Some one-dimensional nature of the co- _3203

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Some one-dimensional nature of the combined

A)-A, by one, there are (A) a (A), P, Pq ~ (A) _A, it is an involution. Theorem 9: Non-identity of the projective transformation is necessary and sufficient condition for cooperation is the existence of pairs of elements should be interchangeable. Proof: Necessity is clear. Sufficient proof of the following: Let the projective transformation of non-identical 'p under a pair of elements of A and A should be commutative, eN, (A)-A, (A)-A; p-~ P' is any projective transformation of the corresponding elements, ie (P) = P,, 36 Order () =, by the projective transformation so that cross-ratio corresponding to the four elements equal to shows: (AA, pp ') = (AA, P,), then the cross ratio of Nature may: (AA, pp ') = (AA, JP) and thus, it is an involution. Theorem 10: The projective transformation of non-identical as the necessary and sufficient condition for cooperation is to reconcile the separate elements of any pair to be two self-corresponding elements. Proof: Necessity: From Theorem 2, on the combined two different elements may wish to set from the corresponding S,, and. s be elements of any pair of dead and P,, due to a projective transformation together, thereby preserving the cross ratio unchanged, eP (S,[link widoczny dla zalogowanych], 5) = (5, SP, P), the nature of the cross ratio was used: (5 , S:, pp ') = l, south or (Sfi ~, PP') ', which (Sfi ~, PP') = or (5,52, pp ') =- 1, ·. '5,, 5P, four different points, (S, Spp ') ≠ J,. *. (5, S2, pp ') =- 1. Sufficiency: If (5, Spp ') = a J, then (. S,. S, P) = a J 『, that should be elements of any pair of p-~ p' are interchangeable, so the involution . Decomposition Theorem 5 on United 11: The involution can be decomposed into the product of the involution (not only). Proof: According to Theorem 3: involution can always be written: D constant), would result in,: -k, beta 0 = 2 ', the following can be derived from the above inference: inference: two hyperbolic involution can be decomposed into a linear elliptic and hyperbolic involution involution of the plot, elliptic involution can be decomposed quadratic and linear hyperbolic hyperbolic involution involution of the plot, linear hyperbolic involution can be decomposed into a quadratic hyperbolic and elliptic involution involution of the plot. Theorem 12: Any projective transformation , if not on the joint, it can be expressed as the product of two pairs together. Proof: Let delete shadow transform ① if (identical transformation), the involution j, so that = ② If J, located l_h,, - and ∞ ≠, then the plant's total non-involutive Zhao capture the first 40. De-prop of a joint nature of Bute Terrace 6 transcripts of a number of projective transformation, and let {} a {, / x, v} as g ( ) = g () = if g is a projective transformation is co-constructed ,): Smile

g ()=,)=)).·. parlous involution, and 'f = l'f = g'g. (, g.: g'T strider projective transformation that the product of two pairs of co-6 on the involution of the product is necessary and sufficient conditions on the co ... Theorem 13: Suppose on the involution: J; aul ~ +6 (u + u) + a = O (ad-b. ≠ D). uu +6 (u + u) + d = D (oral d a b. ≠ D), Tony 0p2J is necessary and sufficient condition for cooperation is the I two,, I_266l - mouth l Proof: Let,: u-u: mu (ab-abuu + (port d-bbu + (66 a adu + (6d-bd) = D is: = (ad'-bb ') u + Li (bd'-b'd) so know by Theorem 5::, for the purpose of together is a necessary and sufficient conditions for a 66 (port, b-bb), order was: ldJ + a'b = 2bb ', $ P: la ... all [
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