By Graham M. L. Gladwell, Antonino Morassi

The papers during this quantity current an outline of the overall features and useful purposes of dynamic inverse tools, in the course of the interplay of numerous themes, starting from classical and complex inverse difficulties in vibration, isospectral platforms, dynamic tools for structural id, energetic vibration keep an eye on and harm detection, imaging shear stiffness in organic tissues, wave propagation, to computational and experimental points proper for engineering difficulties.

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**Example text**

Let A(t) be the solution of dA = [A, S] , dt A(0) = A0 ∈ S, (23) then (ii) A(t) = Q(t)A0 QT (t), where Q is the solution of (22). Proof (i) dQT d T T dQ dt (Q Q) = dt Q + Q dt , = −QT ST Q − QT SQ, = −QT (ST + S)Q = 0. Therefore QT (t)Q(t) = constant = QT (0)Q(0) = I : Q(t) is an orthogonal matrix. (ii) d (QT AQ) = −(QT ST )AQ + QT (AS − SA)Q + QT A(−SQ) = 0. L. Gladwell Hence QT AQ = const = QT (0)A0 Q(0) = A0 , and A(t) = QA0 QT . , ˙ = AS − SA, A (24) where ST = −S is an orthogonal matrix, is an isospectral ﬂow, called a Toda Flow.

An alternative approach is based on functional theoretical methods and speciﬁc properties of the Sturm-Liouville operators, see Titchmarsh (1962). As we will see in the next sections, the asymptotic behavior of eigenpairs plays an important role in inverse spectral theory. With reference to the Dirichlet problem (6)-(7), let us consider the solution y = y(x, λ) to the initial value problem ⎧ (13) ⎪ ⎨ y (x) + λy(x) = q(x)y(x), in (0, 1), y(0) = 0, (14) ⎪ ⎩ y (0) = 1, (15) for some (possibly complex) number λ and for a (possibly complex-valued) L2 potential q.

Linear Algebra and Its Applications, 17:15–51, 1977. L. Gladwell. Inverse Problems in Vibration. Kluwer Academic Publishers, 2004. L. Gladwell. Minimal mass solutions to inverse eigenvalue problems. Inverse Problems, 22:539–551, 2006. L. Gladwell. The inverse problem for the vibrating beam. Proceedings of the Royal Society of London - Series A, 393:277–295, 1984. L. L. Gladwell. Total positivity and the QR algorithm. Linear Algebra and Its Applications, 271:257–272, 1998. L. Gladwell and O. Rojo.