Mathematical Circles (Russian Experience)

Throughout The History Of Mathematics, Maximum And Minimum Problems Have Played An Important Role In The Evolution Of The Field. Many Beautiful And Important Problems Have Appeared In A Variety Of Branches Of Mathematics And Physics, As Well As In Other Fields Of Sciences. The Greatest Scientists Of The Past--euclid, Archimedes, Heron, The Bernoullis, Newton, And Many Others--took Part In Seeking Solutions To These Concrete Problems. The Solutions Stimulated The Development Of The Theory, And, As A Result, Techniques Were Elaborated That Made Possible The Solution Of A Tremendous Variety Of Problems By A Single Method. This Book Presents Fifteen Stories Designed To Acquaint Readers With The Central Concepts Of The Theory Of Maxima And Minima, As Well As With Its Illustrious History. This Book Is Accessible To High School Students And Would Likely Be Of Interest To A Wide Variety Of Readers. In Part One, The Author Familiarizes Readers With Many Concrete Problems That Lead To Discussion Of The Work Of Some Of The Greatest Mathematicians Of All Time. Part Two Introduces A Method For Solving Maximum And Minimum Problems That Originated With Lagrange. While The Content Of This Method Has Varied Constantly, Its Basic Conception Has Endured For Over Two Centuries. The Final Story Is Addressed Primarily To Those Who Teach Mathematics, For It Impinges On The Question Of How And Why To Teach. Throughout The Book, The Author Strives To Show How The Analysis Of Diverse Facts Gives Rise To A General Idea, How This Idea Is Transformed, How It Is Enriched By New Content, And How It Remains The Same In Spite Of These Changes.--publisher. Why Do We Solve Maximum And Minimum Problems? -- The Oldest Problem : Dido's Problem -- Maxima And Minima In Nature (optics) -- Maxima And Minima In Geometry -- Maxima And Minima In Algebra And In Analysis -- Kepler's Problem -- The Brachistochrone -- Newton's Aerodynamical Problem -- What Is A Function? -- What Is An Extremal Problem? -- Extrema Of Functions Of One Variable -- Extrema Of Functions Of Many Variables : The Lagrange Principle -- More Problem Solving -- What Happened Later In The Theory Of Extremal Problems? -- More Acurately, A Discussion. V.m. Tikhomirov ; Translated From The Russian By Abe Shenitzer. Translation Of: Rasskazy O Maksimumakh I Minimumakh. Includes Bibliographical References (p. 187). Some Scientists Claim That Strong Tobacco And Spirits Clear The Head And Spur Creativity. It Would Be Well, However, To Try Other Means: To Exercise, Jog, Swim, Or Learn To Play Games Like Tennis, Basketball, Badminton, Volleyball, And So On...[n]ot Only Checkers, Chess, Cards, Or Billiards Are A Source Of Interesting Problems. Other Sports Provide Them As Well. Mathematical Methods Are Increasingly Applied In Sports. Just Think How Many Yet-unsolved Problems Arise When We Study The Interaction Between Ball And Racket Or Between Ball And Court. - From The Introduction. This Unique Book Presents Simple Mathematical Models Of Various Aspects Of Sports, With Applications To Sports Training And Competitions. Requiring Only A Background In Precalculus, It Would Be Suitable As A Textbook For Courses In Mathematical Modeling And Operations Research At The High School Or College Level. Coaches And Those Who Do Sports Will Find It Interesting As Well. The Lively Writing Style And Wide Range Of Topics Make This Book Especially Appealing. Mathematics And Sports (in Place Of A Foreword) -- What Is Applied Mathematics? -- Why Five Sets? (mathematical Modeling Of Tennis) -- Those Judges! -- Records! Records! -- Linear Programming And Sports -- Game Models -- Organizing Competitions Is An Operations Planning Problem -- Classifications In Sports. L.e. Sadovskiĭ And A.l. Sadovskiĭ ; Translated From The Russian By S. Makar-limanov. Translation Of: Matematika I Sport. Includes Bibliographical References (p. 151-152). "The theory of fixed points finds its roots in the work of Poincaré, Brouwer, and Sperner and makes extensive use of such topological notions as continuity, compactness, homotopy, and the degree of a mapping. Fixed point theorems have numerous applications in mathematics; most of the theorems ensuring the existence of solutions for differential, integral, operator, or other equations can be reduced to fixed point theorems. In addition, these theorems are used in such areas as mathematical economics and game theory. This book presents a readable exposition of fixed point theory. The author focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. Another aim of the book is to show how fixed point theory uses combinatorial ideas related to decomposition (triangulation) of figures into distinct parts called faces (simplexes), which adjoin each other in a regular fashion. All necessary background concepts--such as continuity, compactness, degree of a map, and so on--are explained, making the book accessible even to students at the high school level. In addition, the book contains exercises and descriptions of applications. Readers will appreciate this book for its lucid presentation of this fundamental mathematical topic."--Publisher

The theory of fixed points finds its roots in the work of Poincare, Brouwer, and Sperner and makes extensive use of such topological notions as continuity, compactness, homotopy, and the degree of a mapping. Fixed point theorems have numerous applications in mathematics; most of the theorems ensuring the existence of solutions for differential, integral, operator, or other equations can be reduced to fixed point theorems. In addition, these theorems are used in such areas as mathematical economics and game theory. This book presents a readable exposition of fixed point theory. The author focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. Another aim of the book is to show how fixed point theory uses combinatorial ideas related to decomposition (triangulation) of figures into distinct parts called faces (simplexes), which adjoin each other in a regular fashion. All necessary background concepts-such as continuity, compactness, degree of a map, and so on-are explained, making the book accessible even to students at the high school level. In addition, the book contains exercises and descriptions of applications. Readers will appreciate this book for its lucid presentation of this fundamental mathematical topic.

This volume contains some examples of mathematical applications in sports. Sports discussed include tennis, figure skating, gymnastics, track and field, soccer, skiing, hockey, and swimming. Problems and situations are posed and answers with thorough explanations are provided. Chapters include: (1) Mathematics and Sports; (2) What Is Applied Mathematics?; (3) Why Five Sets? (Mathematical Modeling of Tennis); (4) Those Judges!; (5) Records! Records!; (6) Linear Programming and Sports; (7) Game Models; (8) Organizing Competitions Is an Operations Planning Problem; (9) Classifications in Sports; and (10) Conclusion. (Contains 37 references.) (ASK) "Some scientists claim that strong tobacco and spirits clear the head and spur creativity. It would be well, however, to try other means: to exercise, jog, swim, or learn to play games like tennis, basketball, badminton, volleyball, and so on ... [N]ot only checkers, chess, cards, or billiards are a source of interesting problems. Other sports provide them as well. Mathematical methods are increasingly applied in sports. Just think how many yet-unsolved problems arise when we study the interaction between ball and racket or between ball and court."--The introduction This unique book presents simple mathematical models of various aspects of sports with applications to sports training and competitions. Requiring only a background in precalculus, it would be suitable as a textbook for courses in mathematical modeling and operations research at the high school or college level. Coaches and those who participate in sports will find it interesting as well. The lively writing style and wide range of topics make this book especially appealing. The inaugural volume in the new Mathematical world series. Tikhomirov presents the history, ideas, methods, and mathematicians associated with maximum and minimum problems, and introduces a general method for their solution that originated with Lagrange. Primarily for high school and college students and teachers. Translated from the Russian. Annotation copyright Book News, Inc. Portland, Or. Presents an exposition of fixed point theory. This work focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property. It aims to show how fixed point theory uses combinatorial ideas related to decomposition of figures into distinct parts called faces, which adjoin each other in a regular fashion. Fourteen chapters, translated from the Russian, on topics that continuous mappings of a closed interval and a square, first and second combinatorial lemma, Sperner's lemma, compactness, retraction, and continuous mappings of a sphere. Annotation copyright Book News, Inc. Portland, Or.