Related papers: Three Random Tangents to a Circle
Given an Apollonian Circle Packing $\mathcal{P}$ and a circle $C_0 = \partial B(z_0, r_0)$ in $\mathcal{P}$, color the set of disks in $\mathcal{P}$ tangent to $C_0$ red. What proportion of the concentric circle $C_{\epsilon} = \partial…
We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each ''hole'' is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$,…
We consider the problem of finding integer-sided triangles with R/r an integer, where R and r are the radii of the circumcircle and incircle respectively. We show that such triangles are relatively rare.
We study properties of certain circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the arc of a circle erected internally on the third side.
We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by…
We give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction.…
The discrete unitary (reversible) analogues of the continuous (irreversible) tent maps are numerically investigated, in particular, the lengths probability distribution of their periodic orbits. It is found that its density can be well…
We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding…
We investigate the problem of density estimation on the unit circle and the unit sphere from a computational perspective. Our primary goal is to develop new density estimators that are both rate-optimal and computationally efficient for…
Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis,…
Let p(w) denote the probability that four random circular caps of angular radius 70deg<w<90deg cover the unit sphere S^2. An exact expression for p(w) is unknown. We give nontrivial lower bounds for p(w) when w>84deg; no improvement on the…
This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…
We obtain universal affine type estimates for the location of the geometric medians of triangle perimeters and for the location of the geometric medians of triangular domains. At the end, some alternative implementations of the triangle…
We consider $O(1)$ dense loop model in a square lattice wrapped on a cylinder of odd circumference $L$ and calculate the exact densities of loops. These densities of loops are equal to the densities of critical bond percolation clusters on…
Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and…
In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent…
We have computed a table of the triangle sides of all congruent numbers less than 10,000, which improves and extends the existing public table. We give some background on properties of the triangle sides, and explain how we computed our…
In the present work we derive an analytical expression for the mass density of an object with spherical symmetry, whose corresponding potential allows obtaining a circular velocity around it that is in agreement with the observed rotation…
The $n$-dimensional random twisted hypercube $\mathbf{G}_n$ is constructed recursively by taking two instances of $\mathbf{G}_{n-1}$, with any joint distribution, and adding a random perfect matching between their vertex sets. Benjamini,…
We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a…