6,815 research outputs found
Right triangles with algebraic sides and elliptic curves over number fields
Given any positive integer n, we prove the existence of infinitely many right
triangles with area n and side lengths in certain number fields. This
generalizes the famous congruent number problem. The proof allows the explicit
construction of these triangles; for this purpose we find for any positive
integer n an explicit cubic number field Q(\lambda) (depending on n) and an
explicit point P_\lambda of infinite order in the Mordell-Weil group of the
elliptic curve Y^2=X^3-n^2*X over Q(\lambda).Comment: To appear in Math. Slovac
The Use of Ligands in Copper-Catalyzed [3+2] Azide-Alkyne Cycloaddition: Clicker than Click Chemistry?
Well-defined copper(I) complexes for Click azide-alkyne cycloaddition reactions: one Click beyond
Copper(I)-Phosphinite Complexes in Click Cycloadditions: Three-Component Reactions and Preparation of 5-Iodotriazoles
© 2016 The Authors. Published by Wiley-VCH Verlag GmbH & Co. KGaA.The remarkable activity displayed by copper(I)–phosphinite complexes of general formula [CuBr(L)] in two challenging cycloadditions is reported: a) the one-pot azidonation/cycloaddition of boronic acids, NaN3, and terminal alkynes; b) the cycloaddition of azides and iodoalkynes. These air-stable catalysts led to very good results in both cases and the expected triazoles could be isolated in pure form under ‘Click-suitable’ conditions
The Full Automorphism Group of a Cyclic -gonal Surface
If is prime, a compact Riemann surface of genus is called
cyclic -gonal if it admits a cyclic group of automorphisms of order
such that the quotient space has genus 0. If in addition
is not normal in the full automorphism , then we call a non-normal
cyclic -gonal group. In the following we classify all non-normal -gonal
groups.Comment: 18 pages, 5 figure
[(NHC)(2)Cu]X Complexes as Efficient Catalysts for Azide-Alkyne Click Chemistry at Low Catalyst Loadings
Taming Brønsted acid reactivity: nucleophilic substitutions of propargylic alcohols with n-nucleophiles mediated by phosphorus-based Brønsted acid catalysts
The activity of diethyl phosphite and diphenyl phosphate in propargylation reactions with N-nucleophiles of varying basicity is presented. Careful choice of the reaction conditions minimised undesired rearrangements and arylation processes, typical side reactions with Brønsted acid catalysis. These systems are compatible with technical solvents and presence of air and they are also applicable to C-, O- and S-nucleophile
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