Optimal Mathematical Models for Nonlinear Dynamical Systems

We propose a new method for the optimal causal representation of nonlinear systems. The proposed approach is based on the best constrained approximation of mappings in probability spaces by operators constructed from matrices of special form so that the approximant preserves the causality property. It is supposed that the observable input is contaminated with noise. The approximant minimises the mean-square difference between a desired output signal and the output signal of the approximating model. The method provides a numerically realisable mathematical model of the system. An analysis is given of the error associated with this representation.     

Randomized first order algorithms with applications to ℓ 1-minimization

In this paper we propose randomized first-order algorithms for solving bilinear saddle points problems. Our developments are motivated by the need for sublinear time algorithms to solve large-scale parametric bilinear saddle point problems where cheap online assessment of the solution quality is crucial. We present the theoretical efficiency estimates of our algorithms and discuss a number of applications, primarily to the problem of ? ₁ minimization arising in sparsity-oriented signal processing. We demonstrate, both theoretically and by numerical examples, that when seeking for medium-accuracy solutions of large-scale ? ₁ minimization problems, our randomized algorithms outperform significantly (and progressively as the sizes of the problem grow) the state-of-the art deterministic methods.     

Crystallization in the Al-Si,Al-Ge,and Al-Si-Ge systems at centrifugation

Crystallization in the Al-Si, Al-Ge, and Al-Si-Ge systems at centrifugation is studied. Of them, the Al-Si system is the least prone to sedimentation. In the others, sedimentation considerably changes the structure of the alloys at the bottom of the ingots compared with their top. At certain concentrations of the constituents, the number of crystallites in the lower part of the ingot is larger than in the upper part and the crystallites at the bottom are coarser than at the top. The Si: Ge atomic ratio in the Al-Si-Ge system changes by a factor of 2–12 against the initial ratio (1: 1) when the (Si + Ge) concentration changes as a result of centrifugation. Also, this ratio changes over the crystal surface (in the samples not subjected to centrifugation, this ratio remains unchanged over the surface). Crystallites in the Al-Si-Ge system are covered by Ge.     

Phase space representation of quantum dynamics

We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose-Hubbard model, Dicke model and others.     

Structure and dynamics of water confined in silica nanopores

We report the results of molecular simulation of water in silica nanopores at full hydration and room temperature. The model systems are approximately cylindrical pores in amorphous silica, with diameters ranging from 20 to 40 ?. The filled pores are prepared using grand canonical Monte Carlo simulation and molecular dynamics simulation is used to calculate the water structure and dynamics. We found that water forms two distinct molecular layers at the interface and exhibits uniform, but somewhat lower than bulk liquid, density in the core region. The hydrogen bond density profile follows similar trends, with lower than bulk density in the core and enhancements at the interface, due to hydrogen bonds between water and surface non-bridging oxygens and OH groups. Our studies of water dynamics included translational mean squared displacements, orientational time correlations, survival probabilities in interfacial shells, and hydrogen bond population relaxation. We found that the radial-axial anisotropy in translational motion largely follows the predictions of a model of free diffusion in a cylinder. However, both translational and rotational water mobilities are strongly dependent on the proximity to the interface, with pronounced slowdown in layers near the interface. Within these layers, the effects of interface curvature are relatively modest, with only a small increase in mobility in going from the 20 to 40 ? diameter pore. Hydrogen bond population relaxation is nearly bulk-like in the core, but considerably slower in the interfacial region.     

On the existence of non-monotone non-oscillating wavefronts

We present a monostable delayed reaction–diffusion equation with the unimodal birth function which admits only non-monotone wavefronts. Moreover, these fronts are either eventually monotone (in particular, such is the minimal wave) or slowly oscillating. Hence, for the Mackey–Glass type diffusive equations, we answer affirmatively the question about the existence of non-monotone non-oscillating wavefronts. As it was recently established by Hasik et al. and Ducrot et al., the same question has a negative answer for the KPP-Fisher equation with a single delay.