# 101 short cuts in math anyone can do by Rockmaker G.

By Rockmaker G.

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La science et l'hypothese

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Handbook of Means and Their Inequalities (Mathematics and Its Applications)

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X,, (A) (32) having the degrees \$1 5 82 5 S ep- (33) In general, a fundamental series of solutions is not uniquely determined (to within scalar factors) by the pencil A + AB. However, two distinct fundamental series of solutions always have one and the same series of degrees 61, E2, ... , ea. For let us consider in addition to (32) another funda- mental series of solutions al(1), 22(A), ... with the degrees El, 82, .... Suppose that in (33) and similarly, in the series El, E2, , Obviously, el = El.

0 0 ... 1 1 0 with V(") = . , (A+A,)Met) +H(a)), we take the strictly equivalent block A,,), ... , ZAa)l (55) where Z,ZI)= V(a) [(A + Aj) E(a) + H(a)J 0 0 . . A+Ai;+ 0 1 I1 A + A, . (i =1, 2, jiA+Ai 1 . . , t). 11 in (57) by a transformation of the variables x = Tz (I T In the mission edition the author stated that propo,itions analogous to Theorems ti and 7 hold for hermitinn forms. A. I. Ma)'rev has pointed out to the author that this is not the ease. As regards, singular pencils of lie ratitimi norms, we 1197 111.

T) is expressed linearly (with constant coefficients) in terms of the functions fl(t), . . , 2. The preceding analysis shows that: In general, for the consistency of the system (58) certain well-defined linear dependence relations (with constant coefficients) must hold among the right-hand sides of the equations and the derivatives of these right-hand sides. If these relations are satisfied, then the general solution of the system contains both arbitrary constants and arbitrary functions linearly.