By Allan A MacRae
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Additional resources for An Introduction and response to the higher criticism of the Pentateuch
4 a d c b The square has 8 symmetry operations: doing nothing (I), clockwise rotation through 90◦ (R+ ), rotation through 180◦ (Rπ ), counterclockwise rotation through 90◦ (R− ), and flipping about the vertical axis a (R a ), the horizontal axis b (Rb ), or the diagonal axes c (Rc ) or d (Rd ). I R+ Rπ R− Ra Rb Rc Rd I I R+ Rπ R− Ra Rb Rc Rd R+ R+ Rπ R− I Rd Rc Ra Rb Rπ Rπ R− I R+ Rb Ra Rd Rc R− R− I R+ Rπ Rc Rd Rb Ra Ra Ra Rc Rb Rd I Rπ R+ R− Rb Rb Rd Ra Rc Rπ I R− R+ Rc Rc Rb Rd Ra R− R+ I Rπ Rd Rd Ra Rc Rb R+ R− Rπ I The group is not Abelian (for example, R+ R a = Rc , but R a R+ = Rd ).
Therefore, SO(n) is closed. If A ∈ SO(n), so is A−1 , and I ∈ SO(n); SO(n) is a group. 6 y' y A ay f q ax a x = a cos φ; ay = a sin φ a'y x' a'x x ( a = length of A) a x = a cos(φ − θ ) = a(cos φ cos θ + sin φ sin θ ) = cos θa x + sin θay ay = a sin(φ − θ ) = a(sin φ cos θ − cos φ sin θ ) = cos θay − sin θa x 40 4 Symmetries ∴ ax ay ˜ = RR = = ax . ay cos θ sin θ − sin θ cos θ cos θ − sin θ sin θ cos θ R= cos θ sin θ . − sin θ cos θ cos θ sin θ − sin θ cos θ (cos2 θ + sin2 θ ) (cos θ sin θ − sin θ cos θ ) (sin θ cos θ − cos θ sin θ ) (sin2 θ + cos2 θ ) = 1 0 0 1 so R is orthogonal.
24, probability of 21 h¯ is 1 √ (α + β) 2 2 1 = √ 2 2 1 √ +√ 5 5 2 1 2 √ −√ 5 5 2 = 9 . 10 = 1 . 17, probability of 12 h¯ is 1 √ 2 1 2i √ −√ 5 5 2 = 1 1 1 (1 − 2i )(1 + 2i ) = (1 + 4) = . 10 10 2 Probability of − 12 h¯ is 1 √ 2 1 2i √ +√ 5 5 2 = 1 . 2 45 (c) Probability of 21 h¯ is |α|2 = 1 4 ; probability of − 12 h¯ is | β|2 = . 20 (a) [σi , σj ] = σi σj − σj σi = δij + i But δij = δji , jik =− ijk . ijk σk − δji − i So [σi , σj ] = 2i jik σk ijk σk . , (iθσz )2 (iθσz )3 + +··· 2 3! θ4 θ3 θ5 θ2 + − · · · + iσz θ − + +··· = 1− 2 4!
An Introduction and response to the higher criticism of the Pentateuch by Allan A MacRae