We follow the rules: D is a non empty set, f1, f2 are finite sequences of elements of D, and i, n, n1, n2, n3, n4, n5, n6 are natural numbers. We now state two propositions: (1) If n ¬ len f1, then (f1 a f2)⇂n = ((f1)⇂n) a f2. (2) (f1 a f2)↾(len f1 + i) = f1 a (f2↾i). A Gene-Set is a non-empty non empty finite sequence. Let S be a Gene-Set. We introduce GA− SpaceS as a synonym of UnionS. Let f ...

We prove several results about the relations between injectivity and surjectivity for sand automata. Moreover, we begin the exploration of the dynamical behavior of sand automata proving that the property of ultimate periodicity is undecidable. We believe that the proof technique used for this last result might turn out to be useful for many other results in the same context.

This article presents some theorems about functor structures. We start with some basic lemmata concerning the composition of functor structures. Then, two theorems about the restriction operator are formulated. Later, we show two theorems stating that the properties ’full’ and ’faithful’ of functor structures which are equivalent to the ’onto’ and ’one-to-one’ properties of their morphmaps, res...

background: lead-based radiation shields are widely used in radiology departments to protect both workers and patients from any unnecessary exposure to ionizing radiation. recently there has been a great deal of concern expressed about the toxicity of lead. human lead toxicity is well documented. in that light, production of environmentally-friendly lead-free radiation shields with less weight ...

Abstract In this chapter, we consider basic axiomatic properties of ABC rules. These describe the behaviour such rules and offer insights into nature specific Important include Pareto efficiency, committee monotonicity, candidate consistency as well axioms pertaining to strategic voting.