The multicoloring problem is that given a graph G and integer demands x(v) for every vertex v, assign a set of x(v) colors to vertex v, such that neighboring vertices have disjoint sets of colors. In the preemptive sum multicoloring problem the finish time of a vertex is defined to be the highest color assigned to it. The goal is to minimize the sum of the finish times. The study of this proble...

We prove that the strong polarized relation ( μ μ )

The complete solution in (n, yx, yj) 6 Z3 of the Diophantine equation b„ = ±2yi yi is given, where (bn)nez is Berstel's recurrence sequence defined by />o = Z>i=0, b2 = l, b„+3 = 2bn+2 4b„+l + 4bn.

Some identities between partitions and compositions were obtained in the literature. As a natural extension, we introduce and study modulo-p partitions, where p is a positive integer. Moreover, several recurrence relations and some sufficient conditions for the existence of modulo-p partitions are given, respectively. In addition, we obtain some identities of modulo-p partitions. In the end, us...

Chebyshev polynomials of the first and the second kind in n variables z. , Zt , ... , z„ are introduced. The variables z, , z-,..... z„ are the characters of the representations of SL(n + 1, C) corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami op...

Abstract. Following Poupard’s study of strictly ordered binary trees with respect to two parameters, namely, “end of minimal chain” and “parent of maximum leaf” a true Tree Calculus is being developed to solve a partial difference equation system and then make a joint study of those two statistics. Their joint distribution is shown to be symmetric and to be expressed in the form of an explicit ...

We study the problem of gravitational lensing by binary galaxies, idealized as two isothermal spheres. In a wide binary, each galaxy possesses individual tangential, nearly astroidal, caus-tics and roundish radial caustics. As the separation of the binary is made smaller, the caustics undergo a sequence of metamorphoses. The first metamorphosis occurs when the tangential caustics merge to form ...