causs # & /) & = k'"( 1+ (2n)!/o!') rod p, (16v.),p=4n+l,n=l-2500

provtbility p e 3n:+3n+l y'= x1k Cruss P-n1l y' = x,+k

2nl.2r+y t -y, -xx-21+ y t+ y - lt -

y1 = z1+ (x1-2) z+ I (rr z"+ t, n = ^ 6

f(x) = D., ax = a,+a,x+tux,+... +r,ix',+&,,x"1 f(x) [ql (x) = ( l-x'') (f,,.",."/ ( l-q'x) +a",) eigen value eigen vectors [-r, l, l, r, o] [o,o.o.o. r] [0, -r, o l, o] cigcn valuc [-r, r, r, t, o] [o,o,o,o,r]

value projector ( l+tx-xl-rx!) /4 focl, sensc hop, will family of curvas polynomial Euler,'= 1(x,+qx+r) y'=x(x-l)(x{) F (tn,tn,t,x) P ' 4n+l p - 4n-l

I I r.. I 0 I p p' d' o t p'... p'..p'''" o I pp'p:'rr'.. pd,fr 0 I00..0-r

0, l, 0, 0, 0, x',0,0,0, x',0,0 l,0,0, 0, 0, 0, l, 0,0,0,0,0 0, l, 0, 0, 0, 0, 0, x,0,0,0,0 0, 0, r, 0, 0, 0, 0, 0, x2, 0, 0,0 0, 0. 0, I, 0, 0, 0, 0, 0, xr, 0,0 0, 0, 0, 0, r, 0, 0, 0, 0, 0, xr,0 0, 0, 0, 0, 0, l, 0, 0, 0, 0, 0, x' l,0,0, 0,0,0. x'. 0. 0, 0, 0. 0 0, l, 0, 0, 0, 0, 0, x',0,0,0,0 0, 0. l, 0, 0, 0, 0, 0, xr, 0, 0,0 0, 0, 0, l, 0, 0, 0, 0, 0, x',0,0 0, 0, 0, 0, r, 0, 0, 0, o, 0, x'd, 0 0, 0, 0, 0, 0, l, 0, 0, 0, 0, 0, x" r, 0, 0, 0, 0, 0, l, 0, 0, 0, 0, 0 0, l, 0, 0, 0, 0, 0, x',0, o,0,0 0, 0, l, 0, 0, 0, 0, 0, x', 0, 0.0 I, 0, 0, 0, 0, 0, x', 0, O, 0, 0,0 0, l, 0, 0, 0, 0. 0, xr,0, 0, 0,0 0, 0, l. 0, 0, 0. 0,0, xro,0.0.0 0, 0, 0, l,0,0,0,0.0. r,0,0 0, 0, 0, 0, t, 0, 0, 0, 0, 0, x:,0 0, 0, 0, 0, 0, l, 0, 0, 0. 0,0. x' 0, 0, 0, t. 0, 0, 0, 0, 0, x'. 0, o 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, x" 0 0, 0, 0, 0, 0, r, 0, 0, 0, 0, 0, x' 0, r,0, 0,0, x',0,0,0. x'. 0.0 0, 0, l,0.0.0, r,0,0,0, t,0 0, 0, l, 0,0,0, x',0,0,0, xr,0 0, 0, l, 0, 0,0, x',0,0,0, xr,0 0, 0, 0, t, 0,0,0, l, 0, 0, 0, t 0, 0, 0, 1, 0, 0, 0, x',0,0,0, xl 0, 0, 0, l, 0, 0, 0, x',0,0,0, x' B,- l, 0, 0, 0, t, 0, 0, 0, l, 0,0,0 0, l, 0, 0, 0, xr,0,0, O, x',0,0 l, 0, 0, 0, x', 0,0, 0, xt, O, O, O 0, l, 0, 0, 0, x',0,0,0, 1.0,0 l, 0, 0, 0, xr,0, 0, 0, xr. 0. O. O 0, 1, 0, 0, 0, x'",0.0.0, x'.0,0 0, 0, l,0,0,0, r,0,0,0, l,0

0, 0, 0, 1,0,0,0, x',0,0,0, x' 0, 0, 1, 0, 0, 0, x',0,0,0, x',0 0, 0, 0, l, 0, 0, 0, x",0,0,0, I 0, 0, l, 0, 0, 0, x', 0, 0, 0, x', 0 0, 0, 0, l, 0, 0, 0, xro,0,0,0, x' c,= r, 0, 0, 0, 0, 0, l,0,0,0,0,0 0, I,0,0,0,0,0, x', 0, 0, 0, 0 l, 0, 0, 0, 0, 0, x!,0,0,0,0,0 0, 1, 0, 0, 0, 0, 0, x",0,0,0,0 0, 0, t, 0, 0, 0, 0, 0, 1, 0,0,0 0, 0, 0, r, 0,0, 0,0, 0, xr, o, 0 0, 0, r, 0, 0, 0, 0, 0, x" 0,0,0 0, 0, 0, 1,0,0,0,0,0, xt,0,0 0, 0, 0, 0, 1, 0, 0 0,0, 0, 1, 0 0, 0, 0, 0, 0, t, 0, 0, 0. 0. 0. x' 0, 0, 0, 0, t, 0, 0,0, 0, 0, x" 0 0, 0, 0, 0, 0, t, 0, 0, 0, 0, 0, x' Cr= l, 0, 0, 0, 0, 0, r, 0, 0, 0, 0, 0 l, 0, 0, 0, 0, 0, x',0, 0, 0, 0, 0 0, t, 0, 0, 0,0, 0, t,0, 0, 0, 0 0, l, 0, 0, 0, 0, O, xd, 0,0, 0,0 0, 0, r, 0, 0, 0,0, 0, r, 0, 0, 0 0, 0, l, 0, 0, 0, 0, 0, x" 0,0,0 l, 0, 0, 0, l, 0, 0, 0, 1, 0, 0,0 0, l, 0, 0, O, x, 0, 0, 0, x1, O, O 0, 0, 1,0, 0, 0, x':, 0,0,0, x', 0 0, 0, 0, 1,0,0,0, x.,0,0,0, x' l, 0, 0, 0, x',0,0,0, x',0,0,0 0, l, 0, o, 0, x', 0, 0, o, x'", 0,0 0, 0, 1.0.0,0, x'.0,0,0, r,0 0, 0, 0, I, O, O, O, x', 0, 0, 0, x' l, 0, 0, 0, x',0, 0, 0, x', 0, 0, 0 0, l,0, 0,0, x", 0, 0, 0, x" 0, 0 0, 0, 0, r, 0, 0, 0, 0,0, t, 0, 0 0, 0, 0, l,0,0, 0, 0,0. x". 0. 0 0, 0, 0, 0,!,0, 0, 0, 0, 0. l. 0 0,0,0,0, l,0,0,0,0,0, x" 0 0, q 0, 0, 0, 1,0, 0, 0, 0, 0, I 0, 0, 0, 0, 0, t, 0.0, 0, 0, Q x'

Euclid-Vandcmonde factorization theorcn t-et V - (x"), and x'-l = 0, xtu (n. > I ) b ary r pr sentation of n 'lhen there cxisr degr e n natrix A,, A,,.,A' such thar A,A,.. Au * 'A,. 'A,'A, = V, non zero elcntcnls in each row and column ofa, are n,

ij = i(j,xsxaxl+ j,xax3+jrx3+j,) mod n t95,4,31 t85,4,31 tz,ss:l 192,s,31 iz,s,lfll