On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

Heterocyclisation of Unsaturated Phenylsulfides. Synthesis of a Novel Series of 2 Substituted-2,3-dihydrobenzothiazoles

Aminoarylhydroxy thioalkenes react with tosyl derivatives to provide mainly N substituted -2 vinyl -2,3 dihydrobenzothiazoles. A different regiochemistry of cyclization depending on whether a methyl or a phenyl susbtituent was used, was observed Formation of -2H-3,4-dihydrobenzothiazines and dienes was observed.     

Poly(trimethylene carbonate)/Poly(malic acid) Amphiphilic Diblock Copolymers as Biocompatible Nanoparticles

Amphiphilic polycarbonate–poly(hydroxyalkanoate) diblock copolymers, namely, poly(trimethylene carbonate) (PTMC)‐b‐poly(β‐malic acid) (PMLA), are reported for the first time. The synthetic strategy relies on commercially available catalysts and initiator. The controlled ring‐opening polymerization (ROP) of trimethylene carbonate (TMC) catalyzed by the organic guanidine base 1,5,7‐triazabicyclo[4.4.0]dec‐5‐ene (TBD), associated with iPrOH as an initiator, provided iPrO?PTMC?OH, which served as a macroinitiator in the controlled ROP of benzyl β‐malolactonate (MLABe) catalyzed by the neodymium triflate salt (Nd(OTf)₃). The resulting hydrophobic iPrO?PTMC‐b‐PMLABe?OH copolymers were then hydrogenolyzed into the parent iPrO?PTMC‐b‐PMLA?OH copolymers. A range of well‐defined copolymers, featuring different sizes of segments (M_n,NMR up to 9300 g mol^?1; Ð_M=1.28–1.40), were thus isolated in gram quantities, as evidenced by NMR spectroscopy, size exclusion chromatography, thermogravimetric analysis, differential scanning calorimetry, and contact angle analyses. Subsequently, PTMC‐b‐PMLA copolymers with different hydrophilic weight fractions (11–75 %) self‐assembled in phosphate‐buffered saline upon nanoprecipitation into well‐defined nano‐objects with D_h=61–176 nm, a polydispersity index <0.25, and a negative surface charge, as characterized by dynamic light scattering and zeta‐potential analyses. In addition, these nanoparticles demonstrated no significant effect on cell viability at low concentrations, and a very low cytotoxicity at high concentrations only for PTMC‐b‐PMLA copolymers exhibiting hydrophilic fractions over 47 %, thus illustrating the potential of these copolymers as promising nanoparticles.     

Cryopreservation of Prunus avium L. embryogenic tissues

Embryogenic tissues from wild cherry (Prunus avium L.) were successfully cryopreserved by using a one-step freezing procedure. Cryoprotection consisted of a pretreatment on solid medium with increasing sucrose concentrations (0.25 M for 1 day, 0.5 M for 1 day, 0.75 M for 2 days, and 1.0 M for 3 days), followed by air desiccation to about 20 percent moisture content (fresh weight basis). This method was compared with a pretreatment on solid medium containing 5 percent DMSO and 2 percent proline, followed by immersion in a modified PVS2 cryoprotective solution. Pretreatment on solid medium with increasing concentrations of sucrose led to regrowth of frozen embryogenic tissues, and after 6 weeks of culture, growth was comparable to that of non-dehydrated and non-frozen tissues. By contrast, no regrowth was observed when embryogenic tissues were submitted to the solid/liquid pretreatment with DMSO/proline and a modified PVS2 solution.     

Neue Begründung der Bolyai-Lobatschefskyschen Geometrie

Ohne Zusammenfassung     

Über die Theorie der relativ-Abel'schen Zahlkörper

Ohne Zusammenfassung Mit geringen ?nderungen abgedruckt aus den Nachrichten der Kgl. Ges. der Wiss. zu G?ttingen 1898. Inzwischen sind folgende auf diesen Gegenstand bezügliche Inaugural-Dissertationen in G?ttingen erschienen:Das quadratische Reciprocit?tsgesetz im quadratischen Zahlk?rper mit der Classenzahl 1. vonH. D?rrie 1898,Tafel der Klassenanzahlen für kubische Zahlk?rper vonL. W. Reid 1899,Das allgemeine quadratische Reciprocit?tsgesetz in ausgew?hlten Kreisk?rpern der 2 ^{h ten} Einheitswurzeln vonK. S. Hilbert 1900,Quadratische Reciprocit?tsgesetze in algebraischen Zahlk?rpern vonG. Rückle 1901. Insbesondere die letzte Dissertation enth?lt zahlreiche und interessante Beispiele zu der hier entwickelten Theorie.