The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

We give explicit formulas for the number Un(N) of closed polygonal paths of length N (starting from the origin) whose steps are n roots of unity, as well as asymptotic expressions for these numbers when N → ∞. We also prove that the sequences (Un(N))N≥0 are P -recursive for each fixed n ≥ 1 and leave open the problem of determining the values of N for which the dual sequences (Un(N))n≥1 are P -...

In this paper we study von Neumann un-biasing normalisation for ideal and real quantum random number generators, operating on finite strings or infinite bit sequences. In the ideal cases one can obtain the desired un-biasing. This relies critically on the independence of the source, a notion we rigorously define for our model. In real cases, affected by imperfections in measurement and hardware...

A 10-year study revealed that after severe poaching (>93% killed) of elephants (Loxodonta africana) in Zambia’s North Luangwa National Park (NLNP) during the 1970s and 1980s, the age of reproduction in females was greatly reduced. Fifty-eight per cent of births were delivered by females aged 8.5–14 years, an age at which elephants were reported to be sexually immature in nearby South Luangwa Na...

because of reducing the number of spectators of football premier league, this study is seeking for identifying factors put the most impact on this decline. the statistical population consisted of all spectators in iran football premier league in 2015-16, out of which 395 spectators were randomly selected as the research samples. the data were collected via a researcher-made questionnaire. the f...

Given a planar vector field U which generates the Lie symmetry of some other vector field X , we prove a new criterion to control the stability of the periodic orbits of U . The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenh...

Given a set Un = {0, 1, ..., n−1}, a collectionM of subsets of Un that is closed under intersection and contains Un is known as a Moore family. The set of Moore families for a given n, denoted by Mn, increases very quickly with n, thus |M3| is 61 and |M4| is 2480. In [1] the authors determined the number for n = 6 and stated a 24hcomputation-time. Thus, the number for n = 7 can be considered as...

In 1914 S. Ramanujan recorded a list of 17 series for 1/π. We survey the methods of proofs of Ramanujan’s formulae and indicate recently discovered generalisations, some of which are not yet proven. Let us start with two significant events of the 20th century, in the opposite historical order. At first glance, the stories might be thought of a different nature. In 1978, R. Apéry showed the irra...