John Allen Paulos's A Mathematician Reads the Newspaper PDF

By John Allen Paulos

ISBN-10: 0465050670

ISBN-13: 9780465050673

During this energetic quantity, mathematician John Allen Paulos employs his singular wit to steer us via an not going mathematical jungle—the pages of the day-by-day newspaper. From the Senate and intercourse to celebrities and cults, Paulos takes tales that won't appear to contain math in any respect and demonstrates how mathematical naïveté can positioned readers at a special drawback. no matter if he’s utilizing chaos conception to puncture financial and environmental predictions, employing good judgment to explain the dangers of spin doctoring and information compression, or utilizing mathematics and customary experience to provide us a singular viewpoint on greed and relationships, Paulos by no means fails to entertain and enlighten.

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5) Passing to the associated shemves we find Li_e~] Lemma. as desired. ~*(~(~)) is locally free (and hence commutes with arbitrary base change). Proof: From the exact sequence using the local freeness of 0--~ ~i(~)--+ [~--+ OA--~ 0 , H~p(A) , H*(OA) and the degeneration of Hodge => De Rham, we read the result from the short exact sequences: 0 ~ Knowing Hi(%(~A )) ~ ~(T{QA) ) ~i(~) + Hi(OA ) ~ 0 is a locally free module commuting with base change we obtain the exact sequence of Zariski sheaves on Sch/S.

P Let the subscript ~ > 1 denote restriction to the base = Spec ~/p~. Let E(N(p)et) N(p) et denote the etale quotient of N(p), and let denote the universal extension of N(p)e t We have the diagram E(N) (p) ~ e)t E(N(p Since ~ N(p) ~ g~p )et )~ G = ~/p the image of G in N(p) et Consequently the image of the section It follows that the image of x* in x* section of order p . in N(p) et E(N(p) et) But this is a contradiction because the an ~tale p-divisible group over is nonzero. is nonzero. is nonzero.

In order to prove Ext~(A,Gm) ~ Extrig(A,G m) is an let us construct an inverse. Assume given a 37 rigidified extension 0 ~ G~-~ E J ~ A--* 0 I I(A) defines a section of p: (eAolr) (E)-~ i*(E) via By definition of through Thus Infl(A). i*(E)~ and hence a trivialization e~-~ (idlnfl(A),~). Infl(A), the map Let us write it as p2-p~: Al(A) ~ A ~I(A)---~ I n f l ( A ) C - ~ A • ~*(p): (eAO~Al(A))*(E)-~-~ (p2-Pl)*(E) where VAl{A): &l(A) ~ S is an isomorphism is the structural morphism. 5) p[(E) V' . > p~(E) Our inverse mapping is now defined by associating to the rigidified extension above the ~-extension lying extension and the with the same under- ~-structure defined by V'.

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A Mathematician Reads the Newspaper by John Allen Paulos


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