Fundamental theorems

Inequality often appears in linked pairs of variables. Examples include schooling and income, income and consumption, and wealth and happiness. Consider the famous words of Veblen: “wealth confers honor.” Understanding inequality requires understanding input inequality, outcome inequality, and the relation between the two—in both inequality between persons and inequality between subgroups. This article contributes to the methodological toolkit for studying inequality by developing a framework that makes explicit both input inequality and outcome inequality and by addressing three main associated questions: (1) How do the mechanisms for generating and altering inequality differ across inputs and outcomes? (2) Which have more inequality—inputs or outcomes? (3) Under what conditions, and by what mechanisms, does input inequality affect outcome inequality? Results include the following: First, under specified conditions, distinctive mechanisms govern inequality in inputs and inequality in outcomes. Second, input inequality and outcome inequality can be the same or different; if different, whether inequality is greater among inputs or outcomes depends on the configuration of outcome function, types of inputs, distributional form of and inequality in cardinal inputs, and number of and associations among inputs. Third, the link between input inequality and outcome inequality is multiform; it can be nonexistent, linear, or nonlinear, and if nonlinear, it can be concave or convex. More deeply, this work signals the formidable empirical challenges in studying inequality, but also the fast growing toolbox. For example, even if the outcome distribution is difficult to derive, fundamental theorems on the variance make it possible to analyze the input–outcome inequality connection. Similarly, within specified distributions, the general inequality parameter makes it possible to express results in terms of both measures of overall inequality and measures of subgroup inequality.

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Almost direct summands

Nagoya Mathematical Journal ◽

10.1017/s0027763000010898 ◽

2014 ◽

Vol 214 ◽

pp. 195-204 ◽

Cited By ~ 1

Author(s):

Bhargav Bhatt

Keyword(s):

Direct Summand ◽

Hodge Theory ◽

Fundamental Theorems ◽

Ramification Locus

AbstractWe prove new cases of the direct summand conjecture using fundamental theorems inp-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification locus lies entirely in characteristicp.

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Image development in the framework of ξ –complex fuzzy morphisms

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201261 ◽

2021 ◽

pp. 1-13

Author(s):

Aneeza Imtiaz ◽

Umer Shuaib ◽

Abdul Razaq ◽

Muhammad Gulistan

Keyword(s):

Decision Making ◽

Comparative Analysis ◽

Fuzzy Sets ◽

Unit Disk ◽

Homomorphic Image ◽

Significant Role ◽

Original Form ◽

Mathematical Tool ◽

Fuzzy Subgroups ◽

Fundamental Theorems

The study of complex fuzzy sets defined over the meet operator (ξ –CFS) is a useful mathematical tool in which range of degrees is extended from [0, 1] to complex plane with unit disk. These particular complex fuzzy sets plays a significant role in solving various decision making problems as these particular sets are powerful extensions of classical fuzzy sets. In this paper, we define ξ –CFS and propose the notion of complex fuzzy subgroups defined over ξ –CFS (ξ –CFSG) along with their various fundamental algebraic characteristics. We extend the study of this idea by defining the concepts of ξ –complex fuzzy homomorphism and ξ –complex fuzzy isomorphism between any two ξ –complex fuzzy subgroups and establish fundamental theorems of ξ –complex fuzzy morphisms. In addition, we effectively apply the idea of ξ –complex fuzzy homomorphism to refine the corrupted homomorphic image by eliminating its distortions in order to obtain its original form. Moreover, to view the true advantage of ξ –complex fuzzy homomorphism, we present a comparative analysis with the existing knowledge of complex fuzzy homomorphism which enables us to choose this particular approach to solve many decision-making problems.

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