scholarly journals Two limit theorems for the high-dimensional two-stage contact process

2020 ◽  
Vol 17 (2) ◽  
pp. 825
Author(s):  
Xiaofeng Xue
Keyword(s):  
Limit Theorems ◽  
Contact Process ◽  
High Dimensional ◽  
Two Stage

Super-Brownian Motion and Critical Spatial Stochastic Systems

2004 ◽  
Vol 47 (2) ◽  
pp. 280-297 ◽  
Author(s):  
Ed Perkins
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Contact Process ◽  
Voter Model ◽  
Oriented Percolation ◽  
Short Introduction ◽  
Super Brownian Motion

AbstractThis article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces

1982 ◽  
Vol 19 (01) ◽  
pp. 221-228 ◽  
Author(s):  
A. J. Stam
Keyword(s):  
Normal Distribution ◽  
Uniform Distribution ◽  
Total Variation ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Geometric Probability ◽  
High Dimensional ◽  
Standard Normal Distribution ◽  
Uniform Distributions ◽  
Standard Normal

If X = (X 1, · ··, Xn ) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.

scholarly journals New Results for the Two-Stage Contact Process

2015 ◽  
Vol 52 (01) ◽  
pp. 258-268 ◽  
Author(s):  
Eric Foxall
Keyword(s):  
Contact Process ◽  
Duality Relation ◽  
Two Stage ◽  
Open Questions

In this paper, we continue the work started by Steve Krone on the two-stage contact process. We give a simplified proof of the duality relation and answer most of the open questions posed in Krone (1999). We also fill in the details of an incomplete proof.

Visualizing and Evaluating High-Dimensional Mappings of Sets of High Performance Designs

2019 ◽  
Author(s):  
Clinton B. Morris ◽  
Michael R. Haberman ◽  
Carolyn C. Seepersad
Keyword(s):  
High Performance ◽  
Design Space Exploration ◽  
Design Space ◽  
Space Structure ◽  
Three Dimensions ◽  
High Dimensional ◽  
Underlying Structure ◽  
Visualization Technique ◽  
Two Stage ◽  
Input Design

Abstract Design space exploration can reveal the underlying structure of design problems. In a set-based approach, for example, exploration can map sets of designs or regions of the design space that meet specific performance requirements. For some problems, promising designs may cluster in multiple regions of the input design space, and the boundaries of those clusters may be irregularly shaped and difficult to predict. Visualizing the promising regions can clarify the design space structure, but design spaces are typically high-dimensional, making it difficult to visualize the space in three dimensions. To convey the structure of such high-dimensional design regions, a two-stage approach is proposed to (1) identify and (2) visualize each distinct cluster or region of interest in the input design space. This paper focuses on the visualization stage of the approach. Rather than select a singular technique to map high-dimensional design spaces to low-dimensional, visualizable spaces, a selection procedure is investigated. Metrics are available for comparing different visualizations, but the current metrics either overestimate the quality or favor selection of certain visualizations. Therefore, this work introduces and validates a more objective metric, termed preservation, to compare the quality of alternative visualization strategies. Furthermore, a new visualization technique previously unexplored in the design automation community, t-Distributed Neighbor Embedding, is introduced and compared to other visualization strategies. Finally, the new metric and visualization technique are integrated into a two-stage visualization strategy to identify and visualize clusters of high-performance designs for a high-dimensional negative stiffness metamaterials design problem.

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