An effective economy requires prompt prevention of misconduct of legal entities. With the ever-increasing transaction rate, an important part of this work is finding market collusions based on statistics of electronic traces. We report a solution to this problem based on a quantum-theoretical approach to behavioral modeling. In particular, cognitive states of economic subjects are represented by complex-valued vectors in space formed by the basis of decision alternatives, while decision probabilities are defined by projections of these states to the corresponding directions. Coordination of multilateral behavior then corresponds to entanglement of the joint cognitive state, measured by standard metrics of quantum theory. A high score of these metrics indicates the likelihood of collusion between the considered subjects. The resulting method for collusion discovery was tested with open data on the participation of legal entities in public procurement between 2015 and 2020 available at the federal portal https://zakupki.gov.ru. Quantum models are built for about 80 thousand unique pairs and 10 million unique triples of agents in the obtained dataset. The reliability of collusion discovery was defined by comparison with open data of Federal antimonopoly service available at https://br.fas.gov.ru. The achieved performance allows the discovery of about one-half of known pairwise collusions with a reliability of more than 50%, which is comparable with detection based on classical correlation and mutual information. For three-sided behavior, in contrast, the quantum model is practically the only available option since classical measures are typically limited to the bilateral case. Half of such collusions are detected with a reliability of 40%. The obtained results indicate the efficiency of the quantum-probabilistic approach to modeling economic behavior. The developed metrics can be used as informative features in analytic systems and algorithms of machine learning for this field.

Abstract. "On-surface" synthesis of large molecules and assemblies is a great challenge in molecular electronics, biomedical devices, sensors, energy harnessing and catalysis. Usually, plane surfaces are used for these purposes. We study a possibility of macro molecules formation on the surface of nanotube and inside it. It can be used for synthesis of helix molecules, particularly, organic and biological. Nanotubes of different structures were considered. Point-like approximation is used for atoms. The Lennard-Jones 6-12 potential is taken as the interaction potential. Possibilities of helix molecules and ring-like molecules formations were shown. A hypothesis of such processes influence on the organic molecules formation during the early Earth history is suggested and discussed.

In this paper, a model of 3D Helmholtz resonator with two close point-like windows is considered. The Dirichlet condition is assumed at the boundary. The model is based on the theory of self-adjoint extensions of symmetric operators in Pontryagin space. The model is explicitly solvable and allows one to obtain the equation for resonances (quasi-eigenvalues) in an explicit form. A proper choice of the model parameter leads to the coincidence of the model solution with the main term of the asymptotics (in the window width) of the realistic solution, corresponding to small windows. A regularization is suggested to obtain a realistic limiting result for two merging windows.