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Triangles on planar Jordan $C^1$-curves
Triangles planar Jordan $C^1$-curves Metric Geometry
2012/5/9
We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit co...
Degree conditions for the partition of a graph into triangles and quadrilaterals
degree partition triangle quadrilateral
2011/3/1
For two positive integers r and s with r ≥ 2s−2, if G is a graph of order 3r+4s such that d(x)+d(y) ≥ 4r+4s for every xy 6∈ E(G), then G independently contains r triangles and s quadrilaterals, ...
This is a study of the about structures in one-dimensional cellular automata, with the elementary cellular automaton Rule 54 as example. It uses the formalism of “flexible time” to derive expressions ...
A measure for the description of the chirality of triangles is introduced. The measure $X\delta$ is zero for triangles with at least one mirror axe, i.e. equilateral or isosceles triangles, and positi...
Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty ...
Let be a random spherical triangle (meaning that vertices are independent and uniform on the unit sphere). A closed-form expression for the area density of has been known since 1867; a complicated...
Partitioning the triangles of the cross polytope into surfaces
cross polytope simplicial complexes triangulated surfaces difference cycles.
2010/12/6
We present a constructive proof, that there exists a decomposition of the 2-skeleton of the
k-dimensional cross polytope k into closed surfaces of genus ¤ 1, each with a transitive
automorphism gro...
Supplemental material to the article "Partitions of the triangles of the cross polytope into surfaces''
cross polytope simplicial complexes triangulated surfaces difference cycle
2010/12/6
We present a constructive proof, that there exists a decomposition of the 2-skeleton of the
k-dimensional cross polytope k into closed surfaces of genus ¤ 1, each with a transitive
automorphism gro...
On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry
triangles determined Euclidean plane associated bilinear operators discrete geometry
2010/12/6
We prove that if the Hausdorff dimension of the set E R2 is greater than 74 , then the three-dimensional Lebesgue measure of the set of triangles determined by E is positive. We
establish this fact...
On Isosceles Triangles and Related Problems in a Convex Polygon
Isosceles Triangles and Related Problems Convex Polygon
2010/12/15
Given any convex n-gon, in this article, we: (i) prove that its vertices can form at most n2/2 + (n log n) isosceles trianges with two sides of unit length and show that this bound is optimal in the ...
An Inequality on Ternary Quadratic Forms in Triangles
Positive semidefinite ternary quadratic form arithmetic-mean geometric-mean inequality Cauchy inequality triangle
2010/1/22
In this short note, we give a proof of a conjecture about ternary quadratic orms involving two triangles and several interesting applications.