135,033 research outputs found
Lattice Grids and Prisms are Antimagic
An \emph{antimagic labeling} of a finite undirected simple graph with
edges and vertices is a bijection from the set of edges to the integers
such that all vertex sums are pairwise distinct, where a vertex
sum is the sum of labels of all edges incident with the same vertex. A graph is
called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield
and Ringel conjectured that every connected graph, but , is antimagic. In
2004, N. Alon et al showed that this conjecture is true for -vertex graphs
with minimum degree . They also proved that complete partite
graphs (other than ) and -vertex graphs with maximum degree at least
are antimagic. Recently, Wang showed that the toroidal grids (the
Cartesian products of two or more cycles) are antimagic. Two open problems left
in Wang's paper are about the antimagicness of lattice grid graphs and prism
graphs, which are the Cartesian products of two paths, and of a cycle and a
path, respectively. In this article, we prove that these two classes of graphs
are antimagic, by constructing such antimagic labelings.Comment: 10 pages, 6 figure
Closed form summation of C-finite sequences
We consider sums of the form
in which each is a sequence that satisfies a linear recurrence of
degree , with constant coefficients. We assume further that the
's and the 's are all nonnegative integers. We prove that such a
sum always has a closed form, in the sense that it evaluates to a linear
combination of a finite set of monomials in the values of the sequences
with coefficients that are polynomials in . We explicitly
describe two different sets of monomials that will form such a linear
combination, and give an algorithm for finding these closed forms, thereby
completely automating the solution of this class of summation problems. We
exhibit tools for determining when these explicit evaluations are unique of
their type, and prove that in a number of interesting cases they are indeed
unique. We also discuss some special features of the case of ``indefinite
summation," in which
A sum-product theorem in function fields
Let be a finite subset of \ffield, the field of Laurent series in
over a finite field . We show that for any there
exists a constant dependent only on and such that
. In particular such a result is
obtained for the rational function field . Identical results
are also obtained for finite subsets of the -adic field for
any prime .Comment: Simplification of argument and note that methods also work for the
p-adic
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