scholarly journals Crossings in Grid Drawings

10.37236/3025 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Vida Dujmović ◽  
Pat Morin ◽  
Adam Sheffer
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Crossing Number ◽  
Geometric Graphs ◽  
Vertex Sets

We prove tight crossing number inequalities for geometric graphs whose vertex sets are taken from a $d$-dimensional grid of volume $N$ and give applications of these inequalities to counting the number of crossing-free geometric graphs that can be drawn on such grids.In particular, we show that any geometric graph with $m\geq 8N$ edges and with vertices on a 3D integer grid of volume $N$, has $\Omega((m^2/N)\log(m/N))$ crossings. In $d$-dimensions, with $d\ge 4$, this bound becomes $\Omega(m^2/N)$. We provide matching upper bounds for all $d$. Finally, for $d\ge 4$ the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some $d$-dimensional grid of volume $N$ is $N^{\Theta(N)}$. In 3 dimensions it remains open to improve the trivial bounds, namely, the $2^{\Omega(N)}$ lower bound and the $N^{O(N)}$ upper bound.

scholarly journals On the relation between graph distance and Euclidean distance in random geometric graphs

2016 ◽  
Vol 48 (3) ◽  
pp. 848-864 ◽  
Author(s):  
J. Díaz ◽  
D. Mitsche ◽  
G. Perarnau ◽  
X. Pérez-Giménez
Keyword(s):  
Lower Bound ◽  
Euclidean Distance ◽  
Geometric Graph ◽  
Upper Bounds ◽  
Geometric Graphs ◽  
Graph Distance ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Connectivity Threshold

Abstract Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

Minimization of Bounded Cable Tensions in Cable-Based Parallel Manipulators

2007 ◽  
Author(s):  
Mahir Hassan ◽  
Amir Khajepour
Keyword(s):  
Lower Bound ◽  
Numerical Method ◽  
Upper Bound ◽  
Task Force ◽  
Parallel Manipulators ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Alternating Projection ◽  
Alternating Projection Method ◽  
Cable Forces

In this work, the application of the Dykstra’s alternating projection method to find the minimum-2-norm solution for actuator forces is discussed in the case when lower and upper bounds are imposed on the actuator forces. The lower bound is due to specified pretension desired in the cables and the upper bound is due to the maximum allowable forces in the cables. This algorithm presents a systematic numerical method to determine whether or not a solution exists to the cable forces within these bounds and, if it does exist, calculate the minimum-2-norm solution for the cable forces for a given task force. This method is applied to an example 2-DOF translational cable-driven manipulator and a geometrical demonstration is presented.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

scholarly journals On Bipartite Distinct Distances in the Plane

10.37236/9687 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Surya Mathialagan
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Minimum Number

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem.In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leqslant n^{1/3}$. This is done by adapting Székely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geqslant n^{1/3}$.

Asymptotic bounds for the fluid queue fed by sub-exponential On/Off sources

2000 ◽  
Vol 32 (01) ◽  
pp. 244-255 ◽  
Author(s):  
V. Dumas ◽  
A. Simonian
Keyword(s):  
Lower Bound ◽  
Finite Number ◽  
Upper Bound ◽  
Content Distribution ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Fluid Queue ◽  
Asymptotic Bounds ◽  
Multiplicative Factor ◽  
Buffer Content

We consider a fluid queue fed by a superposition of a finite number of On/Off sources, the distribution of the On period being subexponential for some of them and exponential for the others. We provide general lower and upper bounds for the tail of the stationary buffer content distribution in terms of the so-called minimal subsets of sources. We then show that this tail decays at exponential or subexponential speed according as a certain parameter is smaller or larger than the ouput rate. If we replace the subexponential tails by regularly varying tails, the upper bound and the lower bound are sharp in that they differ only by a multiplicative factor.

Doubling up: Two upper bounds for scalars

Linguistics ◽  
2015 ◽  
Vol 53 (3) ◽  
Author(s):  
Mira Ariel
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Upper Bounds ◽  
Bound Analysis

AbstractMost theories of scalar quantifiers, of whatever persuasion, assume a lexical lower-bound-only, ‘at least’ meaning for scalar quantifiers, offering pragmatic or grammatical mechanisms for deriving the upper bound (Carston 1990; Chierchia 2004; Horn 1972 and onwards). I have challenged the lower-bound analysis in Ariel (2004), proposing instead a

Infection Spread in Random Geometric Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 164-181 ◽  
Author(s):  
Ghurumuruhan Ganesan
Keyword(s):  
Lower Bound ◽  
High Probability ◽  
Contact Process ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Long Time ◽  
Explicit Lower Bound ◽  
Infection Spread

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

scholarly journals Almost simplicial polytopes: the lower and upper bound theorems

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Upper Bound Theorem ◽  
Simplicial Polytopes ◽  
Full Version ◽  
The Face ◽  
International Audience ◽  
Bound Theorem

International audience this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.

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