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Summary. Several aspects of the notion of virtual turning points are discussed; its background, its relevance to the bifurcation phenomena of a Stokes curve, its importance in the analysis of the Noumi-Yamada system (a particular higher order Painlevé equation) and a concrete recipe for locating them. Examples given here make it manifest that virtual turning points are indispensable in WKB analysis of higher order linear ordinary differential equations with a large parameter. IntroductionMicrolocal analysis and the exact WKB analysis are intimately related and they are often complementary. A typical example is the exact steepest descent, where a global version of the quantized Legendre transformation is given in terms of exact steepest descent paths. Here in

Consider a three dimensional piecewise homogeneous anisotropic elastic medium Ω which is a bounded domain consisting of a finite number of bounded subdomains D α , with each D α a homogeneous elastic medium. One typical example is a finite element model with elements with curvilinear interfaces for an ansiotropic elastic medium. Assuming the D α are known and Lipschitz, we are concerned with the uniqueness in the inverse boundary value problem of identifying the anisotropic elasticity tensor on Ω from a localized Dirichlet to Neumann map given on a part of the boundary ∂D α 0 ∩ ∂Ω of ∂Ω for a single α 0 , where ∂D α 0 denotes the boundary of D α 0 . If we can connect each D α to D α 0 by a chain of {D α i } n i=1 such that interfaces between adjacent regions contain a curved portion, we obtain global uniqueness for this inverse boundary value problem. If the D α are not known but are subanalytic subsets of R 3 with curved boundaries, then we also obtain global uniqueness.

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The goal of this work is to investigate the relation of the no-response approach to some other non-iterative reconstruction schemes. We will derive several equivalence statements and dependency results. For simplicity we consider the obstacle reconstruction problem from far field data.In particular, we investigate two versions of the no-response test (NRT) for the inverse scattering problem. The first version is the pure NRT, the second combines the NRT with a rangetest element. We show convergence for these two versions without any eigenvalue assumption about the scatterer.Second, we state the natural formulation of the probe method for far field data and reformulate the singular sources method. We show that these statements of the two methods coincide and they form one face of the first version of the no-response test. Third, we prove that the convergence of the linear sampling method implies the convergence of the second version of the no-response test. Precisely, we show that we can use the blowup sequence of the linear sampling method to create the blowup sequence of the second version of the no response test. Fourth, we show that the two versions of the no response method are equivalent with respect to their convergence. Thus, the convergence of the linear sampling method also implies the convergence of the pure no-response test.