scholarly journals From Characteristic Function to Distribution Function: A Simple Framework for the Theory

1991 ◽  
Vol 7 (4) ◽  
pp. 519-529 ◽  
Author(s):  
N.G. Shephard
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Unified Framework ◽  
New Approach ◽  
Inversion Theorem

A unified framework is established for the study of the computation of the distribution function from the characteristic function. A new approach to the proof of Gurland's and Gil-Pelaez's univariate inversion theorem is suggested. A multivariate inversion theorem is then derived using this technique.

A model for a system subject to random shocks

1993 ◽  
Vol 30 (4) ◽  
pp. 979-984 ◽  
Author(s):  
Eui Yong Lee ◽  
Jiyeon Lee
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Stochastic Model ◽  
Poisson Process ◽  
The State ◽  
Stationary Case ◽  
Random Shocks ◽  
Random Amount ◽  
Explicit Expressions

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

Curvature, a mechanical link between the geometrical complexities of a fault: application to bends, kinks and rough faults

2020 ◽  
Vol 223 (1) ◽  
pp. 211-232
Author(s):  
Pierre Romanet ◽  
Dye SK Sato ◽  
Ryosuke Ando
Keyword(s):  
Numerical Models ◽  
Slip Distribution ◽  
Unified Framework ◽  
New Approach ◽  
Fault Surface ◽  
Earthquake Mechanics ◽  
Analytical Perspective ◽  
The Difference ◽  
Field Lines ◽  
New Interpretation

SUMMARY Many recent studies have tried to determine the influence of geometry of faults in earthquake mechanics. However, it still remains largely unknown, and it is explored mainly with numerical models. In this paper, we will try to understand how exactly does the geometry come into play in the mechanics of an earthquake from analytical perspective. We suggest a new interpretation of the effect of geometry on the stress on a fault, where the curvatures of the fault that multiply the slip play a major role. Starting from the representation theorem, which links the displacement in a medium to the slip distribution on its boundary, and assuming a 3-D, homogeneous, infinite medium, a regularized boundary-element equation can be obtained. Using this equation, it is possible to separate the influence of geometry, as expressed by the curvatures and torsions of the field lines of slip on the fault surface, which multiply the slip, from the effect of the gradient of slip. This allows us to shed new light on the mechanical effects of geometrical complexities on the fault surface, with the key parameters being the curvatures and torsions of the slip field lines. We have used this new approach to show that, in 2-D static in-plane problems, the shear traction along the fault is mainly controlled by the gradient of slip along the fault, while the normal traction is mainly controlled by the slip that multiplies the curvature along the fault. Finally, we applied this new approach to re-interpret the effect of roughness (why there is a need for a minimum lengthscale in linear elasticity, how to study mechanically the difference of roughness measurements with the direction of slip, scaling of slip distribution versus geometry), bends and kinks (what is the difference between the two, are they sometimes equivalent), as well as to explain further the false paradox between smooth-and-abrupt-bends. This unified framework allows us to improve greatly our understanding of the effect of fault geometry on the mechanics of earthquakes.

scholarly journals Mapping structure and morphology of amorphous organic thin films by 4D-STEM pair distribution function analysis

Microscopy ◽  
2019 ◽  
Vol 68 (4) ◽  
pp. 301-309 ◽  
Author(s):  
Xiaoke Mu ◽  
Andrey Mazilkin ◽  
Christian Sprau ◽  
Alexander Colsmann ◽  
Christian Kübel
Keyword(s):  
Distribution Function ◽  
Electron Diffraction ◽  
Energy Loss ◽  
Electron Energy ◽  
Electron Energy Loss Spectroscopy ◽  
Pair Distribution Function ◽  
Phase Distribution ◽  
Function Analysis ◽  
New Approach ◽  
Electron Energy Loss

Abstract Imaging the phase distribution of amorphous or partially crystalline organic materials at the nanoscale and analyzing the local atomic structure of individual phases has been a long-time challenge. We propose a new approach for imaging the phase distribution and for analyzing the local structure of organic materials based on scanning transmission electron diffraction (4D-STEM) pair distribution function analysis (PDF). We show that electron diffraction based PDF analysis can be used to characterize the short- and medium-range order in aperiodically packed organic molecules. Moreover, we show that 4D-STEM-PDF does not only provide local structural information with a resolution of a few nanometers, but can also be used to image the phase distribution of organic composites. The distinct and thickness independent contrast of the phase image is generated by utilizing the structural difference between the different types of molecules and taking advantage of the dose efficiency due to use of the full scattering signal. Therefore, this approach is particularly interesting for imaging unstained organic or polymer composites without distinct valence states for electron energy loss spectroscopy. We explore the possibilities of this new approach using [6,6]-phenyl-C61- butyric acid methyl ester (PC61BM) and poly(3-hexylthiophene-2,5-diyl) (P3HT) as the archetypical and best-investigated semiconductor blend used in organic solar cells, compare our phase distribution with virtual dark-field analysis and validate our approach by electron energy loss spectroscopy.

scholarly journals Approximations to Risk Theory's F(x, t) by Means of the Gamma Distribution

1977 ◽  
Vol 9 (1-2) ◽  
pp. 213-218 ◽  
Author(s):  
Hilary L. Seal
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Gamma Distribution ◽  
Gamma Function ◽  
Function Approximation ◽  
Good Accuracy ◽  
Electronic Computer ◽  
Pocket Computer ◽  
Power Approximation ◽  
Aggregate Claims

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.

scholarly journals Statistical Verification of Parameters Describing Liquid Adsorption on Heterogeneous Solid Surfaces

1999 ◽  
Vol 82 (6) ◽  
pp. 1495-1504 ◽  
Author(s):  
Przemysław Podkościelny ◽  
Andrzej Dąbrowski
Keyword(s):  
Distribution Function ◽  
Solid Surfaces ◽  
Surface Phase ◽  
Adsorption Model ◽  
Exponential Equation ◽  
New Approach ◽  
Liquid Adsorption ◽  
Experimental Systems ◽  
Statistical Verification ◽  
Heterogeneous Solid

Abstract A new approach is presented for determining the surface phase capacity and the distribution function in liquid adsorption on heterogeneous solid surfaces. This approach deals with statistical analysis of the errors made by applying the so-called exponential equation of the adsorption isotherm. It is proved that such an analysis, which is suitable for all types of excess experimental isotherms, gives the answer to the dilemma of the reliability of the results obtained by means of the assumed adsorption model. Several experimental systems were analyzed to demonstrate the advantage of our approach.

scholarly journals Simplified Neutrosophic Sets Based on Interval Dependent Degree for Multi-Criteria Group Decision-Making Problems

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 640 ◽  
Author(s):  
Xu Libo ◽  
Li Xingsen ◽  
Pang Chaoyi ◽  
Guo Yan
Keyword(s):  
Decision Making ◽  
Distribution Function ◽  
Group Decision Making ◽  
Group Decision ◽  
Dynamic Change ◽  
Transformation Operator ◽  
Decision Makers ◽  
Neutrosophic Sets ◽  
New Approach ◽  
Dependent Function

In this paper, a new approach and framework based on the interval dependent degree for multi-criteria group decision-making (MCGDM) problems with simplified neutrosophic sets (SNSs) is proposed. Firstly, the simplified dependent function and distribution function are defined. Then, they are integrated into the interval dependent function which contains interval computing and distribution information of the intervals. Subsequently, the interval transformation operator is defined to convert simplified neutrosophic numbers (SNNs) into intervals, and then the interval dependent function for SNNs is deduced. Finally, an example is provided to verify the feasibility and effectiveness of the proposed method, together with its comparative analysis. In addition, uncertainty analysis, which can reflect the dynamic change of the final result caused by changes in the decision makers’ preferences, is performed in different distribution function situations. That increases the reliability and accuracy of the result.

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