Foteini Kariotou

An algebraic formula for the accelerated computation of the low frequency scattering coefficients

An explicit expression for the rapid computation of the low frequency scattering coefficients is presented. Although the demonstration of the methodology is restricted to the simple case of an acoustically soft spherical scatterer, the introduced main concept can be applied to a wide class of scattering problems. The procedure, based on algebraic calculations only, provides an analytic expression of the corresponding scattering coefficients in a straightforward fashion. The proposed algorithm is readily implemented, furnishing an efficient way for obtaining any order approximation of the total field in the exterior of the scatterer.

Mathematical Foundation of Electroencephalography

Electroencephalography (EEG) has evolved over the years to be one of the primary diagnostic technologies providing information concerning the dynamics of spontaneous and stimulated electrical brain activity. The core question of EEG is to acquire the precise location and strength of the sources inside the human brain by knowledge of an electrical potential measured on the scalp. But in what way is the source recovered? Leaving aside the biological mechanisms on the cellular level responsible for the recorded EEG signals, we pay attention to the mathematical aspects of the narrative. Our goal is to provide a brief and concise introduction of the mathematical terminology associated with the modality of EEG. We start from the very beginning, presenting step by step the mathematical formulation behind EEG in a simple and clear manner, keeping the mathematical notation to a minimum. Whilst we serve only the key relations for the described problems, we focus specifically on the limitations of each modelling approach. In this fashion, the reader can appreciate the beauty of the formulas presented and discover every single piece of information encoded within these formulas.