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In the past ** Andrei Khrennikov** has collaborated on articles with

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Highlights•The problem of interpretation of the wave function is analyzed in the ontic/epistemic framework.•The concrete ontic model, random field model is presented and compared with the psi-ontic/epistemic models.•The notions of superposition of pure quantum states are clarified by using the ontic model.

AbstractIn this work, questions about interpolation and approximation of a continuous function f:Zp→R(f:Zp→Qp) by functions of the form ∑k=1Nλk|x−xk|p are discussed. The theorem about uniform approximation of a continuous function f:Zp→R is proved. Nonexistence of such approximation for a Qp-valued function is shown.

AbstractWe prove that any linear operator with a kernel in a Gelfand–Shilov space is a composition of two operators with kernels in the same Gelfand–Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten–von Neumann properties for such operators.

Highlights•Quantum model for representation of states of the world, knowledge, and common knowledge was elaborated.•Quantum decision making: the possibility to agree on disagree — even with common prior.•Classical Aumann theorem can be recovered under mathematically nontrivial conditions of compatibility.

Highlights•Experimental validation of non-Bayesian updating in decision making.•Quantum probability model of concrete decision making problem.•Investigation of zero-priors trap in decision making.

Highlights •A quantum-like model of the process of decision making is presented.•The process of decision making is modeled as interaction with mental reservoir.•Applications to modeling of voters’ behavior and consumer’s persuasion are presented.

AbstractWe found the asymptotics, p→∞, for the number of cycles for iteration of monomial functions in the fields of p-adic numbers. This asymptotics is closely connected with classical results on the distribution of prime numbers.

AbstractThe scheme of a unified Darwinian evolutionary theory for physical and biological systems is described. Every physical system is methodologically endowed with a classical information processor, which turns every system into an agent being also susceptible to evolution. Biological systems retain this structure as natural extensions of physical systems from which they are built up. Optimization of information flows turns out to be the key element to study the possible emergence of quantum behavior and the unified Darwinian description of physical and biological systems. The Darwinian natural selection scheme is completed by the Lamarckian component in the form of the anticipation of states of surrounding bio-physical systems.

AbstractDifferentiation is a universal process found in various phenomena of nature. As seen in the example of cell differentiation, the creation diversity on individual's character is caused by environmental interactions. In this paper, we try to explain its mechanism, which has been discussed mainly in Biology, by using the formalism of quantum physics. Our approach known as quantum bioinformatics shows that the temporal change of statistical state called decoherence fits to describe non-local phenomena like differentiation.

Applying the rules of quantum mechanics to psychology and economics can help us understand the brain and how people make decisions, two physicists argue

AbstractWe present a contextualist statistical realistic model for quantum-like representations in physics, cognitive science, and psychology. We apply this model to describe cognitive experiments to check quantum-like structures of mental processes. The crucial role is played by interference of probabilities for mental observables. Recently one such experiment based on recognition of images was performed. This experiment confirmed our prediction on the quantum-like behavior of mind. In our approach “quantumness of mind” has no direct relation to the fact that the brain (as any physical body) is composed of quantum particles. We invented a new terminology “quantum-like (QL) mind.” Cognitive QL-behavior is characterized by a nonzero coefficient of interference λ. This coefficient can be found on the basis of statistical data. There are predicted not only cosθ-interference of probabilities, but also hyperbolic coshθ-interference. This interference was never observed for physical systems, but we could not exclude this possibility for cognitive systems. We propose a model of brain functioning as a QL-computer (there is a discussion on the difference between quantum and QL computers).

AbstractWe propose a mathematical model of the memory retrieval process based on dynamical systems over a metric space of p-adic numbers representing a configuration ‘space of ideas’ in which two ideas are close if they have a sufficiently long common root. Our aim is to suggest a new way of conceptualizing human memory retrieval that might be useful for simulation purposes or for the construction of artificial intelligence devices, as well as for a deeper understanding of the process itself. The dynamical system is assumed to be located in a blackbox processing unit (the ‘subconscious’) and controlled by an interface control unit (the ‘conscious’) that fixes parameters in the dynamical system and starts its iteration by sending an initial generating idea to it. We show that even simple p-adic dynamical systems admit behavioral scenarios that could explain some of the essential features of the human memory retrieval process.

AbstractIn the present paper we consider countable state of p-adic Potts model on the tree. Under some condition on weights we establish uniqueness of Gibbs measures for the model. Note that this condition does not depend on values of the prime p. An analogous fact is not true when the number of spins is finite.

AbstractThis Letter is an attempt to go beyond QM. In our approach density operators of QM can be represented as covariance operators of classical random fields. Born's rule can be obtained from measurement theory for classical random field under the assumption that the probability of detection of field is proportional to the power of this field.

AbstractWe consider dynamics of financial markets as dynamics of expectations and discuss such a dynamics from the point of view of phenomenological thermodynamics. We describe a financial Carnot cycle and the financial analog of a heat machine. We see, that while in physics a perpetuum mobile is absolutely impossible, in economics such mobile may exist under some conditions.

AbstractIn this paper we demonstrate that the probabilistic quantum-like (QL) behavior–the Born’s rule, interference of probabilities, violation of Bell’s inequality, representation of variables by in general noncommutative self-adjoint operators, Schrödinger’s dynamics–can be exhibited not only by processes in the micro world, but also in economics. In our approach the QL-behavior is induced not by properties of systems. Here systems (commodities) are macroscopic. They could not be superpositions of two different states. In our approach the QL-behavior of economical statistics is a consequence of the organization of the process of production as well as investments. In particular, Hamiltonian (“financial energy”) is determined by rate of return.

AbstractIn cognitive psychology, some experiments for games were reported, and they demonstrated that real players did not use the “rational strategy” provided by classical game theory and based on the notion of the Nasch equilibrium. This psychological phenomenon was called the disjunction effect. Recently, we proposed a model of decision making which can explain this effect (“irrationality” of players) Asano et al. (2010, 2011) [23], [24]. Our model is based on the mathematical formalism of quantum mechanics, because psychological fluctuations inducing the irrationality are formally represented as quantum fluctuations Asano et al. (2011) [55]. In this paper, we reconsider the process of quantum-like decision-making more closely and redefine it as a well-defined quantum dynamics by using the concept of lifting channel, which is an important concept in quantum information theory. We also present numerical simulation for this quantum-like mental dynamics. It is non-Markovian by its nature. Stabilization to the steady state solution (determining subjective probabilities for decision making) is based on the collective effect of mental fluctuations collected in the working memory of a decision maker.

AbstractWe show that the cornerstone of QM—Born's rule—can be obtained from “subquantum detection theory” (SDT). This detection theory is based on purely classical field model for “quantum particles”. Electrons, photons, etc., neutrons are represented as classical waves fluctuating on a very fine time scale, constituting subquantum random fields. A crucial point is that SDT not only reproduce the quantum probabilities for detection, but it provides a possibility to go beyond QM, cf. [Th.M. Nieuwenhuizen, B. Mehmani, V. Spicka, M.J. Aghdami, A.A.Yu. Khrennikov (Eds.), Beyond the Quantum, WSP, Singapore, 2007]. We extend SDT by modelling measurements with the aid of detectors which are sensitive to nonquadratic influence of the subquantum random field. The difference between probabilistic predictions of QM and SDT is estimated.

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