Inequalities

This work is about inequalities which play an important role in mathematical Olympiads. It contains 175 solved problems in the form of exercises and, in addition, 310 solved problems. The book also covers the theoretical background of the most important theorems and techniques required for solving inequalities. It is written for all middle and high-school students, as well as for graduate and undergraduate students. School teachers and trainers for mathematical competitions will also gain benefit from this book.

A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.

This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians.

By examining environmental change through the lens of conflicting social agendas, Andrew Hurley uncovers the historical roots of environmental inequality in contemporary urban America. Hurley's study focuses on the steel mill community of Gary, Indiana, a city that was sacrificed, like a thousand other American places, to industrial priorities in the decades following World War II. Although this period witnessed the emergence of a powerful environmental crusade and a resilient quest for equality and social justice among blue-collar workers and African Americans, such efforts often conflicted with the needs of industry. To secure their own interests, manufacturers and affluent white suburbanites exploited divisions of race and class, and the poor frequently found themselves trapped in deteriorating neighborhoods and exposed to dangerous levels of industrial pollution. In telling the story of Gary, Hurley reveals liberal capitalism's difficulties in reconciling concerns about social justice and quality of life with the imperatives of economic growth. He also shows that the power to mold the urban landscape was intertwined with the ability to govern social relations.

NEW YORK TIMES BESTSELLER • “An impassioned book, laced with anger and indignation, about how our public education system scorns so many of our children.”—The New York Times Book Review In 1988, Jonathan Kozol set off to spend time with children in the American public education system. For two years, he visited schools in neighborhoods across the country, from Illinois to Washington, D.C., and from New York to San Antonio. He spoke with teachers, principals, superintendents, and, most important, children. What he found was devastating. Not only were schools for rich and poor blatantly unequal, the gulf between the two extremes was widening—and it has widened since. The urban schools he visited were overcrowded and understaffed, and lacked the basic elements of learning—including books and, all too often, classrooms for the students. In Savage Inequalities, Kozol delivers a searing examination of the extremes of wealth and poverty and calls into question the reality of equal opportunity in our nation’s schools. Praise for Savage Inequalities “I was unprepared for the horror and shame I felt. . . . Savage Inequalities is a savage indictment. . . . Everyone should read this important book.”—Robert Wilson, USA Today “Kozol has written a book that must be read by anyone interested in education.”—Elizabeth Duff, Philadelphia Inquirer “The forces of equity have now been joined by a powerful voice. . . . Kozol has written a searing exposé of the extremes of wealth and poverty in America’s school system and the blighting effect on poor children, especially those in cities.”—Emily Mitchell, Time “Easily the most passionate, and certain to be the most passionately debated, book about American education in several years . . . A classic American muckraker with an eloquent prose style, Kozol offers . . . an old-fashioned brand of moral outrage that will affect every reader whose heart has not yet turned to stone.”—Entertainment Weekly

The need for informed analyses of health policy is now greater than ever. The twelve essays in this volume show that public debates routinely bypass complex ethical, sociocultural, historical, and political questions about how we should address ideals of justice and equality in health care. Integrating perspectives from the humanities, social sciences, medicine, and public health, this volume illuminates the relationships between justice and health inequalities to enrich debates. Understanding Health Inequalities and Justice explores three questions: How do scholars approach relations between health inequalities and ideals of justice? When do justice considerations inform solutions to health inequalities, and how do specific health inequalities affect perceptions of injustice? And how can diverse scholarly approaches contribute to better health policy? From addressing patient agency in an inequitable health care environment to examining how scholars of social justice and health care amass evidence, this volume promotes a richer understanding of health and justice and how to achieve both. The contributors are Judith C. Barker, Paula Braveman, Paul Brodwin, Jami Suki Chang, Debra DeBruin, Leslie A. Dubbin, Sarah Horton, Carla C. Keirns, J. Paul Kelleher, Nicholas B. King, Eva Feder Kittay, Joan Liaschenko, Anne Drapkin Lyerly, Mary Faith Marshall, Carolyn Moxley Rouse, Jennifer Prah Ruger, and Janet K. Shim.

In contrast to orthodox theory, Howard Botwinick uses a classical Marxist analysis of real capitalist competition to show that substantial patterns of wage disparity among similar workers can persist despite high levels of competition in both capital and labor markets.

This volume presents a comprehensive compendium of classical and new inequalities as well as some recent extensions to well-known ones. Variations of inequalities ascribed to Abel, Jensen, Cauchy, Chebyshev, Hölder, Minkowski, Stefferson, Gram, Fejér, Jackson, Hardy, Littlewood, Po'lya, Schwarz, Hadamard and a host of others can be found in this volume. The more than 1200 cited references include many from the last ten years which appear in a book for the first time. The 30 chapters are all devoted to inequalities associated with a given classical inequality, or give methods for the derivation of new inequalities. Anyone interested in equalities, from student to professional, will find their favorite inequality and much more.