Proofs from THE BOOKSpringer Science & Business Media, 17 d’abr. 2013 - 215 pàgines The (mathematical) heroes of this book are "perfect proofs": brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems from Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. Thirty beautiful examples are presented here. They are candidates for The Book in which God records the perfect proofs - according to the late Paul Erdös, who himself suggested many of the topics in this collection. The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background. For this revised and expanded second edition several chapters have been revised and expanded, and three new chapters have been added. |
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 |
Geometry | 37 |
Cotangent and the Herglotz trick | 119 |
Buffons needle problem | 125 |
Combinatorics | 129 |
Three famous theorems on finite sets | 143 |
Lattice paths and determinants 149 | 148 |
Cayleys formula for the number of trees | 155 |
Completing Latin squares | 161 |
The Dinitz problem | 167 |
Lines in the plane and decompositions of graphs | 47 |
The slope problem 53 | 52 |
Three applications of Eulers formula | 59 |
Cauchys rigidity theorem | 65 |
Touching simplices | 69 |
Every large point set has an obtuse angle | 73 |
Borsuks conjecture | 79 |
Analysis | 85 |
In praise of inequalities | 99 |
A theorem of Pólya on polynomials | 107 |
On a lemma of Littlewood and Offord | 115 |
Altres edicions - Mostra-ho tot
Frases i termes més freqüents
A₁ adjacent angles assume Bertrand's postulate bijection binomial coefficients bipartite graph Book Proof Borsuk's conjecture bound cardinal cells Chapter chromatic number color column combinatorial complete complex numbers configuration congruent conjecture consider contains corresponding countable crossing move denote diagonal dimension Dinitz problem dual graph eigenvalues elements equal Euler's formula example finite set function graph G graph theory hence Hilbert's third problem hyperplane implies induction inequality infinite integers intersect interval Latin square Lemma length linear Math Mathematical matrix multiplicity n-set needle number of edges obtain ordinal number P₁ pairs partial Latin square path Paul Erdős permutations plane graph points Pólya polygon polyhedra polynomial prime number prove result roots sequence solution subgraph subset Suppose Sylvester-Gallai theorem tetrahedra theorem trees triangle Turán values vectors vertex set vertices yields