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."

Synopsis: At the same time as Catholic and evangelical Christians have increasingly come to agree on issues that divided them during the sixteenth-century reformations, they seem increasingly to disagree on issues of contemporary "morality" and "ethics." Do such arguments doom the prospects for realistic full communion between Catholics and evangelicals? Or are such disagreements a new opportunity for Catholics and evangelicals to convert together to the triune God's word and work on the communion of saints for the world? Or should our hope be different than simple pessimism or optimism? In this volume, eight authors address different aspects of these questions, hoping to move Christians a small step further toward the visible unity of the church. Endorsements: "Christians are often divided by the justification of homosexuality or some other controversial moral issue rather than the doctrine of justification. This excellent collection of essays helps us think through the ways in which moral differences have reshaped the ecumenical task." --R. R. Reno, Creighton University "This book identifies the chief cause of internal strife and division afflicting most mainline Protestant denominations. Separating faith from works is an old heresy that always breeds schism. The authors are leading theologians of their respective traditions who write from a wealth of experience in church service and with profound knowledge of classical Christianity. Anyone engaged in ecumenical dialogues and the quest for Christian unity needs to read and heed the message of this book." --Carl E. Braaten, Lutheran School of Theology at Chicago "It is a historical fact that moral disagreement has divided the church. This is not possible unless certain kinds of moral disagreements are, in fact, doctrinal disagreements . . . and other kinds of moral disagreements are, in fact, tolerable divergences owing to context and judgment. The offerings in this excellent collection go a long way toward recognizing this difference and sorting it out for us today." --Paul R. Hinlicky, Roanoke College Author Biography: James J. Buckley is Professor of Theology at Loyola University Maryland. He has contributed to and edited (with Frederick Bauerschmidt and Trent Pomplun) The Blackwell Companion to Catholicism (2007). He is associate director of the Center for Catholic and Evangelical Theology. Michael Root is Professor of Systematic Theology at The Catholic University of America and Executive Director of the Center for Catholic and Evangelical Theology. He was formerly the Director of the Institute for Ecumenical Research, Strasbourg, France.

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.

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.