scholarly journals On Voronoi Diagrams on the Information-Geometric Cauchy Manifolds

Entropy ◽  
2020 ◽  
Vol 22 (7) ◽  
pp. 713 ◽  
Author(s):  
Frank Nielsen
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Information Geometry ◽  
Square Root ◽  
Chi Square ◽  
Scale Parameters ◽  
Leibler Divergence ◽  
Finite Set ◽  
Metric Distance ◽  
Rao Distance

We study the Voronoi diagrams of a finite set of Cauchy distributions and their dual complexes from the viewpoint of information geometry by considering the Fisher-Rao distance, the Kullback-Leibler divergence, the chi square divergence, and a flat divergence derived from Tsallis entropy related to the conformal flattening of the Fisher-Rao geometry. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, and that the dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Cauchy hyperbolic Voronoi diagrams. The dual Voronoi diagrams with respect to the dual flat divergences amount to dual Bregman Voronoi diagrams, and their dual complexes are regular triangulations. The primal Bregman Voronoi diagram is the Euclidean Voronoi diagram and the dual Bregman Voronoi diagram coincides with the Cauchy hyperbolic Voronoi diagram. In addition, we prove that the square root of the Kullback-Leibler divergence between Cauchy distributions yields a metric distance which is Hilbertian for the Cauchy scale families.

CHEMICAL TESSELLATIONS — RESULTS OF BINARY AND TERTIARY REACTIONS BETWEEN METAL IONS AND FERRICYANIDE OR FERROCYANIDE LOADED GELS

2010 ◽  
Vol 20 (07) ◽  
pp. 2241-2252 ◽  
Author(s):  
B. P. J. DE LACY COSTELLO ◽  
I. JAHAN ◽  
P. HAMBIDGE ◽  
K. LOCKING ◽  
D. PATEL ◽  
...  
Keyword(s):  
Metal Ions ◽  
Voronoi Diagram ◽  
Metal Salt ◽  
Voronoi Diagrams ◽  
Reaction Rates ◽  
Cross Reactivity ◽  
Diffusion Reaction ◽  
Reactive Metal ◽  
Simple Chemical ◽  
Pattern Forming

In our recent letter [de Lacy Costello et al., 2009] we described the formation of spontaneous complex tessellations of the plane constructed in simple chemical reactions between drops of metal salts and ferricyanide or ferrocyanide loaded gels. In this paper, we provide more examples of binary tessellations and extend our analysis to tessellations constructed via tertiary mixtures of reactants. We also provide a classification system which describes the tessellation based on the reactivity of the metal salt with the substrate and also the cross-reactivity of the primary products. This results in balanced tessellations where both reactants have equal reactivity or unbalanced tessellations where one reactant has a lower reactivity with the gel. The products can also be partially or fully cross reactive which gives a highly complex tessellation. The tessellations are made up of colored cells (corresponding to different metal ferricyanides or ferrocyanides) separated by bisectors of low precipitate concentration. The tessellations constructed by these reactions constitute generalized Voronoi diagrams. In the case of certain binary or tertiary combinations of reactants where the diffusion/reaction rates differ, then multiplicatively weighted crystal growth Voronoi diagrams are constructed. Where one reactant has limited or no reactivity with the gel (or the products are cross reactive) then the fronts originating from the reactive metal ions cross the fronts originating from the partially reactive metal ions. The fronts can annihilate in the formation of a second Voronoi diagram relating to the relative positions of the reactive drops. Therefore, two or more generalised or weighted Voronoi diagrams can be calculated in parallel by these simple chemical systems. However when these reactions were used to calculate an additively weighted Voronoi diagram (the reaction was initiated at different time intervals) the diagram constructed did not correspond to the theoretical calculation. We use the failure of these reactions to construct an additively weighted Voronoi diagram to prove a mechanism of substrate competition for bisector formation. These tessellations are an important class of pattern forming reactions and are useful in modeling natural pattern forming phenomena in addition to being a great resource for scientific demonstrations.

scholarly journals Square roots in finite full transformation semigroups

1982 ◽  
Vol 23 (2) ◽  
pp. 137-149 ◽  
Author(s):  
Mary Snowden ◽  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Square Root ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Square Roots ◽  
Finite Set ◽  
Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

Discrete Construction of Compoundly Weighted Voronoi Diagram

2013 ◽  
Vol 467 ◽  
pp. 545-548
Author(s):  
Hui Wang
Keyword(s):  
Production Process ◽  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
High Potential ◽  
Graphic Display ◽  
Potential Value ◽  
Discrete Algorithms ◽  
Traditional Algorithm ◽  
Definition Of

Compoundly weighted Voronoi diagram is difficult to construct because the bisector is fairly complex. In traditional algorithm, production process is always extremely complex and it is more difficult to graphic display because of the complex definition of mathematic formula. In this paper, discrete algorithms are used to construct compoundly weighted Voronoi diagrams. The algorithm can get over all kinds of shortcomings that we have just mentioned. So it is more useful and effective than the traditional algorithm. The results show that the algorithm is both simple and useful, and it is of high potential value in practice.

VORONOI DIAGRAMS FOR A TRANSPORTATION NETWORK ON THE EUCLIDEAN PLANE

2006 ◽  
Vol 16 (02n03) ◽  
pp. 117-144 ◽  
Author(s):  
SANG WON BAE ◽  
KYUNG-YONG CHWA
Keyword(s):  
Voronoi Diagram ◽  
Euclidean Plane ◽  
Transportation Network ◽  
Voronoi Diagrams ◽  
The Other ◽  
Constant Number ◽  
Time Path ◽  
The Given

This paper investigates geometric and algorithmic properties of the Voronoi diagram for a transportation network on the Euclidean plane. In the presence of a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm2+ m3+ nm log n) algorithm to compute the Voronoi diagram for a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm + m2+ n log n) time with O(m(n + m)) space and the other O(nm log n + m2log m) time with O(n + m) space.

A ROBUST TOPOLOGY-ORIENTED INCREMENTAL ALGORITHM FOR VORONOI DIAGRAMS

1994 ◽  
Vol 04 (02) ◽  
pp. 179-228 ◽  
Author(s):  
KOKICHI SUGIHARA ◽  
MASAO IRI
Keyword(s):  
Numerical Computation ◽  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
The Other ◽  
Geometric Algorithms ◽  
Incremental Algorithm ◽  
Single Precision ◽  
Careful Choice ◽  
Numerical Errors ◽  
Incremental Method

The paper presents a robust algorithm for constructing Voronoi diagrams in the plane. The algorithm is based on an incremental method, but is quite new in that it is robust against numerical errors. Conventionally, geometric algorithms have been designed on the assumption that numerical errors do not take place, and hence they are not necessarily valid for real computers where numerical errors are inevitable. The algorithm to be proposed in this paper, on the other hand, is designed with the recognition that numerical errors are inevitable in real computation; i.e., in the proposed algorithm higher priority is placed on topological structures than on numerical values. As a result, the algorithm is "completely" robust in the sense that it always gives some output however poor the precision of numerical computation may be. In general, the output cannot be more than an approximation to the true Voronoi diagram which we should have got by infinite-precision computation. However, the algorithm is asymptotically correct in the sense that the output converges to the true diagram as the precision becomes higher. Moreover, careful choice of the way of numerical computation makes the algorithm stable enough; indeed the present version of the algorithm can construct in single-precision arithmetic a correct Voronoi diagram for one million generators randomly placed in the unit square in the plane.

Information-Geometric Measure for Neural Spikes

2002 ◽  
Vol 14 (10) ◽  
pp. 2269-2316 ◽  
Author(s):  
Hiroyuki Nakahara ◽  
Shun-ichi Amari
Keyword(s):  
Information Geometry ◽  
Higher Order ◽  
Neural Firing ◽  
Firing Patterns ◽  
Neural Spikes ◽  
Geometric Measure ◽  
Leibler Divergence ◽  
Novel Method ◽  
Account Information ◽  
Spike Firing

This study introduces information-geometric measures to analyze neural firing patterns by taking not only the second-order but also higher-order interactions among neurons into account. Information geometry provides useful tools and concepts for this purpose, including the orthogonality of coordinate parameters and the Pythagoras relation in the Kullback-Leibler divergence. Based on this orthogonality, we show a novel method for analyzing spike firing patterns by decomposing the interactions of neurons of various orders. As a result, purely pairwise, triple-wise, and higher-order interactions are singled out. We also demonstrate the benefits of our proposal by using several examples.

Dynamic Construction of Voronoi Diagram for a Set of Points and Straight Line Segments

2014 ◽  
Vol 533 ◽  
pp. 264-267
Author(s):  
Xin Liu
Keyword(s):  
Production Process ◽  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
High Potential ◽  
Line Segments ◽  
Potential Value ◽  
Straight Line ◽  
Traditional Algorithm ◽  
Set Of Points

Voronoi Diagram for a set of points and straight line segments is difficult to construct because general figures have uncertain shapes[. In traditional algorithm, when generator of general figure changes, production process will be extremely complex because of the change of regions neighboring with those generator changed. In this paper, we use dynamicconstruction of Voronoi diagrams. The algorithm can get over all kinds of shortcomings that we have just mentioned. So it is more useful and effective than the traditional algorithm[2]. The results show that the algorithm is both simple and useful, and it is of high potential value in practice.

scholarly journals The Voronoi theory of the normal liver lobular architecture and its applicability in hepatic zonation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
C. Lau ◽  
B. Kalantari ◽  
K. P. Batts ◽  
L. D. Ferrell ◽  
S. L. Nyberg ◽  
...  
Keyword(s):  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
Normal Liver ◽  
Digital Tools ◽  
Liver Pathology ◽  
Two Dimensional ◽  
Lobular Architecture ◽  
Precise Characterization ◽  
Human Livers

AbstractThe precise characterization of the lobular architecture of the liver has been subject of investigation since the earliest historical publications, but an accurate model to describe the hepatic lobular microanatomy is yet to be proposed. Our aim was to evaluate whether Voronoi diagrams can be used to describe the classic liver lobular architecture. We examined the histology of normal porcine and human livers and analyzed the geometric relationships of various microanatomic structures utilizing digital tools. The Voronoi diagram model described the organization of the hepatic classic lobules with overall accuracy nearly 90% based on known histologic landmarks. We have also designed a Voronoi-based algorithm of hepatic zonation, which also showed an overall zonal accuracy of nearly 90%. Therefore, we have presented evidence that Voronoi diagrams represent the basis of the two-dimensional organization of the normal liver and that this concept may have wide applicability in liver pathology and research.

scholarly journals MULTI-AGENT SIMULATION OF ALLOCATING AND ROUTING AMBULANCES UNDER CONDITION OF STREET BLOCKAGE AFTER NATURAL DISASTER

2017 ◽  
Vol XLII-4/W4 ◽  
pp. 325-332 ◽  
Author(s):  
S. Azimi ◽  
M. R. Delavar ◽  
A. Rajabifard
Keyword(s):  
Natural Disasters ◽  
Voronoi Diagram ◽  
Smart City ◽  
Voronoi Diagrams ◽  
Multi Agent System ◽  
Medical Assistance ◽  
Agent System ◽  
Constraint Network ◽  
Multi Agent ◽  
The Difference

In response to natural disasters, efficient planning for optimum allocation of the medical assistance to wounded as fast as possible and wayfinding of first responders immediately to minimize the risk of natural disasters are of prime importance. This paper aims to propose a multi-agent based modeling for optimum allocation of space to emergency centers according to the population, street network and number of ambulances in emergency centers by constraint network Voronoi diagrams, wayfinding of ambulances from emergency centers to the wounded locations and return based on the minimum ambulances travel time and path length implemented by NSGA𝜫 and the use of smart city facilities to accelerate the rescue operation. Simulated annealing algorithm has been used for minimizing the difference between demands and supplies of the constrained network Voronoi diagrams. In the proposed multi-agent system, after delivering the location of the wounded and their symptoms, the constraint network Voronoi diagram for each emergency center is determined. This process was performed simultaneously for the multi-injuries in different Voronoi diagrams. In the proposed multi-agent system, the priority of the injuries for receiving medical assistance and facilities of the smart city for reporting the blocked streets was considered. Tehran Municipality District 5 was considered as the study area and during 3 minutes intervals, the volunteers reported the blocked street. The difference between the supply and the demand divided to the supply in each Voronoi diagram decreased to 0.1601. In the proposed multi-agent system, the response time of the ambulances is decreased about 36.7%.

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