A First Course in Infinitesimal Calculus by Daniel A. Murray

By Daniel A. Murray

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In our consideration an important point is that estimates of this kind exist. In this case, choosing ae = max(asi,ae2) and assuming zero initial data: x(0) = 0 or 77(^(0) = 0, i = 0 , 1 , . . 14) L(x(t)) = L{Ax{t) + b£(t)) \ L(a(t)) = L(c*x(t)) ] PL{x(t)) = AL(x(t)) + bL(t(t)) L(a(t))=c'L(x(t)) \ j L(x(t))=-{A-pI)-1bL(S(t)) L(*(t))=c*L(x(t)) \ j L(a(t)) = L(

Ak(t). 3) has the following geometrical interpretation: to each of vectors aj(t) assign a curve in M n . 3) implies that all these curves cannot be placed on a linear subspace of dimension less than n. Let us also remark that a symmetric matrix K is said to be a positively definite, K > 0, if the corresponding quadratic form z*Kz is positively definite: z*Kz > 0 V z £ t " , z ^ 0 . 1. At first we note that condition 1') is simply another formulation of condition 1). It can easily be seen that condition 2') yields condition 2).

28) are satisfied. Here i = 0 , 1 , . . , n — 1. For proving this fact it is sufficient to show that the first inequality holds. The latter follows directly from estimates )dr < KOI < |e *(0)l + < /3ext\x(0)\+ f \eA^-^\m(T)\dT o < t A A 0. 28) imply the existence of p$ such that a(t)\ 0. 29) can be obtained by different methods, they can also be improved, and so on. In our consideration an important point is that estimates of this kind exist.

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