scholarly journals Strict Inequalities of Critical Values in Continuum Percolation

2011 ◽  
Vol 142 (3) ◽  
pp. 460-486 ◽  
Author(s):  
Massimo Franceschetti ◽  
Mathew D. Penrose ◽  
Tom Rosoman
Keyword(s):  
Critical Values ◽  
Continuum Percolation ◽  
Strict Inequalities

scholarly journals Homological Percolation: The Formation of Giant k-Cycles

2020 ◽  
Author(s):  
Omer Bobrowski ◽  
Primoz Skraba
Keyword(s):  
Thermodynamic Limit ◽  
Exponential Decay ◽  
Algebraic Topology ◽  
Percolation Model ◽  
Critical Values ◽  
Giant Component ◽  
Connected Components ◽  
Continuum Percolation ◽  
Dimensional Torus ◽  
Higher Dimensional

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

Critical Values

2008 ◽  
Vol 3 (3) ◽  
pp. 201-202
Author(s):  
Keith Krehbiel
Keyword(s):  
Critical Values

scholarly journals Detecting Heterogeneity of Substitution Along DNA and Protein Sequences

Genetics ◽  
1996 ◽  
Vol 143 (1) ◽  
pp. 589-602 ◽  
Author(s):  
Peter J E Goss ◽  
R C Lewontin
Keyword(s):  
Null Hypothesis ◽  
Nonuniform Distribution ◽  
Critical Values ◽  
Third Order ◽  
Variance Test ◽  
Data Set ◽  
Significance Level ◽  
Uniformly Most Powerful Test ◽  
Interval Lengths ◽  
Uniformly Most Powerful

Abstract Regions of differing constraint, mutation rate or recombination along a sequence of DNA or amino acids lead to a nonuniform distribution of polymorphism within species or fixed differences between species. The power of five tests to reject the null hypothesis of a uniform distribution is studied for four classes of alternate hypothesis. The tests explored are the variance of interval lengths; a modified variance test, which includes covariance between neighboring intervals; the length of the longest interval; the length of the shortest third-order interval; and a composite test. Although there is no uniformly most powerful test over the range of alternate hypotheses tested, the variance and modified variance tests usually have the highest power. Therefore, we recommend that one of these two tests be used to test departure from uniformity in all circumstances. Tables of critical values for the variance and modified variance tests are given. The critical values depend both on the number of events and the number of positions in the sequence. A computer program is available on request that calculates both the critical values for a specified number of events and number of positions as well as the significance level of a given data set.

Critical Values in Cytology

2021 ◽  
Author(s):  
Allison M. Onken ◽  
Paul A. VanderLaan ◽  
Matthew W. Rosenbaum
Keyword(s):  
Critical Values

scholarly journals Mechanical criterion for coal and gas outburst: a perspective from multiphysics coupling

2021 ◽  
Author(s):  
Ting Liu ◽  
Baiquan Lin ◽  
Xuehai Fu ◽  
Ang Liu
Keyword(s):  
Control Method ◽  
Risk Identification ◽  
Controlling Factors ◽  
Gas Pressure ◽  
Critical Values ◽  
Coal And Gas Outburst ◽  
Strong Interactions ◽  
Gas Outburst ◽  
Multiphysics Coupling ◽  
Coal Strength

AbstractAlthough a series of hypotheses have been proposed, the mechanism underlying coal and gas outburst remains unclear. Given the low-index outbursts encountered in mining practice, we attempt to explore this mechanism using a multiphysics coupling model considering the effects of coal strength and gas mass transfer on failure. Based on force analysis of coal ahead of the heading face, a risk identification index Cm and a critical criterion (Cm ≥ 1) of coal instability are proposed. According to this criterion, the driving force of an outburst consists of stress and gas pressure gradients along the heading direction of the roadway, whereas resistance depends on the shear and tensile strengths of the coal. The results show that outburst risk decreases slightly, followed by a rapid increase, with increasing vertical stress, whereas it decreases with increasing coal strength and increases with gas pressure monotonically. Using the response surface method, a coupled multi-factor model for the risk identification index is developed. The results indicate strong interactions among the controlling factors. Moreover, the critical values of the factors corresponding to outburst change depending on the environment of the coal seams, rather than being constants. As the buried depth of a coal seam increases, the critical values of gas pressure and coal strength decrease slightly, followed by a rapid increase. According to its controlling factors, outburst can be divided into stress-dominated, coal-strength-dominated, gas-pressure-dominated, and multi-factor compound types. Based on this classification, a classified control method is proposed to enable more targeted outburst prevention.

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