Approximate Distribution Research Articles

Learning Kernel Stein Discrepancy for Training Energy-Based Models

The primary challenge in unsupervised learning is training unnormalized density models and then generating similar samples. Few traditional unnormalized models know what the quality of the trained model is, as most models are evaluated by downstream tasks and often involve complex sampling processes. Kernel Stein Discrepancy (KSD), a goodness-of-fit test method, can measure the discrepancy between the generated samples and the theoretical distribution; therefore, it can be employed to measure the quality of trained models. We first demonstrate that, under certain constraints, KSD is equal to Maximum Mean Discrepancy (MMD), a two-sample test method. PT KSD GAN (Kernel Stein Discrepancy Generative Adversarial Network with a Pulling-Away Term) is produced to compel generated samples to approximate the theoretical distribution. The generator, functioning as an implicit generative model, employs KSD as loss to avoid tedious sampling processes. In contrast, the discriminator is trained to identify the data manifold, also known as an explicit energy-based model. To demonstrate the effectiveness of our approach, we undertook experiments on two-dimensional toy datasets. Our results highlight that our generator adeptly captures the accurate density distribution, while the discriminator proficiently recognizes the unnormalized approximate distribution shape. When applied to linear Independent Component Analysis datasets, the log likelihoods of PT KSD GAN improve by about 5‰ over existing methods when the data dimension is less than 30. Furthermore, our tests on image datasets reveal that the PT KSD GAN excels in navigating high-dimensional challenges, yielding authentically genuine samples.

Red Giants Search Method Based on Convolutional Neural Networks

Red giants play a crucial role in astronomical exploration. However, the current search for these stars heavily relies on spectrum analysis, making it a cumbersome process. In this paper, we propose a novel red giant search approach called CBAMResNets. CBAMResNets enhances feature extraction from images by incorporating the Convolutional Block Attention Module into the ResNets model. To train and validate our model, we utilize accurately labeled stellar image data obtained by combining the Sky Mapper Southern Survey (SMSS) and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope. Our experiments demonstrate that CBAMResNets outperforms other networks such as VGG16 and TPC across most performance metrics and remains competitive with traditional machine-learning methods. Specifically, for stars within the magnitude range of 11–19 in the u band, our model achieves a precision rate of 0.92 and a recall rate of 0.9194. Similarly, for stars within the magnitude range of 11–15 in the u band, the precision rate is 0.92, and the recall rate is 0.9813. Furthermore, we apply CBAMResNets to the SMSS subregion, resulting in the identification of 20,243 potential red giant candidates out of the 304,477 observed stars. We validate the authenticity of these candidates by analyzing their stellar absolute magnitudes and temperatures, estimating a contamination rate of approximately 6.4%. Additionally, we examine the approximate distribution of their metallicity. The catalog containing the identified red giant candidates can be accessed at Zenodo. 4 4 doi:10.5281/zenodo.8352420

Multi-Hypothesis Marginal Multi-Target Bayes Filter for a Heavy-Tailed Observation Noise

A multi-hypothesis marginal multi-target Bayes filter for heavy-tailed observation noise is proposed to track multiple targets in the presence of clutter, missed detection, and target appearing and disappearing. The proposed filter propagates the existence probabilities and probability density functions (PDFs) of targets in the filter recursion. It uses the Student’s t distribution to model the heavy-tailed non-Gaussian observation noise, and employs the variational Bayes technique to acquire the approximate distributions of individual targets. K-best hypotheses, obtained by minimizing the negative log-generalized-likelihood ratio, are used to establish the existence probabilities and PDFs of targets in the filter recursion. Experimental results indicate that the proposed filter achieves better tracking performance than other filters.

Estimation of Hardness and Residual Stress on End-Milled Surfaces Using Linear Regression Model

This study presents a novel method for estimating the surface integrity of end-milled workpieces. It is well known that the mechanical properties of machined surfaces in cutting affect the quality of the final product. In particular, hardness and residual stress often require strict control; however, nondestructive inspection remains a challenge. This study proposes a method to estimate the hardness and residual stress of end-milled surfaces by analyzing cutting forces and images of the tool during machining to obtain approximate temperature and stress distributions. These state quantities are highly correlated with the dislocation density and its distribution on the machined surface, which in turn is strongly correlated with residual stress and surface hardness. Despite this strong correlation, few research studies have been conducted on the topic. In the proposed method, cutting forces, measured by a dynamometer, are analyzed to estimate the specific cutting forces and edge force coefficients. Simultaneously, the flank wear width is recorded using an image-based on-machine measuring device installed in the machine tool. From this information, the average stresses at the primary and tertiary cutting zones are estimated, while the cutting temperature in the primary cutting zone is roughly estimated by considering the traditional shear-angle prediction theory. Using these estimations, hardness and residual stress are calculated based on a simple linear regression model. Parameter identification for the model is performed based on measured hardness and residual stress in end-milling experiments. The model was validated against experimental measurements, which showed that the proposed method can accurately estimate hardness and residual stress, although it was observed that the selection of explanatory variables has a significant effect on accuracy.

Effect of Prolong-life-with-nine-turn-method (Yan Nian Jiu Zhuan) Qigong on fatigue and gastrointestinal function in patients with chronic fatigue syndrome: Study protocol for a randomized controlled trial.

Chronic fatigue syndrome (CFS) is a physical and mental disorder in which long-term fatigue is the main symptom. CFS patients are often accompanied by functional gastrointestinal diseases (FGIDs), which lead to decreased quality of life and increased fatigue. Prolong-life-with-nine-turn-method (PLWNT) is a kind of physical and mental exercise. Its operation includes adjusting the mind, breathing and cooperating with eight self-rubbing methods and one upper body rocking method. PLWNT was used to improve the digestive function in ancient China and to treat FGIDs such as functional dyspepsia and irritable bowel syndrome in modern times. Previous studies have shown that PLWNT can reduce fatigue in patients with CFS. But it is unclear whether the effect of PLWNT on CFS fatigue is related to gastrointestinal function. The aim of this study was to explore the relationship between PLWNT and fatigue and gastrointestinal function in patients with CFS. This study is a non-inferiority randomized controlled trial (RCT). The whole study period is 38 weeks, including 2 weeks of baseline evaluation, 12 weeks of intervention and 6 months of follow-up. Ninety-six CFS patients will be stratified random assigned to the intervention group (PLWNT) and the control group (cognitive behavior treatment) in the ratio of 1:1 through the random number table generated by SPSS. In the evaluation of results, Multidimensional Fatigue Inventory-20 (MFI-20), Gastrointestinal Symptom Rating Scale (GSRS), Bristol Stool Form Scale (BSFS), and Short Form 36 item health survey (SF-36) will be evaluated at week 0 (baseline), week 6 (midterm), week 12 (endpoint) and month 9 (follow up). The intestinal flora will be evaluated at week 0 (baseline) and week 12 (endpoint). The data results will be processed by statistical experts. The data analysis will be based on the intention to treat principle and per-protocol analysis. In the efficacy evaluation, repeated measurement analysis of variance will be used for data conforming to normal distribution or approximate normal distribution. The data which do not conform to the analysis of repeated measurement variance will be analyzed by the generalized estimation equation Linear discriminant analysis will be used to clarify the difference species of intestinal flora. The significance level sets as 5%. The safety of interventions will be evaluated after each treatment session. This trial will provide evidence to PLWNT exerting positive effects on fatigue and gastrointestinal function of CFS. It will further explore whether the improvement of PLWNT on CFS fatigue is related to gastrointestinal function. The trial was registered at Chinese Clinical Trial Registry http://www.chictr.org.cn/showproj.aspx?proj=151456 (Registration No.: ChiCTR2200056530). Date: 2022-02-07.

Hadron transverse momentum distributions in the Tsallis statistics with escort probabilities

Abstract The exact and approximate hadron transverse momentum distributions for the Fermi–Dirac, Bose–Einstein and Maxwell-Boltzmann statistics of particles in the framework of the Tsallis statistics with escort probabilities (the Tsallis-3 statistics) have been derived. The classical and quantum transverse momentum distributions in the zeroth term approximation and the quantum transverse momentum distributions in the factorization approximation introduced in the zeroth term approximation were found. The transverse momentum distributions in the zeroth term approximation and in the factorization approximation of the zeroth term approximation are the same in the Tsallis-3, Tsallis-2 and q-dual statistics. The well-known classical phenomenological Tsallis distribution exactly coincides with the classical transverse momentum distribution of the Tsallis-3 statistics in the zeroth term approximation for which the entropy of the system is zero in the whole range of state variables. However, the quantum phenomenological Tsallis distribution does not coincide with either the exact or approximate transverse momentum distributions of the Tsallis-3 statistics. The exact Tsallis-3 classical distribution and the classical phenomenological Tsallis distribution were applied to describe the experimental spectra of the charged pions produced in the proton-proton collisions at high energies. The values of the parameters (T, q) for both these model distributions differ in the whole energy range. Thus, the classical phenomenological Tsallis distribution is an unsatisfactory approximation for the exact classical transverse momentum distribution of the Tsallis-3 statistics.

Multilevel State-Space Models Enable High Precision Event Related Potential Analysis.

During cognitive tasks, the elicited brain responses that are time-locked to the stimulus presentation are manifested in electroencephalogram (EEG) as Event Related Potentials (ERPs). In general, ERPs are signals embedded in the background of much stronger neural oscillations, and thus they are traditionally extracted by averaging hundreds of trial responses so that the neural oscillations can cancel out each other. However, often in cognitive science experiments, it is difficult to administer large number of trials due to physical constraints. Additionally, these excessive averaging can also blur fine structures of the ERPs signals, which might otherwise be indicative of various intrinsic factors. Here we propose to model the background oscillations using a novel oscillation state-space representation and identify their time-traces in a data-driven way. This allows us to effectively separate the oscillations from the response signals of interest, thus improving the signal-to-noise of the evoked response, and eventually increasing trial fidelity. We also consider a random-walk like continuity constraint for the ERP waveforms to recover smooth, de-noised estimates. We employ a generalized expectation maximization algorithm for estimating the model parameters, and then infer the approximate posterior distribution of ERP waveforms. We demonstrate the reduced reliance of our proposed ERP extraction technique via a simulation study. Finally, we showcase how the extracted ERPs using our method can be more informative than the traditional average-based ERPs when analyzing EEG data in cognitive task settings with fewer trials.

Asymptotic behavior of a system of two coupled queues when the content of one queue is very high

We consider a system of two parallel discrete-time single-server queues, queue 1 and queue 2. The service time of any customer in either queue is equal to 1 time slot. Arrivals during consecutive slots occur independently from slot to slot. However, the arrival streams into both queues are possibly mutually interdependent, i.e., during any slot, the numbers of arrivals in queue 1 and queue 2 need not be statistically independent. Their joint probability generating function (pgf) A(x, y) fully characterizes the queueing model. As a consequence of the possible intra-slot correlation in the arrival process, the numbers of customers present (“system contents”) in queues 1 and 2, at any given slot boundary, are not necessarily independent either. In a previous paper, we have already discussed the mathematical difficulty of computing their steady-state joint pgf U(z_1,z_2); explicit closed-form results can only be obtained for specific choices of A(x, y). In this paper, we therefore look at the problem from an other angle. Specifically, we study the (asymptotic) conditional steady-state behavior of the system under the condition that the content of queue 1 is (temporarily) very high (goes to infinity). For ease of terminology, we refer to the system as the “asymptotic system” in these circumstances. We prove that the asymptotic system is nearly identical to the original (unconditional) system, but with a modified joint arrival pgf A^*(x,y) that can be computed explicitly from A(x, y). This fundamental result allows us to determine the stability condition of queue 2 in the asymptotic system, and explicitly compute the classical queueing performance metrics of queue 2, such as the pgf, the moments and the approximate tail distribution of its system content, when this condition is fulfilled. It also leads to accurate approximative closed-form expressions for the joint tail distribution of the system contents in both queues, in the original (unconditional) system. We extensively illustrate our methodology by means of various specific (popular) choices of A(x, y). In some examples, where an explicit solution for U(z_1,z_2) or for the (approximative) joint tail distribution is known, we retrieve the known results easily. In other cases, new results are found for arrival pgfs A(x, y) for which no explicit results were known until now.

Combining unsupervised deep learning and Monte Carlo dropout for seismic data reconstruction and its uncertainty quantification

Many methods, such as multichannel singular spectrum analysis (MSSA) and deep seismic prior (DSP), have been developed for seismic data reconstruction, but they do not quantify the uncertainty of reconstructed traces, relying on the subjective visual inspection of results. Our goal is to quantify the reconstructed uncertainty while recovering missing traces. We develop a framework including an unsupervised deep-learning-based seismic data reconstruction method and the existing Monte Carlo dropout method to achieve this goal. The only information required by our framework is the original incomplete data. A convolutional neural network trained on the original nonmissing traces can simultaneously denoise and reconstruct seismic data. For uncertainty quantification, the Monte Carlo dropout method treats the well-known dropout technique as Bayesian variational inference. This refers to the fact that the dropout technique can be regarded as an approximation to the probabilistic Gaussian process and thus can be used to obtain an approximate distribution (Bernoulli variational distribution) of the posterior distribution. The reconstructed result and uncertainty of the trained model are yielded through multiple Monte Carlo dropout simulations. The analysis of the reconstructed uncertainty quantifies the confidence to use reconstructed traces. Tests on synthetic and field data illustrate that our framework outperforms the MSSA and DSP methods on reconstructed accuracy and quantifies the reconstructed uncertainty as an objective benchmark to guide decision making.

Research on a Model for an Empirical Analysis of Inherent Defect Insurance Based on Ruin Theory

The system of inherent defect insurance is an important measure to serve the real economy through financial means and improve the quality of construction projects, which is the future development direction of China’s construction industry. However, the related research is not perfect due to the short implementation of the insurance. This could bring risks to the promotion of insurance for companies. Insurance ruin theory is an important method in risk management theory, so adopting it to manage the risk inherent defect insurance from the perspective of insurance companies is vital. The research starts with the classic insurance ruin theory and determines the coefficient of premium collection from the perspective of claim settlement distribution expectations. Furthermore, the approximate distribution of claim settlement is deduced, and a comprehensive risk assessment model is constructed. Finally, based on the data of insurance actuarial practice in Shanghai, both the ruin probability of inherent defect insurance in each insuring term and its average required initial reserve are calculated, which provides the analyses on the risks and main subitems of inherent defect insurance as well as relevant suggestions. Finally, the sensitivity analysis is used to further analyse the risk of different insurance stages of IDI, and the relevant measures are proposed. The research can provide theoretical assistances for insurance companies to carry out effective risk management and provide model tools to make scientific decisions.

Probabilistic fatigue cumulative damage model considering the randomness of load and material properties

A probabilistic fatigue cumulative damage model that can be applied to the fatigue life prediction of aluminum alloy under spectrum loading is proposed in this paper. The model is constructed by a novel nonlinear cumulative damage model, the P–S–N curve, and the probabilistic load–exceedance curve. It introduces the randomness of the loading process and fatigue resistance of the material to the deterministic model. In this model, it is derived that the cumulative damage under fleet spectrum loading exhibits an approximate lognormal distribution, and a method to calculate the distribution parameters is provided. Finally, the validity of the proposed probabilistic model is verified through fatigue tests on 2297-T87 aluminum–lithium alloy specimens under random loading, demonstrating its capability to capture the probabilistic nature of the fatigue behavior of aluminum alloy effectively.

Variational Approximation for Adaptive Extended Target Tracking in Clutter With Random Matrix

As a computationally efficient framework, the random matrix approach can simultaneously estimate the kinematic state and extent of the extended target. For the extended target tracking in clutter, the measurement origin uncertainty, the unknown detection probability and measurement rate challenge the existing methods. In this paper, we propose the Beta Gamma Gaussian inverse Wishart filter based on the variational approximation. The proposed method takes the association event as an unknown parameter with a prior distribution. Following a more rigorous path, we derive an approximate posterior distribution of the unknowns using the analytical techniques of variational Bayesian inference. The joint estimations of the kinematic state, extent, detection probability, measurement rate and association events are obtained in this work. The simulation results illustrate the effectiveness and robustness of the proposed approach.

CHILDREN WITH A MIGRATION HISTORY IN THE EDUCATIONAL SYSTEM OF ORENBURG REGION

Schools and kindergartens of Orenburg region, as well as many other regions of Russia, from the early 2000s began to accept children from migrant families of Central Asia and Transcaucasia for education. In the 2010s, this trend intensified, and classes began to be filled with already significant groups of children whose families had moved to the region over the past ten years. Orenburg Region is multinational in itself, so the influx of non-ethnic migrants only increased the multicultural nature of children’s and pedagogical collectives.
 
 The purpose of the study is to determine the level of linguistic and socio-cultural adaptation of children with a migration history in the region, the degree of readiness of Orenburg region educational organizations to receive foreign minors and to form ideas about the attitudes of migrant families and students themselves for full socio-cultural integration into the Russian society.
 
 Methods and materials. The source base of the study was made by statistical materials on migration processes, in-depth interviews and conversations with teachers and administration of educational organizations in Orenburg region, diagnostic examination of non-native speaking schoolchildren in three educational organizations of the region using a special technique.
 
 The scientific novelty of the research lies in combination of applied ethnographic tools with the study of the current situation in the educational system of the region. The authors answer the problematic question: how to organize the work on linguistic and socio-cultural adaptation of foreign-speaking students, depending on the proportion of children with a migration history at school, the level of teachers’ education and qualification on the example of specific educational institutions in the cities Orenburg and Orsk, a suburban district of Orenburg region.
 
 Study results. The data collected in a number of Russian cities in 2021–2022, including in the region under study, are analyzed, which give the opportunity to assert that the proportion of children from migrant families in school and the native language of foreign-speaking children are not the determining factors in the progress of linguistic and socio-cultural adaptation by means of education. The article gives a description of an approximate distribution of students with a migration history participating in the study in four groups, depending on the recommended model of language adaptation. The factor that significantly complicates the activities on linguistic and socio-cultural adaptation of migrant children in the conditions of the Orenburg region is emphasized: this is a “pendulum” migration, as well as long summer and winter holidays, when parents send their children to their homeland “to grandparents”, not really considering the need to support language practice. The data obtained during the study indicate that the majority of students with a migration history in Orenburg region associate their future with Russia: it is very important to support such children at the initial stage of adaptation and training in a Russian school, because, as a rule, they will be motivated to successfully study and move up the social ladder if the language barrier is overcome.
 
 Conclusions. In general, the experience of many years of positive interethnic interaction accumulated in Orenburg region and reasonably structured national and migration policies create the basis for successful adaptation and integration of migrant families arriving in the region. However, if we talk about the educational system of Orenburg region, then in order to successfully solve the tasks of linguistic and socio-cultural adaptation of children with a migration history, Orenburg region schools need additional staff and financial support.

Effect of molecular weight distribution on the structure and properties of polypropylene cast film and stretched microporous membrane

Abstract Two isotactic polypropylene (iPP) resins with approximate weight-average molecular weight (Mw) and different molecular weight distribution (MWD) were chosen to prepare cast films by the melt-stretching method. The lamellar morphology and mechanical properties of the initial precursor films and stretched microporous membranes were characterized. It was found that PP1 resin with a broader MWD exhibits a higher relaxation time of around 5.5 s, a larger entanglement density of 24.7 mol/m3 and fast crystalline ability. The lamellar lateral size, orientation degree and entanglement density of the corresponding cast film are significantly increasing, resulting in higher elastic recovery and more apparent strain-hardening behavior. The PP resin with broader MWD also benefits the pore formation, resulting in a larger pore size and better air permeability of the stretched microporous membrane.

An Autonomous Soaring for Small Drones Using the Extended Kalman Filter Thermal Updraft Center Prediction Method Based on Ordinary Least Squares

Many birds in the natural world are capable of engaging in sustained soaring within thermal updrafts for extended periods without flapping their wings. Autonomous soaring has the potential to greatly improve both the range and endurance of small drones. In this paper, the extended Kalman filter (EKF) thermal updraft center prediction method based on ordinary least squares (OLS) is proposed to develop the autonomous soaring system for small drones, and an adaptive step size update strategy is incorporated into the EKF. The proposed method is compared with EKF thermal updraft prediction methods through simulated experiments. The results indicate that the proposed prediction method has low computational complexity and fast convergence speed and performs more stably in weak thermal updrafts. The above advantages stem from the OLS providing an approximate distribution of the thermal updraft around the drone for the EKF. This empowers the EKF algorithm with ample information to dynamically update the thermal updraft center in real time. The adaptive step size update strategy further accelerates the convergence speed of this process. In addition, flight experiments were conducted on the Talon fixed-wing drone platform to test the autonomous soaring system. During the flight experiment, the drone successfully engaged in static soaring within thermal updrafts, effectively hovering and gaining energy. Throughout the approximately 40 min flight duration, the drone only utilized its propulsion for about 8 min. This demonstrated the effectiveness of the autonomous soaring system using the EKF thermal updraft center prediction method based on OLS. Finally, by analyzing and discussing the differences between the simulation experiment results and the flight experiment results, some improvement strategies for the current work are proposed.

Laplace based Bayesian inference for ordinary differential equation models using regularized artificial neural networks

Parameter estimation and associated uncertainty quantification is an important problem in dynamical systems characterised by ordinary differential equation (ODE) models that are often nonlinear. Typically, such models have analytically intractable trajectories which result in likelihoods and posterior distributions that are similarly intractable. Bayesian inference for ODE systems via simulation methods require numerical approximations to produce inference with high accuracy at a cost of heavy computational power and slow convergence. At the same time, Artificial Neural Networks (ANN) offer tractability that can be utilized to construct an approximate but tractable likelihood and posterior distribution. In this paper we propose a hybrid approach, where Laplace-based Bayesian inference is combined with an ANN architecture for obtaining approximations to the ODE trajectories as a function of the unknown initial values and system parameters. Suitable choices of customized loss functions are proposed to fine tune the approximated ODE trajectories and the subsequent Laplace approximation procedure. The effectiveness of our proposed methods is demonstrated using an epidemiological system with non-analytical solutions—the Susceptible-Infectious-Removed (SIR) model for infectious diseases—based on simulated and real-life influenza datasets. The novelty and attractiveness of our proposed approach include (i) a new development of Bayesian inference using ANN architectures for ODE based dynamical systems, and (ii) a computationally fast posterior inference by avoiding convergence issues of benchmark Markov Chain Monte Carlo methods. These two features establish the developed approach as an accurate alternative to traditional Bayesian computational methods, with improved computational cost.

Interdisciplinary Fundamental Concepts in STEM: Solid State Physics and COVID-19 Pandemic Evolution

The rapid development of computer-aided design tools, such as MATLAB or Octave, or Mathematica enabled students to solve many complicated problems focusing less on underlying STEM-related concepts that are interdisciplinary. Therefore, it is important to demonstrate how it could be done using specific examples that could be linked to different subjects or even to their everyday life experience. This paper reports on using the COVID-19 pandemic evolution model (Shur, 2022) in my class on the physics of advanced semiconductors devices. I use this model to show how the concepts, such as the Born-Oppenheimer approximation and Fermi-Dirac distribution function could be used in a completely different STEM field. In solid-state physics, the Born-Oppenheimer approximation is used to separate rapid electronic motion, relevant to the electronic states and much slower nuclei motion (since nuclei are thousands of times heavier than electrons). Likewise, the COVID-19 model uses a relatively fast pandemic evolution growth or decay constant, a slow function of time itself. In solid-state physics, the Fermi-Dirac distribution function describes the transition from the occupied electronic states to empty electron states with the temperature determining the transition interval. The COVID-19 model uses the generalized Fermi-distribution function to describe the mitigation measures that determine the transition from a high to a lower infection rate. A more accurate COVID-19 evolution model requires a generalized Fermi-Dirac function that accounts for a slow variation of the effect of the mitigation measures with time. In turn, this generalization could be used in solid-state physics to describe the electron temperature increase in the electric field.

EM-test for homogeneity in location-scale mixture model with a structural scale

The testing problem for the homogeneity of a location-scale mixture model is fundamental for subsequent statistical inference. In this paper, we employ the EM-test method to test the homogeneity of two commonly encountered location-scale mixture models with a structural scale, and we also establish the theoretical properties of test statistics. Further, to enhance the reliability of the test, based on the computer experiments approach, we develop a Bartlett-corrected approximate distribution for the proposed test statistics. Through Monte Carlo studies and real data example, the effectiveness of the proposed testing procedure is compared with that of the classical one, especially for small or moderate sample sizes.

Statistical analysis of stochastic magnetic fluctuations in space plasma based on theMMSmission

ABSTRACTBased on the Magnetospheric Multiscale (MMS) mission we look at magnetic field fluctuations in the Earth’s magnetosheath. We apply the statistical analysis using a Fokker–Planck equation to investigate processes responsible for stochastic fluctuations in space plasmas. As already known, turbulence in the inertial range of hydromagnetic scales exhibits Markovian features. We have extended the statistical approach to much smaller scales in space, where kinetic theory should be applied. Here we study in detail and compare the characteristics of magnetic fluctuations behind the bow shock, inside the magnetosheath, and near the magnetopause. It appears that the first Kramers–Moyal coefficient is linear and the second term is a quadratic function of magnetic increments, which describe drift and diffusion, correspondingly, in the entire magnetosheath. This should correspond to a generalization of Ornstein–Uhlenbeck process. We demonstrate that the second-order approximation of the Fokker–Planck equation leads to non-Gaussian kappa distributions of the probability density functions. In all cases in the magnetosheath, the approximate power-law distributions are recovered. For some moderate scales, we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this shape is reduced to the normal Gaussian distribution. It is worth noting that for smaller kinetic scales the rescaled distributions exhibit a universal global scale invariance, consistently with the stationary solution of the Fokker–Planck equation. These results, especially on kinetic scales, could be important for a better understanding of the physical mechanism governing turbulent systems in space and astrophysical plasmas.

Numerical analysis on the transport properties and residence time distribution of ribbon biomass particles in a riser reactor based on CFD-DEM approach

A bended ribbon biomass particle model was developed to explore the dynamic transport properties inside a riser reactor. Residence time distribution (RTD) of the particles was analyzed by using the Eulerian-Lagrange method. The effects of sampling height, particle density, particle size and gas-to-solid mass ratio on RTD were investigated. The coupled Computational Fluid Dynamics and Discrete Element Method (CFD-DEM) model was verified firstly by experimental data on pressure drop and residence time distribution density function. The simulation results demonstrated that the ribbon biomass particles display a typical annular-core spatial distribution during transportation. The RTD of particles exhibit an approximate single-peaked normal distribution. The mean residence time (MRT) can reach up to 0.7 s when the particle density is 1200 kg/m3. Particle with higher density has longer mean residence time. The flow patterns are closer to plug flow if particle length over 12 mm. The particle flow pattern is not sensitive to changes in particle density and size, while the gas-to-material mass ratio has a significant impact on it.