A finite ring with an identity whose lattice of ideals forms a unique chain is called ring. Let R be commutative invariants p,n,r,k,m. It known that Eisenstein extension degree k Galois S=GR(pn,r). If p−1 does not divide k, the structure unit group U(R) known. The case (p−1)∣k was partially considered by M. Luis (1991) providing counterexamples demonstrated results Ayoub failed to capture direc...

In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, o...

Convergence is an issue being widely concerned about. Thus, in this paper, we mainly put forward two types of concepts of convergence in mean and convergence in distribution for the sequence of uncertain random variables. Then some of theorems are proved to show the relations among the three convergence concepts that are convergence in mean, convergence in measure and convergence in distributio...

We empirically searched for cycling and chaotic behavior in the innnite population Simple Genetic Algorithm. We found examples of period 2 cycling (which we expected) and long period cycling (which we didn't expect). These examples had mutation and crossover distributions which do not correspond to the way that mutation and crossover are normally used in practice. We also searched unsuccessfull...