How can Christians committed to the classical Christian tradition address the issues raised by contemporary Islam? Before a much-needed dialogue between Christians and Muslims is established, Christians need to ask themselves how their Scriptures and traditions might come to bear on such a dialogue. Do the divisions among Catholic and Evangelical Christians fracture the classical Christian tradition in ways that undercut Christian-Muslim dialogue before it has even begun? Or could the classical tradition provide invaluable resources for resolving divisions between Catholic and Evangelical Christians in ways that would prepare them for meaningful conversation with Muslim brothers and sisters? And what does it have to teach us about what Christians can and must learn from Muslims about their own traditions? The scholarly essays compiled in Christian Theology and Islam consider these and further questions, offering valuable insight for concerned Christians and academics in the fields of theology and religion.

Simulating Fuzzy Systems demonstrates how many systems naturally become fuzzy systems and shows how regular (crisp) simulation can be used to estimate the alpha-cuts of the fuzzy numbers used to analyze the behavior of the fuzzy system. This monograph presents a concise introduction to fuzzy sets, fuzzy logic, fuzzy estimation, fuzzy probabilities, fuzzy systems theory, and fuzzy computation. It also presents a wide selection of simulation applications ranging from emergency rooms to machine shops to project scheduling, showing the varieties of fuzzy systems.

In probability and statistics we often have to estimate probabilities and parameters in probability distributions using a random sample. Instead of using a point estimate calculated from the data we propose using fuzzy numbers which are constructed from a set of confidence intervals. In probability calculations we apply constrained fuzzy arithmetic because probabilities must add to one. Fuzzy random variables have fuzzy distributions. A fuzzy normal random variable has the normal distribution with fuzzy number mean and variance. Applications are to queuing theory, Markov chains, inventory control, decision theory and reliability theory.

The primary purpose of this book is to present information about selected topics on the interactions and applications of fuzzy + neural. Most of the discussion centers around our own research in these areas. Fuzzy + neural can mean many things: (1) approximations between fuzzy systems and neu ral nets (Chapter 4); (2) building hybrid neural nets to equal fuzzy systems (Chapter 5); (3) using neura.l nets to solve fuzzy problems (Chapter 6); (4) approximations between fuzzy neural nets and other fuzzy systems (Chap ter 8); (5) constructing hybrid fuzzy neural nets for certain fuzzy systems (Chapters 9, 10); or (6) computing with words (Chapter 11). This book is not intend to be used primarily as a text book for a course in fuzzy + neural because we have not included problems at the end of each chapter, we have omitted most proofs (given in the references), and we have given very few references. We wanted to keep the mathematical prerequisites to a minimum so all longer, involved, proofs were omitted. Elementary dif ferential calculus is the only prerequisite needed since we do mention partial derivatives once or twice."

In probability and statistics we often have to estimate probabilities and parameters in probability distributions using a random sample. Instead of using a point estimate calculated from the data we propose using fuzzy numbers which are constructed from a set of confidence intervals. In probability calculations we apply constrained fuzzy arithmetic because probabilities must add to one. Fuzzy random variables have fuzzy distributions. A fuzzy normal random variable has the normal distribution with fuzzy number mean and variance. Applications are to queuing theory, Markov chains, inventory control, decision theory and reliability theory.

1.1 Introduction This book is written in the following divisions: (1) the introductory chapters consisting of Chapters 1 and 2; (2) introduction to fuzzy probability in Ch- ters3-5; (3)introductiontofuzzyestimationinChapters6-11; (4)fuzzy/crisp estimatorsofprobabilitydensity(mass)functionsbasedonafuzzymaximum entropyprincipleinChapters12-14; (5)introductiontofuzzyhypothesiste- ing in Chapters 15-18; (6) fuzzy correlation and regression in Chapters 19-25; (7) Chapters 26 and 27 are about a fuzzy ANOVA model; (8) a fuzzy esti- tor of the median in nonparametric statistics in Chapter 28; and (9) random fuzzy numbers with applications to Monte Carlo studies in Chapter 29. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. For a beginning introduction to fuzzysetsandfuzzylogicsee 8]. Oneotheritemrelatingtofuzzysets, needed infuzzyhypothesistesting, isalsoinChapter2: howwewilldeterminewhich of the following three possibilities is trueM N or M? N, for two fuzzy numbers M, N. TheintroductiontofuzzyprobabilityinChapters3-5isbasedonthebook 1] and the reader is referred to that book for more information, especially applications. Whatisnewhereis: (1)usinganonlinearoptimizationprogram in Maple 13] to solve certain optimization problems in fuzzy probability, where previously we used a graphical method; and (2) a new algorithm, suitable for using only pencil and paper, for solving some restricted fuzzy arithmetic problems. The introduction to fuzzy estimation is based on the book 3] and we refer the interested reader to that book for more about fuzzy estimators.

Monte Carlo Methods in Fuzzy Optimization is a clear and didactic book about Monte Carlo methods using random fuzzy numbers to obtain approximate solutions to fuzzy optimization problems. The book includes various solved problems such as fuzzy linear programming, fuzzy regression, fuzzy inventory control, fuzzy game theory, and fuzzy queuing theory. The book will appeal to engineers, researchers, and students in Fuzziness and applied mathematics.

1.1 Introduction This book is written in five major divisions. The first part is the introduc tory chapters consisting of Chapters 1-3. In part two, Chapters 4-10, we use fuzzy probabilities to model a fuzzy queuing system . We switch to employ ing fuzzy arrival rates and fuzzy service rates to model the fuzzy queuing system in part three in Chapters 11 and 12. Optimization models comprise part four in Chapters 13-17. The final part has a brief summary and sug gestions for future research in Chapter 18, and a summary of our numerical methods for calculating fuzzy probabilities, values of objective functions in fuzzy optimization, etc., is in Chapter 19. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. Two other items relating to fuzzy sets, needed in Chapters 13-17, are also in Chapter 2: (1) how we plan to handle the maximum/minimum of a fuzzy set; and (2) how we will rank a finite collection of fuzzy numbers from smallest to largest."

The book aims at surveying results in the application of fuzzy sets and fuzzy logic to economics and engineering. New results include fuzzy non-linear regression, fully fuzzified linear programming, fuzzy multi-period control, fuzzy network analysis, each using an evolutionary algorithm; fuzzy queuing decision analysis using possibility theory; fuzzy differential equations; fuzzy difference equations; fuzzy partial differential equations; fuzzy eigenvalues based on an evolutionary algorithm; fuzzy hierarchical analysis using an evolutionary algorithm; fuzzy integral equations. Other important topics covered are fuzzy input-output analysis; fuzzy mathematics of finance; fuzzy PERT (project evaluation and review technique). No previous knowledge of fuzzy sets is needed. The mathematical background is assumed to be elementary calculus.

1. 1 Introduction This book is written in two major parts. The ?rst part includes the int- ductory chapters consisting of Chapters 1 through 6. In part two, Chapters 7-26, we present the applications. This book continues our research into simulating fuzzy systems. We started with investigating simulating discrete event fuzzy systems ([7],[13],[14]). These systems can usually be described as queuing networks. Items (transactions) arrive at various points in the s- tem and go into a queue waiting for service. The service stations, preceded by a queue, are connected forming a network of queues and service, until the transaction ?nally exits the system. Examples considered included - chine shops, emergency rooms, project networks, bus routes, etc. Analysis of all of these systems depends on parameters like arrival rates and service rates. These parameters are usually estimated from historical data. These estimators are generally point estimators. The point estimators are put into the model to compute system descriptors like mean time an item spends in the system, or the expected number of transactions leaving the system per unit time. We argued that these point estimators contain uncertainty not shown in the calculations. Our estimators of these parameters become fuzzy numbers, constructed by placing a set of con?dence intervals one on top of another. Using fuzzy number parameters in the model makes it into a fuzzy system. The system descriptors we want (time in system, number leaving per unit time) will be fuzzy numbers.

Monte Carlo Methods in Fuzzy Optimization is a clear and didactic book about Monte Carlo methods using random fuzzy numbers to obtain approximate solutions to fuzzy optimization problems. The book includes various solved problems such as fuzzy linear programming, fuzzy regression, fuzzy inventory control, fuzzy game theory, and fuzzy queuing theory. The book will appeal to engineers, researchers, and students in Fuzziness and applied mathematics.

1.1 Introduction This book is written in the following divisions: (1) the introductory chapters consisting of Chapters 1 and 2; (2) introduction to fuzzy probability in Ch- ters3-5; (3)introductiontofuzzyestimationinChapters6-11; (4)fuzzy/crisp estimatorsofprobabilitydensity(mass)functionsbasedonafuzzymaximum entropyprincipleinChapters12-14; (5)introductiontofuzzyhypothesiste- ing in Chapters 15-18; (6) fuzzy correlation and regression in Chapters 19-25; (7) Chapters 26 and 27 are about a fuzzy ANOVA model; (8) a fuzzy esti- tor of the median in nonparametric statistics in Chapter 28; and (9) random fuzzy numbers with applications to Monte Carlo studies in Chapter 29. First we need to be familiar with fuzzy sets. All you need to know about fuzzy sets for this book comprises Chapter 2. For a beginning introduction to fuzzysetsandfuzzylogicsee 8]. Oneotheritemrelatingtofuzzysets, needed infuzzyhypothesistesting, isalsoinChapter2: howwewilldeterminewhich of the following three possibilities is trueM N or M? N, for two fuzzy numbers M, N. TheintroductiontofuzzyprobabilityinChapters3-5isbasedonthebook 1] and the reader is referred to that book for more information, especially applications. Whatisnewhereis: (1)usinganonlinearoptimizationprogram in Maple 13] to solve certain optimization problems in fuzzy probability, where previously we used a graphical method; and (2) a new algorithm, suitable for using only pencil and paper, for solving some restricted fuzzy arithmetic problems. The introduction to fuzzy estimation is based on the book 3] and we refer the interested reader to that book for more about fuzzy estimators.

1. 1 Introduction This book is written in two major parts. The ?rst part includes the int- ductory chapters consisting of Chapters 1 through 6. In part two, Chapters 7-26, we present the applications. This book continues our research into simulating fuzzy systems. We started with investigating simulating discrete event fuzzy systems ([7],[13],[14]). These systems can usually be described as queuing networks. Items (transactions) arrive at various points in the s- tem and go into a queue waiting for service. The service stations, preceded by a queue, are connected forming a network of queues and service, until the transaction ?nally exits the system. Examples considered included - chine shops, emergency rooms, project networks, bus routes, etc. Analysis of all of these systems depends on parameters like arrival rates and service rates. These parameters are usually estimated from historical data. These estimators are generally point estimators. The point estimators are put into the model to compute system descriptors like mean time an item spends in the system, or the expected number of transactions leaving the system per unit time. We argued that these point estimators contain uncertainty not shown in the calculations. Our estimators of these parameters become fuzzy numbers, constructed by placing a set of con?dence intervals one on top of another. Using fuzzy number parameters in the model makes it into a fuzzy system. The system descriptors we want (time in system, number leaving per unit time) will be fuzzy numbers.

In probability and statistics we often have to estimate probabilities and parameters in probability distributions using a random sample. Instead of using a point estimate calculated from the data we propose using fuzzy numbers which are constructed from a set of confidence intervals. In probability calculations we apply constrained fuzzy arithmetic because probabilities must add to one. Fuzzy random variables have fuzzy distributions. A fuzzy normal random variable has the normal distribution with fuzzy number mean and variance. Applications are to queuing theory, Markov chains, inventory control, decision theory and reliability theory.

Simulating Fuzzy Systems demonstrates how many systems naturally become fuzzy systems and shows how regular (crisp) simulation can be used to estimate the alpha-cuts of the fuzzy numbers used to analyze the behavior of the fuzzy system. This monograph presents a concise introduction to fuzzy sets, fuzzy logic, fuzzy estimation, fuzzy probabilities, fuzzy systems theory, and fuzzy computation. It also presents a wide selection of simulation applications ranging from emergency rooms to machine shops to project scheduling, showing the varieties of fuzzy systems.

1. 1 Introduction This book is written in four major divisions. The first part is the introductory chapters consisting of Chapters 1 and 2. In part two, Chapters 3-11, we develop fuzzy estimation. For example, in Chapter 3 we construct a fuzzy estimator for the mean of a normal distribution assuming the variance is known. More details on fuzzy estimation are in Chapter 3 and then after Chapter 3, Chapters 4-11 can be read independently. Part three, Chapters 12- 20, are on fuzzy hypothesis testing. For example, in Chapter 12 we consider the test Ho : /1 = /10 verses HI : /1 f=- /10 where /1 is the mean of a normal distribution with known variance, but we use a fuzzy number (from Chapter 3) estimator of /1 in the test statistic. More details on fuzzy hypothesis testing are in Chapter 12 and then after Chapter 12 Chapters 13-20 may be read independently. Part four, Chapters 21-27, are on fuzzy regression and fuzzy prediction. We start with fuzzy correlation in Chapter 21. Simple linear regression is the topic in Chapters 22-24 and Chapters 25-27 concentrate on multiple linear regression. Part two (fuzzy estimation) is used in Chapters 22 and 25; and part 3 (fuzzy hypothesis testing) is employed in Chapters 24 and 27. Fuzzy prediction is contained in Chapters 23 and 26. A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data.

This book presents important applications of soft computing and fuzziness to the growing field of web planning. A new method of using fuzzy numbers to model uncertain probabilities and how these can be used to model a fuzzy queuing system is demonstrated, as well as a method of modeling fuzzy queuing systems employing fuzzy arrival rates and fuzzy service rates. All the computations needed to get to the fuzzy numbers for system performance are described starting for the one server case to more than three servers. A variety of optimization models are discussed with applications to the average response times, server utilization, server and queue costs, as well as to phenomena identified with web sites such as "burstiness" and "long tailed distributions".

The book aims at surveying results in the application of fuzzy sets and fuzzy logic to economics and engineering. New results include fuzzy non-linear regression, fully fuzzified linear programming, fuzzy multi-period control, fuzzy network analysis, each using an evolutionary algorithm; fuzzy queuing decision analysis using possibility theory; fuzzy differential equations; fuzzy difference equations; fuzzy partial differential equations; fuzzy eigenvalues based on an evolutionary algorithm; fuzzy hierarchical analysis using an evolutionary algorithm; fuzzy integral equations. Other important topics covered are fuzzy input-output analysis; fuzzy mathematics of finance; fuzzy PERT (project evaluation and review technique). No previous knowledge of fuzzy sets is needed. The mathematical background is assumed to be elementary calculus.

This book is an excellent starting point for any curriculum in fuzzy systems fields such as computer science, mathematics, business/economics and engineering. It covers the basics leading to: fuzzy clustering, fuzzy pattern recognition, fuzzy database, fuzzy image processing, soft computing, fuzzy applications in operations research, fuzzy decision making, fuzzy rule based systems, fuzzy systems modeling, fuzzy mathematics. It is not a book designed for researchers - it is where you really learn the "basics" needed for any of the above-mentioned applications. It includes many figures and problem sets at the end of sections.

The primary purpose of this book is to present information about selected topics on the interactions and applications of fuzzy + neural. Most of the discussion centers around our own research in these areas. Fuzzy + neural can mean many things: (1) approximations between fuzzy systems and neu ral nets (Chapter 4); (2) building hybrid neural nets to equal fuzzy systems (Chapter 5); (3) using neura.l nets to solve fuzzy problems (Chapter 6); (4) approximations between fuzzy neural nets and other fuzzy systems (Chap ter 8); (5) constructing hybrid fuzzy neural nets for certain fuzzy systems (Chapters 9, 10); or (6) computing with words (Chapter 11). This book is not intend to be used primarily as a text book for a course in fuzzy + neural because we have not included problems at the end of each chapter, we have omitted most proofs (given in the references), and we have given very few references. We wanted to keep the mathematical prerequisites to a minimum so all longer, involved, proofs were omitted. Elementary dif ferential calculus is the only prerequisite needed since we do mention partial derivatives once or twice."

1. 1 Introduction This book is written in four major divisions. The first part is the introductory chapters consisting of Chapters 1 and 2. In part two, Chapters 3-11, we develop fuzzy estimation. For example, in Chapter 3 we construct a fuzzy estimator for the mean of a normal distribution assuming the variance is known. More details on fuzzy estimation are in Chapter 3 and then after Chapter 3, Chapters 4-11 can be read independently. Part three, Chapters 12- 20, are on fuzzy hypothesis testing. For example, in Chapter 12 we consider the test Ho : /1 = /10 verses HI : /1 f=- /10 where /1 is the mean of a normal distribution with known variance, but we use a fuzzy number (from Chapter 3) estimator of /1 in the test statistic. More details on fuzzy hypothesis testing are in Chapter 12 and then after Chapter 12 Chapters 13-20 may be read independently. Part four, Chapters 21-27, are on fuzzy regression and fuzzy prediction. We start with fuzzy correlation in Chapter 21. Simple linear regression is the topic in Chapters 22-24 and Chapters 25-27 concentrate on multiple linear regression. Part two (fuzzy estimation) is used in Chapters 22 and 25; and part 3 (fuzzy hypothesis testing) is employed in Chapters 24 and 27. Fuzzy prediction is contained in Chapters 23 and 26. A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data.