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Register a new userG.N. Chebotarevs On the Problem of Resolvents
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Alexander J. Sutherland
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This is an English translation of G.N. Chebotarevs classical paper On the Problem of Resolvents, which was originally written in Russian and published in Vol. 114, No. 2 of the Scientific Proceedings of the V.I. Ulyanov-Lenin Kazan State University. In this paper, Chebotarev extends the method in Wimans On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations to argue that the general polynomial of degree 21 admits a solution using algebraic functions of at most 15 variables. However, his and Wimans proofs assume that certain intersections in affine space are generic without proof.
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This is an English translation of Anders Wimans classical paper {U}ber die Anwendung der Tschirnhausen Transformation auf die Reduktion algebraischer Gleichungen from 1927. The original work first appeared in the 1927 extraordinary edition of Nova Acta Regiae Societatis Scientiarum Upsaliensis. In this paper, Wiman gives an argument that the general polynomial of degree nine can be solved using one algebraic function of four variables and accessory irrationalities of degree at most five. However, his argument assumes certain intersections in affine space are generic without proof.
This is an English translation of Felix Kleins paper Ueber die Transformation elfter Ordnung der elliptischen Functionen from 1879.
The Secret Santa ritual, where in a group of people every member presents a gift to a randomly assigned partner, poses a combinatorial problem when considering the probabilities involved in the formation of pairs, where two persons exchange gifts mutually. We give different possible derivations for such probabilities by counting fixed-point-free permutations with certain numbers of 2-cycles.
We present updates to the problems on Hirzebruchs 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about contact structures, and about (complementary pairs of) foliations.
The Coupon Collectors Problem is one of the few mathematical problems that make news headlines regularly. The reasons for this are on one hand the immense popularity of soccer albums (called Paninimania) and on the other hand that no solution is known that is able to take into account all effects such as replacement (limited purchasing of missing stickers) or swapping. In previous papers we have proven that the classical assumptions are not fulfilled in practice. Therefore we define new assumptions that match reality. Based on these assumptions we are able to derive formulae for the mean number of stickers needed (and the associated standard deviation) that are able to take into account all effects that occur in practical collecting. Thus collectors can estimate the average cost of completion of an album and its standard deviation just based on elementary calculations. From a practical point of view we consider the Coupon Collectors problem as solved. ----- Das Sammelbilderproblem ist eines der wenigen mathematischen Probleme, die regelma{ss}ig in den Schlagzeilen der Nachrichten vorkommen. Dies liegt einerseits an der gro{ss}en Popularitat von Fu{ss}ball-Sammelbildern (Paninimania genannt) und andererseits daran, dass es bisher keine Losung gibt, die alle relevanten Effekte wie Nachkaufen oder Tauschen berucksichtigt. Wir haben bereits nachgewiesen, dass die klassischen Annahmen nicht der Realitat entsprechen. Deshalb stellen wir neue Annahmen auf, die die Praxis besser abbilden. Darauf aufbauend konnen wir Formeln fur die mittlere Anzahl benotigter Bilder (sowie deren Standardabweichung) ableiten, die alle in der Praxis relevanten Effekte berucksichtigen. Damit konnen Sammler die mittleren Kosten eines Albums sowie deren Standardabweichung nur mit Hilfe von elementaren Rechnungen bestimmen. Fur praktische Zwecke ist das Sammelbilderproblem damit gelost.
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