Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Regulation of Spermatogenesis: Differentiation of GFP-labeled Stem Cells and the Function of Cytoplasmic Bridges

During spermatogenesis, different genes are expressed in a strictly coordinated fashion providing an excellent model to study cell differentiation. Recent identification of testis specific genes and the development of green fluorescence protein (GFP) transgene technology and an in vivo system for studying the differentiation of transplanted male germ cells in infertile testis has opened new possibilities for studying the male germ cell differentiation at molecular level. We have employed these techniques in combination with transillumination based stage recognition (Parvinen and Vanha-Perttula, 1972) and squash preparation techniques (Parvinen and Hecht, 1981) to study the regulation of male germ cell differentiation. By using transgenic mice expressing enhanced-(E)GFP as a marker we have studied the expression and hormonal regulation of beta-actin and acrosin proteins in the developmentally different living male germ cells. Beta-actin was demonstrated in all male germ cells, whereas acrosin was expressed only in late meiotic and in postmeiotic cells. Follicle stimulating hormone stimulated b-actin-EGFP expression at stages I-VI and enhanced the formation of microtubules in spermatids and this way reduced the size of the acrosomic system. When EGFP expressing spermatogonial stem cells were transplanted into infertile mouse testis differentiation and the synchronized development of male germ cells could be observed during six months observation time. Each colony developed independently and maintained typical stage-dependent cell associations. Furthermore, if more than two colonies were fused, each of them was adjusted to one stage and synchronized. By studying living spermatids we were able to demonstrate novel functions for Golgi complex and chromatoid body in material sharing between neighbor spermatids. Immunosytochemical analyses revealed a transport of haploid cell specific proteins in spermatids (TRA54 and Shippo1) and through the intercellular bridges (TRA54). Cytoskeleton inhibitor (nocodazole) demonstrated the importance of microtubules in material sharing between spermatids and in preserving the integrity of the chromatoid body. Golgi complex inhibitor, brefeldin A, revealed the great importance of Golgi complex i) in acrosomic system formation ii) TRA54 translation and in iii) granule trafficking between spermatids.