mathpix

1 Please use the Snipping Tool ( ) on one single equation at a time! Performance is highest when the images are well zoomed in. 0. 1 (x )2. f (x) = e 2 2. 2 x 1. L(q) = Eq( ) [log p( )] + Eq( ) [log p( )] + Eq(z)q( ) [log p(z| )]. 2.. X p(x). H(Y |X) = p(x, y) log x X ,y Y. p(x, y). 3. ct0.. 0 0 ct x0 0 0 x . 0 = . y 0 0 1 0 y . 0. z 0 0 0 1 z 4. p B = D pq q D pq q R . pq q , Ap = 0. 5.. se2 F (1 F ) (1 + F )(1 + Av F ) (2 F )(1 + A 1 v 1 F ). Z. (I) dv = . 12 2 |m| 1 v v 1 Av F + 2 + A 1 v 1 F. 6.. 2 1 x Y 1 exp( 2 (n + 1)). (x) = [1 e ]. n=0. 1 exp( 2 (x + n)). 7.. 1 0 0 0. 0 cos r sin r 0 . Tx ( r ) =.

2 0 sin r cos r . 0 . 0 0 0 1. 8. Wb = n( b )0 V b ( b ). 9. (t) (t) (t). (t) (t). j f (xi ; j , j ). Tj,i := P(Zi = j|Xi = xi ; ) = (t) (t) (t) (t) (t) (t). 1 f (xi ; 1 , 1 ) + 2 f (xi ; 2 , 2 ). 1. 10. (D )2 D D 1 . ( 2. 2 ) + (1 ) (1 ) /2 F 2 = T X. 4 4 2. 11. 0. dk eik( ). Z.. hvI,AA ( )vJ,B B . ( 0 )i IJ AB A B . = 2 2 2. IJ AB A B ( 0 ). 2 k + r / . 12. (b) (a). K (bz)/K (az). a (az, bz) = (a) (b) (b) (a). K (az)I (bz) K (bz)I (az). 13. Z Z. S d 4. xns mv 2 ` t(ns /m) d2 xr 2 (` t/`2P ) log(R0 /a0 ) ` t/`2P. 14. Z ~2 c0 + + c2 + + . H = dr +. m m + m m + F m m F m m m m 2MB 2 m m1 1 2 m1 m2 1 4 2 3 3 4.

3 15. d2 ( ) 2 1 d ( ) q + r(1 2 )2. + ( ) = 0. d 2 ( 1) d 2 ( 1)2. 16. Dtt Dtb (Q 1 )bc Dct Dtj Dtb (Q 1 )bc Dcj Dtb . D = det Dit Dib (Q 1 )bc Dct Dij Dib (Q 1 )bc Dcj Dib . 0 0 Qab 17. Z Z.. Z[A+ , , ] = D D exp{i d2 x[ i + A+ + + + + ]}. 18. P+ = |h+| i|2. = h+| i h+| i = h |+ih+| i = h |P+ | i 19. Z. sech udu = sin 1 (tanh u) or 2 tan 1 eu 2. 20. 1 1. P (e) = P (e|s0 ) + P (e|s1 ). 2 2. Z Z 1 . 1 1 2A T. = p(r|s0 )dr + p(r|s1 )dr 2 21 A T 2 . 21. Z 1. W (x, C) = P exp[i d (q A (x + ( )) + q i X i (x + ( )))]. 0. 22. 1 1. = h[A, B]} + h{ A, B}. 2 | {z } 2| {z }. purely imaginary purely real 23.. (0) (0) T (0) 2.

4 W w(0).. argmin w + w w w + 2. + (w). w 2. 3.