Points on a line, shoelace and dominoes
In this survey we consider numerous known and unknown combinatorial realizations of the sequence A079487 and basic facts about it.
math.CO↗
arXiv subjects
K. Kokhas
Publications and source records attributed to K. Kokhas.
How to sum up triangles
We prove configuration theorems that generalize the Desargues, Pascal, and Pappus theorems. Our generalization of the Desargues theorem allows us to introduce the structure of an Abelian group on the (properly extended) set of triangles which are perspective from a point. In barycentric coordinates, the corresponding group operation becomes the addition in R^3.
math.AG↗
Spectral estimations for Laplace operator for the discrete Heisenberg group
Let H be the discrete 3-dimensional Heisenberg group with the standard generators x, y, z. The element Delta of the group algebra for H of the form Delta= (x+x^{-1}+y+y^{-1})/4 is called the Laplace operator. This operator can also be defined as transition operator for random walk on the group. The spectrum of Delta in the regular representation of H is the interval [-1,1]. Let E(A), where A is a subset of [-1,1], be a family of spectral projectors for Delta and m(A)=(E(A)e, e) be the corresponding spectral measure. Here e is the characteristic function of the unit element of the group H. We estimate the value m([-1,-1+t] \cup [1-t,1]) when t tends to 0. More precisely we prove the inequality m([-1,-1+t] \cup [1-t,1]) > const t^{2+alpha} for any positive alpha.
math.SP↗