Excursions in the History of Mathematics (Operator Theory, Advances and Applications)

This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians—Dedekind, Euler, Gauss, Hilbert, and Weierstrass—who played major roles in the historical events described in the first four parts of the work. __Excursions in the History of Mathematics__ was written with several goals in mind: to arouse mathematics teachers’ interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses. This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians who played major roles in the historical events described in the first four parts of the work. Each of the first three parts—on number theory, calculus/analysis, and proof—begins with a survey of the respective subject and is followed in more depth by specialized themes. Among the specialized themes are: Fermat as the founder of modern number theory, Fermat’s Last Theorem from Fermat to Wiles, the history of the function concept, paradoxes, the principle of continuity, and an historical perspective on recent debates about proof. The fourth part contains essays describing mathematics courses inspired by history. The essays deal with numbers as a source of ideas in teaching, with famous problems, and with the stories behind various "great" quotations. The last part gives an account of five mathematicians—Dedekind, Euler, Gauss, Weierstrass, and Hilbert—whose lives and work we hope readers will find inspiring. Key features of the work include: * A preface describing in some detail the author's ideas on teaching mathematics courses, in particular, the role of history in such courses; * Explicit comments and suggestions for teachers on how history can affect the teaching of mathematics; * A description of a course in the history of mathematics taught in an In-Service Master's Program for high school teachers; * Inclusion of issues in the philosophy of mathematics; * An extensive list of relevant references at the end of each chapter. Excursions in the History of Mathematics was written with several goals in mind: to arouse mathematics teachers’ interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses. This book comprises five parts. The first three contain ten historical essays on important topics: number theory, calculus/analysis, and proof, respectively. Part four deals with several historically oriented courses, and Part five provides biographies of five mathematicians who played major roles in the historical events described in the first four parts of the work. Each of the first three parts-on number theory, calculus/analysis, and proof-begins with a survey of the respective subject and is followed in more depth by specialized themes. Among the specialized themes are: Fermat as the founder of modern number theory, Fermat's Last Theorem from Fermat to Wiles, the history of the function concept, paradoxes, the principle of continuity, and an historical perspective on recent debates about proof. The fourth part contains essays describing mathematics courses inspired by history. The essays deal with numbers as a source of ideas in teaching, with famous problems, and with the stories behind various "great" quotations. The last part gives an account of five mathematicians-Dedekind, Euler, Gauss, Hilbert, and Weierstrass-whose lives and work we hope readers will find inspiring. Key features of the work include: * A preface describing in some detail the author's ideas on teaching mathematics courses, in particular, the role of history in such courses; * Explicit comments and suggestions for teachers on how history can affect the teaching of mathematics; * A description of a course in the history of mathematics taught in an In-Service Master's Program for high school teachers; * Inclusion of issues in the philosophy of mathematics; * An extensive list of relevant references at the end of each chapter. Excursions in the History of Mathematics was written with several goals in mind: to arouse mathematics teachers' interest in the history of their subject; to encourage mathematics teachers with at least some knowledge of the history of mathematics to offer courses with a strong historical component; and to provide an historical perspective on a number of basic topics taught in mathematics courses This Book Was Written With Several Goals In Mind: To Arouse Mathematics Teachers' Interest In The History Of Their Subject; To Encourage Mathematics Teachers With At Least Some Knowledge Of The History Of Mathematics To Offer Courses With A Strong Historical Component; And To Provide An Historical Perspective On A Number Of Basic Topics Taught In Mathematics Courses. Pt. A. Number Theory -- Highlights In The History Of Number Theory : 1700 Bc-2008 -- Fermat : The Founder Of Modern Number Theory -- Fermat's Last Theorem : From Fermat To Wiles -- Pt. B. Calculus/analysis -- A History Of The Infinitely Small And The Infinitely Large In Calculus, With Remarks For The Teacher -- A Brief History Of The Function Concept -- More On The History Of Functions, With Remarks On Teaching -- Pt. C. Proof -- Highlights In The Practice Of Proof : 1600 Bc-2009 -- Paradoxes : What Are They Good For? -- Principle Of Continuity : Sixteenth-nineteenth Centuries -- Proof : A Many-splendored Thing -- Pt. D. Courses Inspired By History -- Numbers As A Source Of Mathematical Ideas -- History Of Complex Numbers, With A Moral For Teachers -- A History-of-mathematics Course For Teachers, Based On Great Quotations -- Famous Problems In Mathematics -- Pt. E. Brief Biographies Of Selected Mathematicians -- The Biographies. Israel Kleiner. Includes Bibliographical References And Index. Front Matter....Pages i-xxi Front Matter....Pages 1-1 Highlights in the History of Number Theory: 1700 BC– 2008....Pages 3-30 Fermat: The Founder of Modern Number Theory....Pages 31-45 Fermat’s Last Theorem: From Fermat to Wiles....Pages 47-64 Front Matter....Pages 65-65 History of the Infinitely Small and the Infinitely Large in Calculus, with Remarks for the Teacher....Pages 67-101 A Brief History of the Function Concept....Pages 103-124 More on the History of Functions, with Remarks on Teaching....Pages 125-150 Front Matter....Pages 151-151 Highlights in the Practice of Proof: 1600 BC–2009....Pages 153-180 Paradoxes: What Are They Good For?....Pages 181-196 Principle of Continuity: Sixteenth–Nineteenth Centuries....Pages 197-214 Proof: A Many-Splendored Thing....Pages 215-236 Front Matter....Pages 237-237 Numbers as a Source of Mathematical Ideas....Pages 239-259 History of Complex Numbers, with a Moral for Teachers....Pages 261-272 A History-of-Mathematics Course for Teachers, Based on Great Quotations....Pages 273-284 Famous Problems in Mathematics....Pages 285-302 Front Matter....Pages 303-303 The Biographies....Pages 305-342 Back Matter....Pages 343-347 "The present book deals with factorization problems for matrix and operator functions. The problems originate from, or are motivated by, the theory of non-selfadjoint operators, the theory of matrix polynomials, mathematical systems and control theory, the theory of Riccati equations, inversion of convolution operators, theory of job scheduling in operations research." "The book systematically employs a geometric principle of factorization which has its origins in the state space theory of linear input-output systems and in the theory of characteristic operator functions. This principle allows one to deal with different factorizations from one point of view. Covered are canonical factorization, minimal and non-minimal factorizations, pseudo-canonical factorization, and various types of degree one factorization." "Considerable attention is given to the matter of stability of factorization which in terms of the state space method involves stability of invariant subspaces."--Jacket