A Century of Mathematics: Through the Eyes of the Monthly by John Ewing

By John Ewing

This can be the tale of yankee arithmetic in the past century. It comprises articles and excerpts from a century of the yank Mathematical per thirty days, giving the reader a chance to skim all 100 volumes of this well known arithmetic journal with out truly commencing them. It samples arithmetic 12 months by way of yr and decade through decade. The reader can glimpse the mathematical group on the flip of the century, the talk approximately Einstein and relativity, the debates approximately formalism in good judgment, the immigration of mathematicians from Europe, and the frantic attempt to arrange because the warfare started. more moderen articles take care of the appearance of pcs and the adjustments they introduced, and with many of the triumphs of contemporary examine.

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A. a. a. est quasi-int6grable. a. ~tag@e X = ~ c i IA. a. par exemple int~grable X 4tant E(XIB) avec P X-l(dxlB) du borelien = f x P x-l(dxIB) = P {X-I(dx)~B}/P(B) "infinit@simal" l'@v~nement B dx est r@alis@. pris darts une sous Dans la pratique, O -alg@bre ~ . est la O -alg@bre engendr@e par E(XIB) et al@atoire, E(XlB c) sur E(XIB) B et sur Bc = E(XIB) B et & E(x[~) est m a i n t e n a n t B, B c, ~ ~ ou tionnelle l'@v@nement B doit pouvoir @tre Le cas le plus simple est celui o~ B , c~d ou ~ = {B,BC,~,~} , les esp@rances en ~ iB + E(X{B c) IBC , E(XIB c) sur et @tant par consequent dP = I B B x-l(dx) sous la condition que p e u v e n t alors atre regroup4es p o u r constituer une variable sur I o~ de l'image r@ciproque ~ savoir E(X{~) @gale ~ = probabilit@ de la droite r@elle, Bc ,donc constante ~-mesurable, s@par@ment et telle que de plus x dP B un ~l~ment quelconque ~).

V~nements A d'~v~nements Am Les i n d i c a t e u r s et d'apr~s A m e s t r~alis~, des 4 p r e u v e s ~ pour c&d constitu~ des ~ p r e u v e s ~ p a r t i r d ' u n c e r t a i n rang d 4 p e n d a n t m e o n s t i t u 4 des ~ p r e u v e s r 4 a l i s a n t t o u s l e s de de ces 4 v 4 n e m e n t s = lim inf mgn constitu~ , c~d en fait c o n s t i t u ~ n (n~ 1) U A m = lira + ( ~ Am ) mgn n m~n m~n ~v~nements A . Le c o n t r a i r e llim inf A n n , Aml lim+<,, est l ' 4 v 4 n e m e n t est l'~v~nement V A m~ n m = mgn l'un des ~ v ~ n e m e n t s r~alisant 1A des 4 p r e u v e s l i m inf A n v~rifient , est r4alisant de ~ une infinit4 l i m sup A nc ' et inversement.

A. e. a. e. ayant X Xn(n ~i) choisie pour limite). a. q. e. ) lim % E ( Z m) = 0 . a. a. e. alors X (XntkY m) (w) = rain (Xn(W), Ym(0J))) E(Xn ) = lira + E ( X n A Y m) ~ m puisque Zm = (XnA Ym ) - Xn + O et i n v e r s e m e n t si (mesurables) Etape 2 (X X lim % X et Y n sont m positives, et Y m n >11 fix@ lira % E ( Y m ) m lorsque = lim + Y Xn et donc pour tout entier m +~ , de sorte que lira + E ( X n ) ~ i m % E ( Y m) , . a. alors E(X+Y) = E(X) + X- = - (X/~O) aVb . Si donc d @ s i g n e la plus grande + E(Y).

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