In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are "Neben type" modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM.
In the present work, we prepared core-corona-type nanospheres bearing a thermoresponsive polymer with a controlled chain length on their surface. The corona layers were composed of poly(N-isopropylacrylamide) (PNIPAAm) chains (Mn = 3000-18,000) with a narrow polydispersity index prepared by atom-transfer radical polymerization (ATRP). Nanospheres were prepared by dispersion copolymerization of styrene with the PNIPAAm macromonomer in a polar solvent. The obtained nanospheres were monodisperse in diameter. The diameter of the nanospheres was regulated either by the number or chain length of the PNIPAAm macromonomers. In fact, the nanosphere diameter was regulated from ca. 100 to 1000 nm. When two types of PNIPAAm macromonomers are used, the obtained nanospheres have two different kinds of PNIPAAm on their surface. The surface of the nanospheres was observed to be thermoresponsive nanosphere in 0, 50, 100 mmol L(-1) NaCl aqueous solution. The nanosphere diameter and the surface-grafted polymer were concurrently adjusted for use in biomedical applications.
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