Zobrazeno 1 - 10
of 182
pro vyhledávání: '"Cairns, Grant"'
Autor:
Aebi, Christian, Cairns, Grant
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 Pe
Externí odkaz:
http://arxiv.org/abs/2312.10866
Autor:
Aebi, Christian, Cairns, Grant
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.
Externí odkaz:
http://arxiv.org/abs/2309.13551
Autor:
Aebi, Christian, Cairns, Grant
We show that there are only two equable triangles having vertices on the Eisenstein lattice, up to Euclidean motions.
Externí odkaz:
http://arxiv.org/abs/2309.04476
Autor:
Cairns, Grant, Nikolayevsky, Yuri
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
Externí odkaz:
http://arxiv.org/abs/2207.03068
Autor:
Aebi, Christian, Cairns, Grant
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.
Externí odkaz:
http://arxiv.org/abs/2202.05156
Autor:
Aebi, Christian, Cairns, Grant
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 t
Externí odkaz:
http://arxiv.org/abs/2111.06453
Autor:
Aebi, Christian, Cairns, Grant
A surprising simple result about quadrilaterals is given as an application of the vector triple product identity.
Externí odkaz:
http://arxiv.org/abs/2106.11860
Autor:
Aebi, Christian, Cairns, Grant
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 equa
Externí odkaz:
http://arxiv.org/abs/2105.00919
Autor:
Aebi, Christian, Cairns, Grant
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
Externí odkaz:
http://arxiv.org/abs/2006.07566
We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.
Comment: 7 pages
Comment: 7 pages
Externí odkaz:
http://arxiv.org/abs/1904.08237