Asymptotic values of quasiregular maps

Suslin analytic sets characterize the sets of asymptotic values of entire holomorphic functions. By a theorem of Ahlfors, the set of asymptotic values is finite for a function with finite order of growth. Quasiregular maps are a natural generalization of holomorphic functions to dimensions n ≥ 3 and, in fact, many of the properties of holomorphic functions have counterparts for quasiregular maps. It is shown that analytic sets also characterize the sets of asymptotic values of quasiregular maps in Rn, even for those with finite order of growth. Our construction is based on Drasin's quasiregular sine function

Application of the Boundary Method to the determination of the properties of the beam cross-sections

Using the 3-D equations of linear elasticity and the asylllptotic expansion methods in terms of powers of the beam cross-section area as small parameter different beam theories can be obtained, according to the last term kept in the expansion. If it is used only the first two terms of the asymptotic expansion the classical beam theories can be recovered without resort to any "a priori" additional hypotheses. Moreover, some small corrections and extensions of the classical beam theories can be found and also there exists the possibility to use the asymptotic general beam theory as a basis procedure for a straightforward derivation of the stiffness matrix and the equivalent nodal forces of the beam. In order to obtain the above results a set of functions and constants only dependent on the cross-section of the beam it has to be computed them as solutions of different 2-D laplacian boundary value problems over the beam cross section domain. In this paper two main numerical procedures to solve these boundary value pf'oblems have been discussed, namely the Boundary Element Method (BEM) and the Finite Element Method (FEM). Results for some regular and geometrically simple cross-sections are presented and compared with ones computed analytically. Extensions to other arbitrary cross-sections are illustrated.