On the algebraic closure in rings
The "algebraic closure" of a subset K C A of a ring is an algebraic analogue of topological closure.
Invertibility and Singularity for Bounded Linear Operators
This introduction to functional analysis focuses on the types of singularity that prevent an operator from being invertible. The presentation is based on the open mapping theorem, Hahn-Banach theorem, dual space construction, enlargement of normed space, and Liouville's theorem. Suitable for advanced undergraduate and graduate courses in functional analysis, this volume is also a valuable resource for researchers in Fredholm theory, Banach algebras, and multiparameter spectral theory. The treatment develops the theory of open and almost open operators between incomplete spaces. It builds the enlargement of a normed space and of a bounded operator and sets up an elementary algebraic framework for Fredholm theory. The approach extends from the definition of a normed space to the fringe of modern multiparameter spectral theory and concludes with a discussion of the varieties of joint spectrum. This edition contains a brief new Prologue by author Robin Harte as well as his lengthy new Epilogue, "Residual Quotients and the Taylor Spectrum." Dover republication of the edition published by Marcel Dekker, Inc., New York, 1988.
Participation Improves Collective Decisions (When It Involves Deliberation): Experimental Evidence From Kenya
Abstract Citizen participation in decision making has been widely lauded as a method for improving societal outcomes. Deliberative discussion, in particular, is believed to be more transformative than a mere aggregation of individual preferences, leading to more socially optimal decision making and behavior. I report the results from a laboratory experiment with 570 subjects in Nairobi, directly testing the effect of participation in deliberative group decision making on collective outcomes. Participants engage in a group task to earn compensation toward a shared group fund. Randomly assigned treatments vary according to whether decision making over the task to be completed involves: (1) external assignment; (2) non-deliberative majority voting; or (3) consensus through deliberative discussion. I find that deliberation improves collective decision making. Deliberation is also associated with changes in preferences, greater agreement with decision outcomes, and greater perceived fairness. Evidence for behavior change is weaker, but there is some support for further research into the relationship between preference change and behavior change.