By Steve Slavin

Likelihood is is the 1st e-book to make data obtainable to each person, despite how a lot math you bear in mind from tuition.

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**Example text**

In+ I) is some (n+1]-multi some more notations : index and n a function on W we w r i t e alil u ~lu= iI x1 i 3 s ... " xs a ms+l i Ys+l "'" ~ n 9 Yn 3 ln+1 If X = (X 1 ..... Xn,Xn+ 1) is some vector field on W we w r i t e 3ix = [3ix I ..... QiXn+1). differential Analogously, of order p of u and by DPX the one o~ X. We consider the f o l l o w i n g norms IIX[z)ll = P and IIoPX(z)II = : sup lil=p j=l ..... [z]l J n+l IlOPX[z)Cu I . . . Up)ll = Ilupll=l the norm at the right hand side of previous already i n t r o d u c e d We also take UXII F i n s l e r norm on the t a n g e n t b u n d l e is the TW.

Y being any r e a l i z a t i o n of ~ X we d e f i n e n-1 X = " --' [ ~ n i=O (Rij,yJ. 31 As X is R - i n v a r i a n t In particular, for X it is also R k - i n v a r i a n t if y is a c h a r a c t e r i s t i c then Rk(y] is also V k EIN. line a characteristic [of f i n i t e line type) [of f i n i t e type) for X. a) In case R is the m a p p i n g necessarily condition invariant under {y = O, x > characteristic lines X is c o m p l e t e l y {x,y) § k o of f i n i t e is of the t o j a s i e w i c z as {y = O, x < O} are type.

Using the h Whitney extension Pi theorem [I] we may consider a diffeomorphism defined on some neighbourhood h]V' = h and j~ (h-I) Pi Pi As allready remarked C~ diffeomorphism [~ o h[x] = Hor V' of 9A inside A satisfying = 0 along 3A. this h may be blown down to give the desired H :[~2,0) ~ d e f i n e d V x E on @(V') point taking in consideration = 0. serves for all characteristic however that if in some singular [resp. singular are) for ~ on orbits arrive that the associated away from 0. = V with j~ [H-I)[0) V'].