Volume 54, Nº 2 (2018)

Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,

The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space PG(N, q) is a complete (q+1)-arc, or equivalently, that no [q+1,N+1, q−N+1]_q generalized doubly-extended Reed–Solomon code can be extended to a [q + 2,N + 1, q − N + 2]_q maximum distance separable code.

We propose a test to distinguish between two classes of distribution tails using only higher order statistics of a sample and prove its consistency. We do not assume the corresponding distribution functions to belong to any maximum domain of attraction.

We substantially improve a presently known explicit exponentially growing lower bound on the chromatic number of a Euclidean space with forbidden equilateral triangle. Furthermore, we improve an exponentially growing lower bound on the chromatic number of distance graphs with large girth. These refinements are obtained by improving known upper bounds on the product of cardinalities of two families of homogeneous subsets with one forbidden cross-intersection.

We study the maximum remaining service time in infinite-server queues of type M|G|∞ (at a given time and in a stationary regime). The following cases for the arrival flow rate are considered: (1) time-independent, (2) given by a function of time, (3) given by a random process. As examples of service time distributions, we consider exponential, hyperexponential, Pareto, and uniform distributions. In the case of a constant rate, we study effects that arise when the average service time is infinite (for power-law distribution tails). We find the extremal index of the sequence of maximum remaining service times. The results are extended to queues of type M^X|G|∞, including those with dependent service times within a batch.