-planes with a boundary.In particular, interesting differences arise for square lattices with
straight boundaries between the case where the boundary has the same
direction of the lattice and the one where the boundary is slanted
at an angle of 45 degrees to the direction of the lattice. This
is joint work with Y. Kreimer.