B. Mazur's Current Developments in Mathematics 1998 PDF

By B. Mazur

ISBN-10: 1571460772

ISBN-13: 9781571460776

Those are the complaints of the joint seminar by way of M.I.T. and Harvard at the present advancements in arithmetic for the yr 1998. confirmed in 1995, this seminar has been endured at the 3rd weekend of November each year. The organizing committee for the seminar consisted of extraordinary mathematicians from the math departments of either associations: Barry Mazur, Wilfried Schmid, and Shing-Tung Yau from Harvard, and David Jerison, Isadore Singer and Daniel Stroock from M.I.T.. We belief that those court cases may be of curiosity to many mathematicians, and may motivate destiny advancements and examine targets in arithmetic.

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Dann h¨atte man das Horner-Schema 1 −8 22 −23 3 −15 21 1 −5 7 −2 urde ergeben: Also ist p∗ (3) = −2. Die Polynomdivision w¨ p∗ (x) a = p(x) + x−3 x−3 mit einem geeigneten Polynom p und einer Konstanten a. Multipliziert man die Gleichung mit x − 3, so erh¨alt man a = p ∗ (x) − p(x) · (x − 3). Setzt man = 3 ein, so folgt: a = p∗ (3) = −2. Tats¨achlich erh¨alt man im vorliegenden Fall: −2 p∗ (x) : (x − 3) = x2 − 5x + 7 + . x−3 Dies ist der typische Fall einer Polynom-Division mit Rest, und die l¨asst sich auch allgemein mit dem Horner-Schema durchf¨ uhren.

Das w¨are hier die Ungleichung x + xy < y + xy. Allerdings ist diese Argumentation die ber¨ uchtigte falsche Schlussrichtung“. Um die zu ” vermeiden, fangen wir gleich von der richtigen Seite aus an: 0 < x 1 < x2 =⇒ =⇒ =⇒ x 1 + x1 x 2 < x2 + x 1 x 2 x1 (1 + x2 ) < x2 (1 + x1 ) x1 x2 < . 1 + x1 1 + x2 Die Divisionen beim letzten Schritt sind zul¨assig, denn es handelt sich um Multiplikationen mit positiven Zahlen. Damit ist gezeigt, dass f streng monoton wachsend und damit erst recht injektiv ist.

An der Menge R kann man die Transitivit¨at allerdings nur m¨ uhsam erkennen. Beispiele 1. Die einfachste Relation ist die Gleichheit“: Dabei ist R := {(x, x) : x ∈ A} ” die Diagonale“ ΔA ⊂ A × A, und die Beziehung x ∼ y bedeutet einfach, ” dass x = y ist. Die Gleichheit besitzt alle oben beschriebenen Eigenschaften: (a) ∀ x ∈ A : x = x (reflexiv). (b) ∀ x, y ∈ A : x = y =⇒ y = x (symmetrisch). (c) ∀ x, y, z ∈ A : x = y ∧ y = z =⇒ x = z (transitiv). Eine Relation, die zugleich reflexiv, symmetrisch und transitiv ist, nennt man ¨ ¨ eine Aquivalenzrelation (nicht zu verwechseln mit der logischen Aquiva¨ lenz).

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Current Developments in Mathematics 1998 by B. Mazur

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