Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

Effects of straw and silicon soil amendments on some foliar and stem-base diseases in pot-grown winter wheat

Four foliar and two stem-base pathogens were inoculated onto wheat plants grown in different substrates in pot experiments. Soils from four different UK locations were each treated in three ways: (i) straw incorporated in the field at 10 t ha−1 several months previously; (ii) silicon fertilization at 100 mg L−1 during the experiment; and (iii) no amendments. A sand and vermiculite mix was used with and without silicon amendment. The silicon treatment increased plant silica concentrations in all experiments, but incorporating straw was not associated with raised plant silica concentrations. Blumeria graminis and Puccinia recondita were inoculated by shaking infected plants over the test plants, followed by suitable humid periods. The silicon treatment reduced powdery mildew (B. graminis) substantially in sand and vermiculite and in two of the soils, but there were no effects on the slight infection by brown rust (P. recondita). Phaeosphaeria nodorum and Mycosphaerella graminicola were inoculated as conidial suspensions. Leaf spot caused by P. nodorum was reduced in silicon-amended sand and vermiculite; soil was not tested. Symptoms of septoria leaf blotch caused by M. graminicola were reduced by silicon amendment in a severely infected sand and vermiculite experiment but not in soil or a slightly infected sand and vermiculite experiment. Oculimacula yallundae (eyespot) and Fusarium culmorum (brown foot rot) were inoculated as agar plugs on the stem base. Severity of O. yallundae was reduced by silicon amendment of two of the soils but not sand and vermiculite; brown foot rot symptoms caused by F. culmorum were unaffected by silicon amendment. The straw treatment reduced severity of powdery mildew but did not detectably affect the other pathogens. Both straw and silicon treatments appeared to increase plant resistance to all diseases only under high disease pressure.