The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development. of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these constructions are impossible. Our approach introduces all the relevant concepts about fields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. We have used this text to teach second-year students at La Trobe University over a period of many years, each time refining the material in the light of student performance.
Like a cool drink on a summer day, “Hope When the World Shut Down” offers an oasis of hope from a world that often leaves us parched and exhausted. A compilation of short stories from believers in all walks of life, these snapshots, written during the heart of the COVID-19 pandemic, will leave you laughing and crying as you read how ordinary people held onto God’s promises. You’ll be sure to identify with these easy-to-read stories by doctors, educators, students, ministers, homemakers and grandparents, packed with uplifting Scripture. Whether reading it as a daily devotional or all in one sitting, you will come away refreshed, feeling like you just spent an evening with an old friend.
This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction. Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach. Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.
Portuguese Agriculture in Transition represents the synthesis of a six-year study undertaken by nine social scientists from the University of Arizona, Stanford University, Göttingen University, and the University of Lisbon, aimed at improving the efficiency and productivity of Portuguese agriculture. The fourteen essays seek to explain the constraints that affect the making of agricultural policy in Portugal, the sources of comparative advantage within the agricultural sector, and the technical innovations that have recently begun to change farming in the northwest of the country.
6 strategic principles for reinventing your products, your services—and your company's future The digital age has completely transformed business—and marketing has not kept up. From research frameworks and traditional concept development to planning to budgeting to distribution channels and media placement, marketing has not advanced—which may be why Chief Marketing Officers often don't get a seat at the table. In order to have a future, marketing must play a direct role in driving profitable sales and increasing revenues. The Old Rules of Marketing Are Dead offers the new rules for reinventing your brand, including: Defining the product's essence Creating metrics to ensure accountability Developing a core message Disseminating the brand Marketing needs to lead, not follow. The Old Rules of Marketing are Dead shows how to reinvent marketing and position it as a strategic business partner for any organization. Table of Contents Rule 1: The Core is Everything; Rule 2: You Have Nothing Without the Foundation; Rule 3: There are Many Choices But Only One Customer; Rule 4: Do the Right Things for the Right Reasons; Rule 5: Infrastructure is More Than Just Pipes; Rule 6: Lead and Others Will Follow
First published in 1998, this volume features specialists in agricultural economics who have provided case studies on small farms in northern and central Portugal and southern and central Italy. The collaboration is a result of an early 1990s research project on small farm agriculture in Portugal and Italy and the likely impacts of the Common Agricultural Policy. It recognises that small farms have become an unexpected yet durable aspect of the agricultural landscape since World War II. As small farms represent 95% of the number of farms in Portugal and Italy, the contributors provide some much needed analysis of an often overlooked aspect of the agricultural sector.
This book examines the dramatic work of Dickens, Browning, Collins, and Tennyson, their interaction with the theatrical world, and their attempts to develop their reputations as playwrights. These major Victorian writers each authored several professional plays, but why has their achievement been overlooked?
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