Foreword In severa l chapter s w e develo p th e necessar y tool s i n th e proces s investigation without eve n alluding to the fact tha t thes e are stand results i n variou s area s o f mathematics. W e ai m t o ge t acros s t o reader tha t mathematic s i s not discovere d b y reproduction an d sli modification o f technique s foun d i n textbooks , bu t rathe r throug painful an d ofte n comica l proces s o f discovery an d rediscovery . While not al l the problems in this book are at th e cutting edg modern mathematics , th e technique s ar e selecte d precisel y fo r th importance an d ubiquit y i n mathematica l research . Connection s tween differen t technique s an d area s o f mathematics ar e emphasiz throughout an d constitut e on e o f th e mos t importan t lesson s t book attempt s t o impart . The studen t i s expecte d t o wor k har d whil e readin g thi s bo This i s no t be d tim e reading , no r i s i t a fantas y novel . Yo u m have a pe n an d plent y o f pape r handy , an d expec t t o fill u p abo ten pages of calculations fo r ever y page you read. Mathematic s i s a spectato r sport , s o creat e i n additio n t o readin g an d computin Every time you see a theorem or a calculation, tr y t o formulate a n one. Ever y tim e yo u se e a proof, tr y t o find a better one . An d m importantly, hav e fun ! My goal is to make this book interesting and accessibl e to anyo with th e basi c knowledg e o f hig h schoo l mathematic s wh o i s cu ous about researc h mathematics . Severa l chapter s requir e knowled of calculu s o f severa l variables , an d thi s i s clearl y indicate d i n t beginning o f each o f those chapters . On the other hand, many topics of this book may even be of int est to graduate students in mathematics an d professional researche While the vast majorit y o f techniques describe d i n this boo k ar e w known t o professiona l mathematicians , th e perspectiv e an d interl ing of topics and ideas may turn out to be unusual and even surprisi

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