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A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of functions such as machine learning. Based on available finite-dimensional approximate solution manifolds, this paper proposes a novel model reduction framework for kinetic equations. The method employs projections onto tangent bundles of approximate manifolds, naturally resulting in first-order hyperbolic systems. Under certain conditions on the approximate manifolds, the reduced models preserve several crucial properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability. For the first time, this paper rigorously discusses the relation between the H-theorem of kinetic equations and the linear stability conditions of reduced systems, determining the choice of Riemannian metrics involved in the model reduction. The framework is widely applicable for the model reduction of many models in kinetic theory. [ABSTRACT FROM AUTHOR]

In this paper, we present a model describing the collective motion of birds. The model introduces spontaneous changes in direction which are initialized by few agents, here referred to as leaders, whose influence acts on their nearest neighbors, in the following referred to as followers. Starting at the microscopic level, we develop a kinetic model that characterizes the behavior of large flocks with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a k-nearest neighbors search algorithm with a binary tree. Finally, we conduct various numerical experiments for different scenarios to validate the algorithm's effectiveness and investigate collective dynamics in both two and three dimensions. [ABSTRACT FROM AUTHOR]

We propose a kinetic model for understanding the link between opinion formation phenomena and epidemic dynamics. The recent pandemic has brought to light that vaccine hesitancy can present different phases and temporal and spatial variations, presumably due to the different social features of individuals. The emergence of patterns in societal reactions permits to design and predict the trends of a pandemic. This suggests that the problem of vaccine hesitancy can be described in mathematical terms, by suitably coupling a kinetic compartmental model for the spreading of an infectious disease with the evolution of the personal opinion of individuals, in the presence of leaders. The resulting model makes it possible to predict the collective compliance with vaccination campaigns as the pandemic evolves and to highlight the best strategy to set up for maximizing the vaccination coverage. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic. [ABSTRACT FROM AUTHOR]

We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels. [ABSTRACT FROM AUTHOR]

A hypothesis is formulated and substantiated that quasi-static deformation in an amorphous metal alloy is a complex relaxation multi-stage process, which is a hierarchical sequence of interrelated structural transitions of the first order ordered in time. These nonequilibrium processes sequentially proceed at different scale space-time levels, starting from the lowest level—a cluster of atoms of the first coordination sphere with a relaxation time τη, then the middle level—a nanocluster of atoms of the fifth coordination sphere with a relaxation time τφ, spatial scale of 10 nm and relaxation time τ, and τ τφ τη. They are accompanied by transformations of various types of potential energy of atoms (elastic, inelastic, plastic deformation, ZST) into each other. A mechanism and a model of a nonequilibrium transition from an elastic mechanical state to a state with shear transformation zones, a mechanism and a model of localized plastic deformation in an amorphous metal alloy are constructed. In the interval of non-uniqueness, in response to a locally introduced perturbation, a traveling autowave arises, which transfers the slip band from the inelastic deformation regime to the plastic deformation regime. Model parameters are estimated and important physical properties of plastic deformation are calculated. [ABSTRACT FROM AUTHOR]

In models of opinion dynamics, many parameters -- either in the form of constants or in the form of functions -- play a critical role in describing, calibrating, and forecasting how opinions change with time. When examining a model of opinion dynamics, it is beneficial to infer its parameters using empirical data. In this paper, we study an example of such an inference problem. We consider a mean-field bounded-confidence model with an unknown interaction kernel between individuals. This interaction kernel encodes how individuals with different opinions interact and affect each other's opinions. Because it is often difficult to quantitatively measure opinions as empirical data from observations or experiments, we assume that the available data takes the form of partial observations of a cumulative distribution function of opinions. We prove that certain measurements guarantee a precise and unique inference of the interaction kernel and propose a numerical method to reconstruct an interaction kernel from a limited number of data points. Our numerical results suggest that the error of the inferred interaction kernel decays exponentially as we strategically enlarge the data set. [ABSTRACT FROM AUTHOR]

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: proper orthogonal decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD's precision improves as more modes are considered. Notably, our work recognizes that the classical POD model order reduction approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation. [ABSTRACT FROM AUTHOR]

We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker–Planck equations, and one-dimensional functional inequalities of the type of Poincaré and Wirtinger, with weight, for probability densities in the form of one-dimensional Barenblatt solutions of the porous medium equation. It it also shown that Hardy's classical one-dimensional inequality can be obtained resorting to the differential expression of the Cauchy-type density, as given by a related Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

The relevance of the work is related to the solution of environmental problems arising from an increase in the amount of industrial waste water contaminated with organic dyes. The aim of the work was to optimize the process of wastewater treatment from organic dyes - methylene blue, methylviolet and Congo red, when using copper-zinc ferrite, taking into account the change in its mechanism over time and varying the process parameters: the concentration of dye solutions or ferrite mass. Quantitative kinetic characteristics of the process of cleaning solutions from organic dyes with a ferrite composite material of the composition Zn0.875Cu0.1Fe4.42O4 at various mass ratios n = "ferritic composite material: organic dye" are determined. It was proved that the purification of wastewater from organic dyes proceeds through the mechanisms of photocatalytic transformations and sorption and the kinetic characteristics of photocatalytic processes are an order of magnitude higher than those for the adsorption of dyes. An experimental database on wastewater treatment from dyes was created for each mechanism separately. The kinetic equations for the dependence of the concentration of dyes on time and the ratio n are calculated. The cleaning process was optimized for one or two process mechanisms progress while varying the process time, the initial dye concentration or ferrite mass, depending on the need to achieve certain process rates and the depth of cleaning. A scheme of a method for optimizing the process of wastewater treatment from organic dyes using a copper-zinc ferrite composite is proposed. The results are aimed at improving the efficiency and completeness of the purification process. A significant increase in the economic effect of the introduction of the cleaning process is predicted, since it is proposed to use waste from galvanic production for the production of copper-zinc ferrites. [ABSTRACT FROM AUTHOR]

In this work we define a kinetic model for understanding the impact of heterogeneous opinion formation dynamics on epidemics. The considered many-agent system is characterized by nonsymmetric interactions which define a coupled system of kinetic equations for the evolution of the opinion density in each compartment. In the quasi-invariant limit we may show positivity and uniqueness of the solution of the problem together with its convergence towards an equilibrium distribution exhibiting bimodal shape. The tendency of the system towards opinion clusters is further analyzed by means of numerical methods, which confirm the consistency of the kinetic model with its moment system whose evolution is approximated in several regimes of parameters. [ABSTRACT FROM AUTHOR]

The study included two soil sites of different order (Inceptisols and Vertisols) within Nineveh Governorate, Iraq. Soil samples were taken from the surface depth (30 cm) naturally by means of columns. The experiment was carried out by adding two types of water (river water and well water) to each column with a volume equivalent to the pore size, and alternating wetting and drying were done for ten cycles, and the period between one cycle and another was 10 days. Soil samples were analyzed after the first cycle, the fifth cycle and the tenth cycle to find out the effect of the number of hydration cycles on the exchanged ions, the relative activity RA and the Gapone’s constant KG. The results indicated that the amount released from the exchanged sodium and potassium ions in vertisols was generally greater than the amount released in the soil of inceptisols when using both types of water, while the relative activity values of sodium were higher when wetting with well water except for the first cycle in the soil of inceptisols, while the relative activity of potassium in the two soils of the study, the values were higher when wetting with river water and for all cycles except the fifth cycle in inceptisols, while the Gapone’s constant for sodium in the two study soils had higher values, when wetting with river water in the first and fifth cycles, but in the tenth cycle the values were higher when wetting with well water, while the Gapone’s constant for potassium in the two study soils, the values were higher when wetting with well water and for all cycles except for the fifth cycle in inceptisols. The following kinetic equations were applied: Zero order equation, First order equation, Diffusion equation, Eluvich equation and Power function equation, the power function equation was the best equation that describes the process of release sodium and potassium ions in the study soil. [ABSTRACT FROM AUTHOR]

IntroductionPhosphorus (P) is one of the most important elements necessary for plant growth and production of agricultural products. In calcareous soils, phosphorus deficiency is a general issue due to high pH, high soil calcium carbonate content, lack of organic matter and moisture. Phosphorus absorption capacity depends on different soil reactions such as: adsorption, sedimentation, stabilization and release. The speed and amount of plant available P depends on the soil reactions. Studying the kinetics of P release from soil is a good indicator to check the status of P uptake by plant. The kinetics of P release in soils is a subject of importance in soil and environmental sciences. The aim of this research was to investigate the kinetics of P release and derive the most suitable equation to describe the release of P from a calcareous soil when subjected to the acidification of rock phosphate and the addition of vermicompost. Materials and MethodsIn order to investigate the ability of acidified rock phosphate and vermicompost in P release, an experiment was conducted with 2 replications on a light-textured soil with low OC and Olsen-P (1.2 mg/kg). One hundred grams air dried calcareous soil was transferred into special containers and 5 treatments including: 1- control (soil), 2- rock phosphate, 3- acidified rock phosphate (20 CC nitric acid 0.1 N and 5 g rock phosphate), 4- vermicompost, and 5- acidified vermicompost (20 CC nitric acid 0.1 N and 5 g vermicompost) were applied. The treatments incubated two weeks in 20±2℃ temperature. The Kinetics of P release was studied by adding 20 mL of 0.5N NaHCO3 to, one gram of air dried treatments. Extraction times were considered to be 0.25 h to 256 h (in 11 times) based on the time of adding the NaHCO3 extractant until filtering. After adding the extractant, the samples were shaken and centrifuged. After filtering, the concentration of released P in samples were determined by spectrophotometer (Model: CE 292 Series2, ultraviolet). For higher accuracy in the measurements, acid-washed containers were adjusted based on the amount of soil moisture which was dried in the oven (105℃). Finally, the P release data were fitted to different kinetic equations. The effect of different fertilizer treatments on P release in specified times and then kinetics parameters were investigated and compared with the control. Results and DiscussionAddition of acidified and non acidified rock phosphate and vermicompost increased the amount and speed of P release in the calcareous soil. Six kinetic equations were fitted to describe the release of P in the period of 0.25 h to 256 h from the soil to evaluate the effect of the treatments. The highest release of P was in vermicompost and acidified rock phosphate treatment, which were an organic fertilizer and a source for preparing phosphate fertilizers. To describe the release rate, kinetic equations were used. The best equations were chosen by highest coefficient of determination (R2) and the least of standard error (SE). The zero, first, second order equations could not describe the release of P in the studied calcareous soil. The R2 value decreased from the zero to second order equation. The simplified Elovich equation described well the release of P from the soil with the average R2 of 0.79 and with the average SE of 0.4. Comparison of the average effect of the studied treatments with the control showed that the acidifed vermicompost and rock phosphate treatments increased the capacity and speed of P release compared to the control. On the other hand, acid addition has increased the capacity and speed of P release in the calcareous soil. ConclusionThe findings indicated an initial rapid release of P, which then decreased over time. Notably, the application of vermicompost and the acidification of the soil with rock phosphate resulted in a pronounced and accelerated release of P. Generally, organic fertilizer treatments exhibited a higher release of P compared to chemical fertilizer treatments. This observation is in accordnce with the findings of the data presented by Ghorbanzadeh et al. (2009), who explored the P release potential of bone meal. Their data demonstrated that the acidification of bone meal accelerated and enhanced P release. To further enhance the practical relevance of these results, it is recommended to conduct this research in the presence of plants.

In this paper, the propagation law of online teaching knowledge is studied by constructing a learner online teaching interaction network to characterize the position of each learner in this social network. The SI model is used to portray the knowledge dissemination process in the online learning interactive network, fully considering the two types of nodes of teachers and learners as well as the different transfer rates of knowledge dissemination of the two types of nodes, the SI model is improved, and the kinetic equations of knowledge dissemination are constructed. Finally, with the help of Matlab software to simulate the knowledge propagation process in the online English learning of teachers and students, the model was examined, corrected and analyzed, respectively, so as to put forward the optimization strategy of online English teaching interaction mode in colleges and universities. The results show that the path process from t0 to full dissemination is almost the same in the two online learning interaction networks with N1 =2000 and N2 =1000. The higher the degree of connection and participation among learners, the better the interactive learning effect. The optimization strategy for online English teaching interaction modes in colleges and universities proposed in this paper provides certain reference suggestions for improving learners’ interactive learning effectiveness.

Abstract—A new approach to the solution of kinetic equations describing the process of formation of anhysteretic remanent magnetization (ARM) is proposed, which made it possible to accelerate the numerical calculation of the process of formation of ARM by two orders of magnitude for uniaxial oriented non-interacting single-domain particles, while practically not yielding in accuracy to a strict numerical solution. It follows from the calculation results that the susceptibility of ARM is entirely determined by the magnitude of the particle's coercivity parameter . The data of the previous approximate calculations of ARM value are compared with the exact results presented here and it is shown that the discrepancy between the exact data and the approximate estimates increases with the growth of g, but remains relatively small, within 23%. The proposed algorithm for the rapid calculation of kinetic equations allows us to analyze with physical rigor the method of pseudo-Thellier estimation of paleointensity for an ensemble of single-domain particles, which is supposed to be done in subsequent works. [ABSTRACT FROM AUTHOR]

Introduction Phosphorus (P) is one of the most important elements necessary for plant growth and production of agricultural products. In calcareous soils, phosphorus deficiency is a general issue due to high pH, high soil calcium carbonate content, lack of organic matter and moisture. Phosphorus absorption capacity depends on different soil reactions such as: adsorption, sedimentation, stabilization and release. The speed and amount of plant available P depends on the soil reactions. Studying the kinetics of P release from soil is a good indicator to check the status of P uptake by plant. The kinetics of P release in soils is a subject of importance in soil and environmental sciences. The aim of this research was to investigate the kinetics of P release and derive the most suitable equation to describe the release of P from a calcareous soil when subjected to the acidification of rock phosphate and the addition of vermicompost. Materials and Methods In order to investigate the ability of acidified rock phosphate and vermicompost in P release, an experiment was conducted with 2 replications on a light-textured soil with low OC and Olsen-P (1.2 mg/kg). One hundred grams air dried calcareous soil was transferred into special containers and 5 treatments including: 1-control (soil), 2- rock phosphate, 3- acidified rock phosphate (20 CC nitric acid 0.1 N and 5 g rock phosphate), 4- vermicompost, and 5- acidified vermicompost (20 CC nitric acid 0.1 N and 5 g vermicompost) were applied. The treatments incubated two weeks in 20±2℃ temperature. The Kinetics of P release was studied by adding 20 mL of 0.5N NaHCO3 to, one gram of air dried treatments. Extraction times were considered to be 0.25 h to 256 h (in 11 times) based on the time of adding the NaHCO3 extractant until filtering. After adding the extractant, the samples were shaken and centrifuged. After filtering, the concentration of released P in samples were determined by spectrophotometer (Model: CE 292 Series2, ultraviolet). For higher accuracy in the measurements, acid-washed containers were adjusted based on the amount of soil moisture which was dried in the oven (105℃). Finally, the P release data were fitted to different kinetic equations. The effect of different fertilizer treatments on P release in specified times and then kinetics parameters were investigated and compared with the control. Results and Discussion Addition of acidified and non acidified rock phosphate and vermicompost increased the amount and speed of P release in the calcareous soil. Six kinetic equations were fitted to describe the release of P in the period of 0.25 h to 256 h from the soil to evaluate the effect of the treatments. The highest release of P was in vermicompost and acidified rock phosphate treatment, which were an organic fertilizer and a source for preparing phosphate fertilizers. To describe the release rate, kinetic equations were used. The best equations were chosen by highest coefficient of determination (R2) and the least of standard error (SE). The zero, first, second order equations could not describe the release of P in the studied calcareous soil. The R2 value decreased from the zero to second order equation. The simplified Elovich equation described well the release of P from the soil with the average R2 of 0.79 and with the average SE of 0.4. Comparison of the average effect of the studied treatments with the control showed that the acidifed vermicompost and rock phosphate treatments increased the capacity and speed of P release compared to the control. On the other hand, acid addition has increased the capacity and speed of P release in the calcareous soil. Conclusion The findings indicated an initial rapid release of P, which then decreased over time. Notably, the application of vermicompost and the acidification of the soil with rock phosphate resulted in a pronounced and accelerated release of P. Generally, organic fertilizer treatments exhibited a higher release of P compared to chemical fertilizer treatments. This observation is in accordnce with the findings of the data presented by Ghorbanzadeh et al. (2009), who explored the P release potential of bone meal. Their data demonstrated that the acidification of bone meal accelerated and enhanced P release. To further enhance the practical relevance of these results, it is recommended to conduct this research in the presence of plants. [ABSTRACT FROM AUTHOR]

In previous papers of the authors, a generalized evolutionary approach was developed for the analysis of popular inspection and corruption games. Namely, a two-level hierarchy was studied, where a local inspector I of a pool of agents (that may break the law) can be corrupted and is further controlled by the higher authority A. Here, we extend this two-level modeling by answering the following questions: (i) what levels of illegal profit r of violators and what level of bribes α (fraction of illegal profit asked as a bribe from a violator) of an inspector are feasible, that is, realizable in stable equilibria of generalized replicator dynamics; and (ii) what α can be optimal for a corrupted inspector that aims at maximizing the total profit. Concrete settings that we have in mind are illegal logging, the sales of products with substandard quality, and tax evasion. [ABSTRACT FROM AUTHOR]

Kinetic equations model the position-velocity distribution of particles subject to transport and collision effects. Under a diffusive scaling, these combined effects converge to a diffusion equation for the position density in the limit of an infinite collision rate. Despite this well-defined limit, numerical simulation is expensive when the collision rate is high but finite, as small time steps are then required. In this work, we present an asymptotic-preserving multilevel Monte Carlo particle scheme that makes use of this diffusive limit to accelerate computations. In this scheme, we first sample the diffusive limiting model to compute a biased initial estimate of a quantity of interest, using large time steps. We then perform a limited number of finer simulations with transport and collision dynamics to correct the bias. The efficiency of the multilevel method depends on being able to perform correlated simulations of particles on a hierarchy of discretization levels. We present a method for correlating particle trajectories and present both an analysis and numerical experiments. We demonstrate that our approach significantly reduces the cost of particle simulations in highcollisional regimes, compared with prior work, indicating significant potential for adopting these schemes in various areas of active research. [ABSTRACT FROM AUTHOR]

This paper is devoted to the comparison of discrete velocity models used for simulation of compressible flows with arbitrary specific heat ratios in the lattice Boltzmann method. The stability of the governing equations is analyzed for the steady flow regime. A technique for the construction of stability domains in parametric space based on the analysis of eigenvalues is proposed. A comparison of stability domains for different models is performed. It is demonstrated that the maximum value of macrovelocity, which defines instability initiation, is dependent on the values of relaxation time, and plots of this dependence are constructed. For double-distribution-function models, it is demonstrated that the value of the Prantdl number does not seriously affect stability. The off-lattice parametric finite-difference scheme is proposed for the practical realization of the considered kinetic models. The Riemann problems and the problem of Kelvin–Helmholtz instability simulation are numerically solved. It is demonstrated that different models lead to close numerical results. The proposed technique of stability investigation can be used as an effective tool for the theoretical comparison of different kinetic models used in applications of the lattice Boltzmann method. [ABSTRACT FROM AUTHOR]

Based on some recent experimental evidence, kinetics of cluster growth in a fullerene C60 low-polar solution are considered within a model based on oxides C60Ox as aggregation initializers. A system of equations for the time-evolution of the cluster-size distribution function is developed. First stage of clusters growth consists in formation of packed small C60 clusters around separate fullerene oxides. For this stage analytical solutions of equations were obtained and compared with numerical results. During the next stages, cluster-cluster aggregation events start to play a role. First numerical results for stage II are obtained and discussed. The results are compared with experimental study of clusters growth in the saturated C60/benzene solution by dynamic light scattering. [ABSTRACT FROM AUTHOR]

We explore the critical and scaling behavior of delayed bifurcations in systems with dynamic disorder well-known in kinetics when the control parameter in the rate is made to sweep stochastically across the critical point. The analysis is based on an extended Hamiltonian system that includes noise as an additional degree of freedom conjugate to the relevant dynamical variable. Appropriate quantifiers for measuring the time delay in reaching the bifurcation point have been introduced. We show that the time delay (i.e., the time difference between the static and dynamic bifurcation times) exhibits a power law decay with the deviation of the control parameter from its critical value, the decay constant being close to unity and is independent of the nature of bifurcations. The characteristic time to reach the zero solution state decreases algebraically with the deviation of the dynamical variable from its critical value and the decay exponent scales as the highest power of nonlinearity characterizing the nature of bifurcations. Bifurcations are essential features of various chemical kinetics; biological and ecological evolution. Critical and scaling behavior of delayed bifurcations in such systems with dynamic disorder has been explored when the control parameter in the rate is varied stochastically across the critical point. [ABSTRACT FROM AUTHOR]

In this series of works, we develop a discrete-velocity-direction model (DVDM) with collisions of BGK-type for simulating rarefied flows. Unlike the conventional kinetic models (both BGK and discrete-velocity models), the new model restricts the transport to finite fixed directions but leaves the transport speed to be a 1-D continuous variable. Analogous to the BGK equation, the discrete equilibriums of the model are determined by minimizing a discrete entropy. In this first paper, we introduce the DVDM and investigate its basic properties, including the existence of the discrete equilibriums and the H-theorem. We also show that the discrete equilibriums can be efficiently obtained by solving a convex optimization problem. The proposed model provides a new way in choosing discrete velocities for the computational practice of the conventional discrete-velocity methodology. It also facilitates a convenient multidimensional extension of the extended quadrature method of moments. We validate the model with numerical experiments for two benchmark problems at moderate computational costs. [ABSTRACT FROM AUTHOR]

We study a regularized version of the Landau equation, which was recently introduced in [J. A. Carrillo, J. Hu, L. Wang and J. Wu, A particle method for the homogeneous Landau equation, J. Comput. Phys. X7 (2020) 100066, 24] to numerically approximate the Landau equation with good accuracy at reasonable computational cost. We develop the existence and uniqueness theory for weak solutions, and we reinforce the numerical findings in the above-mentioned paper by rigorously proving the validity of particle approximations to the regularized Landau equation. [ABSTRACT FROM AUTHOR]

Understanding the impact of collective social phenomena in epidemic dynamics is a crucial task to effectively contain the disease spread. In this work, we build a mathematical description for assessing the interplay between opinion polarization and the evolution of a disease. The proposed kinetic approach describes the evolution of aggregate quantities characterizing the agents belonging to epidemiologically relevant states and will show that the spread of the disease is closely related to consensus dynamics distribution in which opinion polarization may emerge. In the present modelling framework, microscopic consensus formation dynamics can be linked to macroscopic epidemic trends to trigger the collective adherence to protective measures. We conduct numerical investigations which confirm the ability of the model to describe different phenomena related to the spread of an epidemic. [ABSTRACT FROM AUTHOR]

Discrete kinetic equations describing binary processes of agglomeration and fragmentation are considered using formal equivalence between the kinetic equations and the geodesic equations of some affinely connected space A associated with the kinetic equation and called the kinetic space of affine connection. The geometric properties of equations are treated locally in some coordinate chart (x ; U) . The peculiarity of the space A is that in the coordinates (x) of some selected local chart, the Christoffel symbols defining the affine connection of the space A are constant. Examples of the Smoluchowski equation for agglomeration processes without fragmentation and the exchange-driven growth equation are considered for small dimensions in terms of geodesic equations. When fragmentation is taken into account, the kinetic equations can be written as equations of quasigeodesics. Particular cases of spaces with symmetries are discussed. [ABSTRACT FROM AUTHOR]

Deterministic models for radiation transport describe the density of radiation particles moving through a background material. In radiation therapy applications, the phase space of this density is composed of energy, spatial position and direction of flight. The resulting six-dimensional phase space prohibits fine numerical discretizations, which are essential for the construction of accurate and reliable treatment plans. In this work, we tackle the high dimensional phase space through a dynamical low-rank approximation of the particle density. Dynamical low-rank approximation (DLRA) evolves the solution on a low-rank manifold in time. Interpreting the energy variable as a pseudo-time lets us employ the DLRA framework to represent the solution of the radiation transport equation on a low-rank manifold for every energy. Stiff scattering terms are treated through an efficient implicit energy discretization and a rank adaptive integrator is chosen to dynamically adapt the rank in energy. To facilitate the use of boundary conditions and reduce the overall rank, the radiation transport equation is split into collided and uncollided particles through a collision source method. Uncollided particles are described by a directed quadrature set guaranteeing low computational costs, whereas collided particles are represented by a low-rank solution. It can be shown that the presented method is L2-stable under a time step restriction which does not depend on stiff scattering terms. Moreover, the implicit treatment of scattering does not require numerical inversions of matrices. Numerical results for radiation therapy configurations as well as the line source benchmark underline the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension d and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties. [ABSTRACT FROM AUTHOR]

In this paper we deal with two aspects of the Covid epidemic. The first is a phase change during the epidemic. The empirical observation is that once a certain threshold of active infections is reached, the rate of infection is increasing significantly. This threshold depends, among others, also on the season. We model this phenomenon as a jump in the coefficient of the virus exposition, giving the force of infection. In a chemical mass action law this coefficient corresponds to the reaction rate. We get a free boundary problem in time, which exhibits deterministic ‘metastability’. In a population which is in a state of herd immunity, still, if the number of imported infections is large enough, an epidemic wave can start. The second aspect is the two scale nature of the infection network. On one hand side, there is always a finite number of reoccuring–deterministic–contacts, and on the other hand there is a large number of possible random contacts. We present a simple example, where the group size of deterministic contacts is two, and the graph of random contacts is complete. [ABSTRACT FROM AUTHOR]

We study a hierarchy of models based on kinetic equations for the descriptions of traffic ow in presence of autonomous and human-driven vehicles. The autonomous cars considered in this paper are thought of as vehicles endowed with some degree of autonomous driving which decreases the stochasticity of the drivers' behavior. Compared to the existing literature, we do not model autonomous cars as externally controlled vehicles. We investigate whether this feature is enough to provide a stabilization of traffic instabilities such as stop and go waves. We propose two indicators to quantify traffic instability and we find, with analytical and numerical tools, that traffic instabilities are damped as the penetration rate of the autonomous vehicles increases. [ABSTRACT FROM AUTHOR]

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform an L² stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector-splitting integrator [C. Lubich and I. V. Oseledets, BIT, 54 (2014), pp. 171-188] when first discretizing the equations and then applying the DLRA. Based on this we propose a projector-splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator [G. Ceruti and C. Lubich, BIT, 62 (2021), pp. 23--44] has more favorable stability properties. Moreover, we explain why the projector-splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented. [ABSTRACT FROM AUTHOR]

Rumor, as an important form of information dissemination, has always been a research hotspot in the field of complex networks. How to better understand the rules of rumor propagation and establish a practical dissemination model is a significant challenge. To further study the state transfer in information transmission, this paper established the Ignorant-Spreader-Stifler-Transition (ISRT) model, introduced different influence mechanisms and calculate the influence rate accurately by function. Specifically, (1) Based on SIR model, this paper introduces the transition state, considering that transition may awaken spontaneously to spread rumors due to individual cognition. In this paper, the ratio of the current communicator and the degree of doubt of the transition are introduced into the spontaneous arousal function. (2) This paper redefines the propagation probability function and the forgetting probability function, and introduces the time function to describe the rate from the propagator to the restorer. (3) Due to the presence of highly emotional leader propagators in the network who would awaken the immune to spread rumors again, the model added a link from the recovering person to the infected person. Finally, the nonpropagation equilibrium point E 0 and propagation equilibrium point E 1 are obtained by establishing the mean field equation. The experimental results show that different influencing mechanisms can more accurately locate the stage change of rumor transmission, which provides theoretical support for more effective control of information transmission. [ABSTRACT FROM AUTHOR]

A mathematical model is proposed for studying the processes in a stationary plasma thruster (SPT), taking into account the ionization of the working substance, xenon, based on the hybrid equations of electromagnetic hydrodynamics (EMHD) of the plasma, which fully take into account the inertia of electrons. The choice of an EMHD model for studying plasma processes is predetermined by their small scale and low concentration of plasma particles in an SPT. The 1D2V case of plane symmetry is considered in detail, for which a numerical algorithm for studying solutions of hybrid equations of EMHD equations based on the method of macroparticles is constructed. A number of fundamental questions are solved: calculation of average values, interpolation, construction of the initial distribution of macroparticles, the choice of boundary conditions for the electric field, etc. The results of calculations with and without allowance for induction fields in a plasma thruster are presented. The effect of induction fields generated by plasma currents on processes in an SPT and the role of electron inertia have not been studied before, and the results obtained are original. In particular, a new nontraditional scheme for calculating the electric field based on the generalized Ohm's law is proposed, which in EMHD is reduced to a boundary value problem for an elliptic system of equations for the components of the electric field and, among other things, requires setting boundary conditions. The need for the spatial and temporal averaging of electromagnetic fields when calculating the acceleration of the thruster plasma, taking into account the induction field, is an important result. [ABSTRACT FROM AUTHOR]

We present a hybrid method for time-dependent particle transport problems that combines Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For spatial discretizations, the MC algorithm computes a piecewise constant solution and the discrete ordinates use bilinear discontinuous finite elements. From the hybridization of the problem, the resulting problem solved by Monte Carlo is scattering free, resulting in a simple, efficient solution procedure. Between time steps, we use a projection approach to "relabel" collided particles as uncollided particles. From a series of standard 2-D Cartesian test problems we observe that our hybrid method has improved accuracy and reduction in computational complexity of approximately an order of magnitude relative to standard discrete ordinates solutions. [ABSTRACT FROM AUTHOR]

[Display omitted] • Chcl-CA based UAE showed high efficiency in the green extraction of GSP. • Kinetic model is established to describe the extraction process of GSPs. • UAE-NADES enhanced the antifungal ability of GSPs. Ultrasonic-assisted extraction using a natural deep eutectic solvent (UAE-NADES) is an efficient method for extracting grape seed polyphenols (GSPs). In this study, response surface methodology was used to optimize the extraction of GSPs with UAE-NADES, and the theoretical extraction rate of GSPs was 139.014 mg GAE/g, the actual extraction rate was 135.78 ± 1.3 mg GAE/g. A pseudo-second-order kinetic extraction fitting was established to simulate the extraction process and mechanism (R2 > 0.99). Analysis of antioxidant capacity, Fourier-infrared spectroscopy and scanning electron microscopy revealed that UAE-NADES works synergetically to maintain the stability of extracted GSPs. The results of high-performance liquid chromatography showed that catechin (41.14 mg/g) is the main component of GSPs in the extract. The UAE-NADES extraction of GSPs can inhibit the growth of Alternaria alternata at 0.25 mg GAE/g, while the GSPs extracted by other methods can effectively inhibit the growth of Alternaria alternata at 0.35 mg GAE/g. Thus, this study demonstrates that UAE-NADES is a high-efficiency means of extracting GSPs and, in a wider sense, is a promising extraction technology for the green utilization of waste resources. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic Lévy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability and convergence of the scheme. Along the way, we introduce new tools of discrete functional analysis, such as discrete non-local Poincaré and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations. [ABSTRACT FROM AUTHOR]

A system of kinetic equations describing the changes in the concentration of reactive oxygen species (ROS) in aqueous solutions of proteins was obtained from the analysis of chemical reactions involving singlet oxygen. Applying the condition of the stationarity of the intermediate products to the system, we determined the functional dependence of the hydrogen peroxide concentration on the protein concentration under the action of thermal and laser radiation. An approximate analytical solution to the nonlinear system of differential equations that define the ROS concentration dynamics was found. For aqueous solutions of bovine serum albumin (BSA) and bovine gamma globulin (BGG), the orders and rate constants of the reactions describing the ROS conversions were determined by minimizing the sum of squared deviations of the functions found by solving both the static and dynamic problems from experimentally measured dependences. When solving the optimization problem, the Levenberg–Marquardt algorithm was used. [ABSTRACT FROM AUTHOR]

A subsonic rarefied gas flow past a rectangular cylinder of infinite span is studied by applying a numerical method based on the solution of the S-model kinetic equation. The effect exerted on the resulting flow field by the Reynolds number Re∞ ranging from 10 to 200 is analyzed. At Re∞ = 200 the effect of the cylinder geometry on the flow field is investigated by varying the aspect ratio AR of the cylinder from 1 to 8. The results are presented in terms of the drag, lift, and pressure coefficients and the Strouhal number. The flow patterns downstream of the cylinder exhibit a recirculation region whose size and shape depend on both Re∞ and AR. In the case of steady flow, it is found that the flow-characterizing coefficients depend strongly on the Reynolds number. In the case of unsteady flow, this dependence becomes weaker. As AR increases, the recirculation region downstream of the cylinder is reduced, which leads to a decrease in the drag coefficient. Additionally, the reliability of the approach applied to a class of similar problems is estimated by comparing the present results with data available in the literature. [ABSTRACT FROM AUTHOR]

First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this article we construct global in time weak solutions to a first order mean field games system involving kinetic transport operators, where the costs are local (hence non-regularizing) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. This is achieved by characterizing solutions through two convex optimization problems in duality. As part of our approach, we develop tools for the analysis of mean field games on a non-compact domain by variational methods. We introduce a notion of 'reachable set', built from the initial measure, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L1-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we prove higher order Sobolev estimates of the solutions. [ABSTRACT FROM AUTHOR]

We study the propagation of a disease in a population where agents are characterized by their awareness level, representing the measures they take to avoid the infection. We introduce another agent, the government, which is constantly sending a message to the population trying to steer the mean awareness to a value which should ensure the extinction of the disease. We propose three levels to analyze this model. First, an agent-based model, which we use later to derive a mean-field system of ordinary differential equations; and finally, we propose a kinetic approach to model the evolution of the distribution of agents on the awareness levels. We obtain nonlinear systems of different dimension, first an ODE-Boltzmann system and later an ODEs-PDE system, where a Boltzmann or a first order, non-local partial differential equation are coupled with two ordinary differential equations that describe the evolution of the epidemic and the response of the government. We prove the existence and uniqueness of solutions in an abstract setting. Finally, we consider stubborn agents that are not willing to apply protection measures. [ABSTRACT FROM AUTHOR]

A laboratory experiment was conducted to study the release of phosphorous. the study included taking three different samples in their content of gypsum (243,138.67) g kg-1 observations, symbolized by (G1, G2, and G3) and different in their physical and chemical properties, ... Bacillus cereus ,Pseudomonas putida to find out the efficiency of bacteria in releasing phosphorous, the amount of phosphorous released at the functional periods of 2, 4, 8, 24 hours and 2, 4, 7, 14, 21 days was calculated and the amount of phosphorous for each time and in order to obtain the best equation to describe the kinetics of phosphorus liberation in different in its content From gypsum, kinematics equations were used (zero order equation, first order equation, diffusion, forces function, elovig)The results showed that the amount of liberated phosphorous decreases with the increase in the amount of gypsum and increases with the increase in the incubation period. The lowest amount of liberated phosphorous in the G3 sample at the time of 2 hours was 1.7 mg p L-1 for the uninoculated treatment. The highest amount of liberated phosphorus was in the G1 sample at the time of 21 days. It was 6.9 mg p L-1 for the treatment inoculated with P putida it had the advantage in increasing the liberated amount of phosphorous, and the best kinetic equation described phosphorous liberation for each soil of the study was the equation of the force function, and the values ​​of the coefficient of liberation speed for sample G1 amounted to (0.025, 0.89, 0.089) mg kg day-1 and for sample G2 it amounted to (0.025, 0.094, 0.078) mg kg day-1, while the sample G3 was (0.044, 0.134, 0.139) mg kg day-1 for unvaccinated treatments, B. cereus and P. putida respectively