Documenta Mathematica Optimization Stories - Mathematics

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32 Eberhard Knobloch Figure 1: Leonhard Euler (1707–1783) (L. Euler, Opera omnia, series I, vol. 1, Leipzig – Berlin 1911, Engraving after p. I) and soul is false, without mentioning the name of his illustrious predecessor. Another statement prescribed the construction of a ship mast. The third statement considered a thought experiment: What would happen at the centre of the earth if a stone were dropped into a straight tunnel drilled to the centre of the earth and beyond to the other side of the planet? Euler distinguished between exactly three possiblities: Either the stone will rest at the centre or will at once proceed beyond it or it will immediately return from the centre to us. There is no mention of speed. Euler just stated that the last case will take place. No justification or explanation is given, though none of these three possibilities had the slightest evidence. What is worse, a far better answer had been already given by Galileo in 1632. In the second day of his Dialogue about the two main world systems Galileo discussed this thought experiment in order to refute Aristotle’s distinction between natural and unnatural motions. The natural motion of heavy bodies is the straight fall to the centre of the earth. But what about a cannon ball Documenta Mathematica · Extra Volume ISMP (2012) 31–35
Euler and Infinite Speed 33 Figure 2: Title page of Euler’s Physical dissertation on sound (L. Euler, Opera omnia, series III, vol. 1, Leipzig – Berlin 1926, p. 181) that has dropped into such an earth tunnel? Even the Aristotelian Simplicio avowed that the cannon ball would reach the same height from which it had dropped into the tunnel in the other half of the tunnel. The natural motion would change into an unnatural motion. Galileo erroneously presupposed a constant gravitation. But he rightly deduced an oscillating motion of the cannon ball. Euler did not mention the Italian mathematician. Presumably he did not know his solution of the thought experiment. Nine years later he came back to this question in his Mechanics or the science of motion set forth analytically. Now he explicitly concluded that the speed of the falling stone will become infinitely large in the centre of the earth. Nevertheless it will immediately return to the starting-point. Euler admitted: This seems to differ from truth because hardly any reason is obvious why a body, having infinitely large speed that it has acquired in C, should proceed to any other region than to CB, especially because the direction of the infinite speed turns to this region. However that may be, here we have to be confídent more in the calculation than in our judgement and to confess that we do not understand at all the jump if it is done from the infinite into the finite. Documenta Mathematica · Extra Volume ISMP (2012) 31–35
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Page 5 and 6: Richard W. Cottle William Karush an
Page 7 and 8: Documenta Math. 1 Preface When in d
Page 9 and 10: Documenta Math. 3 Introduction When
Page 11 and 12: Introduction 5 Finally, I would lik
Page 13 and 14: Documenta Math. 7 Stories about the
Page 15 and 16: Documenta Math. 9 Jiu Zhang Suan Sh
Page 17 and 18: Jiu Zhang Suan Shu and the Gauss Al
Page 19 and 20: Jiu Zhang Suan Shu and the Gauss Al
Page 21 and 22: Documenta Math. 15 Leibniz and the
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Page 25 and 26: Documenta Math. 19 Leibniz and the
Page 27 and 28: Leibniz and the Infinite 21 finally
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Page 35 and 36: A Short History of Newton’s Metho
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Page 83 and 84: 2 A first counterexample Who Solved
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Who Solved the Hirsch Conjecture? 8
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Who Solved the Hirsch Conjecture? 8
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Documenta Math. 87 Pope Gregory, th
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Pope Gregory, the Calendar, and Con
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Documenta Math. 95 Löwner-John Ell
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2 The ellipsoids Löwner-John Ellip
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Löwner-John Ellipsoids 99 For maxi
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H + Löwner-John Ellipsoids 101 t P
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Löwner-John Ellipsoids 103 any odd
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Löwner-John Ellipsoids 105 [23] P.
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Documenta Math. 107 A Brief History
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A Brief History 109 first-generatio
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A Brief History 111 developments ar
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A Brief History 113 in practice. Ka
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A Brief History 115 that LPs with s
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Version-to-Version Speedup 10 9 8 7
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A Brief History 119 [10] Crowder, H
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A Brief History 121 Robert E. Bixby
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Documenta Math. 123 Discrete Optimi
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Discrete Optimization Stories 125 I
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Documenta Math. 127 The Origins of
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Minimal Spanning Tree Algorithm 129
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Documenta Math. 143 The Coming of t
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The Sixties The Coming of the Matro
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The Coming of the Matroids 147 Figu
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The Coming of the Matroids 149 desc
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The Coming of the Matroids 151 In f
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The Coming of the Matroids 153 [10]
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Documenta Math. 155 On the History
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On the History of the Shortest Path
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Documenta Math. 169 On the History
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Transportation and Maximum Flow Pro
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Documenta Math. 181 Edmonds, Matchi
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The Birth of Polyhedral Combinatori
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Documenta Math. 199 Flinders Petrie
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The travelling salesman problem Fli
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Flinders Petrie 203 accuracy. The r
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Flinders Petrie 205 Figure 3: Paper
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Flinders Petrie 207 Figure 4: Petri
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Flinders Petrie 209 [8] F. R. Hodso
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Documenta Math. 211 D. Ray Fulkerso
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D. Ray Fulkerson and Project Schedu
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G 1 2 D. Ray Fulkerson and Project
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Documenta Math. 221 The Ongoing Sto
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The Ongoing Story of Gomory Cuts 22
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Acknowledgement The Ongoing Story o
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Documenta Math. 227 Markowitz and M
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Markowitz and Manne + Eastman + Lan
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Documenta Math. 239 Ronald Graham:
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Ronald Graham: Foundations of Onlin
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Machines Ronald Graham: Foundations
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Ronald Graham: Foundations of Onlin
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Documenta Math. 247 Continuous Opti
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Figure 1: Minkowski’s grave ( c
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Documenta Math. 251 Cauchy and the
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Cauchy and the Gradient Method 253
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Documenta Math. 255 William Karush
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William Karush and the KKT Theorem
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Documenta Math. 271 Nelder, Mead, a
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Nelder, Mead, and the Other Simplex
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Nelder, Mead, and the Other Simplex
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Documenta Math. 277 Subgradient Opt
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Subgradient Optimization in Nonsmoo
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Documenta Math. 291 A Science Ficti
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A Science Fiction Story in Nonsmoot
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Documenta Math. 301 Broyden Updatin
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Broyden Updating, the Good and the
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Documenta Math. 317 Carathéodory o
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Carathéodory on the Road to the Ma
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Documenta Math. 331 The Cold War an
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The Cold War and the Maximum Princi
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Documenta Math. 345 The Princess an
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yB y The Princess and Infinite-Dime
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x2 The Princess and Infinite-Dimens
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The Princess and Infinite-Dimension
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Documenta Math. 357 Computing Stori
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Documenta Math. 359 A Brief History
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A Brief History of NP-Completeness
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Documenta Math. 377 On the Evolutio
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Optimization Modeling Systems 379 T
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Optimization Modeling Systems 381 m
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Optimization Modeling Systems 383 C
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Optimization Modeling Systems 385 a
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Optimization Modeling Systems 387 e
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Documenta Math. 389 Who Invented th
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Who Invented the Reverse Mode of Di
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Documenta Math. 401 Gordon Moore an
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Gordon Moore and His Law 403 Figure
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Gordon Moore and His Law 405 jector
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!$ ? !$ > !$ & Gordon Moore and His
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Gordon Moore and His Law 415 [14] M
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Documenta Math. 417 More Optimizati
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Documenta Math. 419 Voronoi Diagram
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Voronoi Diagrams and Delaunay Trian
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a a b d Voronoi Diagrams and Delaun
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Documenta Math. 433 Around Hilbert
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Around Hilbert’s 17th Problem 435
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Around Hilbert’s 17th Problem 437
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Documenta Math. 439 From Kepler to
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From Kepler to Hales 441 Figure 1:
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From Kepler to Hales 443 assertions
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From Kepler to Hales 445 |- !A. con
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Documenta Math. 447 Vilfredo Pareto
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Vilfredo Pareto and Multi-objective
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Documenta Math. 455 Optimisation an
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Optimisation and Utility Functions
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Optimisation and Utility Functions
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