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

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Reinforced hybrid interval fuzzy neural networks architecture: Design and analysis

Chinese word segmentation is the basis of the Chinese natural language processing (NLP). With the development of the deep learning, various neural network models are applied to the Chinese word segmentation. However, current neural network models have the characteristics of artificial feature extraction, nonstandard word-weight, inability to effectively use long-distance information and long training time of models in Chinese word segmentation. To solve a series of problems, this article presents a CNN-Bidirectional GRU-CRF neural network model (CNN Bidirectional GRU CRF Network, CBiGCN), which breaks through the limit of conventional method window, truly realizes end-to-end processing and applies to the neural network model by the five-Tag set method, bias-variable-weight greedy strategy and supplements by Goldstein-Armijo guidelines. Besides, this model, with simple structure, is easy to be operated. And it can automatically learn features, reduces large amounts of tasks on specific knowledge in the form of handcrafted features and data pre-processing, makes use of context information effectively. The authors set an experiment with two data corpuses for Chinese word segmentation to evaluate their system. The experiment verified their new model can obtain better Chinese word segmentation results and greatly reduce training time.

Learning Chinese Word Segmentation Based on Bidirectional GRU-CRF and CNN Network Model

An adaptive neuro-complex-fuzzy-inferential modeling mechanism for generating higher-order TSK models

A convergent smoothing algorithm for training max –min fuzzy neural networks

An online gradient-based parameter identification algorithm for the neuro-fuzzy systems

In this paper, a smoothing algorithm with constant learning rate is presented for training two kinds of fuzzy neural networks (FNNs): max-product and max-min FNNs. Some weak and strong convergence results for the algorithm are provided with the error function monotonically decreasing, its gradient going to zero, and weight sequence tending to a fixed value during the iteration. Furthermore, conditions for the constant learning rate are specified to guarantee the convergence. Finally, three numerical examples are given to illustrate the feasibility and efficiency of the algorithm and to support the theoretical findings.

Long Li

Papers

A convergent smoothing algorithm for training max –min fuzzy neural networks

An online gradient-based parameter identification algorithm for the neuro-fuzzy systems

A Smoothing Algorithm with Constant Learning Rate for Training Two Kinds of Fuzzy Neural Networks and Its Convergence

Boundedness and convergence of split complex gradient descent algorithm with momentum and regularizer for TSK fuzzy models