Approximation Method Research Articles

A modified perturbation method for global dynamic analysis of generalized mixed Rayleigh–Liénard oscillator with cubic and quintic nonlinearities

Deriving analytical relationship between the system parameters and amplitude of the limit cycle is a meaningful and challenging task. Currently, numerous existing analytical approximate methods struggle to achieve this goal when expressions of restoring force or nonlinear damping is complicated. To overcome this shortcoming, this study proposes a modified generalized harmonic function perturbation method. Using the proposed method, a generalized mixed RayleighLiénard oscillator with cubic and quintic nonlinearities was investigated. The analytical relationships between the system parameters and amplitude of the limit cycle, as well as the expression of its characteristic quantity, were derived. By employing these analytical relationships, the existence, stability, number, position, and amplitude of each limit cycle are quantitatively analysed. The homoclinic and heteroclinic bifurcations were also predicted using the above analytical relationships. Additionally, analytical approximate solutions for this oscillator were calculated using the proposed method. All results obtained in this study were subsequently confirmed numerically to demonstrate their feasibility and validity. Consequently, the proposed method can be considered an effective supplement to perturbation-based methods. This also implies that the work presented in this paper has a certain theoretical significance and application value in the research area of quantitative analysis methods for strongly nonlinear oscillators.

Coverage Path Planning with Adaptive Hyperbolic Grid for Step-Stare Imaging System

Step-stare imaging systems are widely used in aerospace optical remote sensing. In order to achieve fast scanning of the target region, efficient coverage path planning (CPP) is a key challenge. However, traditional CPP methods are mostly designed for fixed cameras and disregard the irregular shape of the sensor’s projection caused by the step-stare rotational motion. To address this problem, this paper proposes an efficient, seamless CPP method with an adaptive hyperbolic grid. First, we convert the coverage problem in Euclidean space to a tiling problem in spherical space. A spherical approximate tiling method based on a zonal isosceles trapezoid is developed to construct a seamless hyperbolic grid. Then, we present a dual-caliper optimization algorithm to further compress the grid and improve the coverage efficiency. Finally, both boustrophedon and branch-and-bound approaches are utilized to generate rotation paths for different scanning scenarios. Experiments were conducted on a custom dataset consisting of 800 diverse geometric regions (including 2 geometry types and 40 samples for 10 groups). The proposed method demonstrates comparable performance of closed-form path length relative to that of a heuristic optimization method while significantly improving real-time capabilities by a minimum factor of 2464. Furthermore, in comparison to traditional rule-based methods, our approach has been shown to reduce the rotational path length by at least 27.29% and 16.71% in circle and convex polygon groups, respectively, indicating a significant improvement in planning efficiency.

Adaptive Fuzzy Control for Fractional-Order Networked Control Systems with Input Time Delay and Data Loss

This paper considers the adaptive fuzzy control of fractional-order nonlinear networked control systems subjected to network-induced input delay and data loss. To approximate unknown functions, fuzzy logic systems are employed. Furthermore, the Pade approximation method and an intermediate variable are introduced to eliminate the impact of input delay, and an adaptive fuzzy controller is designed using backstepping technology. Based on fractional-order Lyapunov stability theory, the proposed method can ensure that all signals are uniformly ultimately bounded, and the tracking error can converge to a small region of the origin. Two simulation examples are provided to verify the viability of the control method.

Study of the timelike electromagnetic form factors of the Λc

In this paper, the reaction of electron-positron annihilation into Λc+Λ¯c− is investigated. The Λc+Λ¯c− scattering amplitudes are obtained by solving the Lippmann-Schwinger equation. The contact, annihilation, and two pseudoscalar-exchange potentials are taken into account in the spirit of the chiral effective field theory. The amplitudes of e+e−→Λc+Λ¯c− are constructed by the distorted wave Born approximation method, with the final state interactions of the Λc+Λ¯c− rescattering implemented. By fitting to the experimental data, the unknown couplings are fixed, and high-quality solutions are obtained. With these amplitudes, the individual electromagnetic form factors in the timelike region, GEΛc, GMΛc, and their ratio, GEΛc/GMΛc, are extracted. Both modulus and phases are predicted. These individual electromagnetic form factors reveal new insights into the properties of the Λc. The separated contributions of the Born term, contact, annihilation, as well as the two pseudoscalar-exchange potentials to the electromagnetic form factors are isolated. It is found that the Born term dominates the whole energy region. The contact term plays a crucial role in the enhancement near the threshold, and the annihilation term is essential in generating the fluctuation of the electromagnetic form factors. Published by the American Physical Society 2024

MODELING OF GROUND WATER DYNAMICS AND ITS POLLUTION

Problem statement. Large accumulators of liquid waste (e.g., mine water ponds, tailing ponds, etc.) are long-term sources that change the hydrological regime. A negative consequence of this process is flooding of the territory. In addition, the infiltration of contaminated water from such hazardous sources changes the quality of groundwater. Therefore, it is important to analyze the impact of such anthropogenic sources on the process of flooding and deterioration of groundwater quality. To solve this problem, it is very important to use the method of mathematical modeling as an effective mean of researching problems of this class, since the use of physical modeling is practically impossible within the scope of problems of this class. The purpose of the article. Development of numerical models for predicting changes in the hydrological regime (flooding of the territory) and groundwater quality under the influence of anthropogenic pollution sources. Methodology. To assess the dynamics of changes in the hydrological regime, a two-dimensional equation of filtration of a non-pressure groundwater flow is used. A two-dimensional geomigration equation (planned model) is used to analyze changes in groundwater quality during infiltration of contaminated water from the settling pond. This equation takes into account the convective transfer of contaminants in the filtration flow, dispersion, and the intensity of contaminant infiltration into the groundwater flow. The method of total approximation is used for numerical integration of the filtration equation. For the numerical integration of the geomigration equation, an implicit splitting scheme is used. Scientific novelty. Effective numerical models for rapid assessment of changes in groundwater dynamics and quality under the influence of anthropogenic sources that change the hydrological regime are proposed. The constructed numerical models take into account a set of important physical factors that affect the process of geomigration and flooding of the territory, namely: filtration coefficient, variable depth of free-flowing groundwater, dispersion, intensity of the source of impurity emission into the groundwater flow. This makes it possible to obtain a comprehensive assessment of the process of flooding and groundwater pollution.. Practical significance. A computer code has been created that allows practical usage of the developed numerical models. This code is an effective tool for theoretical study of non-stationary processes of territory flooding and anthropogenic groundwater pollution. Conclusions. A numerical model for calculating groundwater dynamics has been developed. The model allows to predict the level of groundwater rise under the influence of a man-made source of wastewater infiltration from a settling pond. A numerical model for calculating the process of geomigration from an anthropogenic source of emissions has been developed. The model makes it possible to predict the dynamics of contamination zone formation in a non-pressure groundwater flow. The developed numerical models take into account the most important parameters that affect the formation of flooding zones and groundwater contamination.

Numerical modelling of large elasto-plastic multi-material deformations on Eulerian grids

Abstract The paper deals with an Eulerian model for heterogeneous multiphase (multi-material) elasto-plastic media based on the diffuse interface method in the one-dimensional uniaxial strain approximation. In this approximation, an equilibrium hypoelastic Wilkins bi-material model is derived. This is carried out on the basis of an analogy between the elasto-plastic Wilkins and hydrodynamic Euler models. For the obtained model, a Godunov-type numerical method is developed based on the approximate Riemann solver HLLC. The results of the proposed Eulerian diffuse interface model are compared with reference solutions on moving grids with explicit tracking interfaces. It is shown that the present diffuse interface numerical model describes with good accuracy strong shock-wave processes in heterogeneous multiphase elasto-plastic media.

New Lie Symmetries and Exact Solutions of a Mathematical Model Describing Solute Transport in Poroelastic Materials

A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.

Latent event history models for quasi-reaction systems

Various processes, such as cell differentiation and disease spreading, can be modelled as quasi-reaction systems of particles using stochastic differential equations. The existing Local Linear Approximation (LLA) method infers the parameters driving these systems from measurements of particle abundances over time. While dense observations of the process in time should in theory improve parameter estimation, LLA fails in these situations due to numerical instability. Defining a latent event history model of the underlying quasi-reaction system resolves this problem. A computationally efficient Expectation-Maximization algorithm is proposed for parameter estimation, incorporating an extended Kalman filter for evaluating the latent reactions. A simulation study demonstrates the method's performance and highlights the settings where it is particularly advantageous compared to the existing LLA approaches. An illustration of the method applied to the diffusion of COVID-19 in Italy is presented.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

Recently, some researchers have revisited the analysis of chlorine transportation in cylindrical pipes by deploying a coupling between the Laplace transform method and the complex analysis’ residue approach for inverting complex integrals. This method yielded interesting results after the incorporation of root-finding numerical schemes. Thus, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of a bi-layered pipe using the hybrid of the Laplace transform method and the asymptotic approximations method. The need for asymptotic approximations for the modified Bessel functions, which arise in the reduced ordinary differential equations, necessitates the quest for closed-form analytical solutions, which are largely considered benchmark solutions for numerical investigations. Moreover, the obtained closed-form asymptotic solutions have been examined graphically; where it was observed that both the radial diffusion coefficient η and the spatial radial variable are contributory in the transport of chorine concentration in the media.

Adaptive continuation based smooth -norm approximation for compressed sensing MR image reconstruction.

There are a number of algorithms for smooth -norm (SL0) approximation. In most of the cases, sparsity level of the reconstructed signal is controlled by using a decreasing sequence of the modulation parameter values. However, predefined decreasing sequences of the modulation parameter values cannot produce optimal sparsity or best reconstruction performance, because the best choice of the parameter values is often data-dependent and dynamically changes in each iteration. We propose an adaptive compressed sensing magnetic resonance image reconstruction using the SL0 approximation method. The SL0 approach typically involves one-step gradient descent of the SL0 approximating function parameterized with a modulation parameter, followed by a projection step onto the feasible solution set. Since the best choice of the parameter values is often data-dependent and dynamically changes in each iteration, it is preferable to adaptively control the rate of decrease of the parameter values. In order to achieve this, we solve two subproblems in an alternating manner. One is a sparse regularization-based subproblem, which is solved with a precomputed value of the parameter, and the second subproblem is the estimation of the parameter itself using a root finding technique. The advantage of this approach in terms of speed and accuracy is illustrated using a compressed sensing magnetic resonance image reconstruction problem and compared with constant scale factor continuation based SL0-norm and adaptive continuation based -norm minimization approaches. The proposed adaptive estimation is found to be at least twofold faster than automated parameter estimation based iterative shrinkage-thresholding algorithm in terms of CPU time, on an average improvement of reconstruction performance 15% in terms of normalized mean squared error. An adaptive continuation-based SL0 algorithm is presented, with a potential application to compressed sensing (CS)-based MR image reconstruction. It is a data-dependent adaptive continuation method and eliminates the problem of searching for appropriate constant scale factor values to be used in the CS reconstruction of different types of MRI data.

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook.

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

A fast analysis algorithm for structural vibration modal sensitivity

Vibration modal sensitivity analysis has a wide range of applications in structural optimization design, model correction, fault detection, and reliability analysis. The existing methods for calculating modal sensitivity mainly include modal superposition method, Nelson’s method, and some improved algorithms derived from them. However, the existing algorithms have the drawback of low computational efficiency when applied to modal sensitivity analysis of the large-scale structures. In view of this, the objective of this work is to develop a fast modal sensitivity analysis method that can more efficiently calculate the modal sensitivity of the large structures with thousands of degrees of freedom. This new sensitivity analysis technique integrates Sherman-Morrison-Woodbury formula, subspace iteration and forward-difference to achieve the goal of fast calculation. The innovations of the proposed method mainly include the following aspects. Firstly, the Sherman-Morrison-Woodbury formula is used to quickly calculate the perturbed flexibility matrix due to the minor modification in the structure. Then, the perturbed flexibility matrix is introduced into the subspace iteration method to calculate the perturbed modal parameters. Finally, the forward-difference algorithm is used to calculate the modal sensitivity. Two numerical examples are used to verify the superiority of the proposed modal sensitivity algorithm. Taking the 1952-bar structure as an example, the calculation time of the proposed algorithm is only 8.3 % and 18.2 % of that of the existing approximate modal superposition method and the improved Nelson’s method, respectively. In terms of calculation accuracy, the results obtained by the proposed method are exactly the same as the exact solution when the perturbation scale is set to 10−5. It is found that the proposed method significantly reduces computation time compared to the existing methods on the premise that the calculation accuracy is basically unchanged.

Global Existence and Blow-up of Solutions for a Class of Singular Parabolic Equations with Viscoelastic Term

In this paper, we consider the initial boundary value problem for a class of singular parabolic equations with viscoelastic term and logarithmic term. By using the technique of cut-off and the method of Faedo-Galerkin approximation, the local existence of the weak solution is established. Based on the potential well method, the global existence of the weak solution is derived. Furthermore, we prove that the weak solution blows up in finite time by taking the concavity analysis method.

A Hybrid Finite Element Method – Kirchhoff Approximation Method for Modeling Acoustic Scattering from an Underwater Vehicle Model with Alberich Coatings with Periodic Internal Cavities

Anechoic tiles can significantly reduce the echo intensity of underwater vehicles, thereby increasing the difficulty of detecting such vehicles. However, the computational efficiency of conventional methods such as the finite element method (FEM) and the boundary element method (BEM) has its limitations. A fast hybrid method for modeling acoustic scattering from underwater vehicles with anechoic tiles with periodic internal cavities, is developed by combining the Kirchhoff approximation (KA) and FEM. The accuracy and rapidity of the KA method were validated by FEM. According to the actual situation, the reflection coefficients of rubber materials with two different structures under rigid backing are simulated by FEM. Using the KA method, the acoustic scattering characteristics of the underwater vehicle with anechoic tiles are obtained by inputting the reflection coefficients and the target’s geometric grid. Experiments on the monostatic target strength (TS) in the frequency range of 1 to 20 kHz and time domain echo characteristics of acoustic scattering on a benchmark scale model with anechoic tiles are conducted. The research results indicate that the TS values and echo characteristic curves of the KA solutions closely approximate the experimental results, which verifies the accuracy of the KA method in calculating the TS and echo characteristics of underwater vehicles with anechoic tiles.

Assessing the survival time of hospitalized patients in Eastern Ethiopia during 2019-2020 using the Bayesian approach: A retrospective cohort study.

Severe acute malnutrition remains a significant health challenge, particularly in low- and middle-income countries. The aim of this study was to determine the survival time of under-five children with severe acute malnutrition. A retrospective cohort study was conducted at a hospital, focusing on under-five children with severe acute malnutrition. The study included 322 inpatients admitted to the Chiro hospital in Chiro, Ethiopia, between September 2019 and August 2020, whose data was obtained from medical records. Survival functions were analysed using Kaplan‒Meier plots and log-rank tests. The survival time of severe acute malnutrition was further analysed using the Cox proportional hazards model and Bayesian parametric survival models, employing integrated nested Laplace approximation methods. Among the 322 patients, 118 (36.6%) died as a result of severe acute malnutrition. The estimated median survival time for inpatients was found to be 2 weeks. Model selection criteria favored the Bayesian Weibull accelerated failure time model, which demonstrated that age, body temperature, pulse rate, nasogastric(NG) tube usage, hypoglycemia, anemia, diarrhea, dehydration, malaria, and pneumonia significantly influenced the survival time of severe acute malnutrition. This study revealed that children below 24 months, those with altered body temperature and pulse rate, NG tube usage, hypoglycemia, and comorbidities such as anemia, diarrhea, dehydration, malaria, and pneumonia had a shorter survival time when affected by severe acute malnutrition under the age of five. To reduce the death rate of children under 5 years of age, it is necessary to design community management for acute malnutrition to ensure early detection and improve access to and coverage for children who are malnourished.

Resolving nonuniform flow in gas turbines: challenges, progress, and moving forward

Abstract The flow in gas turbines exhibits a highly unsteady, complex and nonuniform manner, which presents two main challenges. Firstly, it introduces instrumentation errors, contributing to uncertainties when calculating one-dimensional performance metrics during rig or engine tests using fixed-location rakes. Secondly, it raises mechanical concerns, including high-cycle fatigue due to blade row interactions in turbomachines and thermal fatigue caused by hot-streaks at the combustor exit. Experimental characterisation of the flow nonuniformity in gas turbines is highly challenging due to the confined space and harsh environment for instrumentation. This paper presents recent efforts to address this issue by resolving the nonuniform flow in gas turbines using spatially under-sampled measurements. The proposed approach utilises discrete probe data and leverages a ‘Fourier-based approximation’ method developed by the author. The technique has undergone preliminary experimental validation involving reconstruction of the total pressure distribution in a multi-stage axial compressor and the total temperature field at the exit of the combustor and high-pressure turbine. Results show that, in the multi-stage axial compressor environment, reconstruction of inter-stage total pressure is achieved using a reduced dataset covering less than 20% of the annulus with reasonably good accuracy. The reconstructed total pressure yields almost identical mean total pressure values at all spanwise locations, with a maximum deviation of less than 0.02%. Additionally, reconstruction of the total temperature distribution at an engine-representative full annulus combustor is achieved using measurements at ten carefully selected circumferential locations. Results show that the reconstructed temperature field successfully captures the primary features associated with combustor exit temperature flow. The reconstructed temperature field yields excellent agreement in the magnitude of radial temperature distribution factor (RTDF) and overall temperature distribution factor (OTDF) predictions to the experiment, with a deviation of less than 0.5% for RTDF and less than 2.5% for OTDF. Lastly, reconstruction of the total temperature distribution at the exit of the GE E3 high-pressure turbine (HPT) is achieved using measurements at eight carefully selected circumferential locations. Results demonstrate remarkable robustness in resolving the temperature profile at the HPT exit with high fidelity, irrespective of the HPT inlet conditions. The initial validation results are promising, demonstrating that the new probe layout scheme and the Fourier-based approximation method enable effective characterisation of flow nonuniformity in gas turbines, thereby providing valuable insights into the complex flow of gas turbine engines.

From multiplicity of infection to force of infection for sparsely sampled Plasmodium falciparum populations at high transmission.

High multiplicity of infection or MOI, the number of genetically distinct parasite strains co-infecting a single human host, characterizes infectious diseases including falciparum malaria at high transmission. It accompanies high asymptomatic Plasmodium falciparum prevalence despite high exposure, creating a large transmission reservoir challenging intervention. High MOI and asymptomatic prevalence are enabled by immune evasion of the parasite achieved via vast antigenic diversity. Force of infection or FOI, the number of new infections acquired by an individual host over a given time interval, is the dynamic sister quantity of MOI, and a key epidemiological parameter for monitoring the impact of antimalarial interventions and assessing vaccine or drug efficacy in clinical trials. FOI remains difficult, expensive, and labor-intensive to accurately measure, especially in high-transmission regions, whether directly via cohort studies or indirectly via the fitting of epidemiological models to repeated cross-sectional surveys. We propose here the application of queuing theory to obtain FOI on the basis of MOI, in the form of either a two-moment approximation method or Little's law. We illustrate these methods with MOI estimates obtained under sparse sampling schemes with the recently proposed " var coding" method, based on sequences of the var multigene family encoding for the major variant surface antigen of the blood stage of malaria infection. The methods are evaluated with simulation output from a stochastic agent-based model, and are applied to an interrupted time-series study from Bongo District in northern Ghana before and immediately after a three-round transient indoor residual spraying (IRS) intervention. We incorporate into the sampling of the simulation output, limitations representative of those encountered in the collection of field data, including under-sampling of var genes, missing data, and usage of antimalarial drug treatment. We address these limitations in MOI estimates with a Bayesian framework and an imputation bootstrap approach. We demonstrate that both proposed methods give good and consistent FOI estimates across various simulated scenarios. Their application to the field surveys shows a pronounced reduction in annual FOI during intervention, of more than 70%. The proposed approach should be applicable to the many geographical locations where cohort or cross-sectional studies with regular and frequent sampling are lacking but single-time-point surveys under sparse sampling schemes are available, and for MOI estimates obtained in different ways. They should also be relevant to other pathogens of humans, wildlife and livestock whose immune evasion strategies are based on large antigenic variation resulting in high multiplicity of infection.

Artificial neural network-based sequential approximate optimization of metal sheet architecture and forming process

Abstract This paper introduces a sequential approximate optimization method that combines the finite element method (FEM), dynamic differential evolution (DDE), and artificial neural network (ANN) surrogate models. The developed method is applied to address two optimization problems. The first involves metamaterial design optimization for metal sheet architecture with binary design variables. The second pertains to optimizing process parameters in multi-stage metal forming, where the discrete nature arises owing to changing tool geometries across stages. This process is highly nonlinear, accumulating contact, geometric, and material nonlinear effects discretely through forming stages. The efficacy of the proposed optimization method, utilizing ANN surrogate models, is compared with traditionally used polynomial response surface (PRS) surrogate models, primarily based on low-order polynomials. Efficient learning of ANN surrogate models is facilitated through the FEM and Python integration framework. Initial data for surrogate model training is collected via Latin hypercube sampling and FEM simulations. DDE is employed for sequential approximate optimization, optimizing ANN or PRS surrogate models to determine optimal design variables. PRS surrogate models encounter challenges in dealing with nonlinear changes in sequential approximate optimization concerning discrete characteristics such as binary design variables and discrete nonlinear behavior found in multi-stage metal forming processes. Owing to the discrete nature, PRS surrogate models require more data and iterations for optimal design variables. In contrast, ANN surrogate models adeptly predict nonlinear behavior through the activation function's characteristics. In the optimization problem of Metal Sheet Architecture for design target C, the ANN surrogate model required an average of 4.6 times fewer iterations to satisfy stopping criteria compared to the PRS surrogate model. Furthermore, in the optimization of multi-stage deep drawing processes, the ANN surrogate model required an average of 6.1 times fewer iterations to satisfy stopping criteria compared to the PRS surrogate model. As a result, the sequential global optimization method utilizing ANN surrogate models achieves optimal design variables with fewer iterations than PRS surrogate models. Further confirmation of the method's efficiency is provided by comparing Pearson correlation coefficients and locus plots.

Comparative study of some rigorous and approximate methods for calculating the energy loss straggling

Some exact and approximate methods commonly used to calculate corrections to the Bethe stopping formula are modified and adapted by the authors to the calculation of the energy loss straggling (ELS). An intercomparison is carried out for results of approaches developed in the present work. An excellent agreement obtained between the results for ELS, calculated with an exact method previously proposed by one of the authors and the results of the Lindhard–Sørensen method for moderate relativistic energies. It is shown that some modified by authors approximate methods for the ELS calculating have the higher accuracy than the conventional approximate method of Lijian−Qing−Zhengming (LQZ). So, in all the cases considered, the accuracy of the twice modified LQZ method in the ELS calculations is higher than the accuracy of the usual LQZ method. The velocity averages of the relative error for Z = 92–118 of LQZ varies from 0.08% to 0.33%, and of LQZm2 – from 0.03% to 0.09%. A thrice modified LQZ method was also suggested in this work. The average error of LQZm3 becomes lower than the average error of LQZ, starting from Z = 99, and lower than the error of LQZm2, starting from Z = 109. A number of shortened LQZ methods for the ELS calculating have been proposed and developed too.

Bekenstein–Hawking entropy products for NUT class of black holes in AdS space

In this paper, we derive the entropy product rule for Taub–Newman–Unti–Tamburino (Taub–NUT)–de Sitter black hole (BH) and Taub–NUT–anti-de Sitter BH. We show that the entropy products in terms of both the physical horizons are mass-independent. Both perturbative approximation and direct method have been considered. By introducing the cosmological horizon, we show that for Taub–NUT–de Sitter BH, there exists a mass-independent entropy functional relation in terms of three horizons namely event horizon (EH), Cauchy horizon (CH) and cosmological horizon (CHH) which depends on cosmological parameter ([Formula: see text]) and the NUT parameter (N). For Taub–NUT–anti-de Sitter BHs, we determine the mass-independent entropy functional relations in terms of two physical horizons (namely EH and CH) which depends on only NUT parameter. Sometimes some complicated functions of EH entropy and CH entropy are also strictly mass-independent. This is plausible only due to the new formalism developed in [S. Wu and D. Wu, Phys. Rev. D 100, 101501(R) (2019)] for NUT class of BHs. The formalism states that a generic four dimensional Taub–NUT spacetime should be described completely in terms of three or four different types of thermodynamic hairs. They could be defined as the Komar mass ([Formula: see text]), the angular momentum ([Formula: see text]), the gravitomagnetic charge ([Formula: see text]), the dual (magnetic) mass [Formula: see text]. Finally, we could say that this universality is mainly due to the presence of new conserved charges [Formula: see text] which is closely analogue to the Kerr-like angular momentum [Formula: see text].