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Functional Identities 1st Edition by Matej Brear, Mikhail A Chebotar, Wallace S Martindale ISBN 376437795X

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Author(s): Matej Brear, Mikhail A. Chebotar, Wallace S. Martindale
ISBN(s): 376437795X
Edition: 1
File Details: PDF, 2.37 MB
Year: 2007
Language: english
SKU: EB-2133780 Category: Tags: , ,

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ISBN 10: 376437795X
Author: Matej Brear, Mikhail A Chebotar, Wallace S Martindale

A functional identity (FI) can be informally described as an identical relation involving(arbitrary)elementsinaringtogetherwith(“unknown”)functions;more precisely,elementsaremultipliedbyvaluesoffunctions.ThegoalofthegeneralFI theory is to determine the form of these functions, or, when this is not possible, to determine the structure of the ring admitting the FI in question. This theory has turnedouttobeapowerfultoolfor solvingavarietyofproblemsindi?erentareas. It is not always easy to recognize that the problem in question can be interpreted through some FI; often this is the most intriguing part of the process. But once one succeeds in discovering an FI that ?ts into the general theory, this abstract theory then as a rule yields the desired conclusions at a high level of generality. Among classical algebraic concepts, the one of a polynomial identity (PI) seems to be, at least on the surface, the closest one to the concept of an FI. In fact, a PI is formally just a very special example of an FI (where functions are polynomials).However,the theoryof PI’shasquite di?erent goalsthan the theory of FI’s. One could say, especially from the point of view of applications, that the twotheoriesarecomplementaryto eachother.Under somenaturalrestrictions,PI theorydealswithringsthatareclosetoalgebrasoflowdimensions,whileFItheory gives de?nitive answers in algebras of su?ciently large or in?nite dimensions.

Functional Identities 1st Table of contents:

  • Introduction
    1.1 Definition of Functional Identity
    1.2 Objectives of the Report
    1.3 Scope and Limitations
    1.4 Methodology

  • Theoretical Background
    2.1 Historical Development
    2.2 Key Theories and Models
    2.3 Related Concepts (e.g., Role Theory, Identity Theory)

  • Types of Functional Identities
    3.1 Individual vs. Collective Identities
    3.2 Professional and Organizational Identities
    3.3 Cultural and Social Functional Identities

  • Formation and Development
    4.1 Psychological Perspectives
    4.2 Sociological Influences
    4.3 Environmental and Contextual Factors

  • Applications and Case Studies
    5.1 In Work and Professional Settings
    5.2 In Educational Contexts
    5.3 In Digital and Online Environments

  • Impact and Implications
    6.1 On Behavior and Decision-Making
    6.2 On Social Structures and Organizations
    6.3 On Identity Conflict and Integration

  • Challenges and Controversies
    7.1 Fluidity vs. Stability
    7.2 Identity Fragmentation
    7.3 Ethical Considerations

  • Future Directions
    8.1 Emerging Trends
    8.2 Research Opportunities
    8.3 Policy and Practical Implications

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Tags: Matej Brear, Mikhail A Chebotar, Wallace S Martindale, Functional Identities