Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCauchys Arm Lemma on a Growing Sphere
51
0
0.0
(
0
)
Added by
Sebastien Collette
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We propose a variant of Cauchys Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogovs Comparison Theorem, Legendres Theorem and Cauchys Arm Lemma.
rate research
Read More
Given a convex polyhedron $P$ of $n$ vertices inside a sphere $Q$, we give an $O(n^3)$-time algorithm that cuts $P$ out of $Q$ by using guillotine cuts and has cutting cost $O((log n)^2)$ times the optimal.
We introduce a model for random geodesic drawings of the complete bipartite graph $K_{n,n}$ on the unit sphere $mathbb{S}^2$ in $mathbb{R}^3$, where we select the vertices in each bipartite class of $K_{n,n}$ with respect to two non-degenerate probability measures on $mathbb{S}^2$. It has been proved recently that many such measures give drawings whose crossing number approximates the Zarankiewicz number (the conjectured crossing number of $K_{n,n}$). In this paper we consider the intersection graphs associated with such random drawings. We prove that for any probability measures, the resulting random intersection graphs form a convergent graph sequence in the sense of graph limits. The edge density of the limiting graphon turns out to be independent of the two measures as long as they are antipodally symmetric. However, it is shown that the triangle densities behave differently. We examine a specific random model, blow-ups of antipodal drawings $D$ of $K_{4,4}$, and show that the triangle density in the corresponding crossing graphon depends on the angles between the great circles containing the edges in $D$ and can attain any value in the interval $bigl(frac{83}{12288}, frac{128}{12288}bigr)$.
Motivated by map labeling, Funke, Krumpe, and Storandt [IWOCA 2016] introduced the following problem: we are given a sequence of $n$ disks in the plane. Initially, all disks have radius $0$, and they grow at constant, but possibly different, speeds. Whenever two disks touch, the one with the higher index disappears. The goal is to determine the elimination order, i.e., the order in which the disks disappear. We provide the first general subquadratic algorithm for this problem. Our solution extends to other shapes (e.g., rectangles), and it works in any fixed dimension. We also describe an alternative algorithm that is based on quadtrees. Its running time is $Obig(n big(log n + min { log Delta, log Phi }big)big)$, where $Delta$ is the ratio of the fastest and the slowest growth rate and $Phi$ is the ratio of the largest and the smallest distance between two disk centers. This improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EuroCG 2017]. Finally, we give an $Omega(nlog n)$ lower bound, showing that our quadtree algorithms are almost tight.
We observe that Whiteheads lemma is an immediate consequence of Stallings folds.
Asaoka & Irie recently proved a $C^{infty}$ closing lemma of Hamiltonian diffeomorphisms of closed surfaces. We reformulated their techniques into a more general perturbation lemma for area-preserving diffeomorphism and proved a $C^{infty}$ closing lemma for area-preserving diffeomorphisms on a torus that is isotopic to identity. i.e., we show that the set of periodic orbits is dense for a generic diffeomorphism isotopic to identity area-preserving diffeomorphism on torus. The main tool is the flux vector of area-preserving diffeomorphisms which is, different from Hamiltonian cases, non-zero in general.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
