97 research outputs found
A parametrization of equilateral triangles having integer coordinates
We study the existence of equilateral triangles of given side lengths and
with integer coordinates in dimension three. We show that such a triangle
exists if and only if their side lengths are of the form
for some integers . We also show a similar characterization for the sides
of a regular tetrahedron in : such a tetrahedron exists if and only if
the sides are of the form , for some . The classification of
all the equilateral triangles in contained in a given plane is studied
and the beginning analysis is presented. A more general parametrization is
proven under a special assumption. Some related questions are stated in the
end.Comment: 3 fugures, 17 pages, submitted to Integer
Half domination arrangements in regular and semi-regular tessellation type graphs
We study the problem of half-domination sets of vertices in vertex transitive
infinite graphs generated by regular or semi-regular tessellations of the
plane. In some cases, the results obtained are sharp and in the rest, we show
upper bounds for the average densities of vertices in half-domination sets.Comment: 14 pages, 12 figure
A variation on bisecting the binomial coefficients
In this paper, we present an algorithm which allows us to search for all the
bisections for the binomial coefficients and
include a table with the results for all . Connections with previous
work on this topic is included. We conjecture that the probability of having
only trivial solutions is . \end{abstract}Comment: 14 pages, four tables, two figure
Counting all equilateral triangles in {0,1,2,...,n}^3
We describe a procedure of counting all equilateral triangles in the three
dimensional space whose coordinates are allowed only in the set
. This sequence is denoted here by ET(n) and it has the entry
A102698 in "The On-Line Encyclopedia of Integer Sequences". The procedure is
implemented in Maple and its main idea is based on the results in \cite{eji}.
Using this we calculated the values ET(n) for n=1..55 which are included here.
Some facts and conjectures about this sequence are stated. The main of them is
that \ds \lim_{n\to \infty} \frac{\ln ET(n)}{\ln n+1} exists.Comment: 12 pages, 1 figure, Maple cod
Simultaneous Translational and Multiplicative Tiling and Wavelet Sets in R^2
Simultaneous tiling for several different translational sets has been studied
rather extensively, particularly in connection with the Steinhaus problem. The
study of orthonormal wavelets in recent years, particularly for arbitrary
dilation matrices, has led to the study of multiplicative tilings by the powers
of a matrix. In this paper we consider the following simultaneous tiling
problem: Given a lattice in \L\in \R^d and a matrix A\in\GLd, does there
exist a measurable set such that both \{T+\alpha: \alpha\in\L\} and
are tilings of ? This problem comes directly from the
study of wavelets and wavelet sets. Such a is known to exist if is
expanding. When is not expanding the problem becomes much more subtle.
Speegle \cite{Spe03} exhibited examples in which such a exists for some
\L and nonexpanding in . In this paper we give a complete solution
to this problem in .Comment: 16 pages, no figure
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