We call the integer N a kth-power numerical centre for n if 1 + 2 + · · · + N = N + (N + 1) + · · · + n. We prove, using the explicit lower bounds on linear forms in elliptic logarithms, that there are no nontrivial fifth-power numerical centres for any n, and demonstrate that there are only finitely many pairs (N, n) satisfying the above for any given k > 1. The problem of finding kth-power ce...

Cross-plane thermal conductivity kth measurements of vertical stacks of FePt/C were used to estimate the in-plane thermal conductivity of Heat Assisted Magnetic Recording (HAMR) media that consist of columnar FePt grains segregated by thin C grain boundaries. FePt/C multilayers with varied repeat units and FePt layer thicknesses (chosen to represent HAMR media grain sizes) were measured using F...

By some extremely simple arguments, we point out the following: (i) If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n. (ii) Let OK be the ring of algebraic integers in a quadratic field K = Q( √ d) with d ∈ {−1,−2,−3,−7,−11}. Then, for any irreducible π ∈ OK and positive integer k no...

This paper deals with the kth nearest rectilinear distance of two regular point patterns: square and diamond lattices. The probability density functions of the kth nearest rectilinear distance are theoretically derived for k = 1, 2, . . . , 8. Upper and lower bounds of the kth nearest distance are also derived. As an application of the kth nearest distance, we consider a facility location probl...

In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ≤ D < 4k. In...

Presence of kth-order cyclostationarity is defined in terms of nonvanishing cyclic-cumulants or polyspectr a. Relying upon the asymptotic normality and consistency of kth-order cyclic statistics, asymptotically optimal x2 tests are developed to detect presence of cycles in the kth-order cyclic cumulants or polyspectra, without assuming any specific distribu ion on the data. Constant false alarm...