Related papers: Circles of Apollonius two ways
The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial…
If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle…
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic…
We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of…
According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two…
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…
Paul Erdos asked if, among sufficiently many points in general position, there are always $k$ points such that all the circles through $3$ of these $k$ points have different radii. He later proved that this is indeed the case. However, he…
We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many…
Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we…
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many…
The entanglement entropy of an annulus is examined in a three-dimensional system with or without a gap. For a free massive scalar field theory, we numerically calculate the mutual information across an annulus. We also study the holographic…
When p divides the ordering of Galois group, the distribution of the Sylow p-subgroup of Cl(K) is closely related to the problem of counting fields with certain specifications. Moreover, different orderings of number fields affect the…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these…
A closed plane meander of order n is a closed self-avoiding loop intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on…
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…
If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.