Download e-book for kindle: Numerical Analysis and Parallel Processing by Peter R. Turner

By Peter R. Turner

ISBN-10: 354051645X

ISBN-13: 9783540516453

Every one week of this 3 week assembly was once a self-contained occasion, even if every one had an analogous underlying subject matter - the impact of parallel processing on numerical research. each one week supplied the chance for in depth examine to develop individuals' study pursuits or deepen their realizing of issues of which they already had a few wisdom. there has been additionally the chance for carrying on with person study within the stimulating surroundings created by means of the presence of numerous specialists of overseas stature. This quantity comprises lecture notes for many of the main classes of lectures offered on the assembly; they conceal subject matters in parallel algorithms for giant sparse linear platforms and optimization, an introductory survey of level-index mathematics and superconvergence within the finite aspect approach.

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Download e-book for kindle: Numerical Analysis and Parallel Processing by Peter R. Turner

Every one week of this 3 week assembly was once a self-contained occasion, even though every one had an identical underlying subject matter - the influence of parallel processing on numerical research. each one week supplied the chance for in depth research to increase contributors' examine pursuits or deepen their realizing of subject matters of which they already had a few wisdom.

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Bn+1 bn n+1 bn (n + 1) Dieses neue Intervall kann h¨ ochstens die Zahl n + 1 zu unserer Primzahlenliste hinzuf¨ ugen. Ist nun n + 1 tats¨ achlich eine Primzahl, dann l¨aßt sich bn+1 = urzen. Damit haben wir alles, was wir brauchen, bn (n + 1) nicht gegen an+1 k¨ und das Ergebnis gilt aufgrund des Induktionsschrittes. Die (als zutreffend bewiesene) Bertrandsche Vermutung garantiert, daß die Intervalle [p, 2p − 1] f¨ ur p ≥ 7 u ¨berlappen und deswegen alle ganzen Zahlen n ≥ 7 enthalten, da zwischen jedem Paar p und 2p eine Primzahl liegt.

Angenommen die Reihe konvergiert. Dann muß es einen Reihenrest geben, dessen Summe kleiner als 12 ist, das heißt es muß ein i geben, so daß 1 pi+1 2 + 1 pi+2 Auch M¨ obius-Funktion genannt. + 1 pi+3 + ··· < 1 . 2 40 3 Subharmonische Reihen Es sei nun Ni (x) die Anzahl der positiven ganzen Zahlen, die kleiner als x sind und sich nur durch die ersten i Primzahlen teilen lassen. Ist n eine dieser positiven ganzen Zahlen, dann k¨ onnen wir – wie wir das bereits getan hatten – die Darstellung n = k × m2 verwenden, bei der k eine quadratfreie Zahl ist.

Dann ist P entweder eine Primzahl (im Widerspruch zu unserer Annahme, daß N die gr¨ oßte Primzahl ist) oder P ist eine zusammengesetzte Zahl und somit durch Primzahlen teilbar. Dividiert man P durch jede der auftretenden Primzahlen, dann erh¨ alt man jeweils den Rest 1. Also muß es mindestens eine weitere Primzahl geben, die gr¨ oßer als N ist. Das ist erneut ein Widerspruch zu der Annahme, daß die Anzahl der Primzahlen endlich ist und N die gr¨oßte dieser Primzahlen bezeichnet. Es gibt also unendlich viele Primzahlen, was zu beweisen war.

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Numerical Analysis and Parallel Processing by Peter R. Turner


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