In this paper we analyze the gradient blow-up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core-shell geometry. Assuming that the core and shell have circular boundaries that are nearly touching, we derive an asymptotic formula for the solution in terms of the single and double layer potentials with image line charges. We also deduce an integral formula with image line charges for the problem relating to two nearly touching separated conductors.Mathematics subject classification (MSC2000): 35J25, 73C40

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We consider the conductivity problem with a simply connected or multi-coated inclusion in two dimensions. The potential perturbation due to an inclusion admits a classical multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). The GPTs have been fundamental building blocks in conductivity inclusion problems. In this paper, we present a new concept of geometric multipole expansion and its expansion coefficients, named the Faber polynomial polarization tensors (FPTs), using the conformal mapping and the Faber polynomials associated with the inclusion. The proposed expansion leads us to a series solution method for a simply connected or multi-coated inclusion of general shape, while the classical expansion leads us to a series solution only for a single- or multilayer circular inclusion. We also provide matrix expressions for the FPTs using the Grunsky matrix of the inclusion. In particular, for the simply connected inclusion with extreme conductivity, the FPTs admit simple formulas in terms of the conformal mapping associated with the inclusion. As an application of the concept of the FPTs, we construct semi-neutral inclusions of general shape that show relatively negligible field perturbations for low-order polynomial loadings. These inclusions are of the multilayer structure whose material parameters are determined such that some coefficients of geometric multipole expansion vanish.

Real-time quantification of the patient's consciousness level during anesthesia is an important issue to avoid intraoperative awareness and post-operative side effects. A depth-of-anesthesia (DoA) monitoring method called Bispectral Index (BIS) is generally used for this purpose. However, BIS is known to be inaccurate at the transitory state, and also shows a critical time delay in quantifying the patient's consciousness level. This paper introduces a novel method to reduce the response time in the quantification process. This thesis develops a new index called HDoA by analyzing EEG using Hidden Markov Model. The proposed approach is composed by two steps, training and testing. In the training step, two HMM, awakened and anesthetized model are learned based on each training set. In the testing step, by evaluating the probability of producing the testing EEG from two models respectively, the index HDoA is derived. Since the evaluation of DoA using HMM is training based method, it have better performance with more training process. Experiments show that HDoA has a high correlation with BIS at a steady state, and outperforms BIS in two ways: (1) shorter delay time in transition state, and (2) higher Fisher Score. The validity of HDoA has been tested by 8 real clinical data.

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