In this paper we study the
-chromatic number of the cartesian product of two graphs. The
-chromatic number of a graph
is defined as the maximum number
of colors that can be used to color the vertices of
, such that we obtain a proper coloring and each color
has at least one representative
adjacent to a vertex of every color
, 1 ≦
. In this paper we get ρ(
) ≦ ρ(
+ 1) + Δ(
) + 1, when the girth of
is assumed to be greater than or equal to 7.
The basis number of a graph G is defined to be the least positive integer d such that G has a d-fold basis for the cycle space of G. We investigate the basis number of the cartesian product of stars and wheels with ladders, circular ladders and Möbius ladders.
Letp=(p1,p2,...) be a vector with an infinite number of coordinates, 1≦pk≦,k=1,2,... On the set of random functions depending on infinite number of variables, a mixed norm ∥.∥p is introduced, and thus the spacesLp with mixed norm are defined. Part 1 contains observations of general properties of those spaces (in particular, convergence properties depending on the behaviour of the exponentspk ask→ ∞). Part 2 contains the proof of infinite-dimensional version of S. L. Sobolev's theorem (in mixed norm) for potentials of Wiener semigroup on infinite dimensional torusT∞.
We refine a method introduced in  and  for studying the number of distinct values taken by certain polynomials of two
real variables on Cartesian products. We apply it to prove a "gap theorem", improving a recent lower bound on the number of
distinct distances between two collinear point sets in the Euclidean space.
Authors:S. Hassi, Z. Sebestyén, H. De Snoo, and F. Szafraniec
An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be
the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition
can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts
of a relation are characterized metrically and in terms of Stone’s characteristic projection onto the closure of the linear