Robert Pertsch Gilbert

An explicit formula for the polyharmonic Green function of any order for the unit disc of the complex plane based on the work of E. Almansi is widely known through I.N. Vekua’s 1967 book. A simple, however, not easy way to prove the formula is by verification of the defining properties. P.R. Garabedian has suggested a constructive way for this function in his 1964 pde book but only has applied it for the biharmonic case. His construction is not smoothly working for higher order as is explained here in the triharmonic case. The general case will demand very concentrated calculations. The procedure should similarly work also for the half plane.

In 1972, I came upon the idea to begin a journal on Applied Analysis. It was not to be another engineering mathematics journal, nor we want it to be just another analysis journal. The stress was to...

Andro Mikelic passed away on 28th November 2020, after a sudden and short illness. He was a professor of Mathematics at the Institut Camille Jordan, the Mathematics Department of the University of ...

The present contribution focuses on the receptor driven endocytosis typical of viral entry into a cell. The process is characterized by a local increase in receptor density necessary to establish contact between the cell and the virus. While the receptors of the virus are fixed on its surface, the receptors of the cell are able to move over its membrane, which leads to a local change in their concentration. In the model developed, the receptor motion is described by the diffusion equation along with two boundary conditions. The boundary conditions represent the balance of fluxes at the front of the contact area, where the velocity is assumed to be proportional to the gradient of the chemical potential, and the energy balance behind and before the front, causing the fronts movement. The model provides a basis to incorporate different phenomena such as for example cooperativity. This property strongly influences the effective binder density necessary to establish contact. The moving boundary problem describing the process is numerically solved by using the finite difference method and applied to study the change of receptor density over the membrane as well as the motion of the adhesion front. Two features are investigated in particular: The process initiation is analyzed in order to obtain information on possibilities for inhibiting the viral entry, and the non‐dimensional analysis is performed to minimize the number of necessary process parameters and to check their priority.

We outline the mathematical model of the ultrasonic response of cancellous bone and its time harmonic formulation. In contrast to the Biot model, the fluid is not inviscid. Our fluid is viscous, but does not interact with the solid components.

This chapter considers the definition of G-convergence and some basic facts about it. It introduces a corrector operator approach which is related to the method of oscillating test functions and also to the ”condition N”. The chapter utilizes the corrector approach to derive effective equations of viscoelastic mixtures with time-varying interface. It shows that effective constitutive equations contain long-memory terms, and that the effective material parameters satisfy certain auxiliary problems which resemble cell problems of periodic homogenization. The chapter aims to derive homogenized models for flows of incompressible viscoelastic mixtures with time-varying interface. Both phases are modeled as Kelvin-Voight materials: the elastic stresses satisfy Hook's law written in the spatial formulation, and the viscous stresses obey Newton's law. The chapter considers the fluid-structure interaction problem, where one phase is a solid-like viscoelastic material, and the other is a viscous Newtonian fluid.

This study focuses on the receptor driven endocytosis typical of viral entry into a cell. A locally increased density of receptors at the time of contact between the cell and the virus is necessary in this case. The virus is considered as a substrate with fixed receptors on its surface, while the receptors of the host cell are free to move over its membrane, allowing a local change in their concentration. In the contact zone the membrane inflects and forms an envelope around the virus. The created vesicle imports its cargo into the cell. The described process is simulated by the diffusion equation accompanied by two boundary conditions. The first boundary condition states that the conservation of binders expressed as the local rate of change of density has to be equal to the negative of the local flux divergence. The second boundary condition represents the energy balance condition with contributions due to the binding of receptors, the free energy of the membrane, its curvature and the kinetic energy due to the motion of the front. The described moving boundary problem in terms of the binder density and the velocity of the adhesion front is well posed and relies on biomechanically motivated assumptions. The problem is numerically solved by using the finite difference method, and the illustrative examples have been chosen to show the influence of the mobility of the receptors and of their initial densities on the velocity of the process. SIGNIFICANCE: The receptor driven endocytosis represents one of the most important mechanisms for the viral entry into a cell. However, the high velocities and small characteristic length scale cause many difficulties during the experimental investigation of such a process. This calls upon the application of virtual computer simulations investigating the process parameters and identifying factors inhibiting or completely ceasing the viral entry into cells. The development of methods for the optimization of the cell immunity system is aimed to as the final goal.