Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end by arguing that it would be inconsistent for mathematicians to suspend judgement on the truth of mathematical propositions on the basis of worries about the reliability of randomized algorithms.
Simple Synthetic Transformations of Highly Enantio-Enriched 4-Alkyl-2,5,7-cyclooctatrienols into Functionalized Bicyclo[4.2.0]octa-2,4-dienes and 2,6-Cyclooctadienones
作者:Mauro Pineschi、Federica Del Moro、Paolo Crotti、Franco Macchia
DOI:10.1002/ejoc.200400355
日期:2004.11
4-Alkyl-2,5,7-cyclooctatrienols with high enantiomeric purities (93−96% ee), obtained by an improved copper-phosphoramidite-catalyzed addition of dialkylzinc reagents to cyclooctatetraene monoepoxide, have been subjected to simple and practical manipulations. Simple esterification reactions are able to deliver new functionalized isomeric 8-alkyl-7-acyloxybicyclo[4.2.0]octa-2,4-dienes through a domino
The synthesis of the endoperoxide of cyclooctatetraene via photosensitized singlet oxygenation
作者:Waldemar Adam、Günter Klug
DOI:10.1016/s0040-4039(00)88583-4
日期:1982.1
7,8-Dioxabicyclo[4.2.2.]deca-2,4,9-triene (3), the hitherto unknown endoperoxide of cyclooctatetraene, was prepared via cycloaddition of singlet oxygen and characterized by spectroscopic and chemical means.
The Reaction of Cyclooctatetraene Oxide with Grignard Reagents
作者:Tsutomu Matsuda、Masahiko Sugishita
DOI:10.1246/bcsj.40.174
日期:1967.1
The reaction of cyclooctatetraeneoxide with ethylmagnesium bromide, phenylmagnesium bromide and t-butylmagnesium bromide was found to yield 2, 4, 6-cycloheptatrienyl-1-ethyl-, phenyl- and t-butylcarbinol respectively. The treatment of cyclooctatetraeneoxide with a catalytic amount of magnesium bromide in ether afforded phenyl acetaldehyde in a 76% yield. The dehydration of the carbinols with a catalytic
Copper-Catalyzed Highly Enantioselective Synthesis of Cyclic Allylic and Homoallylic Alcohols with Dialkylzinc Reagents
作者:Mauro Pineschi、Federica Del Moro、Paolo Crotti、Valeria Di Bussolo、Franco Macchia
DOI:10.1055/s-2004-834864
日期:——
The copper-phosphoramidite catalyzed addition of dialkylzinc reagents to racemic or meso allylic epoxides can be synthetically exploited in different ways, depending on the substrates and reaction conditions used.