Theory of fuzzy integrals and its applications

In this innovative approach to the practice of social science, Charles Ragin explores the use of fuzzy sets to bridge the divide between quantitative and qualitative methods. Paradoxically, the fuzzy set is a powerful tool because it replaces an unwieldy, "fuzzy" instrument—the variable, which establishes only the positions of cases relative to each other, with a precise one—degree of membership in a well-defined set. Ragin argues that fuzzy sets allow a far richer dialogue between ideas and evidence in social research than previously possible. They let quantitative researchers abandon "homogenizing assumptions" about cases and causes, they extend diversity-oriented research strategies, and they provide a powerful connection between theory and data analysis. Most important, fuzzy sets can be carefully tailored to fit evolving theoretical concepts, sharpening quantitative tools with in-depth knowledge gained through qualitative, case-oriented inquiry. This book will revolutionize research methods not only in sociology, political science, and anthropology but in any field of inquiry dealing with complex patterns of causation.

Fuzzy-Set Social Science

Is there a need for fuzzy logic

Part I: Background and Fundamentals Introduction, Mohamed Gad-el-Hak, University of Notre Dame Scaling of Micromechanical Devices, William Trimmer, Standard MEMS, Inc., and Robert H. Stroud, Aerospace Corporation Mechanical Properties of MEMS Materials, William N. Sharpe, Jr., Johns Hopkins University Flow Physics, Mohamed Gad-el-Hak, University of Notre Dame Integrated Simulation for MEMS: Coupling Flow-Structure-Thermal-Electrical Domains, Robert M. Kirby and George Em Karniadakis, Brown University, and Oleg Mikulchenko and Kartikeya Mayaram, Oregon State University Liquid Flows in Microchannels, Kendra V. Sharp and Ronald J. Adrian, University of Illinois at Urbana-Champaign, Juan G. Santiago and Joshua I. Molho, Stanford University Burnett Simulations of Flows in Microdevices, Ramesh K. Agarwal and Keon-Young Yun, Wichita State University Molecular-Based Microfluidic Simulation Models, Ali Beskok, Texas A&M University Lubrication in MEMS, Kenneth S. Breuer, Brown University Physics of Thin Liquid Films, Alexander Oron, Technion, Israel Bubble/Drop Transport in Microchannels, Hsueh-Chia Chang, University of Notre Dame Fundamentals of Control Theory, Bill Goodwine, University of Notre Dame Model-Based Flow Control for Distributed Architectures, Thomas R. Bewley, University of California, San Diego Soft Computing in Control, Mihir Sen and Bill Goodwine, University of Notre Dame Part II: Design and Fabrication Materials for Microelectromechanical Systems Christian A. Zorman and Mehran Mehregany, Case Western Reserve University MEMS Fabrication, Marc J. Madou, Nanogen, Inc. LIGA and Other Replication Techniques, Marc J. Madou, Nanogen, Inc. X-Ray-Based Fabrication, Todd Christenson, Sandia National Laboratories Electrochemical Fabrication (EFAB), Adam L. Cohen, MEMGen Corporation Fabrication and Characterization of Single-Crystal Silicon Carbide MEMS, Robert S. Okojie, NASA Glenn Research Center Deep Reactive Ion Etching for Bulk Micromachining of Silicon Carbide, Glenn M. Beheim, NASA Glenn Research Center Microfabricated Chemical Sensors for Aerospace Applications, Gary W. Hunter, NASA Glenn Research Center, Chung-Chiun Liu, Case Western Reserve University, and Darby B. Makel, Makel Engineering, Inc. Packaging of Harsh-Environment MEMS Devices, Liang-Yu Chen and Jih-Fen Lei, NASA Glenn Research Center Part III: Applications of MEMS Inertial Sensors, Paul L. Bergstrom, Michigan Technological University, and Gary G. Li, OMM, Inc. Micromachined Pressure Sensors, Jae-Sung Park, Chester Wilson, and Yogesh B. Gianchandani, University of Wisconsin-Madison Sensors and Actuators for Turbulent Flows. Lennart Loefdahl, Chalmers University of Technology, and Mohamed Gad-el-Hak, University of Notre Dame Surface-Micromachined Mechanisms, Andrew D. Oliver and David W. Plummer, Sandia National Laboratories Microrobotics Thorbjoern Ebefors and Goeran Stemme, Royal Institute of Technology, Sweden Microscale Vacuum Pumps, E. Phillip Muntz, University of Southern California, and Stephen E. Vargo, SiWave, Inc. Microdroplet Generators. Fan-Gang Tseng, National Tsing Hua University, Taiwan Micro Heat Pipes and Micro Heat Spreaders, G. P. "Bud" Peterson, Rensselaer Polytechnic Institute Microchannel Heat Sinks, Yitshak Zohar, Hong Kong University of Science and Technology Flow Control, Mohamed Gad-el-Hak, University of Notre Dame) Part IV: The Future Reactive Control for Skin-Friction Reduction, Haecheon Choi, Seoul National University Towards MEMS Autonomous Control of Free-Shear Flows, Ahmed Naguib, Michigan State University Fabrication Technologies for Nanoelectromechanical Systems, Gary H. Bernstein, Holly V. Goodson, and Gregory L. Snider, University of Notre Dame Index

The MEMS Handbook

The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...

Fuzzy sets and systems

Topology has its roots in geometry and analysis. From a geometric point of view, topology was the study of properties preserved by a certain group of transformations, namely the homeomorphisms. Certain notions of topology are also abstractions of classical concepts in the study of real or complex functions. These concepts include open sets, continuity, connectedness, compactness, and metric spaces. They were a basic part of analysis before being generalized in topology.

Fuzzy Topological Spaces

Fuzzy Subsets: Fuzzy Relations.- Fuzzy Equivalence Relations.- Pattern Classification.- Similarity Relations.- References.- Fuzzy Graphs: Paths and Connectedness. Bridges and Cut Vertices. Forests and Trees. Trees and Cycles. A Characterization of Fuzzy Trees. (Fuzzy) Cut Sets. (Fuzzy Chords, (Fuzzy) Cotrees, and (Fuzzy) Twigs. (Fuzzy) 1-Chain with Boundary 0, (Fuzzy) Coboundary, and (Fuzzy) Cocycles. (Fuzzy) Cycle Set and (Fuzzy) Cocycle Set.- Fuzzy Line Graphs.- Fuzzy Interval Graphs. Fuzzy Intersection Graphs. Fuzzy Interval Graphs. The Fulkerson and Gross Characterization. The Gilmore and Hoffman Characterization.- Operations on Fuzzy Graphs: Cartesian Product and Composition. Union and Join.- On Fuzzy Tree Definition.- References.- Applications of Fuzzy Graphs: Clusters.- Cluster Analysis. Cohesiveness. Slicing in Fuzzy Graphs.- Application to Cluster Analysis.- Fuzzy Intersection Equations. Existence of Solutions.- Fuzzy Graphs in Database Theory. Representation of Dependency Structure r(X,Y) by Fuzzy Graphs.- A Description of Strengthening and Weakening Members of a Group. Connectedness Criteria. Inclusive Connectedness Categories. Exclusive Connectedness Categories.- An Application of Fuzzy Graphs to the Problem Concerning Group Structure. Connectedness of a Fuzzy Graph. Weakening and Strenghtening Points of a Fuzzy Directed Graph.- References.- Fuzzy Hypergraphs: Fuzzy Hypergraphs.- Fuzzy Transversals of Fuzzy Hypergraphs. Properties of Tr(H). Construction of H3.- Coloring of Fuzzy Hypergraphs. beta-degree Coloring Procedures. Chromatic Values of Fuzzy Colorings.- Intersecting Fuzzy Hypergraphs. Characterization of Strongly Intersecting Hypergraphs. Simply Ordered Intersecting Hypergraphs. H-dominant Transversals.- Hebbian Structures.- Additional Applications.- References.

Fuzzy Graphs and Fuzzy Hypergraphs

INTRODUCTION Sets Relations Functions Fuzzy Subsets Semigroups Finite-State Machines Finite State Automata Languages and Grammars Nondeterministic Finite-State Automata Relationships Between Languages and Automata Pushdown Automata MAX-MIN AUTOMATA Max-Min Automata General Formulation of Automata Classes of Automata Behavior of Max-Min Automata Equivalences and Homomorphisms of Max-Min Automata Reduction of Max-Min Automata Definite Max-Min Automata Reduction of Max-Min Machines Equivalences Irreducibility and Minimality Nondeterministic and Deterministic Case FUZZY MACHINES, LANGUAGES, AND GRAMMARS Max-Product Machines Equivalences Irreducibility and Minimality Max-Product Grammars and Languages Weak Regular Max-Product Grammars Weak Regular Max-Product Languages Properties of GBP Exercises FUZZY LANGUAGES AND GRAMMARS Fuzzy Languages Types of Grammars Fuzzy Context-Free Grammars Context-Free Max-Product Grammars Context-Free Fuzzy Languages On the Description of the Fuzzy Meaning of Context-Free Languages Trees and Pseudoterms Fuzzy Dendrolanguage Generating Systems Normal Form of F-CFDS Sets of Derivation Trees of Fuzzy Context-Free Grammars Fuzzy Tree Automaton Fuzzy Tree Transducer Fuzzy Meaning of Context-Free Languages PROBABILISTIC AUTOMATA AND GRAMMARS Probabilistic Automata and their Approximation e-Approximating by Nonprobability Devices e-Approximating by Finite Automata Applications The Pe Relation Fuzzy Stars Acceptors and Probabilistic Acceptors Characterizations and the Re -Relation Probabilistic and Weighted Grammars Probabilistic and Weighted Grammars of Type 3 Interrelations with Programmed and Time-variant Grammars Probabilistic Grammars and Automata Probabilistic Grammars Weakly Regular Grammars and Asynchronous Automata Type-0 Probabilistic Grammars and Probabilistic Turing Machines Context-Free Probabilistic Grammars and Pushdown Automata Realization of Fuzzy Languages by Various Automata Properties of Lk, k - 1,2,3 Further Properties of L3 ALGEBRAIC FUZZY AUTOMATA THEORY Fuzzy Finite State Machines Semigroups of Fuzzy Finite State Machines Homomorphisms Admissible Relation Fuzzy Transformation Semigroups Products of Fuzzy Finite State Machines Submachines of a Fuzzy Finite State Machine Retrievability, Separability, and Connectivity Decomposition of Fuzzy Finite State Machines Subsystems of a Fuzzy Finite State Machine Strong Subsystems Cartesian Composition of Fuzzy Finite State Machines Cartesian Composition Admissible Partitions Coverings of Products of Fuzzy Finite State Machines Associative Properties of Products Covering Properties of Products Fuzzy Semiautomaton over a Finite Group MORE ON FUZZY LANGUAGES Fuzzy Regular Languages On Fuzzy Recognizers Minimal Fuzzy Recognizers Fuzzy Recognizers and Recognizable Sets Operation on (Fuzzy) Subsets Construction of Recognizers and Recognizable Sets Accessible and Coaccessible Recognizers Complete Fuzzy Machines Fuzzy Languages on a Free Monoid Algebraic Character and Properties of Fuzzy Regular Languages Deterministic Acceptors of Regular Fuzzy Languages MINIMIZATION OF FUZZY AUTOMATA Equivalence, Reduction, and Minimization of Finite Fuzzy Automata Equivalence of Fuzzy Automata: An Algebraic Approach Reduction and Minimization of Fuzzy Automata Minimal Fuzzy Finite State Automata Behavior, Reduction, and Minimization of Finite L-Automata Matrices over a Bounded Chain Systems of Linear Equivalences over a Bounded Chain Finite L-Automata-Behavior Matrix e-Equivalence e-Irreducibility Minimization L-FUZZY AUTOMATA, GRAMMARS, AND LANGUAGES Fuzzy Recognition of Fuzzy Languages Fuzzy Languages Fuzzy Recognition by Machines Cutpoint Languages Fuzzy Languages not Fuzzy Recognized by Machines in DT2 Rational Probabilistic Events Recursive Fuzzy Languages Closure Properties Fuzzy Grammars and Recursively Enumerable Fuzzy Languages Recursively Enumerable L-Subsets Various Kind of Automata with Weights APPLICATIONS A Formulation of Fuzzy Automata and its Application as a Model of Learning Systems Formulation of Fuzzy Automata Special Cases of Fuzzy Automata Fuzzy Automata as Models of Learning Systems Applications and Simulation Results Properties of Fuzzy Automata Fractionally Fuzzy Grammars with Application to Pattern Recognition Fractionally Fuzzy Grammars A Pattern Recognition Experiment General Fuzzy Acceptors for Syntactic Pattern Recognition e-Equivalence by Inputs Fuzzy-State Automata: Their Stability and Fault Tolerance Relational Description of Automata Fuzzy-State Automata Stable and Almost Stable Behavior of Fuzzy-State Automata Fault Tolerance of Fuzzy-State Automata Clinical Monitoring with Fuzzy Automata Fuzzy Systems REFERENCES INDEX Each chapter also includes a section of exercises

Fuzzy Automata and Languages: Theory and Applications

Operations on fuzzy graphs

L-subsets and L-subgroups L-subgroups of Abelian groups L-submodules L-subrings and L-ideals L-subfields structure of L-subrings and L-ideals algebraic L-varieties and intersection equations L-subspaces Galois theory and group L-subalgebras.

John N. Mordeson

Papers

Fuzzy Topological Spaces

Fuzzy Graphs and Fuzzy Hypergraphs

Fuzzy Automata and Languages: Theory and Applications

Operations on fuzzy graphs

Fuzzy commutative algebra