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Neoclassical Analysis:
Calculus Closer to the Real World by Mark Burgin

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Neoclassical Analysis: Calculus Closer to the Real World by Mark Burgin , Nova Science Publishers, Inc., New York.
Neoclassical analysis is a new direction that suggests an original solution to these problems of imprecision and inexactness in real life. Neoclassical analysis is a synthesis of three mathematical fields: the classical calculus, set-valued analysis (in a broad sense that includes interval analysis), and fuzzy set theory. The aim of this synthesis is to extend the powerful technique of the classical calculus to a much broader scope, and to make this technique more relevant to situations in physics and computation. Neoclassical analysis has common features with all three fields that it synthesizes, as well as essential distinctions from them. Thus, in contrast to set-valued analysis and fuzzy set theory, neoclassical analysis studies ordinary structures of classical calculus, such as numbers, functions, sets, sequences, series, and operators. Methods of set-valued analysis are oriented at sets. Methods of fuzzy set theory are oriented at fuzzy sets. Neoclassical analysis, following the classical calculus, is interested in individual objects: numbers, points, functions, curves, etc.
At the same time, in neoclassical analysis, ordinary structures of the classical calculus are studied by means of fuzzy concepts: fuzzy limits, fuzzy continuity, fuzzy derivatives, etc. For example, continuous functions studied in the classical calculus become a part of the set of fuzzy continuous functions studied in neoclassical analysis.
One more goal of neoclassical analysis is to extend the powerful calculus technology to a broader scope of exact (crisp) objects, e.g., for functions and curves that are discontinuous. This new technology developed in neoclassical analysis allows one to build adequate mathematical models for discrete spaces and functions in such spaces, introducing flexible and scalable concepts of discrete continuity, differentiation, and integration. This is especially important for computation as it operates only with discrete sets.

Neoclassical Analysis: Calculus Closer to the Real World by Mark Burgin cover the following topics.

  • 1. Introduction
    2. Fuzzy Limits
    3. Fuzzy Continuous Functions
    4. Fuzzy Differentiation
    5. Monotone and Fuzzy Monotone Functions
    6. Fuzzy Maxima and Minima of Real Functions
    7. Fuzzy Integration
    8. Fuzzy Dynamical Systems
    9. Conclusion

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