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Extra resources for Algebraic Topology, Aarhus 1978
Hint: Consider a holomorphic extension F W ! ) 4. Constant-coefficient linear systems; exponentiation of matrices Let A be an n n matrix, real or complex. 3) 1 k X t k A : kŠ kD0 We will establish estimates implying the convergence of this infinite series for all real t, indeed for all complex t. 1). 3), we need the notion of the norm of a matrix. This is a special case of results discussed in Appendix A, Functional Analysis. u; v/j Ä kuk kvk; 4. u; v/ D u1 v 1 C Cun v n . 8). u; v/j Ä t 2 kuk2 C t 2 kvk2 ; for any t > 0.
0/ D x on a parameter , assuming F is smooth jointly in ; y. t; ; x/ in . ; x/. As one special case, let F . y/. t0 ; ; x/ D y. t; x/: It is also true that if F is analytic, then one has the analytic dependence of solutions on parameters, especially on t, so that power-series techniques work in that case. One approach to the proof of this is given in the exercises below, and another at the end of 9. Exercises 1. Let be open in R2n , identified with C n , via z D x C iy. Let X W ! x; y/ are realvalued.
0/k; for t 0: Proof. 1). s/ ds: 0 This identity is known as Duhamel’s principle. 28 1. 3. 0/ ; where there are n factors on the right. Proof. To prove this at t D T , divide the interval Œ0; T into n equal parts. j C 1/T =n ; requiring continuity across each endpoint of these intervals. t/k; ın ! 0 as n ! t/k Ä CT ky0 k on Œ0; T . t/. t/k Ä K T CT ın ky0 k ! 0 as n ! T / ! T / as n ! 1. 9). Exercises 1. t/. X/t . h2 /:) 2. 1). t/=2. 0/. s/ ds: 0 Prove this. t/. 3. Generalize the Gronwall inequality of Exercise 2 as follows.