# Operator algebras and topology by Schick T. By Schick T.

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Example text

H. L INH AND V. M EHRMANN, Lyapunov, Bohl and Sacker-Sell spectral intervals for differential-algebraic equations, J. Dynam. Differential Equations, 21 (2009), pp. 153–194.  S. E. M ATTSSON AND G. S ÖDERLIND, Index reduction in differential-algebraic equations using dummy derivatives, SIAM J. Sci. Statist. , 14 (1993), pp. 677–692.  C. C. PANTELIDES, The consistent initialization of differential-algebraic systems, SIAM J. Sci. Statist. , 9 (1988), pp. 213–231.  J. W. P OLDERMAN AND J.

Z (μ+1) = r on Lμ . 3. ,z (μ) = v on Lμ , where the corank is the dimension of the corange and the convention is used that corank of F−1;z is 0. 4. ,z (μ+1) = 0, rank Z 2T Fμ;z = a, and Z 2T Fμ;z T2 = 0 on Lμ . 5. We have rank Fz˙ T2 = d = L − a − v on Lμ such that there exists a smooth full rank matrix function Z 1 of size N × d satisfying rank Z 1T Fz˙ T2 = d. Note that not every system will satisfy Hypothesis 1 but many of those from applications do. In order to reduce the computational costs, μ should be chosen as small as possible.

9) along a function x ∗ ∈ C 1D (I, Rm ) with graph in G is also an index-one DAE. 1) has index one on G, where D(t)+ denotes the Moore–Penrose inverse of D(t). Proof. The first part is a direct consequence of the definition and the second one follows from continuity arguments. 13) and consider the solvability of initial value problems (IVPs). 5. 1) have a properly stated leading term, t0 ∈ I ⊆ I f , and let x ∗ ∈ C 1D (I, Rm ) have values in D f . 12). 13) is uniquely solvable. Proof. Here we drop the argument t of the functions.