For given integers a, b and j ≥ 1 we determine the set R a,b of integers n for which an − bn is divisible by nj . For j = 1, 2, this set is usually infinite; we determine explicitly the exceptional cases for which a, b the set R a,b (j = 1, 2) is finite. For j = 2, we use Zsigmondy’s Theorem for this. For j ≥ 3 and gcd(a, b) = 1, R a,b is probably always finite; this seems difficult to prove, h...

Settling depth of apple was determined by a water column and a digital camera. The depth was experimentally modeled using multivariate regression using a coded program in MATLAB software. The best models were based on the density, dropping height volume/mass with coefficient of determination and mean square error of 0.90 and 4.08, respectively.

We give a characterization theorem for non-degenerate plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree r of a non-degenerate foliation as above provides the minimum number, r+ 1, of points in the projective plane through which pass infinitely many algebraic leaves of the foliation.

It is shown that associated with each metric space (X, d) there is a compactification udX of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of udX are presented, and a detailed study of the structure of udX is undertaken. This culminates in a topol...

Let S ⊂ R3 be a C4-smooth relatively compact orientable surface with a sufficiently regular boundary. For β ∈ R+, let Ej(β) denote the jth negative eigenvalue of the operator associated with the quadratic form H(R) ∋ u 7→ ∫∫∫

The usual ordering “≤" on R is a total ordering, that is, for any two real numbers in R, we can determine their order without difficulty. However, for any two closed intervals in R, there is not a natural ordering among the set of all closed intervals in R. Several methods have been developed to compare two intervals. In this paper, we define the μ-ordering which is a new method for ordering cl...