A sequence of rational integers {An} is said to be a divisibility sequence if Am | An whenever m | n. If the divisibility sequence {An} also satisfies a linear recurrence relation of order k, it is said to be a linear divisibility sequence. The best known example of a linear divisibility sequence of order 2 is the Lucas sequence {un}, one particular instance of which is the famous Fibonacci seq...

where u0 = 05 U\ 13 and a and b are integers. We will call a and b the parameters of the recurrence. We will denote such a sequence as u(a3 b) . Two of the most important questions concerning these sequences are: For a given PFLS w(a, 2?), which odd primes have a maximal rank of apparition? and For which odd primes does the PFLS u(a5 b) have a maximal period modulo p? No definitive results are ...

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in H. This is due to...

is called the characteristic polynomial of the sequence U. In the case where P = -g = 1, the sequence U is the Fibonacci sequence and we denote its terms by F0, Fl9 F2, ... . Let p be an odd prime with p\Q and let e > 1 be an integer. The positive integer u = u(p) is called the rank of apparition of p in the sequence U if p\Uu and p\Um for 0 < m < u; furthermore, u = u(p) is called the period o...

This is the second part of a two-part article that provides in-depth background information and a critical examination of a phenomenon that has importance not only to transpersonal psychology but also to psychology and science more generally – the protracted Marian apparition at Medjugorje. Part 1 presented an outer history of the case and examined the phenomenon from the perspectives of critic...