arXiv subjects
Publications and source records attributed to Grant Cairns.
We show that for an integer $\ell$, there exists an acute integer lattice triangle of lattice perimeter $\ell$ such that its orthocenter is an integer lattice point, if and only if $\ell=6 $ or $\ell\ge 8$. Analogous results are obtained for the circumcenter and the centroid, and the results are contrasted with those for obtuse and right triangles.
This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circumcenter, centroid and orthocenter of such triangles never all lie on the integer lattice. Several further observations are made concerning the circumcenter, centroid and orthocenter.
This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose vertices have degree 4, bar the root which has degree 3. This study naturally complements the authors' previous study of equable parallelograms whose vertices lie on the integer lattice.
We classify perimeter dominant triangles whose side lengths are in $\sqrt3\mathbb N$ and whose area is in $\frac{\sqrt3}4\mathbb N$. There is one exceptional example, which is equilateral, and three infinite families determined by certain Pell, or Pell-like, equations.
Paul Yiu proved that all Heron triangles are realizable on the integer lattice. We give an analogous result for triangles with vertices on the Eisenstein lattice.
We show that there are only two equable triangles having vertices on the Eisenstein lattice, up to Euclidean motions.
One of Pierre Molino's principal mathematical achievements was his theory of Riemannian foliations. One of his last papers, published in 2001, showed that his theory could be extended to a large class of non-integrable distributions. The key example here is that of a \emph{respectful decomposition} of a Lie algebra $\mathfrak{g}$; this is vector space decomposition $\mathfrak{g}=H+V$ such that $[V,H]\subseteq H$. This paper will examine the basic properties of respectful decompositions.
We give a vector identity for $n+2$ points in $\mathbb R^n$. It follows as a corollary that when $n$ is odd the sum of the signed volumes of the $n$-simplices is zero, and when $n$ is even, the alternating sum of the signed volumes is zero.
A lattice equable quadrilateral is a quadrilateral in the plane whose vertices lie on the integer lattice and which is equable in the sense that its area equals its perimeter. This paper treats the tangential and extangential cases. We show that up to Euclidean motions, there are only 6 convex tangential lattice equable quadrilaterals, while the concave ones are arranged in 7 infinite families, each being given by a well known diophantine equation of order 2 in 3 variables. On the other hand, apart from the kites, up to Euclidean motions there is only one concave extangential lattice equable quadrilateral, while there are infinitely many convex ones.
A surprising simple result about quadrilaterals is given as an application of the vector triple product identity.
This paper studies equable parallelograms whose vertices lie on the integer lattice. Using Rosenberger's Theorem on generalised Markov equations, we show that the g.c.d. of the side lengths of such parallelograms can only be 3, 4 or 5, and in each of these cases the set of parallelograms naturally forms an infinite tree all of whose vertices have degree 4, bar the root. The paper then focuses on what we call Pythagorean equable parallelograms. These are lattice equable parallelograms whose complement in a circumscribing rectangle consists of two Pythagorean triangles. We prove that for these parallelograms the shortest side can only be 3, 4, 5, 6 or 10, and there are five infinite families of such parallelograms, given by solutions to corresponding Pell-like equations.
We show that there are 4 infinite families of lattice equable kites, given by corresponding Pell or Pell-like equations, but up to Euclidean motions, there are exactly 5 lattice equable trapezoids (2 isosceles, 2 right, 1 singular) and 4 lattice equable cyclic quadrilaterals. We also show that, with one exception, the interior diagonals of lattice equable quadrilaterals are irrational.
We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.
We show that for $K_5$ (resp.~ $K_{3,3}$) there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, provided $i$ is odd with $1\le i\le 15$ (resp.~ $1\le i\le 17$). Conversely, using the deleted product cohomology, we show that for $K_5$ and $K_{3,3}$, if $A$ is any set of pairs of independent edges, and $A$ has odd cardinality, then there is a drawing in the plane for which each element in $A$ cross an odd number of times, while each pair of independent edges not in $A$ cross an even number of times. For $K_6$ we show that there is a drawing with $i$ independent crossings, and no pair of independent edges cross more than once, if and only if $3\le i\le 40$.
We establish an elementary, but rather striking pattern concerning the quartic residues of primes $p$ that are congruent to 5 modulo 8. Let $g$ be a generator of the multiplicative group of $\mathbb Z_p$ and let $M$ be the $4\times 4$ matrix whose $(i+1),(j+1)-$th entry is the number of elements $x$ of $\mathbb Z_p$ of the form $x\equiv g^k \pmod p$ where $k\equiv i \pmod 4$ and $\lfloor 4x/p \rfloor = j$, for $i,j=0,1,2,3$. We show that $M$ is a Latin square, provided the entries in the first row are distinct, and that $M$ is essentially independent of the choice of $g$. As an application, we prove that the sum in $\mathbb Z$ of the quartic residues is $\frac{p}5(M_{11}+2M_{12}+3M_{13}+4M_{14})$.
Children, sitting in a circle, each have a nonnegative number of candies in front of them. A whistle is blown and each child with more than one candy passes one candy to the left and one to the right. The sharing process is repeated until a fixed state is attained, or the system enters a periodic cycle. This paper treats the case where the total number of candies equals the number of children. For a given initial distribution of candies, a necessary and sufficient condition is given for the system to ultimately attain the equitable distribution in which each child has one candy.
We present a sufficient condition for a pair of finite integer sequences to be degree sequences of a bipartite graph, based only on the lengths of the sequences and their largest and smallest elements.
It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.