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This work examines asteroid’s motion with orbital eccentricity e in the range (0.1, 0.2) across the two-body mean motion resonance (MMR) with Jupiter due to the Yarkovsky effect. We calculated time delays dtr caused by the resonance on the mobility of an asteroid with the Yarkovsky drift speed. Our results considered only asteroids that successfully cross over the resonance without close encounters with planets. We found a functional relation that accurately describes dependence between the average time lead/lag dtr, the strength of the resonance (SR), and the semimajor axis drift speed da/dt with asteroids' orbital eccentricities in the range (0.1, 0.2). We analysed average values of dtr using this functional relation comparing with obtained values of dtr from the numerical integrations, which were performed in an ORBIT9 integrator with a very large number of test asteroids. We checked the validity of our previous functional relation, derived for asteroids’ orbital eccentricities in the range (0, 0.1) (Figure 1), on the present results for eccentricities in the range (0.1, 0.2) (Figure 2). Also, we tried to find a unique functional relation for the whole interested interval of asteroids' orbital eccentricities (0, 0.2) and discussed it.

We establish a way to handle main collective fluctuations in correlated quantum systems based on a fluctuating local field concept. This technique goes beyond standard mean-field approaches, such as Hartree-Fock and dynamical mean-field theories (DMFT), as it includes a fluctuating classical field that acts on the leading order parameter of the system. Effective model parameters of this theory are determined from the variational principle, which allows one to resolve the Fierz ambiguity in decoupling of the local interaction term. In the saddle-point approximation for the fluctuating field our method reproduces the mean-field result. The exact numerical integration over this field allows one to consider nonlinear fluctuations of the global order parameter of the system while local correlations can be accounted for by solving the DMFT impurity problem. We apply our method to the magnetic susceptibility of finite Hubbard systems at half-filling and demonstrate that the introduced technique leads to a superior improvement of results with respect to parental mean-field approaches, without significant numerical complications. We show that the fluctuating local field method can be used in a very broad range of temperatures substantially below the Néel temperature of DMFT, which remains a major challenge for all existing theoretical approaches. © 2020 American Physical Society. The authors are very grateful to Maria Bandelmann for help with graphics. The work of E.A.S. is supported by the Russian Science Foundation, Grant No. 18-12-00185. A.I.L. acknowledges support by the Cluster of Excellence “Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG), Project No. ID390715994.

The pricing of financial options is usually based on statistical sampling of the evolution of the underlying under a chosen model, using a suitable numerical scheme. It is widely accepted that using low-discrepancy sequences instead of pseudorandom numbers in most cases increases the accuracy. It is important to understand and quantify the reasons for this effect. In this work, we use Global Sensitivity Analysis in order to study one widely used model for pricing of options, namely the Heston model. The Heston model is an important member of the family of the stochastic volatility models, which have been found to better describe the observed behaviour of option prices in the financial markets. By using a suitable numerical scheme, like those of Euler, Milstein, Kahl-Jäckel, Andersen, one has the flexibility needed to compute European, Asian or exotic options. In any case the problem of evaluating an option price can be considered as a numerical integration problem. For the purposes of modelling and complexity reduction, one should make the distinction between the model nominal dimension and its effective dimension. Another notion of “average dimension” has been found to be more practical from the computational point of view. The definitions and methods of evaluation of effective dimensions are based on computing Sobol’ sensitivity indices. A classification of functions based on their effective dimensions is also known. In the context of quantitative finance, Global Sensitivity Analysis (GSA) can be used to assess the efficiency of a particular numerical scheme. In this work we apply GSA based on Sobol sensitivity indices in order to assess the interactions of the various dimensions in using the above mentioned schemes. We observe that the GSA offers useful insight on how to maximize the advantages of using QMC in these schemes.

We discuss a constitutive model for stochastically distributed fiber reinforced tissues, where the active behavior of the fibers depends on the relative orientation of the electric field. Unlike other popular approaches, based on numerical integration over the unit sphere, or on the use of second order structure tensors, for the passive behavior we adopt a second order approximation of the strain energy density of the distribution. The purely mechanical quantities result to be dependent on two (second and fourth order, respectively) averaged structure tensors. In line with the approximation used for the passive behavior, we model the active behavior accounting for the statistical fiber distribution. We extend the Helmholtz free energy density by introducing a directional active potential, dependent on a stochastic permittivity tensor associated to a particular direction, and approximate the total active potential through a second order Taylor expansion of the permittivity tensor. The approximation allows us to derive explicitly the active stress and the active constitutive tensors, which result to be dependent on the same two averaged structure tensors that characterize the passive response. Active anisotropy follows from the distribution of the fibers and inherits its stochastic parameters. Examples of passive and active behaviors predicted by the model in terms of response to biaxial testing are presented, and comparisons with passive experimental data are provided.