are not particularly noted for the pleasure which they give to their recipients. One meaning of the word dummy is a "counterfeit object." This seems to me the right word to describe a form of treatment which is intended to have no effect and I follow those who use it. A placebo is something which is intended to act through a psychological mechanism. It is an aid to therapeutic suggestion, but t...

We prove that every recursively enumerable language can be generated by a programmed grammar with context-free core rules using unconditional transfer with leftmost derivation of type 3 or type 2. We show that every recursively enumerable tally language can be generated by a programmed grammar with context-free core rules using unconditional transfer or by some k-limited ET0L system, where k 2 ...

We give a complete characterization of so called powerful arithmetic progressions, i.e. of progressions whose kth term is a kth power for all k. We also prove that the length of any primitive arithmetic progression of powers can be bounded both by any term of the progression different from 0 and ±1, and by its common difference. In particular, such a progression can have only finite length.

This paper suggests a exible framework for n-dimensional crossover, consisting of cutting, classi cation, and copying of genes. We prove that under this framework, any cutting strategy generates two equivalence classes of genes, making the framework appropriate as a crossover scheme. Three notable features of this framework are: (i) it enables more e ective use of genes' geographical linkage, (...