New Classes of Random Tessellations Arising from Iterative Division of Cells

2010 ◽  
Vol 42 (1) ◽  
pp. 26-47 ◽  
Author(s):  
Richard Cowan
Keyword(s):  
Probability Distribution ◽  
Mathematical Formalism ◽  
Simulation Methods ◽  
Vast Number ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Typical Cell ◽  
New Ideas ◽  
Successive Division

We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

scholarly journals New Classes of Random Tessellations Arising from Iterative Division of Cells

2010 ◽  
Vol 42 (01) ◽  
pp. 26-47 ◽  
Author(s):  
Richard Cowan
Keyword(s):  
Probability Distribution ◽  
Mathematical Formalism ◽  
Simulation Methods ◽  
Vast Number ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Typical Cell ◽  
New Ideas ◽  
Successive Division

We present new ideas about the type of random tessellation which evolves through successive division of its cells. These ideas are developed in an intuitive way, with many pictures and only a modicum of mathematical formalism–so that the wide application of the ideas is clearly apparent to all readers. A vast number of new tessellation models, with known probability distribution for the volume of the typical cell, follow from the concepts in this paper. There are other interesting models for which results are not presented (or presented only through simulation methods), but these models have illustrative value. A large agenda of further research is opened up by the ideas in this paper.

Random mosaics with cells of general topology

1996 ◽  
Vol 28 (02) ◽  
pp. 332
Author(s):  
Richard Cowan ◽  
Albert K. L. Tsang
Keyword(s):  
Euler Characteristic ◽  
Ergodic Theory ◽  
General Topology ◽  
Traditional Theory ◽  
Random Partition ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Partition Process

This paper considers a structure, named a ‘random partition process’, which is a generalisation of a random tessellation. The cells, possibly multi-part and with holes, have a general topology summarised by the Euler characteristic. Vertices of all orders are allowed. Using the tools of ergodic theory, all of the formulae, from the traditional theory of random tessellations with convex cells, are generalised. Some motivating examples are given.

Further random tessellations with the classic Poisson polygon distributions

1996 ◽  
Vol 28 (02) ◽  
pp. 338-339 ◽  
Author(s):  
Roger E. Miles ◽  
Margaret S. Mackisack
Keyword(s):  
Orientation Distribution ◽  
Convex Polygons ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Random Convex

It is well-known that Poisson lines in the plane, with orientation distribution Θ on [0, π), generate a (random) tessellation M 0 of (random) convex polygons whose characteristics (area, perimeter, etc.) conform, in an ergodic sense, to a certain class {D Θ} of distributions.

scholarly journals Application of Simulation Methods for Evaluating the Sea Waterways Traffic Organisation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Agnieszka Blokus-Roszkowska ◽  
Leszek Smolarek
Keyword(s):  
Cellular Automaton ◽  
Flow Density ◽  
Final Conclusion ◽  
Simulation Methods ◽  
Traffic Intensity ◽  
Sea Transport ◽  
Global Industry ◽  
Maritime Traffic ◽  
New Ideas ◽  
Traffic Organization

Maritime transport is of great importance as it is the basis of international trade. Shipping is a global industry and highly complex business. Further safety development at sea is determined by the growth of maritime traffic intensity. Therefore, new ideas and technologies are needed to optimize the sea transport system. In the paper, the cellular automaton model is suggested to describe traffic flow at grade roundabout and simple crossing. Not only the cell state but also neighborhood as well is defined for presented cellular automaton. The model, describing vessels motion, takes into account vessel’s type, speed and length, vessel’s behavior and maneuverability, lane status, and flow density. It provides the basis for simulation. Two types of traffic schemes, namely, roundabout and simple routes crossing, are considered. Some results of simulation, including ships safety prediction, illustrate possible applications for evaluation of maritime traffic organization. Final conclusion and remarks outline further work development.

SIMULATING QUANTUM PARTICLES IN NON-SIMPLY-CONNECTED SPACE

1991 ◽  
Vol 06 (12) ◽  
pp. 2175-2187
Author(s):  
A.G. GROZIN
Keyword(s):  
Probability Distribution ◽  
Low Temperatures ◽  
Simulation Methods ◽  
Simulation Algorithm ◽  
Connected Space ◽  
Homotopy Classes ◽  
Quantum Particles ◽  
Simply Connected

The usual simulation methods are inapplicable for a particle in a non-simply-connected space, because closed paths are divided into homotopy classes. The probability distribution over these classes is calculated analytically at low temperatures. A simulation algorithm is proposed.

scholarly journals NEW MEAN VALUES FOR HOMOGENEOUS SPATIAL TESSELLATIONS THAT ARE STABLE UNDER ITERATION

2010 ◽  
Vol 29 (3) ◽  
pp. 143 ◽  
Author(s):  
Christoph Thäle ◽  
Viola Weiss
Keyword(s):  
Euclidean Space ◽  
Convex Geometry ◽  
Dimensional Euclidean Space ◽  
Mean Values ◽  
3 Dimensional ◽  
Extremum Problems ◽  
Random Tessellations ◽  
Typical Cell

Homogeneous random tessellations in the 3-dimensional Euclidean space are considered that are stable under iteration – STIT tessellations. A classification of vertices, segments and flats is introduced and a couple of new metric and topological mean values for them and for the typical cell are calculated. They are illustrated by two examples, the isotropic and the cuboid case. Several extremum problems for these mean values are solved with the help of techniques from convex geometry by introducing an associated zonoid for STIT tessellations.

scholarly journals Partículas e interações fundamentais

1992 ◽  
Vol 14 (14) ◽  
pp. 71
Author(s):  
Orimar Antônio Battistel
Keyword(s):  
Quark Model ◽  
Historical Context ◽  
Mathematical Formalism ◽  
Particle Properties ◽  
Fundamental Interactions ◽  
Modern Conception ◽  
New Ideas ◽  
Structure Of Matter

We present a discussion of some of the ideas and facts that led to the modern conception of the structure of matter in terms of basic constituents and fundamental interactions. We discuss the construction of the quark model in a historical context, stressing that the phenomenological knowledge of particle properties induced the mathematical formalism. The interaction between quarks is discussed in the framework of the color force based on the exchange of gluons. For the sake of completeness we mention briefly the microworld structure that emerged from these new ideas.

scholarly journals Compilation by stochastic Hamiltonian sparsification

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 235 ◽  
Author(s):  
Yingkai Ouyang ◽  
David R. White ◽  
Earl T. Campbell
Keyword(s):  
Electronic Structure ◽  
Quantum Computing ◽  
Probability Distribution ◽  
Quantum Chemistry ◽  
Error Bounds ◽  
Simulation Methods ◽  
Physical Systems ◽  
First Order ◽  
Convex Optimisation ◽  
Principal Application

Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.

Random mosaics with cells of general topology

1996 ◽  
Vol 28 (2) ◽  
pp. 332-332
Author(s):  
Richard Cowan ◽  
Albert K. L. Tsang
Keyword(s):  
Euler Characteristic ◽  
Ergodic Theory ◽  
General Topology ◽  
Traditional Theory ◽  
Random Partition ◽  
Random Tessellation ◽  
Random Tessellations ◽  
Partition Process

This paper considers a structure, named a ‘random partition process’, which is a generalisation of a random tessellation. The cells, possibly multi-part and with holes, have a general topology summarised by the Euler characteristic. Vertices of all orders are allowed. Using the tools of ergodic theory, all of the formulae, from the traditional theory of random tessellations with convex cells, are generalised. Some motivating examples are given.

scholarly journals Probabilistic Models, Learning Algorithms, Response Variability: Sampling in Cognitive Development

2018 ◽  
Author(s):  
Elizabeth Bonawitz ◽  
Stephanie Denison ◽  
Tom Griffiths ◽  
Alison Gopnik
Keyword(s):  
Cognitive Development ◽  
Probability Distribution ◽  
Probabilistic Models ◽  
Learning Algorithms ◽  
Response Variability ◽  
Vast Number ◽  
Empirical Results ◽  
Intractable Problems

Although probabilistic models of cognitive development have become increasingly prevalent, one challenge is to account for how children might cope with a potentially vast number of possible hypotheses. We propose that children might address this problem by ‘sampling’ hypotheses from a probability distribution. We discuss empirical results demonstrating signatures of sampling, which offer an explanation for the variability of children's responses. The sampling hypothesis provides an algorithmic account of how children might address computationally intractable problems and suggests a way to make sense of their ‘noisy’ behavior.

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