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By Charles G. Moore

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R. r. r. r. r. r. r. 6/ 14 2 II E6. r. 1C4 C 10 III E6. r. r. r. r. 1 C 14 I E7. r. 1 C 26 Let us choose, for convenience, a basis of coordinates in the moduli space such that the first nf components of r coincide with the ' ˛ , the others being ' k , that is ˛ D ' ˛ , k D ' k . We can move along the ˛ direction through the action of isometries in G0 . We shall consider infinitesimal isometries in G0 whose effect is to shift the ˛-scalars only: Black Holes and First Order Flows in Supergravity g0 2 G0 W r !

K /. Consider now the implications of the G0 -invariance of W , as expressed by (53). 6/-invariance corresponds to the A D u component of the equation, and implies O L1u k . k /V 1 k k / kO . @W D 0: @ k (57) The invariance of W under G0 =H0 -transformations, on the other hand, implies, using (51) and (50): 0 D L1a . k D L1a b . k D L1a b . k b /V 1 r b Ä / V /V 1 1 b b @W D L1a b . @ r @W @ @W @ V ) 1 b k Ä / V 0 u 1 b @W CV @ O L1u k V @W D 0; @ ˛ 1 k kO 1 k b @W @ k @W @ k (58) where we have used (57) and the property that the block L1a b .

Rpjq / ). 2 Super Lie Groups Super Lie groups (SLG) are, by definition, group objects in the category of super manifolds. This means that morphisms , i , and e are defined satisfying the usual commutative diagrams for multiplication, inverse, and unit respectively. From this, e with a Lie group it follows easily that the reduced morphisms e;e i , and e e endow G e is called the reduced (Lie) group associated with G. G e acts in a natural structure. G way on G. G `g WD ı hg; O 11G i W G ! G 48 C.

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An Introduction to continued fractions by Charles G. Moore

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