Real and Complex Analysis of Common Probability Density Functions

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

Uncertainty in surfactant-polymer flooding is an important challenge to the wide scale implementation of this process. Any successful design of this enhanced oil recovery process will necessitate a good understanding of uncertainty. Thus it is essential to have the ability to quantify this uncertainty in an efficient manner. Monte Carlo Simulation is the traditional uncertainty quantification approach that is used for quantifying parametric uncertainty. However, the convergence of Monte Carlo simulation is relatively low requiring a large number of realizations to converge. This study proposes the use of the probabilistic collocation method in parametric uncertainty quantification for surfactant-polymer flooding using four synthetic reservoir models. Four sources of uncertainty were considered: the chemical flood residual oil saturation, surfactant and polymer adsorption and the polymer viscosity multiplier. The output parameter approximated is the recovery factor. The output metrics were the probability density function and the first two moments. These were compared with the results obtained from Monte Carlo simulation over a large number of realizations. Two methods for solving for the coefficients of the output parameter polynomial chaos expansion are compared: Gaussian quadrature and linear regression. The linear regression approach used two types of sampling: Gaussian quadrature nodes and Chebyshev derived nodes. In general, the probabilistic collocation method was applied successfully to quantify the uncertainty in the recovery factor. Applying the method using Gaussian quadrature produced more accurate results compared with using linear regression with quadrature nodes. Applying the method using linear regression with Chebyshev derived sampling also performed relatively well. Possible enhancements to improve the performance of the probabilistic collocation method were discussed. These enhancements include: improved sparse sampling, approximation order independent sampling and using arbitrary random input distribution that could be more representative of reality.

An accurate prediction of the probability density function (PDF) of the mixture fraction is crucial to the prediction of combustion since mixing plays an important role in turbulent non-premixed and partially premixed flames. This work provides an assessment of the large-eddy simulation (LES)/filtered density function (FDF) method for the prediction of the PDF of the mixture fraction. The advantage of the LES/FDF method is that it provides the full predictions of the statistical distribution of scalars including the mixture fraction. The predictive accuracy of the method for the PDF is yet to be fully validated. Assessing the prediction of the PDF of the mixture fraction, a conserved scalar, is an important starting point. The Sydney/Sandia inhomogeneous inlet jet flame is used as a test case. A quick comparison shows that the LES/FDF predicted PDF shapes of the mixture fraction deviate significantly from the commonly presumed Beta-PDF as well as from the experimental data in the flame. To examine the source of the discrepancy, we clarify the different PDF definitions used in the comparison among the predictions, measurements, and presumed shape PDFs. The discrepancy observed from the comparison is largely reconciled by clarifying the difference between the PDFs that are examined. The PDF of the resolved mixture fraction is shown to be close to the Beta-PDF in both the measurements and predictions, while the PDF directly deduced from the LES/FDF particles deviates significantly from the Beta-PDF. A multimodal PDF analysis and a pseudo convergence analysis are conducted to provide plausible evidence to support the predicted multimodal PDF shapes. The sub-filter scale FDF is shown to be close to the Beta-PDF too through the construction of a synthesised PDF, which supports the common presumed Beta-PDF assumption used in the presumed PDF methods when combined with LES.

Due to negative numbers do not have square roots, scientists introduced complex numbers, which are more abstract compared with real numbers. Residue Theorem has a very significant status in complex analysis – it can be used to simplify difficult integrals. In this article, Cauchy’s Residue Theorem is first introduced with definition and proof. Then the author shows two examples of using the theorem to solve problems, which can help readers to better explain the theorem. After that, K-residue Theorem is introduced, which is a special kind of Residue Theorem, with a special function called K-function. Finally, the theorem is used to solve a problem. These problems are difficult to solve without Residue Theorem. Through the analysis, it is easy to find the importance of Residue Theorem: it is very useful, not only in math but also in many other fields like physics, biology, and chemistry. Moreover, it still has more potential for people to develop.

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Combustion instability due to thermo-acoustic interactions in high-speed propulsion devices such as gas turbines and rocket engines result from pressure waves with very large amplitudes propagating back and forth in the combustion chamber. Exposure to the pressure fluctuations over a long period of time can lead to a cataclysmic failure of engines. The underlying physics governing the generation of the thermo-acoustic instability is a complex interaction among heat release, turbulence, and acoustic waves. Currently, it is very difficult to accurately predict the expected level of oscillations in a combustor. Hence development of strategies and engineering solutions to mitigate thermo-acoustic instability is an active area of research in both academia and industry. In this work, we carry out numerical modeling of thermo-acoustic instability in a self-excited, laboratory scale, model rocket combustor developed at Purdue University. Two different turbulent combustion models to account for turbulence-chemistry interactions are considered in this study, the flamelet model and the transported probability density function (PDF) method. <br>In the flamelet modeling approach, detailed chemical kinetics can be easily incorporated at a relatively low cost in comparison to other turbulent combustion models and it also accounts for turbulence-chemistry interactions. The flamelet model study is divided into two parts. In first part, we examine the effect of different numerical approaches for implementing the flamelet model. In advanced modeling and simulations of turbulent combustion, the accuracy of model predictions is affected by physical model errors as well as errors that arise from the numerical implementation of models in simulation codes. Here we are mainly concerned with the effect of numerical implementation on model predictions of turbulent combustion. Particularly, we employ the flamelet/progress variable (FPV) model and examine the effect of various numerical approaches for the flamelet table integration, with presumed shapes of PDF, on the FPV modeling results. Three different presumed-PDF table integration approaches are examined in detail by employing different numerical integration strategies. The effect of the different presumed-PDF table integration approaches is examined on predictions of two real flames, a laboratory-scale turbulent free jet flame, Sandia Flame D and the self-excited resonance model rocket combustor. Significant difference is observed in the predictions both of the flames. The results in this study further support the claims made in previous studies that it is imperative to preserve the laminar flamelet structure during integration while using the flamelet model to achieve better predictions in simulations. In the second part of the flamelet modeling study, computational investigations of the coupling between the transient flame dynamics such as the ignition delay and local extinction and the thermo-acoustic instability developed in a self-excited resonance combustor to gain deep insights into the mechanisms of thermo-acoustic instability. A modeling framework that employs different flamelet models (the steady flamelet model and the flamelet/progress variable approach) is developed to enable the examination of the effect of the transient flame dynamics caused by the strong coupling of the turbulent mixing and finite-rate chemical kinetics on the occurrence of thermo-acoustic instability. The models are validated by using the available experimental data for the pressure signal. Parametric studies are performed to examine the effect of the occurrence of the transient flame dynamics, the effect of artificial amplification of the Damkohler number, and the effect of neglecting mixture fraction fluctuations on the predictions of the thermo-acoustic instability. The parametric studies reveal that the occurrence of transient flame dynamics has a strong influence on the onset of the thermo-acoustic instability. Further analysis is then conducted to localize the effect of a particular flame dynamic event, the ignition delay, on the thermo-acoustic instability. The reverse effect of the occurrence of the thermo-acoustic instability on the transient flame dynamics in the combustor is also investigated by examining the temporal evolution of the local flame events in conjunction with the pressure wave propagation. The above observed two-way coupling between the transient flame dynamics (the ignition delay) and the thermo-acoustic instability provides a plausible mechanism of the self-excited and sustained thermo-acoustic instability observed in the combustor.<br>The second turbulent combustion model considered in this study is the transported PDF method. The transported PDF method is one of the most attractive models because it treats the highly-nonlinear chemical reaction source term without a closure requirement and it is a generalized model for a wide range of turbulent combustion problems.Traditionally, the transported PDF method has been used to model low-Mach number, incompressible flows where the pressure is assumed to be thermodynamically constant. Since there is significant pressure fluctuations in the model rocket combustor, the flow is highly compressible and it is necessary to account for this compressibility in the transported PDF method. In the past there has been very little work to model compressible reactive flows using the transported PDF and no effort has been made to model thermo-acoustic instability using the transported PDF method. There is a pressing need to further examine and develop the transported PDF method for compressible reactive flows to broaden our understanding of physical phenomenon like thermo-acoustic instability, interaction between combustion and strong shock and expansion waves, coupling between acoustic and heat release which are observed in high-speed turbulent combustion problems. To address this, a modeling framework for compressible turbulent reactive flows by the using the transported PDF method is developed. This framework is validated in a series of test cases ranging from pure mixing to a supersonic turbulent jet flame. The framework is then used to study the thermo-acoustic interactions in the self-excited model rocket combustor.

The statistics of laser scintillation have been studied for many years. The random fluctuations are describedby the probability density function (PDF). A rigorous solution for the PDF exists only for weak scattering (lognormal PDF) and very strong scattering (negative exponential PDF). Many models for a general PDF have beenproposed, and a common method of analysis has been the comparison of the moments of the PDF with experi-mental data. The accuracy of these experimental estimates for the moments is difficult to determine theoreti-cally. The statistical behavior of these estimates for the moments is determined by simulation of actual experi-mental procedures assuming various models for the true PDF. Realizations for histograms of digitized data is gen- erated using binomial random deviates. The only parameters required are the total number of statistically independent data samples collected and the average value for each data sample, which depends on the modelPDF. Each realization for the histogram generates a realization for the moments. The PDF of these estimates forthe moments is approximated by producing a histogram of a large number of such realizations. The PDF of theestimates for the moments then provides reliable confidence intervals. The behavior of the confidence limits ispresented for common models for the PDF of laser scintillation.

11 - Are my results significant?

Structural health monitoring has arisen as an important tool for managing and maintaining civil infrastructure. A critical problem for all structural health monitoring systems is data loss or data corruption due to sensor failure or other malfunctions, which bring into question in subsequent structural health monitoring data analysis and decision-making. Probability density functions play a very important role in many applications for structural health monitoring. This article focuses on data loss compensation for probability density function estimation in structural health monitoring using imputation methods. Different from common data, continuous probability density functions belong to functional data; the conventional distribution-to-distribution regression technique has significant potential in missing probability density function imputation; however, extrapolation and directly borrowing shape information from the covariate probability density function are the main challenges. Inspired by the warping transformation of distributions in the field of functional data analysis, a new distribution regression approach for imputing missing correlated probability density functions is proposed in this article. The warping transformation for distributions is a mapping operation used to transform one probability density function to another by deforming the original probability density function with a warping function. The shape mapping between probability density functions can be characterized well by warping functions. Given a covariate probability density function, the warping function is first estimated by a kernel regression model; then, the estimated warping function is used to transform the covariate probability density function and obtain an imputation for the missing probability density function. To address issues with poor performance when the covariate probability density function is contaminated, a hybrid approach is proposed that fuses the imputations obtained by the warping transformation approach with the conventional distribution-to-distribution regression approach. Experiments based on field monitoring data are conducted to evaluate the performance of the proposed approach. The corresponding results indicate that the proposed approach can outperform the conventional method, especially in extrapolation. The proposed approach shows good potential to provide more reliable estimation of distributions of missing structural health monitoring data.

The objective of this work is to numerically estimate the fraction of Reserves assigned to each Reserves category of the PRMS matrix through a cumulative distribution function. We selected 38 wells from a Permian Basin dataset available to Texas A&amp;M University. Previous work has shown that Swanson's Mean, which relates the Reserves categories through a cdf of a normal distribution, is an inaccurate method to determine the relationship of the Reserves categories with asymmetric distributions. Production data are lognormally distributed, regardless of basin type, thus cannot follow the SM concept. The Gaussian Quadrature (GQ) provides a methodology to accurately estimate the fraction of Reserves that lie in 1P, 2P, and 3P categories – known as the weights. Gaussian Quadrature is a numerical integration method that uses discrete random variables and a distribution that matches the original data. For this work, we associate the lognormal cumulative distribution function (CDF) with a set of discrete random variables that replace the production data, and determine the associated probabilities. The production data for both conventional and unconventional fields are lognormally distributed, thus we expect that this methodology can be implemented in any field. To do this, we performed probabilistic decline curve analysis (DCA) using Arps’ Hyperbolic model and Monte Carlo simulation to obtain the 1P, 2P, and 3P volumes, and calculated the relative weights of each Reserves category. We performed probabilistic rate transient analysis (RTA) using a commercial software to obtain the 1P, 2P, and 3P volumes, and calculated the relative weights of each Reserves category. We implemented the 3-, 5-, and 10-point GQ to obtain the weight and percentiles for each well. Once this was completed, we validated the GQ results by calculating the percent-difference between the probabilistic DCA, RTA, and GQ results. We increase the standard deviation to account for the uncertainty of Contingent and Prospective resources and implemented 3-, 5-, and 10-point GQ to obtain the weight and percentiles for each well. This allows us to also approximate the weights of these volumes to track them through the life of a given project. The probabilistic DCA, RTA and Reserves results indicate that the SM is an inaccurate method for estimating the relative weights of each Reserves category. The 1C, 2C, 3C, and 1U, 2U, and 3U Contingent and Prospective Resources, respectively, are distributed in a similar way but with greater variance, incorporated in the standard deviation. The results show that the GQ is able to capture an accurate representation of the Reserves weights through a lognormal CDF. Based on the proposed results, we believe that the GQ is accurate and can be used to approximate the relationship between the PRMS categories. This relationship will aid in booking Reserves to the SEC because it can be recreated for any field. These distributions of Reserves and resources other than Reserves (ROTR) are important for planning and for resource inventorying. The GQ provides a measure of confidence on the prediction of the Reserves weights because of the low percent difference between the probabilistic DCA, RTA, and GQ weights. This methodology can be implemented in both conventional and unconventional fields.

Many students cite using the residue theorem to evaluate real definite integrals and sums as the main factor in their interest in Complex Analysis. This chapter will review the residue theorem of course. But the residue theorem is only useful in the presence of poles and no branch points. To attack very difficult integrals, we will need to consider integrands with poles and branch points, and occasionally the poles may be essential. Accordingly, we seek a more methodical treatment of applying complex integration to evaluation of real definite integrals and sums. Such a treatment is presented here. We will then present several examples. Ultimately, we will demonstrate a unified approach to using complex integrals to evaluate real definite integrals and sums, regardless of the type of singularities occurring in the complex integrand.

The generation of continuous random processes with jointly specified probability density and covariation functions is considered. The proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system, excited by white Gaussian noise and described by a system of a first-order stochastic differential equations (SDE). The authors explore how the statistical characteristics of the equation's solution depends on the form of its operator and on the intensity of the input noise. Some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non-Gaussian continuous processes are considered. The approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its Markov model as well as in simulation of continuous correlated processes with the known probability density function. >

A novel approach to degree-hour calculation: Indoor and outdoor reference temperature based degree-hour calculation

We propose the suitable use of the Post-Widder inversion formula for Laplace transforms – coupled with the Wynn’s ρ-algorithm for accelerating sequences – in order to evaluate (up to the desired accuracy) the probability density function and the distribution function of a large collection of random variables. The method is illustrated on the Tweedie law and the tempered positive Linnik law. In addition, a further application to some laws arising in the context of Brownian motion is considered.

A probability density or distribution function of turbulence has been thought to be symmetric due to the symmetry of the partial differential equations from the first principles. However, the experimental data have been shown otherwise by a so-called Taylor correlation function, and this is an unresolved issue. A recent study shows that this probability density function can be asymmetric analytically by introducing negative eigenvalues. Here, in this work, we show the mathematical basis for this asymmetry, although the partial differential equations follow the symmetry of the Lie groups. We also demonstrate a complete solution of partial differential equations, including the exponential terms and negative eigenvalues, which plays a vital part in transient phenomena. Our analysis shows that the asymmetry is from the partition of velocities of the same or opposing direction, not from the negative eigenvalues. Fundamentally, the loss of rotational symmetry is caused by the exponential terms for the transient solution, which we demonstrate by the derivation of the complete solution. The new correlation produces excellent agreement with the experimental data. The universality and limitations of the correlation function are discussed, and through the parameter study, the variations and statistical nature of the probability function are clarified. The asymmetry probability function should have wider applicability than the symmetric Gaussian distribution, which is the special case of the asymmetry probability function.