Algebraic topology: Proc. conf. Goettingen 1984 by Larry Smith

By Larry Smith

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49 ) Then, the quantities i- will be the coordinates of a single infinity of points of Jtln -i, which shall denote by may speak of this assemblage of points v The ratio of : as a straight line. acquires all values as the M we We ^ ^ point (#*) moves along the line. Clearly such a line may also be defined 2 independent homogeneous linear equations between xlt xn by n . Let 8) (4 , . . 4 S) ) be a third point, which is . not on the line . M^ Then, the quantities xk = ^) + A 4*) + A 4 3) 8 2 > (fc = 1, 2, .

Quot; i 1? , , where ^ (26) whence According to (9) we shall therefore find P is ^- (27) Being a seminvariant, 2 not changed by any transformation = ly y affecting only the dependent variable. we make successively the y Pi is According two transformations = ly, to (6) and (26), if x=*l(x), changed into W If A-FLA + t + -T-V (f>0\ i ^ i 1 1 Suppose that (1) has been reduced to its semi -canonical form, p : 0. Then, as (28) shows, pt will be zero, if and only if so that = CANONICAL FORM OF THE DIFFERENTIAL EQUATION 4.

2), must be the canonical form of an invariant of (1). For, although (1) can be reduced to any one of oo 4 different canonical forms, this totality of canonical forms is the same for any . equation equivalent to (1). A function of the coefficients of the canonical form, which remains unaltered by the transformations (32), has therefore the same significance for (1) as for any equation equi i. e. it is the canonical form of an invariant. To find the canonical form of the invariants of (1) is, therefore, same as to find the invariants of an equation in its canonical form valent to (1), the under the transformations (32).

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