In contrast to single-product pricing models, multi-product pricing models have been much less studied because of the complexity of multi-product demand functions. It is highly non-trivial to construct a multi-product demand function on the entire set of non-negative prices, not to mention approximating the real market demands to a desirable accuracy. Thus, many decision makers use incomplete demand functions which are defined only on a restricted domain, e.g. the set where all components of demand functions are non-negative. In the first part of this paper, we demonstrate the necessity of defining demand functions on the entire set of non-negative prices through some examples. Indeed, these examples show that incomplete demand functions may lead to inferior pricing models. Then we formulate a type of demand functions using a Nonlinear Complementarity Problem (NCP). We call it a Complementarity-Constrained Demand Function (CCDF). We will show that such demand functions possess certain desirable properties, such as monotonicity. In the second part of the paper, we consider an oligopolistic market, where producers/sellers are playing a non-cooperative game to determine the prices of their products. When a CCDF is incorporated into the best response problem of each producer/seller involved, it leads to a complementarity constrained pricing problem facing each producer/seller. Some basic properties of the pricing models are presented. In particular, we show that, under certain conditions, the complementarity constraints in this pricing model can be eliminated, which tremendously simplifies the computation and theoretical analysis.

We study an electrical conduction problem in biological tissues in the radiofrequency range, which is governed by an elliptic equation with memory. We prove the time exponential asymptotic stability of the solution. We provide in this way both a theoretical justification to the complex elliptic problem currently used in electrical impedance tomography and additional information on the structure of the complex coefficients appearing in the elliptic equation. Our approach relies on the fact that the elliptic equation with memory is the homogenisation limit of a sequence of problems for which we prove suitable uniform estimates.

Lie symmetry analysis is applied to study the non-linear rotating shallow-water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow-water equations can be transformed to the classical shallow-water model. The derived symmetries are used to generate new exact solutions of the rotating shallow-water equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.