By Ian Stewart

Symmetry is without doubt one of the strongest and wide-ranging of mathematical rules with contemporary paintings on symmetry breaking, while symmetrical styles are somewhat altered or corrupted, throwing up a rare variety of functions and ordinary examples, from the stripes on a tiger's again to the territorial styles of fish and to the constitution of viruses. This e-book appears on the functions of symmetry and symmetry breaking to matters as varied as climate styles, crystal buildings, the buckling of beams and particle physics.

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G( w, k) e J ( w-kV) t . --- - kzo dk dw e J ( 21T)Z For an impuls e excitation i n time and spac e , s(w, k ) If we define a new fr equency variable WI = W - kV ( 2. 2 7 ) = f(w) g (k) = 1. ( 2 . 16: ( 2 . 29) whe r e 26 +OO Amplifying Wave s and Ab s olute Ins tability = J -00 -j kzo dk G I{WI , k) e 2rr ( 2 . 3l ) The computation of the r e spon s e a s t .... 3, except that w i s r epla c e d by wI. We now note that if V i s s et equal to Vo (Equation 2 . 26 ) , the point k = ko and w = wro - jero i s a saddl e point of wI (k) , a double root of k at that WI, b e cau s e dwl dk = dw - Vo dk at k = ko ( 2 .

2 . 6 Di s cus s ion 2 . 1 Gr oup Velo city o f Pr opagating Wave s . The c r iter ion on am ­ plifying and evane s c ent wave s fixe s the " s en s e o f caus alityll of a wave with a complex wave numb e r k; that i s , it deter mine s the dir e ction in which the s ignal is tran s fe r r e d when the wave is excite d at s ome fixed r eference plane . 1£ the c r iter ion i s applie d to pur e propagating wave s , some inte r e s ting facts about the concept of gr oup velo city b e ­ come evident .

C omplex k for real w (wpp > wp s /2) . plifying wave s and which one s co r r e spond to evane s c ent wav e s ( r e ­ aliz ing that the analys i s in S e c tion 2 . 4 ha s proved that amplifying wave s must exi s t in thi s cas e ) , we might p e r form numerical com­ putations on a particular ca s e . It i s p e rhap s mor e in structive , however , to u s e the s omewhat intr icate but more generally us e ­ ful ar gument now to be outlined . 45 Example s We shall show that the two roots of k that have ki - :1: 00 a s w - wpp a r e amplifying wave s ; that i s , there i s an infinite ampli­ fication rate pr edicted by this model in both the +z and -z dir e c ­ tion s when Wpp > Wp s / 2 .

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