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Register a new userFelix Kleins On the eleventh degree transformation of elliptic functions
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Yonathan Stone
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This is an English translation of Felix Kleins paper Ueber die Transformation elfter Ordnung der elliptischen Functionen from 1879.
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This is an English translation of Felix Kleins classical paper Uber die Auflosung der allgemeinen Gleichungen funften und sechsten Grades (Auszug aus einem Schreiben an Herrn K. Hensel) from 1905 and is put in modern notation. The original work first appeared in the Journal for Pure and Applied Mathematics (Volume 129) and then was reprinted in Mathematische Annalen (Volume 61, Issue 1). Kleins work (including his Lectures on the Icosahedron and the Solution of Equations of Fifth Degree) lies at the heart of the 19th and 20th work on solving generic polynomials. In this paper, Klein summarizes his approach to solving the generic quintic and sextic polynomials. He also lays the foundation for the modern framework of resolvent degree.
We reinterpret ideas in Kleins paper on transformations of degree~$11$ from the modern point of view of dessins denfants, and extend his results by considering dessins of type $(3,2,p)$ and degree $p$ or $p+1$, where $p$ is prime. In many cases we determine the passports and monodromy groups of these dessins, and in a few small cases we give drawings which are topologically (or, in certain examples, even geometrically) correct. We use the Bateman--Horn Conjecture and extensive computer searches to support a conjecture that there are infinitely many primes of the form $p=(q^n-1)/(q-1)$ for some prime power $q$, in which case infinitely many groups ${rm PSL}_n(q)$ arise as permutation groups and monodromy groups of degree $p$ (an open problem in group theory).
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.
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.
Quadratic Unconstrained Binary Optimization (QUBO) is recognized as a unifying framework for modeling a wide range of problems. Problems can be solved with commercial solvers customized for solving QUBO and since QUBO have degree two, it is useful to have a method for transforming higher degree pseudo-Boolean problems to QUBO format. The standard transformation approach requires additional auxiliary variables supported by penalty terms for each higher degree term. This paper improves on the existing cubic-to-quadratic transformation approach by minimizing the number of additional variables as well as penalty coefficient. Extensive experimental testing on Max 3-SAT modeled as QUBO shows a near 100% reduction in the subproblem size used for minimization of the number of auxiliary variables.
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