Related papers: A Simple Proof of Thue's Theorem on Circle Packing
It was shown by James Tung in 2005 that if a sequence $Z=\{z_n\}$ of points in the complex plane satisfies $$\inf_{n\not=m}|z_n-z_m|>2/\sqrt\alpha,$$ then $Z$ is a sequence of interpolation for the Fock space $F^p_\alpha$. Using results…
We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an…
It is shown that a suspension of particles in a partially-filled, horizontal, rotating cylinder is linearly unstable towards axial segregation and an undulation of the free surface at large enough particle concentrations. Relying on the…
We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that…
This paper presents a simple proof of Dekel (1986)'s representation theorem for betweenness preferences. The proof is based on the separation theorem.
The article presents the mathematical sequences describing circle packing densities in four different geometric configurations involving a hexagonal lattice based equal circle packing in the Euclidian plane. The calculated sequences take…
We embed the space of totally real $r$-cycles of a totally real projective variety into the space of complex $r$-cycles by complexification. We provide a proof of the holomorphic taffy argument in the proof of Lawson suspension theorem by…
The Pythagorean Theorem has been proved in hundreds of ways, yet it inspires fresh insights through geometry and trigonometry. In this paper, we offer a new proof based on three circles that circumscribe the sides of a right triangle.…
We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses.
The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong…
We solve Grothendieck's localization problem for certain class of rings arising from the tight closure theory. The idea of the proof depends heavily on the study of the relative version of the Frobenius map.
We present in this work a new and simple proof of the false centre theorem.
We present a new self-contained proof of the well-known fact that the minimal area of a Voronoi cell in a unit circle packing is equal to $2\sqrt{3}$, and the minimum is achieved only on a perfect hexagon. The proof is short and, in our…
A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number $\tau(G)$ is at most twice the triangle packing number $\nu(G)$, where the triangle packing number $\nu(G)$ is the maximum size of a set of edge-disjoint…
This article presents the formal proof of correctness for a plane Delaunay triangulation algorithm. It consists in repeating a sequence of edge flippings from an initial triangulation until the Delaunay property is achieved. To describe…
Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence…
This work investigates dense packings of congruent hard infinitesimally--thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle $\theta \in \left ( \pi, 2\pi \right ]$.…
A clutter is \emph{clean} if it has no delta or the blocker of an extended odd hole minor, and it is \emph{tangled} if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in…
Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its…
In this short note we argue that Thomas-Fermi Theory the simplest of all density functional theories, although failing to explain features such as binding or stability of negative ions, is surprisingly accurate in estimating sizes of atoms.…