Problem Seminar (eBook)

There was once a bumper sticker that read, "e;Remember the good old days when air was clean and sex was dirty?"e; Indeed, some of us are old enough to remember not only those good old days, but even the days when Math was/un(!), not the ponderous THEOREM, PROOF, THEOREM, PROOF, . . . , but the whimsical, "e;I've got a good prob- lem. "e; Why did the mood change? What misguided educational philoso- phy transformed graduate mathematics from a passionate activity to a form of passive scholarship? In less sentimental terms, why have the graduate schools dropped the Problem Seminar? We therefore offer "e;A Problem Seminar"e; to those students who haven't enjoyed the fun and games of problem solving. CONTENTS Preface v Format I Problems 3 Estimation Theory 11 Generating Functions 17 Limits of Integrals 19 Expectations 21 Prime Factors 23 Category Arguments 25 Convexity 27 Hints 29 Solutions 41 FORMAT This book has three parts: first, the list of problems, briefly punctuated by some descriptive pages; second, a list of hints, which are merely meant as words to the (very) wise; and third, the (almost) complete solutions. Thus, the problems can be viewed on any of three levels: as somewhat difficult challenges (without the hints), as more routine problems (with the hints), or as a textbook on "e;how to solve it"e; (when the solutions are read). Of course it is our hope that the book can be enjoyed on any of these three levels.