By Donald L. Miller
From its turbulent beginnings to its present prestige as one of many world's so much dynamic towns, urban of the Century tells the tale of Chicago—and the tale of the USA, writ small. From its many ordinary mess ups, together with the nice hearth of 1871 and several other cholera epidemics to its winner-take-all politics, dynamic enterprise empires, breathtaking structure, its varied cultures, and its diversified inhabitants of writers, reporters, and artists, Chicago's tale is violent, inspiring, passionate, and interesting from the 1st web page to the last.
The winner of the celebrated nice Lakes publication Award, given to the year's most eminent books highlighting the yank heartland, urban of the Century has obtained constant rave experiences and was once made right into a six-hour movie airing on PBS's American event sequence. Written with full of life prose and exacting aspect, it brings Chicago's historical past to brilliant existence.
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Additional resources for City of the Century: The Epic of Chicago and the Making of America
A2 must be abelian, we can apply Theorem 4-19 to find a Cayley color graph for l', providing we know a Cayley color graph for Q. ) Since elementary group theory s h o w s that 4-5. Cayley Graphs A be a generating set for the group following conditions: Let (i) e d A (ii) If fi E A, 0-l z A (unless Also, we adopt the following convention: fi2 = r e). subject to the The C a y l e y C o l o r Graph of a Group P r e s e n t a t i o n 36 E A, (iii) I f h2 = e, (g ,g6) each p a i r and 4 Chapt. (gh ,g) of d i r e c t e d edges a r e coalesced i n t o a s i n g l e u n d i r e c t e d edge [GI ,961.
And, e a c h c h a n g e - r i n g i n g graph of Chapter 1 5 w i l l be a C a y l e y c o l o r g r a p h . 4-1. Def. 4-1. ) word element set of G. f i n i t e product flf2-. fn set tgl , g 2 , g 3 element of .. for , then r. r I A ... 1 . , ... is a is i n the If every c a n b e e x p r e s s e d a s a word i n g 1 ' 4 2 ' 4 3 g1 , g2 I g3 I . . I are s a i d t o be g e n e r a t o r s A r e l a t i o n i s an e q u a l i t y between t w o words i n 91 ' 4 2 I 4 3 Thm. 4 - 2 . Definitions I .
For example, the imbeddinq of K4 in Figure 5-4 has four regions. The followinq theorem is attributed to both Descdrtes and Euler, independently; we perhaps indicate our preference by calling it the euler polyhedral idcntlty: Thm. 5-11. Let Let be a connected graph imbedded in the sphere, S o . have p vertices and q edges, with r the G G number of regions of the imbedding. Proof _ __ (by induction on for = q for then 1, q): p = 2 Then p - + y r = 2. The result is clearly true and r = 1. Now assume the result holds for all connected graphs with €ewer than q edqcs, and let G be a connected graph with q edges, p vertices, and r regions for an imbedding in So.
City of the Century: The Epic of Chicago and the Making of America by Donald L. Miller