scholarly journals On the Union Complexity of Families of Axis-Parallel Rectangles with a Low Packing Number

10.37236/6792 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Chaya Keller ◽  
Shakhar Smorodinsky
Keyword(s):  
Upper Bounds ◽  
Packing Number ◽  
Empty Intersection ◽  
Axis Parallel

Let $\mathcal{R}$ be a family of $n$ axis-parallel rectangles with packing number $p-1$, meaning that among any $p$ of the rectangles, there are two with a non-empty intersection. We show that the union complexity of $\mathcal{R}$ is at most $O(n+p^2)$, and that the $(k-1)$-level complexity of $\mathcal{R}$ is at most $O(n+k p^2)$. Both upper bounds are tight.

scholarly journals Bounds for the Average $L^p$-Extreme and the $L^\infty$-Extreme Discrepancy

10.37236/1951 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Upper Bounds ◽  
Unit Cube ◽  
Point Sets ◽  
Dimensional Unit ◽  
Axis Parallel ◽  
Geometric Discrepancy

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the $d$-dimensional unit cube with respect to the set system of axis-parallel boxes. For $2\leq p < \infty$ we provide upper bounds for the average $L^p$-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the $L^\infty$-extreme discrepancy with optimal dependence on the dimension $d$ and explicitly given constants.

Graph Models for Backbone Sets and Limited Packings in Networks

2021 ◽  
pp. 213-274
Author(s):  
Vadim Zverovich
Keyword(s):  
Fault Tolerant ◽  
Dominating Set ◽  
Facility Location Problem ◽  
Upper Bounds ◽  
Sensor Nodes ◽  
Theoretic Approach ◽  
Dominating Sets ◽  
Packing Number ◽  
Graph Theoretic ◽  
Threshold Functions

Here, a graph-theoretic approach is applied to some problems in networks, for example in wireless sensor networks (WSNs) where some sensor nodes should be selected to behave as a backbone/dominating set to support routing communications in an efficient and fault-tolerant way. Four different types of multiple domination (k-, k-tuple, α‎- and α‎-rate domination) are considered and recent upper bounds for cardinality of these types of dominating sets are discussed. Randomized algorithms are presented for finding multiple dominating sets whose expected size satisfies the upper bounds. Limited packings in networks are studied, in particular the k-limited packing number. One possible application of limited packings is a secure facility location problem when there is a need to place as many resources as possible in a given network subject to some security constraints. The last section is devoted to two general frameworks for multiple domination: <r,s>-domination and parametric domination. Finally, different threshold functions for multiple domination are considered.

RECTANGLE AND BOX VISIBILITY GRAPHS IN 3D

1999 ◽  
Vol 09 (01) ◽  
pp. 1-27 ◽  
Author(s):  
SÁNDOR R. FEKETE ◽  
HENK MEIJER
Keyword(s):  
Complete Graph ◽  
Dimensional Space ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
3 Dimensional ◽  
Visibility Graphs ◽  
Axis Parallel ◽  
Parallel Direction ◽  
Visibility Representations ◽  
Special Case

We discuss rectangle and box visibility representations of graphs in 3-dimensional space. In these representations, vertices are represented by axis-aligned disjoint rectangles or boxes. Two vertices are adjacent if and only if their corresponding boxes see each other along a small axis-parallel cylinder. We concentrate on lower and upper bounds for the size of the largest complete graph that can be represented. In particular, we examine these bounds under certain restrictions: What can be said if we may only use boxes of a limited number of shapes? Some of the results presented are as follows: • There is a representation of K8 by unit boxes. • There is no representation of K10 by unit boxes. • There is a representation of K56, using 6 different box shapes. • There is no representation of K184 by general boxes. A special case arises for rectangle visibility graphs, where no two boxes can see each other in the x- or y-directions, which means that the boxes have to see each other in z-parallel direction. This special case has been considered before; we give further results, dealing with the aspects arising from limits on the number of shapes.

UPPER BOUNDS ON THE LOWER OPEN PACKING NUMBER OF A TREE

1998 ◽  
Vol 21 (3-4) ◽  
pp. 235-245 ◽  
Author(s):  
Michael A. Henning
Keyword(s):  
Upper Bounds ◽  
Packing Number ◽  
Open Packing

scholarly journals Structural Results for Conditionally Intersecting Families and some Applications

10.37236/8894 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Xizhi Liu
Keyword(s):  
Upper Bounds ◽  
Empty Intersection ◽  
Intersecting Families

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$: (a) $(d,2k+d-3)$-conditionally intersecting families with $n\ge 3k^5$; (b) $(k,2k)$-conditionally intersecting families with $n\ge k^2/(k-1)$; (c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\ge 3k\binom{2k}{k}$. Our results for $(c)$ confirms a conjecture of Mammoliti and Britz for the case $d=3$.

The limits of strain and lattice parameter measurements by CBED

1989 ◽  
Vol 47 ◽  
pp. 518-519
Author(s):  
Hamish L. Fraser
Keyword(s):  
Electron Beam ◽  
Mirror Symmetry ◽  
Local Symmetry ◽  
Lattice Parameter ◽  
Growth Direction ◽  
Surface Relaxation ◽  
Strained Layer ◽  
Axis Parallel ◽  
Local Lattice ◽  
Strained Layer Superlattice

The topic of strain and lattice parameter measurements using CBED is discussed by reference to several examples. In this paper, only one of these examples is referenced because of the limitation of length. In this technique, scattering in the higher order Laue zones is used to determine local lattice parameters. Work (e.g. 1) has concentrated on a model strained-layer superlattice, namely Si/Gex-Si1-x. In bulk samples, the strain is expected to be tetragonal in nature with the unique axis parallel to [100], the growth direction. When CBED patterns are recorded from the alloy epi-layers, the symmetries exhibited by the patterns are not tetragonal, but are in fact distorted from this to lower symmetries. The spatial variation of the distortion close to a strained-layer interface has been assessed. This is most readily noted by consideration of Fig. 1(a-c), which show enlargements of CBED patterns for various locations and compositions of Ge. Thus, Fig. 1(a) was obtained with the electron beam positioned in the center of a 5Ge epilayer and the distortion is consistent with an orthorhombic distortion. When the beam is situated at about 150 nm from the interface, the same part of the CBED pattern is shown in Fig. 1(b); clearly, the symmetry exhibited by the mirror planes in Fig. 1 is broken. Finally, when the electron beam is positioned in the center of a 10Ge epilayer, the CBED pattern yields the result shown in Fig. 1(c). In this case, the break in the mirror symmetry is independent of distance form the heterointerface, as might be expected from the increase in the mismatch between 5 and 10%Ge, i.e. 0.2 to 0.4%, respectively. From computer simulation, Fig.2, the apparent monocline distortion corresponding to the 5Ge epilayer is quantified as a100 = 0.5443 nm, a010 = 0.5429 nm and a001 = 0.5440 nm (all ± 0.0001 nm), and α = β = 90°, γ = 89.96 ± 0.02°. These local symmetry changes are most likely due to surface relaxation phenomena.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

Sign in / Sign up

Export Citation Format

Share Document