Mathematical Bridges

Building bridges between classical results and contemporary nonstandard problems, this highly relevant work embraces important topics in analysis and algebra from a problem-solving perspective. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors and motivated mathematics students from high school juniors to college seniors will find the work a useful resource in calculus, linear and abstract algebra, analysis and differential equations. Students with an interest in mathematics competitions must have this book in their personal libraries. Building bridges between classical results and contemporary nonstandard problems, Mathematical Bridges embraces important topics in analysis and algebra from a problem-solving perspective. Blending old and new techniques, tactics and strategies used in solving challenging mathematical problems, readers will discover numerous genuine mathematical gems throughout that will heighten their appreciation of the inherent beauty of mathematics. Most of the problems are original to the authors and are intertwined in a well-motivated exposition driven by representative examples. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors and educators teaching problem-solving courses or organizing mathematics clubs, as well as motivated mathematics students from high school juniors to college seniors, will find Mathematical Bridges a useful resource in calculus, linear and abstract algebra, analysis and differential equations. Students desiring to hone and develop their mathematical skills or with an interest in mathematics competitions must have this book in their personal libraries. "Building bridges between classical and contemporary results, this book presents important topics in analysis and algebra vi a rich set of engaging and creative problems. Blending old and new techniques and strategies, readers will discover numerous genuine mathematical gems throughout that willheighten their appreciation of the inherent beauty of mathematics. Most of the problems are original to the authors and are intertwined in a well-motivated expostion driven by representative examples. The book is structured to assist the reader in formulating and proving conjectures, as well as devising solutions to important mathematical problems by making connections between various concepts and ideas from different areas of mathematics. Instructors and educators teaching problem-solving courses or organizing mathematics clubs, as well as motivated mathematics students from high school juniors to college seniors, will find Mathematical Bridges a useful resource in calculus, linear and abstract algebra, and analysis. Students desiring to hone and develop their mathematical skills or with an interest in mathematics competitions must have this book in their personal libraries." Front Matter....Pages i-viii Mathematical (and Other) Bridges....Pages 1-23 Cardinality....Pages 25-42 Polynomial Functions Involving Determinants....Pages 43-55 Some Applications of the Hamilton-Cayley Theorem....Pages 57-74 A Decomposition Theorem Related to the Rank of a Matrix....Pages 75-92 Equivalence Relations on Groups and Factor Groups....Pages 93-107 Density....Pages 109-129 The Nested Intervals Theorem....Pages 131-148 The Splitting Method and Double Sequences....Pages 149-173 The Number e ....Pages 175-188 The Intermediate Value Theorem....Pages 189-200 The Extreme Value Theorem....Pages 201-211 Uniform Continuity....Pages 213-226 Derivatives and Functions’ Variation....Pages 227-251 Riemann and Darboux Sums....Pages 253-287 Antiderivatives....Pages 289-307 Back Matter....Pages 309-309