scholarly journals Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation

2011 ◽  
Vol 48 (3) ◽  
pp. 597-610
Author(s):  
Jianping Jiang ◽  
Sanguo Zhang ◽  
Tiande Guo
Keyword(s):  
Free Energy ◽  
Euclidean Space ◽  
Continuum Percolation ◽  
Site Percolation ◽  
Number Of Clusters ◽  
Infinite Cluster ◽  
The Mean ◽  
New Formula

A new formula for continuum percolation on the Euclidean space Rd (d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.

scholarly journals Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation

2011 ◽  
Vol 48 (03) ◽  
pp. 597-610 ◽  
Author(s):  
Jianping Jiang ◽  
Sanguo Zhang ◽  
Tiande Guo
Keyword(s):  
Free Energy ◽  
Euclidean Space ◽  
Continuum Percolation ◽  
Site Percolation ◽  
Number Of Clusters ◽  
Infinite Cluster ◽  
The Mean ◽  
New Formula

A new formula for continuum percolation on the Euclidean space R d (d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.

On a continuum percolation model

1991 ◽  
Vol 23 (3) ◽  
pp. 536-556 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Free Energy ◽  
Poisson Process ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Infinite Cluster ◽  
Cluster Density ◽  
Critical Intensity

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting ofkparticles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

On a continuum percolation model

1991 ◽  
Vol 23 (03) ◽  
pp. 536-556 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Free Energy ◽  
Poisson Process ◽  
Percolation Model ◽  
Continuum Percolation ◽  
Infinite Cluster ◽  
Cluster Density ◽  
Critical Intensity

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

scholarly journals Mapping similarities in temporal parking occupancy behavior based on city-wide parking meter data

2018 ◽  
Vol 1 ◽  
pp. 1-5
Author(s):  
Fabian Bock ◽  
Karen Xia ◽  
Monika Sester
Keyword(s):  
Similarity Measure ◽  
Historical Data ◽  
Parking Space ◽  
Number Of Clusters ◽  
Occupancy Patterns ◽  
Silhouette Index ◽  
The Mean ◽  
Occupancy Behavior ◽  
Maximum Occupancy ◽  
Repetitive Pattern

The search for a parking space is a severe and stressful problem for drivers in many cities. The provision of maps with parking space occupancy information assists drivers in avoiding the most crowded roads at certain times. Since parking occupancy reveals a repetitive pattern per day and per week, typical parking occupancy patterns can be extracted from historical data.<br> In this paper, we analyze city-wide parking meter data from Hannover, Germany, for a full year. We describe an approach of clustering these parking meters to reduce the complexity of this parking occupancy information and to reveal areas with similar parking behavior. The parking occupancy at every parking meter is derived from a timestamp of ticket payment and the validity period of the parking tickets. The similarity of the parking meters is computed as the mean-squared deviation of the average daily patterns in parking occupancy at the parking meters. Based on this similarity measure, a hierarchical clustering is applied. The number of clusters is determined with the Davies-Bouldin Index and the Silhouette Index.<br> Results show that, after extensive data cleansing, the clustering leads to three clusters representing typical parking occupancy day patterns. Those clusters differ mainly in the hour of the maximum occupancy. In addition, the lo-cations of parking meter clusters, computed only based on temporal similarity, also show clear spatial distinctions from other clusters.

scholarly journals The Possibility of Many Compensation Points in a Mixed-Spin Ising Ferrimagnetic System

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Hadey K. Mohamad
Keyword(s):  
Free Energy ◽  
Mean Field ◽  
Field Approximation ◽  
Spin Model ◽  
Mean Field Approximation ◽  
Zero Field ◽  
Mixed Spin ◽  
Simple Cubic ◽  
Simple Cubic Lattice ◽  
The Mean

The magnetic properties of a ferrimagnetic mixed spin-3/2 and spin-5/2 Ising model with different anisotropies are investigated by using the mean-field approximation (MFA). In particular, the effect of magnetic anisotropies on the compensation phenomenon, acting on A-atoms and B-ones for the mixed-spin model, has been considered in a zero field. The free energy of a mixed-spin Ising ferrimagnetic system from MFA of the Hamiltonian is calculated. By minimizing the free energy, we obtain the equilibrium magnetizations and the compensation points. The phase diagram of the system in the anisotropy dependence of transition temperature has been discussed as well. Our results of this model predict the existence of many (two or three) compensation points in the ordered system on a simple cubic lattice.

scholarly journals Catalogue of Clusters that are Members of Superclusters

1983 ◽  
Vol 104 ◽  
pp. 185-186
Author(s):  
M. Kalinkov ◽  
K. Stavrev ◽  
I. Kuneva
Keyword(s):  
Standard Deviation ◽  
Correlation Coefficient ◽  
The Other ◽  
Clusters Of Galaxies ◽  
Number Of Clusters ◽  
Abell Clusters ◽  
The Mean ◽  
Abell Cluster ◽  
Image Position

An attempt is made to establish the membership of Abell clusters in superclusters of galaxies. The relation is used to calibrate the distances to the clusters of galaxies with two redshift estimates. One is m10, the magnitude of the ten-ranked galaxy, and the other is the “mean population,” P, defined by: where p = 40, 65, 105 … galaxies for richness groups 0, 1, 2 …, and r is the apparent radius in degrees given by: The first iteration for redshift, z1, is obtained from m10 alone: The standard deviation for Eq. (1) is 0.105, the number of clusters with known velocities is 342 and the correlation coefficient between observed and fitted values is 0.921. With zi from Eq. (1), we define Cartesian galactic coordinates Xi = Rih−1 cosBi cosLi, Yi = Rih−1 cosBi sinLi, Zi = Rih−1 sinBi for each Abell cluster, i = 1, …, 2712, where Ri is the distance to the cluster (Mpc), and Ho = 100 h km s−1 Mpc−1.

scholarly journals Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection

1972 ◽  
Vol 45 ◽  
pp. 139-165 ◽  
Author(s):  
Joseph Erbacher
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Additional Assumption ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Normal Connection ◽  
Isometric Immersions ◽  
The Mean ◽  
Negative Sectional Curvature

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

scholarly journals The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1211 ◽  
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Elliptic Equations ◽  
Constant Mean Curvature ◽  
Euclidean Case ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

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