Proofs from THE BOOKSpringer Science & Business Media, 29 cze 2013 - 199 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. |
Spis treści
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 | 35 |
In praise of inequalities | 101 |
A theorem of Pólya on polynomials | 109 |
On a lemma of Littlewood and Offord | 117 |
Combinatorics | 121 |
Three famous theorems on finite sets | 135 |
Cayleys formula for the number of trees 141 | 140 |
Completing Latin squares | 147 |
The Dinitz problem | 153 |
Lines in the plane and decompositions of graphs | 45 |
The slope problem | 51 |
Three applications of Eulers formula | 57 |
Cauchys rigidity theorem | 63 |
The problem of the thirteen spheres | 67 |
Touching simplices | 73 |
Every large point set has an obtuse angle | 77 |
Borsuks conjecture | 83 |
Analysis | 89 |
Sets functions and the continuum hypothesis 91 | 90 |
Graph Theory | 159 |
Fivecoloring plane graphs | 161 |
How to guard a museum | 165 |
Turáns graph theorem | 169 |
Communicating without errors | 173 |
Of friends and politicians | 183 |
Probability makes counting sometimes easy 187 | 186 |
About the Illustrations | 196 |
197 | |
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a₁ adjacent Amer angles arccos 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 convex corresponding countable crossing move degree denote diagonal dimension Dinitz problem dual graph eigenvalues elements equal Euler's formula example finite set geometric graph G graph theory hence Hilbert's third problem implies induction inequality infinite integers intersect interval Latin square lemma length linear Math matrix multiple n-set obtain ordinal number P₁ pair partial Latin square Paul Erdős permutations plane graph points Pólya polygon polyhedra polynomial polytope prime number prove result roots sequence sphere subgraph subset Suppose Sylvester-Gallai theorem tetrahedron theorem trees triangle Turán Turán's theorem vectors vertex set vertices w₁ well-ordered yields