Stanislav Borysov 🇺🇦

Population synthesis is concerned with the generation of synthetic yet realistic representations of populations. It is a fundamental problem in the modeling of transportation where the synthetic populations of micro-agents represent a key input to most agent-based models. In this paper, a new methodological framework for how to ‘grow’ pools of micro-agents is presented. The model framework adopts a deep generative modeling approach from machine learning based on a Variational Autoencoder (VAE).… Show more

Population synthesis is concerned with the generation of synthetic yet realistic representations of populations. It is a fundamental problem in the modeling of transportation where the synthetic populations of micro-agents represent a key input to most agent-based models. In this paper, a new methodological framework for how to ‘grow’ pools of micro-agents is presented. The model framework adopts a deep generative modeling approach from machine learning based on a Variational Autoencoder (VAE). Compared to the previous population synthesis approaches, including Iterative Proportional Fitting (IPF), Gibbs sampling and traditional generative models such as Bayesian Networks or Hidden Markov Models, the proposed method allows fitting the full joint distribution for high dimensions. The proposed methodology is compared with a conventional Gibbs sampler and a Bayesian Network by using a large-scale Danish trip diary. It is shown that, while these two methods outperform the VAE in the low-dimensional case, they both suffer from scalability issues when the number of modeled attributes increases. It is also shown that the Gibbs sampler essentially replicates the agents from the original sample when the required conditional distributions are estimated as frequency tables. In contrast, the VAE allows addressing the problem of sampling zeros by generating agents that are virtually different from those in the original data but have similar statistical properties. The presented approach can support agent-based modeling at all levels by enabling richer synthetic populations with smaller zones and more detailed individual characteristics. Show less

Machine learning models are capable of capturing the structure-property relationship from a dataset of computationally demanding ab initio calculations. In fact, machine learning models have reached chemical accuracy on small organic molecules contained in the popular QM9 dataset. At the same time, the domain of large crystal structures remains rather unexplored. Over the past two years, the Organic Materials Database (OMDB) has hosted a growing number of electronic properties of previously… Show more

Machine learning models are capable of capturing the structure-property relationship from a dataset of computationally demanding ab initio calculations. In fact, machine learning models have reached chemical accuracy on small organic molecules contained in the popular QM9 dataset. At the same time, the domain of large crystal structures remains rather unexplored. Over the past two years, the Organic Materials Database (OMDB) has hosted a growing number of electronic properties of previously synthesized organic crystal structures. The complexity of the organic crystals contained within the OMDB, which have on average 85 atoms per unit cell, makes this database a challenging platform for machine learning applications. In this paper, we focus on predicting the band gap which represents one of the basic properties of a crystalline material. With this aim, we release a consistent dataset of 12,500 crystal structures and their corresponding DFT band gap freely available for download at https://omdb.diracmaterials.org/dataset. We run two recent machine learning models, kernel ridge regression with the Smooth Overlap of Atomic Positions (SOAP) kernel and the deep learning model SchNet, on this new dataset and find that an ensemble of these two models reaches mean absolute error (MAE) of 0.361 eV, which corresponds to a percentage error of 12% on the average band gap of 3.03 eV. The models also provide chemical insights into the data. For example, by visualizing the SOAP kernel similarity between the crystals, different clusters of materials can be identified, such as organic metals or semiconductors. Finally, the trained models are employed to predict the band gap for 260,092 materials contained within the Crystallography Open Database (COD) and made available online so the predictions can be obtained for any arbitrary crystal structure uploaded by a user. Show less

We propose a new approach for constructing synthetic pseudo-panel data from cross-sectional data. The pseudo panel and the preferences it intends to describe is constructed at the individual level and is not affected by aggregation bias across cohorts. This is accomplished by creating a high-dimensional probabilistic model representation of the entire data set, which allows sampling from the probabilistic model in such a way that all of the intrinsic correlation properties of the original data… Show more

We propose a new approach for constructing synthetic pseudo-panel data from cross-sectional data. The pseudo panel and the preferences it intends to describe is constructed at the individual level and is not affected by aggregation bias across cohorts. This is accomplished by creating a high-dimensional probabilistic model representation of the entire data set, which allows sampling from the probabilistic model in such a way that all of the intrinsic correlation properties of the original data are preserved. The key to this is the use of deep learning algorithms based on the Conditional Variational Autoencoder (CVAE) framework. From a modelling perspective, the concept of a model-based resampling creates a number of opportunities in that data can be organized and constructed to serve very specific needs of which the forming of heterogeneous pseudo panels represents one. The advantage, in that respect, is the ability to trade a serious aggregation bias (when aggregating into cohorts) for an unsystematic noise disturbance. Moreover, the approach makes it possible to explore high-dimensional sparse preference distributions and their linkage to individual-specific characteristics, which is not possible if applying traditional pseudo-panel methods. We use the presented approach to reveal the dynamics of transport preferences for a fixed pseudo panel of individuals based on a large Danish cross-sectional data set covering the period from 2006 to 2016. The model is also utilized to classify individuals into 'slow' and 'fast' movers with respect to the speed at which their preferences change over time. It is found that the prototypical fast mover is a young woman who lives as a single in a large city whereas the typical slow mover is a middle-aged man with high income from a nuclear family who lives in a detached house outside a city. Show less

Large datasets of ab initio calculations have enabled many pioneering studies of machine learning applied to quantum-chemical systems. For example, machine learning models already achieve chemical accuracy on the popular QM9 dataset with small organic molecules. Here, we present a new, more challenging dataset of 12,500 large organic crystal structures and their corresponding DFT band gap, freely available at https://omdb.diracmaterials.org/dataset. The complexity of the organic crystals in… Show more

Large datasets of ab initio calculations have enabled many pioneering studies of machine learning applied to quantum-chemical systems. For example, machine learning models already achieve chemical accuracy on the popular QM9 dataset with small organic molecules. Here, we present a new, more challenging dataset of 12,500 large organic crystal structures and their corresponding DFT band gap, freely available at https://omdb.diracmaterials.org/dataset. The complexity of the organic crystals in this dataset, which have on average 85 atoms per unit cell, makes it a challenging platform for machine learning applications. We run tworecent machine learning models, kernel ridge regression with the Smooth Overlapof Atomic Positions (SOAP) kernel and the deep learning model SchNet, on this new dataset and find that an ensemble of these two models reaches mean absoluteerror (MAE) of 0.361 eV, which corresponds to a percentage error of 12% onthe average band gap of 3.03 eV. The models also provide chemical insights into the data. For example, by visualizing the SOAP kernel similarity between the crystals, different clusters of materials can be identified, such as organic metals or semiconductors. Finally, the trained models are employed to predict the band gap for 260,092 materials contained within the Crystallography Open Database (COD) and made available online so the predictions can be obtained for any arbitrary crystal structure uploaded by a user. Show less

This chapter aims to be a smooth introduction to the basic concepts of machine learning, and, building on them, explain some to the latest advanced techniques. After a brief historical perspective, we overview the two currently most popular machine learning frameworks—deep learning and probabilistic graphical models. We conclude the chapter with practical pieces of advice about machine learning experiments which are necessary to know for a beginner. A good understanding of these fundamentals… Show more

This chapter aims to be a smooth introduction to the basic concepts of machine learning, and, building on them, explain some to the latest advanced techniques. After a brief historical perspective, we overview the two currently most popular machine learning frameworks—deep learning and probabilistic graphical models. We conclude the chapter with practical pieces of advice about machine learning experiments which are necessary to know for a beginner. A good understanding of these fundamentals opens up a wide portfolio of opportunities for predictive models in transportation, and is hopefully a good basis for the remainder of this book. Show less

Many functional materials can be characterized by a specific pattern in their electronic band structure, for example, Dirac materials, characterized by a linear crossing of bands; topological insulators, characterized by a “Mexican hat” pattern or an effectively free electron gas, characterized by a parabolic dispersion. To find material realizations of these features, manual inspection of electronic band structures represents a relatively easy task for a small number of materials. However, the… Show more

Many functional materials can be characterized by a specific pattern in their electronic band structure, for example, Dirac materials, characterized by a linear crossing of bands; topological insulators, characterized by a “Mexican hat” pattern or an effectively free electron gas, characterized by a parabolic dispersion. To find material realizations of these features, manual inspection of electronic band structures represents a relatively easy task for a small number of materials. However, the growing amount of data contained within modern electronic band structure databases makes this approach impracticable. To address this problem, we present an automatic graphical pattern search tool implemented for the electronic band structures contained within the Organic Materials Database. The tool is capable of finding user-specified graphical patterns in the collection of thousands of band structures from high-throughput calculations in the online regime. Using this tool, it only takes a few seconds to find an arbitrary graphical pattern within the ten electronic bands near the Fermi level for 26,739 organic crystals. The source code of the developed tool is freely available and can be adapted to any other electronic band structure database. Show less

Identifying novel functional materials with desired key properties is an important part of bridging the gap between fundamental research and technological advancement. In this context, high-throughput calculations combined with data-mining techniques highly accelerated this process in different areas of research during the past years. The strength of a data-driven approach for materials prediction lies in narrowing down the search space of thousands of materials to a subset of prospective… Show more

Identifying novel functional materials with desired key properties is an important part of bridging the gap between fundamental research and technological advancement. In this context, high-throughput calculations combined with data-mining techniques highly accelerated this process in different areas of research during the past years. The strength of a data-driven approach for materials prediction lies in narrowing down the search space of thousands of materials to a subset of prospective candidates. Recently, the open-access organic materials database OMDB was released providing electronic structure data for thousands of previously synthesized three-dimensional organic crystals. Based on the OMDB, we report about the implementation of a novel density of states similarity search tool which is capable of retrieving materials with similar density of states to a reference material. The tool is based on the approximate nearest neighbor algorithm as implemented in the ANNOY library and can be applied via the OMDB web interface. The approach presented here is wide ranging and can be applied to various problems where the density of states is responsible for certain key properties of a material. As the first application, we report about materials exhibiting electronic structure similarities to the aromatic hydrocarbon p-terphenyl which was recently discussed as a potential organic high-temperature superconductor exhibiting a transition temperature in the order of 120 K under strong potassium doping. Although the mechanism driving the remarkable transition temperature remains under debate, we argue that the density of states, reflecting the electronic structure of a material, might serve as a crucial ingredient for the observed high T_c... Show less

The experiment sought to find correlations between traffic management, traffic flow and air quality by measuring the difference between the air pollution levels when cars are waiting for the traffic light to turn green compared to when they are driving through the intersection. The project succeeded in finding as reliable correlations for NO2 and CO as can be achieved in real-life data gathering. The results showed an increase in local pollution levels, which might be caused by acceleration of… Show more

The experiment sought to find correlations between traffic management, traffic flow and air quality by measuring the difference between the air pollution levels when cars are waiting for the traffic light to turn green compared to when they are driving through the intersection. The project succeeded in finding as reliable correlations for NO2 and CO as can be achieved in real-life data gathering. The results showed an increase in local pollution levels, which might be caused by acceleration of cars starting from a standstill or trying to cross the intersection before it turns red. Show less

We combined the group theory and data mining approach within the Organic Materials Database that leads to the prediction of stable Dirac-point nodes within the electronic band structure of three-dimensional organic crystals. We find a particular space group P212121 (#19) that is conducive to the Dirac nodes formation. We prove that nodes are a consequence of the orthorhombic crystal structure. Within the electronic band structure, two different kinds of nodes can be distinguished: 8-fold… Show more

We combined the group theory and data mining approach within the Organic Materials Database that leads to the prediction of stable Dirac-point nodes within the electronic band structure of three-dimensional organic crystals. We find a particular space group P212121 (#19) that is conducive to the Dirac nodes formation. We prove that nodes are a consequence of the orthorhombic crystal structure. Within the electronic band structure, two different kinds of nodes can be distinguished: 8-fold degenerate Dirac nodes protected by the crystalline symmetry and 4-fold degenerate Dirac nodes protected by band topology. Mining the Organic Materials Database, we present band structure calculations and symmetry analysis for 6 previously synthesized organic materials. In all these materials, the Dirac nodes are well separated within the energy and located near the Fermi surface, which opens up a possibility for their direct experimental observation. Show less

We present an organic materials database (OMDB) hosting thousands of Kohn-Sham electronic band structures, which is freely accessible online at http://omdb.diracmaterials.org. The OMDB focus lies on electronic structure, density of states and other properties for purely organic and organometallic compounds that are known to date. The electronic band structures are calculated using density functional theory for the crystal structures contained in the Crystallography Open Database. The OMDB web… Show more

We present an organic materials database (OMDB) hosting thousands of Kohn-Sham electronic band structures, which is freely accessible online at http://omdb.diracmaterials.org. The OMDB focus lies on electronic structure, density of states and other properties for purely organic and organometallic compounds that are known to date. The electronic band structures are calculated using density functional theory for the crystal structures contained in the Crystallography Open Database. The OMDB web interface allows users to retrieve materials with specified target properties using non-trivial queries about their electronic structure. We illustrate the use of the OMDB and how it can become an organic part of search and prediction of novel functional materials via data mining techniques. As a specific example, we provide data mining results for metals and semiconductors, which are known to be rare in the class of organic materials. Show less

A data mining study of electronic Kohn-Sham band structures was performed to identify Dirac materials within the Organic Materials Database (OMDB). Out of that, the 3-dimensional organic crystal 5,6-bis(trifluoromethyl)-2-methoxy-1H-1,3-diazepine was found to host different Dirac line-nodes within the band structure. From a group theoretical analysis, it is possible to distinguish between Dirac line-nodes occurring due to 2-fold degenerate energy levels protected by the monoclinic crystalline… Show more

A data mining study of electronic Kohn-Sham band structures was performed to identify Dirac materials within the Organic Materials Database (OMDB). Out of that, the 3-dimensional organic crystal 5,6-bis(trifluoromethyl)-2-methoxy-1H-1,3-diazepine was found to host different Dirac line-nodes within the band structure. From a group theoretical analysis, it is possible to distinguish between Dirac line-nodes occurring due to 2-fold degenerate energy levels protected by the monoclinic crystalline symmetry and 2-fold degenerate accidental crossings protected by the topology of the electronic band structure. The obtained results can be generalized to all materials having the space group P21/c (No. 14, C52h) by introducing three distinct topological classes. Show less

While our transport systems are generally designed for habitual behavior, the dynamics of large and mega cities systematically push it to its limits. Particularly, transport planning and operations in large events are well known to be a challenge. Not only they imply stress to the system on an irregular basis, their associated mobility behavior is also difficult to predict. Previous studies have shown a strong correlation between number of public transport arrivals with the semi-structured data… Show more

While our transport systems are generally designed for habitual behavior, the dynamics of large and mega cities systematically push it to its limits. Particularly, transport planning and operations in large events are well known to be a challenge. Not only they imply stress to the system on an irregular basis, their associated mobility behavior is also difficult to predict. Previous studies have shown a strong correlation between number of public transport arrivals with the semi-structured data mined from online announcement websites. However, these models tend to be complex in form and demand substantial information retrieval, extraction and data cleaning work, and so they are difficult to generalize from city to city. In contrast, this paper focuses on enriching previously mined information about special events using automated web search queries. Since this context data comes in unstructured natural language form, we employ supervised topic model to correlate it with real measurements of transport usage. In this way, the proposed approach is more generic and a transit agency can start planning ahead as early as the event is announced on the web. The results show that using information mined from the web search not only shows high prediction accuracy of public transport demand, but also potentially provides interesting insights about popular event categories based on extracted topics. Show less

Public special events, like sports games, concerts and festivals are well known to create disruptions in transportation systems, often catching the operators by surprise. Although these are usually planned well in advance, their impact is difficult to predict, even when organisers and transportation operators coordinate. The problem highly increases when several events happen concurrently. To solve these problems, costly processes, heavily reliant on manual search and personal experience, are… Show more

Public special events, like sports games, concerts and festivals are well known to create disruptions in transportation systems, often catching the operators by surprise. Although these are usually planned well in advance, their impact is difficult to predict, even when organisers and transportation operators coordinate. The problem highly increases when several events happen concurrently. To solve these problems, costly processes, heavily reliant on manual search and personal experience, are usual practice in large cities like Singapore, London or Tokyo. This paper presents a Bayesian additive model with Gaussian process components that combines smart card records from public transport with context information about events that is continuously mined from the Web. We develop an efficient approximate inference algorithm using expectation propagation, which allows us to predict the total number of public transportation trips to the special event areas, thereby contributing to a more adaptive transportation system. Furthermore, for multiple concurrent event scenarios, the proposed algorithm is able to disaggregate gross trip counts into their most likely components related to specific events and routine behavior. Using real data from Singapore, we show that the presented model not only outperforms the best baseline model by up to 41\​% in $R^2$, but also has explanatory power for its individual components. Show less

The paper aims at continuing the discussion on sustainability and attempts to forecast the impossibility of the expanding consumption worldwide due to the planet’s limited resources. As the population of China, India and other developing countries continue to increase, they would also require more natural and financial resources to sustain their growth. We coarsely estimate the volumes of these resources (energy, food, freshwater) and the gross domestic product (GDP) that would need to be… Show more

The paper aims at continuing the discussion on sustainability and attempts to forecast the impossibility of the expanding consumption worldwide due to the planet’s limited resources. As the population of China, India and other developing countries continue to increase, they would also require more natural and financial resources to sustain their growth. We coarsely estimate the volumes of these resources (energy, food, freshwater) and the gross domestic product (GDP) that would need to be achieved to bring the population of India and China to the current levels of consumption in the United States. We also provide estimations for potentially needed immediate growth of the world resource consumption to meet this equality requirement. Given the tight historical correlation between GDP and energy consumption, the needed increase of GDP per capita in the developing world to the levels of the U.S. would deplete explored fossil fuel reserves in less than two decades. These estimates predict that the world economy would need to find a development model where growth would be achieved without heavy dependence on fossil fuels. Show less

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the… Show more

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented results show that binarization preserves the correlation structure of the market. Properties of distributions of external fields and couplings as well as the market interaction network and industry sector clustering structure are studied for different historical dates and moving window sizes. We demonstrate that the observed positive heavy tail in distribution of couplings is related to the sparse clustering structure of the market. We also show that discrepancies between the model’s parameters might be used as a precursor of financial instabilities. Show less

We study historical correlations and lead-lag relationships between individual stock risk (volatility of daily stock returns) and market risk (volatility of daily returns of a market-representative portfolio) in the US stock market. We consider the cross- correlation functions averaged over all stocks, using 71 stock prices from the Standard & Poor’s 500 index for 1994–2013. We focus on the behavior of the cross-correlations at the times of financial crises with significant jumps of market… Show more

We study historical correlations and lead-lag relationships between individual stock risk (volatility of daily stock returns) and market risk (volatility of daily returns of a market-representative portfolio) in the US stock market. We consider the cross- correlation functions averaged over all stocks, using 71 stock prices from the Standard & Poor’s 500 index for 1994–2013. We focus on the behavior of the cross-correlations at the times of financial crises with significant jumps of market volatility. The observed historical dynamics showed that the dependence between the risks was almost linear during the US stock market downturn of 2002 and after the US housing bubble in 2007, remaining at that level until 2013. Moreover, the averaged cross-correlation function often had an asymmetric shape with respect to zero lag in the periods of high correlation. We develop the analysis by the application of the linear response formalism to study underlying causal relations. The calculated response functions suggest the presence of characteristic regimes near financial crashes, when the volatility of an individual stock follows the market volatility and vice versa. Show less

Conventional dynamic atomic force microscopy (AFM) can be extended to bimodal and multimodal AFM in which the cantilever is simultaneously excited at two or more resonance frequencies. Such excitation schemes result in one additional amplitude and phase images for each driven resonance, and potentially convey more information about the surface under investigation. Here we present a theoretical basis for using this information to approximate the parameters of a tip-surface interaction model. The… Show more

Conventional dynamic atomic force microscopy (AFM) can be extended to bimodal and multimodal AFM in which the cantilever is simultaneously excited at two or more resonance frequencies. Such excitation schemes result in one additional amplitude and phase images for each driven resonance, and potentially convey more information about the surface under investigation. Here we present a theoretical basis for using this information to approximate the parameters of a tip-surface interaction model. The theory is verified by simulations with added noise corresponding to room-temperature measurements. Show less

We present a theoretical framework for the dynamic calibration of the higher eigenmode parameters (stiffness and optical lever inverse responsivity) of a cantilever. The method is based on the tip–surface force reconstruction technique and does not require any prior knowledge of the eigenmode shape or the particular form of the tip–surface interaction. The calibration method proposed
requires a single-point force measurement by using a multimodal amplitude of a higher eigenmode.

We propose a theoretical framework for reconstructing tip-surface interactions using the intermodulation technique when more than one eigenmode is required to describe the cantilever motion. Two particular cases of bimodal motion are studied numerically: one bending and one torsional mode, and two bending modes. We demonstrate the possibility of accurate reconstruction of a two-dimensional conservative force field for the former case, while dissipative forces are studied for the latter.

We consider both analytically and numerically the creation conditions of diverse hierarchical trees. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure are studied. We argue that a consistent probabilistic picture requires the use of deformed algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both regular and degenerate ones are studied… Show more

We consider both analytically and numerically the creation conditions of diverse hierarchical trees. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure are studied. We argue that a consistent probabilistic picture requires the use of deformed algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both regular and degenerate ones are studied analytically,
while the creation probabilities of Fibonacci and free-scale trees are determined numerically. We
find a general expression for the creation probability of an arbitrary tree and calculate the sum
of terms of deformed geometrical progression that results from the consideration of the degenerate
tree. Show less

We find the analytic solution of a pair of stochastic equations with arbitrary forces and multiplicative Lévy noises in a steady-state nonequilibrium case. This solution shows that Lévy flights always suppress a quasiperiodic motion related to the limit cycle.

We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure is studied. We argue that a consistent probabilistic picture requires the use of deformed algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both regular and degenerate ones are studied analytically, while… Show more

We consider the creation conditions of diverse hierarchical trees both analytically and numerically. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into a united structure is studied. We argue that a consistent probabilistic picture requires the use of deformed algebra. Our consideration is based on the study of the main types of hierarchical trees, among which both regular and degenerate ones are studied analytically, while the creation probabilities of Fibonacci, scale-free and arbitrary trees are determined numerically. Show less

Within the framework of basic-deformed and finite-difference calculi, as well as deformation procedures proposed by Tsallis, Abe, and Kaniadakis and generalized by Naudts, we develop field-theoretical schemes of statistically distributed fields. We construct a set of generating functionals and find their connection with corresponding correlators for basic-deformed, finite-difference, and Kaniadakis calculi. Moreover, we introduce pair of additive functionals, which expansions into deformed… Show more

Within the framework of basic-deformed and finite-difference calculi, as well as deformation procedures proposed by Tsallis, Abe, and Kaniadakis and generalized by Naudts, we develop field-theoretical schemes of statistically distributed fields. We construct a set of generating functionals and find their connection with corresponding correlators for basic-deformed, finite-difference, and Kaniadakis calculi. Moreover, we introduce pair of additive functionals, which expansions into deformed series yield both Green functions and their irreducible proper vertices. We find as well formal equations, governing by the generating functionals of systems which possess a symmetry with respect to a field variation and are subjected to an arbitrary constrain. Finally, we generalize field-theoretical schemes inherent in concrete calculi in the Naudts manner. From the physical point of view, we study dependences of both one-site partition function and variance of free fields on deformations. We show that within the basic-deformed statistics dependence of the specific partition function on deformation has in logarithmic axes symmetrical form with respect to maximum related to deformation absence; in case of the finite-difference statistics, the partition function takes non-deformed value; for the Kaniadakis statistics, curves of related dependences have convex symmetrical form at small curvatures of the effective action and concave form at large ones. We demonstrate that only moment of the second order of free fields takes non-zero values to be proportional to inverse curvature of effective action. In dependence of the deformation parameter, the free field variance has linearly arising form for the basic-deformed distribution and increases non-linearly rapidly in case of the finite-difference statistics; for more complicated case of the Kaniadakis distribution, related dependence has double-well form. Show less

We consider the effect of stochastic sources on the self-organization process being initiated with creation of the limit cycle. The general expressions obtained are applied to the stochastic Lorenz system to show that offset from the equilibrium steady state can destroy the limit cycle at a certain relation between characteristic scales of temporal variation of principal variables. Noise-induced resonance related to the limit cycle is found analytically to appear in the non-equilibrium steady… Show more

We consider the effect of stochastic sources on the self-organization process being initiated with creation of the limit cycle. The general expressions obtained are applied to the stochastic Lorenz system to show that offset from the equilibrium steady state can destroy the limit cycle at a certain relation between characteristic scales of temporal variation of principal variables. Noise-induced resonance related to the limit cycle is found analytically to appear in the non-equilibrium steady state system if the fastest variations display a principal variable, which is coupled with two different degrees of freedom or more. Show less

Conditions for the creation of a limit cycle, which provide the transition of a nonequilibrium system into a self-organized modulation mode, have been studied. An approach, which allows one to replace the equations of self-consistent evolution for a pair of real-valued variables by a single equation of motion for a complex-valued order parameter, is proposed. The optimum basis has been found, in which the evolution of the complex-valued order parameter is described by the Ginzburg–Landau… Show more

Conditions for the creation of a limit cycle, which provide the transition of a nonequilibrium system into a self-organized modulation mode, have been studied. An approach, which allows one to replace the equations of self-consistent evolution for a pair of real-valued variables by a single equation of motion for a complex-valued order parameter, is proposed. The optimum basis has been found, in which the evolution of the complex-valued order parameter is described by the Ginzburg–Landau equation characterized by a complex-valued non-linearity only. Conditions for the system to transit into the self-organized modulation mode are determined. Show less