A Total Order in (0, 1] Defined Through a ‘Next’ Operator

Order - A ‘next’ operator, σ, is built on the set R 1 =(0, 1] ? {1 ? 1 / e} defining a partial order that, with the help of the axiom of choice, can be extended to a...     

A singular function with a non-zero finite derivative

This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its generalizations and variants—the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative 1 on a subset of the Cantor set.     

On actually computable bijections between ℕ and Q +

We present three different ways of getting an actually computable enumeration of Q ⁺ in the sense of being able to know exactly which rational occupies a given position and vice versa. The first enumeration is based on the Pierce expansion model for representing real numbers. The other two are based on regular continued fractions.The first and third authors were supported by UPF Grant, Support d'Iniciació a la Recerca, # F3087613.     

Photocycloadditions of 2-(trimethylsilyloxy)-1,3-butadiene to 2-cycloalkenones. Access to the basic pentalenolactone skeleton

The [2 + 2] Photocycloaddition of 2-(trimethylsilyloxy)-1,3-butadiene to a number of 2-cycloalkenones proved to be quite a general reaction leading to good yields of the cycloadducts (Table). This finding is surprising since dienes, in general, are better known as quenchers of enone triplets rather than as photochemical reactants. Both the high substrate concentrations, which can be employed in these cycloadditions, and the remarkable regio and stereoselectivity of the processes qualify them as valuable for syntheses. In a first application, the photoproducts 1a, b were transformed in three steps into a viable precursor 4 of the pentalenolactone-G and -H antibiotics.