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Within the framework of a realistic multiband p-d model, we derived an effective Hamiltonian to describe the exchange interaction effects near the spin crossover in magnetic Mott-Hubbard insulators under pressure. It is shown that the single-ion mechanism of spin crossover under the change in the crystal field does not lead to a thermodynamic phase transition; however, at T = 0 a quantum phase transition appears. It has been found that the cooperativity leads to a modification of the quantum phase transition to a first-order phase transition and the appearance of metastable states of the system. The pressure-temperature phase diagram has been obtained to describe the magnetization and high-spin population near the spin crossover of Mott's insulators with d 6 ions.

Diffusion–reaction models are used to describe development processes in the framework of morphogen theory. The images of the concentration fields for the subset of the interacting morphogens are available. In order to interpret this data in terms of the model parameters, the inverse source problem is stated. The sensitivity operator, composed of the independent adjoint problem solutions ensemble, allows transforming the inverse problem to the family of nonlinear ill-posed operator equations. The equations are solved with the Newton–Kantorovich-type algorithm. The approach is applied to the morphogen synthesis region identification problem for the model of regulation of the renewing zone size in biological tissue.

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