scholarly journals The Palm measure and the Voronoi tessellation for the Ginibre process

2010 ◽  
Vol 20 (1) ◽  
pp. 90-128 ◽  
Author(s):  
André Goldman
Keyword(s):  
Voronoi Tessellation ◽  
Palm Measure

scholarly journals Stationary partitions and Palm probabilities

2006 ◽  
Vol 38 (3) ◽  
pp. 602-620 ◽  
Author(s):  
Günter Last
Keyword(s):  
Probability Measure ◽  
Voronoi Tessellation ◽  
Random Partition ◽  
Weighted Version ◽  
Stationary Point Process ◽  
Typical Cell ◽  
Palm Measure ◽  
Special Case ◽  
Palm Probabilities ◽  
Shift Coupling

A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

scholarly journals Stationary partitions and Palm probabilities

2006 ◽  
Vol 38 (03) ◽  
pp. 602-620 ◽  
Author(s):  
Günter Last
Keyword(s):  
Probability Measure ◽  
Voronoi Tessellation ◽  
Random Partition ◽  
Weighted Version ◽  
Stationary Point Process ◽  
Typical Cell ◽  
Palm Measure ◽  
Special Case ◽  
Palm Probabilities ◽  
Shift Coupling

A stationary partition based on a stationary point process N in ℝ d is an ℝ d -valued random field π={π(x): x∈ℝ d } such that both π(y)∈N for each y∈ℝ d and the random partition {{y∈ℝ d : π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

scholarly journals An optical observational cluster mass function at z ∼ 1 with the ORELSE survey

2021 ◽  
Vol 502 (3) ◽  
pp. 3942-3954
Author(s):  
D Hung ◽  
B C Lemaux ◽  
R R Gal ◽  
A R Tomczak ◽  
L M Lubin ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Large Scale ◽  
Near Infrared ◽  
Function Analysis ◽  
Voronoi Tessellation ◽  
Cosmological Parameters ◽  
Mass Range ◽  
Mass Function ◽  
Cluster Mass ◽  
Theoretical Mass

ABSTRACT We present a new mass function of galaxy clusters and groups using optical/near-infrared (NIR) wavelength spectroscopic and photometric data from the Observations of Redshift Evolution in Large-Scale Environments (ORELSE) survey. At z ∼ 1, cluster mass function studies are rare regardless of wavelength and have never been attempted from an optical/NIR perspective. This work serves as a proof of concept that z ∼ 1 cluster mass functions are achievable without supplemental X-ray or Sunyaev-Zel’dovich data. Measurements of the cluster mass function provide important contraints on cosmological parameters and are complementary to other probes. With ORELSE, a new cluster finding technique based on Voronoi tessellation Monte Carlo (VMC) mapping, and rigorous purity and completeness testing, we have obtained ∼240 galaxy overdensity candidates in the redshift range 0.55 < z < 1.37 at a mass range of 13.6 < log (M/M⊙) < 14.8. This mass range is comparable to existing optical cluster mass function studies for the local universe. Our candidate numbers vary based on the choice of multiple input parameters related to detection and characterization in our cluster finding algorithm, which we incorporated into the mass function analysis through a Monte Carlo scheme. We find cosmological constraints on the matter density, Ωm, and the amplitude of fluctuations, σ8, of $\Omega _{m} = 0.250^{+0.104}_{-0.099}$ and $\sigma _{8} = 1.150^{+0.260}_{-0.163}$. While our Ωm value is close to concordance, our σ8 value is ∼2σ higher because of the inflated observed number densities compared to theoretical mass function models owing to how our survey targeted overdense regions. With Euclid and several other large, unbiased optical surveys on the horizon, VMC mapping will enable optical/NIR cluster cosmology at redshifts much higher than what has been possible before.

scholarly journals Modeling the microstructure of biopolymer aerogels using Voronoi tessellation method

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Rajesh Chandrasekaran ◽  
Markus Hillgärtner ◽  
Ameya Rege ◽  
Barbara Milow ◽  
Mikhail Itskov
Keyword(s):  
Voronoi Tessellation

scholarly journals Optimal Random Packing of Spheres and Extremal Effective Conductivity

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1063
Author(s):  
Vladimir Mityushev ◽  
Zhanat Zhunussova
Keyword(s):  
Effective Conductivity ◽  
Dimensional Space ◽  
Elementary Function ◽  
Voronoi Tessellation ◽  
Random Packing ◽  
Periodicity Cell ◽  
Algebraic Equations ◽  
Optimal Packing ◽  
Linear Algebraic Equations ◽  
Initial Location

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

scholarly journals Design and Mechanical Properties Verification of Gradient Voronoi Scaffold for Bone Tissue Engineering

Micromachines ◽  
2021 ◽  
Vol 12 (6) ◽  
pp. 664
Author(s):  
Haiyuan Zhao ◽  
Yafeng Han ◽  
Chen Pan ◽  
Ding Yang ◽  
Haotian Wang ◽  
...  
Keyword(s):  
Mechanical Property ◽  
Tissue Engineering ◽  
Elastic Modulus ◽  
Porous Structure ◽  
Bone Tissue Engineering ◽  
Bone Tissue ◽  
Voronoi Tessellation ◽  
Impact Resistance ◽  
Natural Bone ◽  
The Impact

In order to obtain scaffold that can meet the therapeutic effect, researchers have carried out research on irregular porous structures. However, there are deficiencies in the design method of accurately controlling the apparent elastic modulus of the structure at present. Natural bone has a gradient porous structure. However, there are few studies on the mechanical property advantages of gradient bionic bone scaffold. In this paper, an improved method based on Voronoi-tessellation is proposed. The method can get controllable gradient scaffolds to fit the modulus of natural bone, and accurately control the apparent elastic modulus of porous structure, which is conducive to improving the stress shielding. To verify the designed structure can be fabricated by additive manufacturing, several designed models are obtained by SLM and EBM. Through finite element analysis (FEA), it is verified that the irregular porous structure based on Voronoi-tessellation is more stable than the traditional regular porous structure of the same structure volume, the same pore number and the same material. Furthermore, it is verified that the gradient irregular structure has a better stability than the non-gradient structure. An experiment is conducted successfully to verify the stability performance got by FEA. In addition, a dynamic impact FEA is also performed to simulate impact resistance. The result shows that the impact resistance of the regular porous structure, the irregular porous structure and the gradient irregular porous structure becomes better in turn. The mechanical property verification provides a theoretical basis for the structural design of gradient irregular porous bone tissue engineering scaffolds.

Markov paths on the Poisson-Delaunay graph with applications to routeing in mobile networks

2000 ◽  
Vol 32 (01) ◽  
pp. 1-18 ◽  
Author(s):  
F. Baccelli ◽  
K. Tchoumatchenko ◽  
S. Zuyev
Keyword(s):  
Communication Networks ◽  
Mobile Networks ◽  
Path Length ◽  
Euclidean Distance ◽  
Voronoi Tessellation ◽  
Voronoi Cells ◽  
Ergodic Properties ◽  
Potential Applications ◽  
The Mean ◽  
Mobile Communication Networks

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

Examination of a Polycrystalline Thin-Film Model to Explore the Relation between Probe Size and Structural Correlation Length in Fluctuation Electron Microscopy

2012 ◽  
Vol 18 (1) ◽  
pp. 241-253 ◽  
Author(s):  
M.M.J. Treacy ◽  
J.M. Gibson
Keyword(s):  
Electron Microscopy ◽  
Random Network ◽  
Voronoi Tessellation ◽  
Coherent Scattering ◽  
Specimen Thickness ◽  
Probe Size ◽  
Thin Film Model ◽  
Mean Grain Size ◽  
Fluctuation Electron Microscopy ◽  
Thin Polycrystalline

AbstractWe examine simulated electron microdiffraction patterns from models of thin polycrystalline silicon. The models are made by a Voronoi tessellation of random points in a box. The Voronoi domains are randomly selected to contain either a randomly-oriented cubic crystalline grain or a region of continuous random network material. The microdiffraction simulations from coherent probes of different widths are computed at the ideal kinematical limit, ignoring inelastic and multiple scattering. By examining the normalized intensity variance that is obtained in fluctuation electron microscopy experiments, we confirm that intensity fluctuations increase monotonically with the percentage of crystalline grains in the material. However, anomalously high variance is observed for models that have 100% crystalline grains with no imperfections. We confirm that the reduced normalized variance, V(k,R) − 1, that is associated with four-body correlations at scattering vector k, varies inversely with specimen thickness. Further, for probe sizes R larger than the mean grain size, we confirm that the reduced normalized variance obeys the predicted form given by Gibson et al. [Ultramicroscopy, 83, 169–178 (2000)] for the kinematical coherent scattering limit.

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