Proofs from THE BOOKSpringer Science & Business Media, 14 de març 2013 - 239 pàgines From the Reviews: "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..." LMS Newsletter, January 1999 This third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such as an exciting new way to "enumerate the rationals". |
Continguts
2 | |
3 | |
7 | |
Binomial coefficients are almost never powers | 13 |
Representing numbers as sums of two squares | 17 |
Every finite division ring is a field | 23 |
Some irrational numbers | 27 |
Three times π²6 35 | 34 |
On a lemma of Littlewood and Offord | 123 |
Cotangent and the Herglotz trick | 127 |
Buffons needle problem | 133 |
Combinatorics | 137 |
Pigeonhole and double counting | 139 |
Three famous theorems on finite sets 151 | 150 |
Shuffling cards | 157 |
Lattice paths and determinants | 167 |
Geometry | 43 |
decomposing polyhedra | 45 |
Lines in the plane and decompositions of graphs 53 | 52 |
The slope problem | 59 |
Three applications of Eulers formula 65 | 64 |
Cauchys rigidity theorem | 71 |
Touching simplices | 75 |
Every large point set has an obtuse angle | 79 |
Borsuks conjecture | 85 |
Analysis | 91 |
Sets functions and the continuum hypothesis | 93 |
In praise of inequalities | 109 |
A theorem of Pólya on polynomials 117 | 116 |
Cayleys formula for the number of trees | 173 |
Completing Latin squares | 179 |
The Dinitz problem | 185 |
Identities versus bijections | 191 |
Graph Theory | 197 |
How to guard a museum | 203 |
Turáns graph theorem | 207 |
Communicating without errors | 213 |
Of friends and politicians | 223 |
Probability makes counting sometimes easy 227 | 226 |
About the Illustrations | 236 |
237 | |