Elliptic Algebra and Integrable Models for Solitons on Noncummutative Torus T{\cal T}

We study the algebra ${\cal A}_n$ and the basis of the Hilbert space ${\cal H}_n$ in terms of the $\theta$ functions of the positions of $n$ solitons. Then we embed the Heisenberg group as the quantum operator factors in the representation of the transfer matrice of various integrable models. Finally we generalize our result to the generic $\theta$ case.Comment: Talk given by Bo-Yu Hou at the Joint APCTP-Nankai Symposium. Tianjin (PRC), Oct. 2001. To appear in the proceedings, to be published by Int. J. Mod. Phys. B. 7 pages, latex, no figure

Ultrastructural and Cytochemical Studies of Gonadal Organization, Spermatogenesis and Spermatozoa of Molluscs

The variety of different types of molluscan spermatozoa, spermatogenesis and gonadal organization was demonstrated in this study. The primitive type of spermatozoon was described in Antalis entalis (Scaphopoda); an intermediate type in Rossia macrosoma (Decapoda) and the modified type in Archidoris tuberculata. Tritonia plebeia. Aplysia faciata (Opisthobranchia) and Arion hortensis (Pulmonata). The spermatozoon of Antalis entalis (Scaphopoda) consisted of a head, midpiece and tail. The acrosome surmounted the nucleus and was membrane-bound. It was formed by an acrosomal granule surrounding an acrosomal fossa into which was invaginated the acrosomal and plasma membranes to form a pit extending to half the acrosomal depth. The nucleus was an elongate truncated cone containing a few nuclear cavities, primarily within the posterior portion. Five large, spherical mitochondria were closely apposed to the nuclear base. There was a proximal and distal centriole orientated perpendicularly within the mid-piece. A discrete satellite complex was associated with the distal centriole. The axoneme was of typical "9+2" structure enveloped by the plasmalemma. During spermatogenesis, the spherical proacrosome was secreted by a single Golgi complex, migrated toward the anterior pole of the spermatid, and there differentiated to assume the form of an electron-dense collar. Nuclear condensation did not occur until the proacrosome and mitochondria migrated towards the potential anterior and posterior poles of the spermatid. Chromatin condensation occurred by the formation of heterochromatin granules. No manchette of microtubules was observed in the course of nuclear condensation and elongation. Mitochondria achieved their mature spherical shape by decreasing in number and increasing in diameter at an early stage of spermiogenesis. No inter-mitochondrial or mitochondria-nuclear (or centriolar) junctions occurred. Beta glycogen granules appeared late in the development of the spermatid, and were mostly distributed in the centriolar region. This study is the first ultrastructural investigation of Scaphopod spermatogenesis. The organization of the testis and process of spermiogenesis of the decapod mollusc Rossia macrosoma was investigated by electron microscopy. Cytochemical analysis, by use of the periodic acid-thiosemicarbazide-silver proteinate (PA-TSC-SP) technique, allowed the precise localization of glycogen in the spermatozoon. Nuclear morphogenesis occurred in three stages (termed A, B and C), resulting in the transformation of the nuclear shape from a spherical (stage A) into an elongated form (stages B and C). Prior to elongation of the spermatid nucleus, the sac-like proacrosome, derived from a single Golgi complex, became located at the apex of the nucleus and further transformed into an elongated structure surrounded by longitudinally orientated microtubules as spermatid elongation occurred. A group of mitochondria assumed an interwoven form in the mitochondrial pocket. Large quantities of glycogen granules, beta-particles about 130-145 A in diameter, were demonstrated in the midpiece in connection with the specialized external plasmamembrane of the mitochondrial sleeve and within or around the axoneme in the tail piece. The most striking feature of the distribution of glycogen was nine rows of intra-gamma-fibres particles, at a spacing of 20-26 um, along the longitudinal axis. The pattern of glycogen distribution in the spermatozoon was compared with that in octopods and other molluscan groups. Comparative ultrastructural features of the ovotestes and spermatogenesis of three opisthobranchs, Archidoris tuberculata. Tritonia plebeia and Aplysia faciata. were described. Early stages of spermatogenesis appeared similar among these three species. However, prominent structural differences were demonstrated with respect to later stages of spermiogenesis and spermatozoa. The ovotestis contained two types of somatic cells within the testicular portion: flattened myoepithelial cells defined the outer acinar wall and accessory cells overlay the basal lamina on the luminal aspect of the male compartment. Early spermatogenesis was similar to that described in other opisthobranchs. In contrast, the late stages of spermiogenesis presented great morphological differences. Stages of spermatid development were divided into Pre-cup, Cup, and Elongating stages depending on the general shape of the nucleus. Nuclear differentiation involved the formation of anterior and posterior nuclear plaques, condensation of chromatin fibrils into lamellae, the insertion of the centriole into the nuclear fossa and eventually formation of an elongated sperm head. There is a discrete difference of helical form in the three species. The present study provided evidence to comfirm the existence of an acrosome in opisthobranchs and its origin from the Golgi complex. (Abstract shortened by ProQuest.)