Qualitative analysis for solutions of a class of nonlinear ordinary differential equations
The boundary value problem (BVP) for a class of nonlinear ordinary differential equations is examined. It can be used to describe the eversion problem for a spherical shell composed of a class of transversely isotropic incompressible Mooney-Rivlin materials. The solutions of the nonlinear equation describing the relation among deformed thickness, initial thickness and material parameters of the spherical shell are obtained. The effects of structure parameter and material parameters on the thickness of the everted spherical shell are discussed by numerical examples.
Inflation and eversion of spherical shells of a special compressible material
A study is made of the qualitative features of the inflation of a shell of the so-called material of type II. This material has a strain energy function which is linear in i1, i3 and arbitrary in i2, where i1, i2 and i3 are the three principal invariants of the stretch tensors. For a specific form of the arbitrary function, a closed-form solution to the equations of equilibrium describing spherical eversion can also be found. By using this solution, a comparison can be made between the qualitative features of the regular and everted shells under internal pressure. It is found that the stress distribution in both cases is different and that a larger pressure can be sustained by the everted shell. However eversion has no effect on the other features studied.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.