Application of the response probability density function technique to predicting the probability of sonic‐boom glass breakage

1974 ◽  
Vol 55 (5) ◽  
pp. 1009-1017
Author(s):  
Robert L. Hershey ◽  
Thomas H. Higgins ◽  
Edward B. Magrab
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Response Probability ◽  
Sonic Boom ◽  
Function Technique ◽  
Glass Breakage

scholarly journals Assessment of a three-dimensional line-of-response probability density function system matrix for PET

2012 ◽  
Vol 57 (21) ◽  
pp. 6827-6848 ◽  
Author(s):  
Rutao Yao ◽  
Ranjith M Ramachandra ◽  
Neeraj Mahajan ◽  
Vinay Rathod ◽  
Noel Gunasekar ◽  
...  
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Three Dimensional ◽  
Response Probability ◽  
Function System ◽  
System Matrix

scholarly journals Shaping of molecular weight distribution by iterative learning probability density function control strategies

2008 ◽  
Vol 222 (7) ◽  
pp. 639-653 ◽  
Author(s):  
H Yue ◽  
H Wang ◽  
J Zhang
Keyword(s):  
Molecular Weight ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Molecular Weight Distribution ◽  
Weight Distribution ◽  
Control Strategies ◽  
Iterative Learning ◽  
Function Technique ◽  
B Spline

A mathematical model is developed for the molecular weight distribution (MWD) of free-radical styrene polymerization in a simulated semi-batch reactor system. The generation function technique and moment method are employed to establish the MWD model in the form of Schultz—Zimm distribution. Both static and dynamic models are described in detail. In order to achieve the closed-loop MWD shaping by output probability density function (PDF) control, the dynamic MWD model is further developed by a linear B-spline approximation. Based on the general form of the B-spline MWD model, iterative learning PDF control strategies have been investigated in order to improve the MWD control performance. Discussions on the simulation studies show the advantages and limitations of the methodology.

Path Integral Method for Nonlinear Systems Under Levy White Noise

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Alberto Di Matteo ◽  
Antonina Pirrotta
Keyword(s):  
Probability Density Function ◽  
Nonlinear Systems ◽  
Probability Density ◽  
Density Function ◽  
Path Integral ◽  
Response Probability ◽  
Stability Index ◽  
Integral Solution ◽  
Stable Random Variables ◽  
White Noises

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

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