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The macroscopic physical and mechanical behavior, mesocosmic pore size distribution, and microscopic particle morphology of polyurethane concrete (PUC) during freeze–thaw conditions are thoroughly examined in this research. The results reveal that as the number of freeze–thaw cycles (F-T cycles) increases, the P-wave velocity of PUC drops constantly, the peak shear strength reduces gradually, but the porosity increases. A four-peak exponential function can be used to describe the pore dispersion features of PUC. With the increase in the number of F-T cycles, PUC showed an increasing trend in all characteristic pore radii, but only the volume fraction of macropores increased. The compressive and shear strength of PUC were positively correlated with the volume fraction of nanopores and micropores and negatively correlated with the volume fraction of macropores. It was proved that the macroporous content is the main reason for the decrease in the mechanical properties of PUC. The fractal dimension of PUC decreases with the increase in the number of F-T cycles, indicating that the complexity of the PUC pore structure decreases after F-T cycles. At the micro-meso-macro scale, the damage progression of PUC under freeze–thaw conditions demonstrates a gradual and mutually feeding connection. The damage evolution process can be broadly divided into three stages, beginning with the formation of micropores in the polyurethane agglomeration, progressing to the gradual germination and expansion of cracks in the polyurethane agglomeration, and finally to the connected sum expansion of cracks between the polyurethane agglomeration and the aggregates, resulting in significant deterioration of the macromechanical properties of PUC. [ABSTRACT FROM AUTHOR]

This study proposes a high computationally efficient data-driven predictive entry guidance method for hypersonic vehicles under multiple no-fly zones. The method uses a reduced-order motion-model-based semianalytic guidance framework to obtain a trained neural network that only requires two-dimensional input. First, the sixth-order entry dynamic motion model is simplified to a third-order model by considering height as the independent variable. Second, based on the reduced-order motion model, a novel exponential function is introduced to yield a semianalytic range-to-go expression in longitudinal guidance. Third, to generate sample trajectory data for training the neural network, the semianalytic guidance framework is supported by the reduced-order motion model with the semianalytic range-to-go expression. Then, a new dynamic lateral guidance reversal logic based on a chain mode strategy is employed to avoid no-fly zones with different configurations and numbers. Finally, to obtain real-time trajectory online, a data-driven online predictive guidance method is proposed based on a back propagation neural network trained by sample trajectory data generated by the semianalytic guidance framework. The proposed method overcomes the drawbacks of most predictor–corrector guidance methods; i.e., the corrected guidance parameters are heavily dependent on the initial values of each iteration in each guidance cycle. Advantageously, the proposed method greatly reduces the online command calculation time in one guidance cycle and only requires two input data to train the neural network, i.e., height and range-to-go, thus yielding results that are close to the engineering reality. The effectiveness of the proposed method is verified through simulations. [ABSTRACT FROM AUTHOR]

• Performance of different loading models is compared in reproducing the observed results of typical real-world footbridges. • SMD model with (inconsistent) parameters may result in significant differences in response predictions. • This paper modifies the SMD model and proposes a pedestrian synchronization ratio formula. • Modified SMD can reliably and conveniently reproduce lateral responses as the real-world data. The pedestrian-induced lateral vibration of footbridges has attracted much attention since the London Millennium Bridge incident. Significant investigations were mainly performed by external-excited force models and self-excited force models. The first type models consider pedestrians as harmonic forces and/or spring-mass-damping (SMD) systems. The second type consider the self-excited forces of pedestrians that initialize the instability of footbridges by e.g. applying the inverted pendulum (IP) model. This study compares performance of different models in reproducing the observed results of typical living footbridges: the London Millennium Bridge (UK), T-bridge (Japan) and Pedro e Inês footbridge (Portugal). With varying step frequencies and pedestrian numbers, both the acceleration time history and the discrepancy between the input work and consumed work exhibit a consistent trend for the harmonic force (HF) model and SMD model, yet a notable difference for the IP model. The HF model and SMD model are better suited for assuming synchronization, making them convenient for estimating the serviceability of pedestrian-induced lateral vibration. On the other hand, the IP model is more appropriate for incorporating random step frequencies and can better capture observed instability phenomena realistically. Because the SMD model with (inconsistent) parameters from literature may result in significant differences in predicting the acceleration amplitudes, this paper modifies the SMD model and proposes a pedestrian synchronization ratio formula as an exponential function of the maximum lateral acceleration. It is validated by comparing the analytical and numerical results for the three footbridges. Furthermore, by the modified SMD model, the lateral lock-in phenomenon of footbridges is simulated reliably and conveniently. This work contributes to analyze pedestrian-induced lateral vibration of footbridges. [ABSTRACT FROM AUTHOR]

This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear β -fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of β -fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions ν s , q (z) = (z − 1 2 [ s ] q ) ϱ on [ 0 , ∞) , we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators' uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre's K-functional. [ABSTRACT FROM AUTHOR]

Introduction: The presence of water significantly reduces the mechanical strength of rocks and induces various engineering geological hazards. The water weakening coefficient Kp is used to quantify this effect, defined as the ratio of wet uniaxial compressive strength to the dry value. Methods: A comprehensive physico-mechanical test was conducted on fifteen sandstones under dry and saturated conditions to predict the water weakening coefficient using easily obtainable physical parameters. Multiple linear regression was employed to establish the relationship between these parameters and the saturated water weakening coefficient. Results: The saturated water weakening coefficient decreases with increasing porosity and increases with higher Primary wave velocity (P-wave velocity). Rocks with higher porosity but lower P-wave velocity typically absorb more water. The P-wave velocity and clay mineral content were identified as the best predictors of the saturated water weakening coefficient (R2 = 0.82). Unsaturated water weakening coefficients at any water saturation level were well estimated using a previous exponential function. Discussion: The roles of different clay minerals and P-wave velocity in the water weakening process of rocks are comprehensively discussed. This study enhances the understanding of the water weakening mechanism and provides an improved evaluation model for the water weakening coefficient of sandstones using physical parameters. [ABSTRACT FROM AUTHOR]

The marine atmospheric corrosion behavior of 700L high-strength automotive beam steel exposed for 36 months was investigated by scanning electron microscopy, energy dispersive spectroscopy, X-ray diffraction, and electrochemical technology. The corrosion kinetics of 700L steel followed the exponential function: D = 4.85t1.23. The rust layers were mainly composited of γ-FeOOH, α-FeOOH, γ-Fe2O3, and Fe3O4, regardless of the exposure duration. With an extended exposure time, the porosity, cracking, and spalling of the rust layers increased, and the densification and thickness uniformity decreased. Electrochemical measurements displayed that the corrosion resistance of the rusted 700L steel gradually decreased with increasing exposure time. A good correlation was found between rust layer composition, microstructure, and corrosion resistance. [ABSTRACT FROM AUTHOR]

This study introduces a novel family of probability distributions called the "tangent exponential-G family," which is derived using trigonometric transformations of the exponential distribution. The significance of this research lies in developing new statistical models with versatile hazard functions, including j-shaped, S-shaped, increasing, and bathtub shapes. The primary aim is to construct and analyze the properties of this new distribution family. We utilized the exponential distribution function and applied the tangent transformation to develop the tangent exponential-G family. Key statistical properties and characteristics of this family are explored. Parameter estimation is performed using different estimation methods, and its accuracy is confirmed through simulation studies, which show reduced biases and mean square errors with increasing sample sizes. We tested the proposed distribution with real-world data sets to demonstrate practical applicability. Comparative analyses using model selection criteria and goodness-of-fit tests reveal that the tangent exponential-G family and its sub-model outperform compared to existing models, indicating its robustness and utility in statistical modeling. [ABSTRACT FROM AUTHOR]

Studies of macroscopic speed modeling of bidirectional pedestrian cross-flows have relied heavily on scenario experiments, but the data itself may be deficient because large-scale scenario experiments are not easy to organize and subjects may not be walking under normal conditions. In order to explore the possibility of using microscopic pedestrian flow simulations for macroscopic speed modeling of pedestrian flows, a series of two-way pedestrian cross-flow simulation experiments were designed. Bidirectional pedestrian flows are defined as Peds1 and Peds2. The crossing angle and pedestrian flow rate are used as variables, and a bidirectional pedestrian flows simulation is designed as an orthogonal experiment. The crossing angles range from 15 to 165 degrees, and bidirectional pedestrian flow rate range from 1 ped/s to 8 ped/s. A series of simulations are built and performed on the GIS agent-based modeling architecture (GAMA) platform. By analyzing the flow data of bidirectional flows in the crossing area, it is found that when the Peds1 density falls below a threshold, Peds1 speed is determined by pedestrians themselves and mainly remains in a free flow state; otherwise, the Peds1 speed decreases with density. The clear effects such as Peds2 density on the Peds1 speed cannot be determined. A piecewise function combined with a linear function and an exponential function is constructed as the Peds1 speed model considering the influence of the crossing angle. The calibration results show that the piecewise function should be better than the non-piecewise function. Compared to the results of established studies, the results in this paper have some differences. Therefore, the simulation method cannot completely replace the scene experiments. However, this approach can provide suggestions for subsequent refinement of the experimental program, as well as a feasible direction for the construction of a speed relationship for bidirectional pedestrian flows. [ABSTRACT FROM AUTHOR]

Study on static in-situ curing characteristics of CFRP based on near infrared laser: The quick curing method of carbon fibre reinforced plastics (CFRP) is one of the hotspots in current research. A static in-situ curing method for CFRP prepreg based on near-infrared laser was put forward in this study. The in-situ curing structural characteristics and the mechanism of CFRP were investigated through real-time surface temperature measurement, COMSOL temperature field simulation, 3D measurement of curing morphology and resin curing degree test. The thermal conductivity of the CFRP along the fiber direction is considerably higher than that along the perpendicular fiber direction. As a result, the temperature profile in the plane takes on an elliptical shape. During the transfer, the temperature field gradually decreases, resulting in an ellipsoidal 3D high-temperature distribution. The different shrinkage phenomena in the different curing regions between the layers lead to an irregular ellipsoidal solidification morphology of the unidirectional CFRP. The temperature in the center of the heat affected zone increases as a power exponential function with time. The area and depth of the heat-affected zone increases with the laser power, and the curing area is positively correlated with the degree of curing. As a result, curing temperature governing equations based on laser power and layer thickness have been proposed, while relationship equations based on laser power, curing depth and curing morphology have been developed. In addition, prediction equations based on curing morphology have been developed for curing degree, in order to achieve precise curing of CFRP. [ABSTRACT FROM AUTHOR]

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order δ , k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies. [ABSTRACT FROM AUTHOR]

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive periodic solutions in this paper are general positive functions, not e exponential functions, which generalizes and improves the existing results. We note that due to the symmetrical property of periodic solutions, the results of this paper provide a deeper understanding of the periodic behavior of biological populations. Two numerical examples show the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

This article investigates a class of multi-agent systems (MASs) with known dynamics external disturbances, where the communication graph is directed, and the followers have undirected connections. To eliminate the impacts of external disturbance, the technologies of disturbance observer-based control are introduced into the containment control problems. Additionally, to save communication costs and energy consumption, a distributed disturbance observer-based event-triggered controller is employed to achieve containment control and reject disturbance. Furthermore, designing the event-triggered function using an exponential function is beneficial for a time-dependent term while ensuring the exclusion of Zeno behavior. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

The existence of propane (C3H8) in a CO2-oil mixture has great potential for increasing CO2 solubility and decreasing minimum miscibility pressure (MMP). In this study, the enhanced solubility, reduced viscosity, and lowered MMP of CO2-saturated crude oil in the presence of various amounts of C3H8 have been systematically examined at the reservoir conditions. Experimentally, a piston-equipped pressure/volume/temperature (PVT) cell is first validated by accurately reproducing the bubble-point pressures of the pure component of C3H8 at temperatures of 30, 40, and 50 °C with both continuous and stepwise depressurization methods. The validated cell is well utilized to measure the saturation pressures of the CO2-C3H8-oil systems by identifying the turning point on a P-V diagram at a given temperature. Accordingly, the gas solubilities of a CO2, C3H8, and CO2-C3H8 mixture in crude oil at pressures up to 1600 psi and a temperature range of 25–50 °C are measured. In addition, the viscosity of gas-saturated crude oil in a single liquid phase is measured using an in-line viscometer, where the pressure is maintained to be higher than its saturation pressure. Theoretically, a modified Peng–Robinson equation of state (PR EOS) is utilized as the primary thermodynamic model in this work. The crude oil is characterized as both a single and multiple pseudo-component(s). An exponential distribution function, together with a logarithm-type lumping method, is applied to characterize the crude oil. Two linear binary interaction parameters (BIP) correlations have been developed for CO2-oil binaries and C3H8-oil binaries to accurately reproduce the measured saturation pressures. Moreover, the MMPs of the CO2-oil mixture in the presence and absence of C3H8 have been determined with the assistance of the tie-line method. It has been found that the developed mathematical model can accurately calculate the saturation pressures of C3H8 and/or CO2-oil systems with an absolute average relative deviation (AARD) of 2.39% for 12 feed experiments. Compared to CO2, it is demonstrated that C3H8 is more soluble in the crude oil at the given pressure and temperature. The viscosity of gas-saturated crude oil can decrease from 9.50 cP to 1.89 cP and the averaged MMP from 1490 psi to 1160 psi at 50 °C with the addition of an average 16.02 mol% C3H8 in the CO2-oil mixture. [ABSTRACT FROM AUTHOR]

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined. [ABSTRACT FROM AUTHOR]

The light-use efficiency-based gross primary productivity (LUE-GPP) model is widely utilized for simulating terrestrial ecosystem carbon exchanges owing to its perceived simplicity and reliability. Variations in cloud cover and aerosol concentrations can affect ecosystem LUE, thereby influencing the performance of the LUE-GPP model, particularly in humid regions. In this study, the performance of six big-leaf LUE-GPP models and one two-leaf LUE-GPP model were evaluated in a humid agroforestry ecosystem from 2018–2020. All big-leaf LUE-GPP models yielded GPP values consistent with that derived from the eddy covariance system (GPPEC), with R2 ranging from 0.66–0.73 and RMSE ranging from 1.81–3.04 g C m−2 d−1. Differences in model performance were attributed to the differences in the quantification of temperature (Ts) and moisture constraints (Ws) and their combination forms in the models. The Ts and Ws algorithms in the eddy covariance-light-use efficiency (EF-LUE) model well characterized the environmental constraints on LUE. Simulation accuracy under the common limitation of Ts and Ws (Ts × Ws) was higher than the maximum limitation of Ts or Ws (Min (Ts, Ws)), and the combination of the Ts algorithm in the Carnegie–Ames–Stanford Approach (CASA) and the Ws algorithm in the EF-LUE model was optimized in combination forms, thereby constraining LUE for GPP estimates (GPPBLO, R2 = 0.76). Various big-leaf LUE-GPP models overestimated or underestimated GPP on sunny or cloudy days, respectively, while the two-leaf LUE-GPP model, which considered the transmission of diffuse radiation and the difference in photosynthetic capacity of canopy leaves, performed well (R2 = 0.72, p < 0.01). Nevertheless, the underestimation/overestimation for shaded/sunlit leaves remained under different weather conditions. Then, the clearness index (Kt) was introduced to calculate the dynamic LUE in the big-leaf and two-leaf LUE-GPP models in the form of exponential or power functions, resulting in consistent performance even in different weather conditions and an overall higher simulation accuracy. This study confirmed the potential applicability of different LUE-GPP models and emphasized the importance of dynamic LUE on model performance. [ABSTRACT FROM AUTHOR]

In this work, we are interested in studying the existence of a ground state solution for the problem Δ (w β (x) Δ u) ± div (w β (x) | ∇ u | p − 2 ∇ u) = f (x , u) in B , and u = ∂ u ∂ ν = 0 on ∂ B , where B is the unit ball in R 4 , w β (x) = (log e | x |) β or w β (x) = (log (1 | x |) β for β ∈ (0 , 1) , 2 < p < 4 and f : R → R is a superlinear continuous class function with supercritical exponential growth. We utilize the Nehari manifold method to establish the existence result. [ABSTRACT FROM AUTHOR]

this article's goal is to present the indefinite refined neutrosophic integrals by parts. All situations where integration by parts can be used are covered, including the use of rotating integrals to solve recurring and non-terminating functions like the product of trigonometric and exponential functions. Furthermore, the Tabular method has been implemented in the computation of the indefinite refined neutrosophic integrals. [ABSTRACT FROM AUTHOR]

Load frequency regulation (LFR) is an indispensable scheme in planning electrical power production to provide consumers with stable, reliable and uninterrupted power. In the face of complicated power system (PS) structures with increasing and intricate power demand, new controllers that offer not only good performance, but also easy commissioning in practice are required. To this end, this research introduces an exponential PID (EXP-PID) controller as a new control scheme to ameliorate the LFR performance of PSs. This controller is simple to design and has a nonlinear feature inherited from two tunable exponential functions, which are placed in front of the PID controller and act on the error signal and its time derivative individually. To achieve the utmost performance, the EXP-PID controller's parameters are procured by a corrected variant of the snake optimizer (co-SO). To validate the proposed control scheme, various single-/multi-area single-/multi-source PSs favored in the area are considered as test benches. A thorough comparison with the state-of-the-art approaches is performed to disclose the true efficacy of our proposal. Among the rivals, co-SO tuned EXP-PID controller, despite its simplicity, is found to render credible and promising performance in mitigating frequency and tie-line power deviations effectively. • An EXP-PID controller is introduced for LFR of power systems. • The new controller enjoys advantages of nonlinear feature and adaptive gains. • Corrected variant of snake optimizer is presented for controller optimization. • Several power systems are studied for performance validation. • The dominance of our proposal is vividly affirmed over the state-of-the-art. [ABSTRACT FROM AUTHOR]

Integral inequalities involving exponential convexity are significant in both theoretical and applied mathematics. In this paper, we establish a new Hermite-Hadamard type inequality for the class of exponentially convex functions by using the concept of (α − s) exponentially convex function. Additionally, using the well-known Hermite-Hadamard and Ostrowski inequalities, we establish several new integral inequalities. These newly obtained results contain several well-known results as special cases. Finally, new estimations for the trapezoidal formula have been provided, illustrating the practical applications of the research. [ABSTRACT FROM AUTHOR]

Purpose: To investigate the dose rate dependence of MapCHECK3 and its influence on measurement accuracy, as well as the effect of dose rate correction. Materials and methods: The average and instantaneous dose rate dependence of MapCHECK2 and MapCHECK3 were studied. The accuracy of measurements was investigated where the dose rate differed significantly between dose calibration of the MapCHECK and the measurement. Measurements investigated include: the central axis dose for different fields at different depths, off‐axis doses outside the field, and off‐axis doses along the wedge direction. Measurements using an ion chamber were taken as the reference. Exponential functions were fit to account for average and instantaneous dose rate dependence for MapCHECK3 and used for dose rate correction. The effect of the dose rate correction was studied by comparing the differences between the measurements for MapCHECK (with and without the correction) and the reference. Results: The maximum dose rate dependence of MapCHECK3 is greater than 2.5%. If the dose calibration factor derived from a 10 × 10 cm2 open field at 10 cm depth was used for measurements, the average differences in central diode dose were 0.8% ± 1.0% and 1.0% ± 0.8% for the studied field sizes and measurement depths, respectively. The introduction of wedge would not only induce −1.8% ± 1.3% difference in central diode dose, but also overestimate the effective wedge angle. After the instantaneous dose rate correction, above differences can be changed to 1.9% ± 8.1%, 0.2% ± 0.1%, and 0.0% ± 0.9%. The pass rate can be improved from 98.4% to 98.8%, 98.3%–100.0%, and 96.3%–100.0%, respectively. Conclusion: Compared with MapCHECK2 (SunPoint1 diodes), the more pronounced dose rate dependence of MapCHECK3 (SunPoint2 diodes) should be carefully considered. To ensure highly accurate measurement, it is suggested to perform the dose calibration at the same condition where measurement will be performed. Otherwise, the dose rate correction should be applied. [ABSTRACT FROM AUTHOR]

In this present study, we reduce the fractional reaction-diffusion equation to a traditional differential equation using the fractional complex transformation and consider the Landau Lifshitz (LLG) equation. Moreover, by using the generalized exponential rational function method and Kudryashov's method respectively we extract new exact and solitary wave solutions for these equations. Some plots of some presented new solutions are represented to exhibit wave characteristics. All results in this paper are essential to understand the physical meaning and behavior of the investigated equation that sheds light on the importance of investigating various nonlinear wave phenomena in mathematical physics. [ABSTRACT FROM AUTHOR]

AbstractAsymptotic goodness-of-fit methods in contingency table analysis can struggle with sparse data, especially in multi-way tables where it can be infeasible to meet sample size requirements for a robust application of distributional assumptions. However, algebraic statistics provides exact alternatives to these classical asymptotic methods that remain viable even with sparse data. We apply these methods to a context in psychometrics and education research that leads naturally to multi-way contingency tables: the analysis of differential item functioning (DIF). We explain concretely how to apply the exact methods of algebraic statistics to DIF analysis using the R package algstat, and we compare their performance to that of classical asymptotic methods. [ABSTRACT FROM AUTHOR]

In this work, the author reviews the origination of normalized tails of the Maclaurin power series expansions of infinitely differentiable functions, presents that the ratio between two normalized tails of the Maclaurin power series expansion of the exponential function is decreasing on the positive axis, and proves that the normalized tail of the Maclaurin power series expansion of the exponential function is absolutely monotonic on the whole real axis. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard ( H H ), H H –Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings ( L R I V H convex mappings) with an exponential function in the kernel. We use fractional operators to develop several generalizations, capturing unique outcomes that are currently under investigation, while also introducing a new operator. Generally, we propose two methods that, in conjunction with more generalized fractional integral operators with an exponential function in the kernel, can address certain novel generalizations of increasing mappings under the assumption of L R I V convexity, yielding some noteworthy results. The results produced by applying the suggested scheme show that the computational effects are extremely accurate, flexible, efficient, and simple to implement in order to explore the path of upcoming intricate waveform and circuit theory research. [ABSTRACT FROM AUTHOR]

In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When  x ∈ R  and  t ∈ R , the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and N-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (R ,  Z , q-discrete,  C). [ABSTRACT FROM AUTHOR]

Early detection of cable trench fires by locating the fire source in a timely manner can reduce the risk of fire. However, existing fire warning methods have low accuracy, long calculation times and difficulty coping with sudden fire situations. We established experimental platforms for cable trenches with different structures and combined these with simulation analysis to investigate the relationship between the ignition point position and the temperature distribution at the ceiling. An exponential function for predicting the ignition point position and the maximum temperature rise of tunnels is proposed based on the extreme values of ceiling temperature. The results indicate that the vertical temperature of the ceiling exhibits an exponential function variation pattern. The maximum deviation for identifying the ignition point is 0.098 m, with an average deviation of 0.044 m and an average accuracy of 98.77%. The maximum temperature prediction error for the ceiling is 14 °C, with an average deviation of 12.33 °C and an average accuracy of 98.30%. Compared to traditional fire prediction methods, the method proposed here has higher accuracy and provides a theoretical basis for early prevention and control of cable trench fires. [ABSTRACT FROM AUTHOR]

In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted under its new form, including characteristic functions and statistical properties. It is shown that this distribution is appropriate for modeling over-dispersed count data. Moreover, its probability mass function is right-skewed with different shapes. The unknown model parameter is estimated using the maximum likelihood method, with more attention given to Bayesian estimation methods. The Bayesian estimator is computed based on three different loss functions: a square error loss function, a linear exponential loss function, and a generalized entropy loss function. The simulation study is implemented to examine the distribution's behavior and compare the classical and Bayesian estimation methods, which indicated that the Bayesian method under the generalized entropy loss function with positive weight is the best for all sample sizes with the minimum mean squared errors. Finally, the discrete linear-exponential distribution proves its efficiency in fitting discrete physical and medical lifetime count data in real-life against other related distributions. [ABSTRACT FROM AUTHOR]

The Andrade rheological model is often employed to describe the response of solar system or extra‐solar planets to tidal perturbations, especially when their properties are still poorly constrained. While for uniform planets with steady‐state Maxwell rheology the analytical form of the Love numbers was established long ago, for the transient Andrade rheology no closed‐form solutions have been yet determined, and the planetary response is usually studied either semi‐analitically in the frequency domain or numerically in the time domain. Closed‐form expressions are potentially important since they could provide insight into the dependence of Love numbers upon the model parameters and the time‐scales of the isostatic readjustment of the planet. First, we focus on the Andrade rheological law in 1‐D and we obtain a previously unknown explicit form, in the time domain, for the relaxation modulus in terms of the higher Mittag‐Leffler transcendental function Eα,β(z) that generalizes the exponential function. Second, we consider the general response of an incompressible planetary model — often referred to as the "Kelvin sphere" — studying the Laplace domain, the frequency domain and the time domain Love numbers by analytical methods. Through a numerical approach, we assess the effect of compressibility on the Love numbers in the Laplace and frequency domains. Furthermore, exploiting the results obtained in the 1‐D case, we establish closed‐form — although not elementary — expressions of the time domain Love numbers and we discuss the frequency domain response of the Kelvin sphere with Andrade rheology analytically. Plain Language Summary: Rheology studies how materials flow when subject to stress. To constrain the rheology of planets, it is useful to consider their response to tidal forces or surface loads that allow to probe their interiors. The Andrade rheological model—introduced in 1910—has recently gained popularity in planetary studies. In contrast with the traditional Maxwell model, the Andrade rheology manifests "transient" creep, an initial stage of flow characterized by a time varying rate of strain. Up to now, the response of a planet characterized by an Andrade rheology has been studied numerically, since exact expressions for the Love numbers were only known in the frequency domain. For the first time, we show that in the case of an incompressible homogeneous planet, analytical Love numbers can also be found in the time domain. They are expressed in terms of the Mittag‐Leffler function, which generalizes the exponential. Using analytical methods, we study the Love numbers of a planet with Andrade rheology in the Laplace, frequency and time domains, also giving some numerical results. Key Points: The Andrade rheological law is widely employed in planetary studies since it accounts for transient flow in a simple wayWe obtain a first analytical expression of the time domain Love numbers for a homogeneous incompressible sphere with Andrade rheologyThe solutions found provide new insight into the planetary response and its sensitivity upon model parameters [ABSTRACT FROM AUTHOR]

The mean daily temperature effects on plant development rates and quality of compact container-grown pepper were evaluated. Compact pepper cultivars Fresh Bites Yellow and Hot Burrito were grown in greenhouses at 18 to 26 °C (Expt. 1) and 20 to 30 °C (Expt. 2) under supplemental high-pressure sodium lighting and a 16-hour photoperiod. The number of days to first open flower, to first ripe fruit, and from flower to ripe fruit were measured and the development rates calculated by taking the reciprocal (e.g., 1/day). Temperature effects were predicted by fitting a nonlinear exponential function that included the base temperature (Tmin) and maximum developmental rate (Rmax) parameters. Plant quality attributes were measured during Expt. 2. As the temperature increased, the times to flower and fruit decreased (i.e., developmental rates increased) for both cultivars. The estimated Tmin was 13.3 °C for 'Fresh Bites Yellow', and that for 'Hot Burrito' was 9.3 °C, whereas the Rmax was similar between cultivars (averages of 0.0488 at flower, 0.0190 at fruit, and 0.0252 from flower to fruit). 'Fresh Bites Yellow' and 'Hot Burrito' grown at ≈25 °C had a relatively short crop time, compact canopy, large fruit size, and high number of fruits per plant at finish. Compact peppers are new crops being grown by greenhouse floriculture operations for their ornamental and edible value, and the information from this study can help growers schedule these crops to meet critical market windows and determine the impacts of changing the growing temperature on crop timing and quality. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decision procedure of reachability for a linear system ξ ′ = A ξ + u , where the matrix A ′ s eigenvalues can be arbitrary algebraic number and the input u is a vector of trigonometric-exponential polynomials. If the initial set contains only one point, the reachability problem under consideration is reduced to the decidability of the sign of trigonometric-exponential polynomial and then achieved by being reduced to verification of a series of univariate polynomial inequalities through Taylor expansions of the related exponential functions and trigonometric functions. If the initial set is open semi-algebraic, we will propose a decision procedure based on OpenCAD and an algorithm of real root isolation derived from the sign-deciding procedure for the trigonometric-exponential polynomials. The experimental results indicate the efficiency of our approach. Under the assumption of Schanuel's Conjecture, the above procedures are complete for bounded time except for several cases. [ABSTRACT FROM AUTHOR]

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert W -function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of the generalized k-Leonardo numbers and study their properties. We investigate the Gaussian Leonardo numbers and associated new families of these Gaussian forms. We obtain combinatorial identities like Binet formula, Cassini’s identity, partial sum, etc. in the closed form. Moreover, we give various generating and exponential generating functions. [ABSTRACT FROM AUTHOR]

The complex interactions were performed among non-pharmaceutical interventions, vaccinations, and hosts for all epidemics in mainland China during the spread of COVID-19. Specially, the small-scale epidemic in the city described by SVEIR model was less found in the current studies. The SVEIR model with control was established to analyze the dynamical and epidemiological features of two epidemics in Jinzhou City led by Omicron variants before and after Twenty Measures. In this study, the total population (N) of Jinzhou Citywas divided into five compartments: the susceptible (S), the vaccinated (V), the exposed (E), the infected (I), and the recovered (R). By surveillance data and the SVEIR model, three methods (maximum likelihood method, exponential growth rate method, next generation matrix method) were governed to estimate basic reproduction number, and the results showed that an increasing tendency of basic reproduction number from Omicron BA.5.2 to Omicron BA.2.12.1. Meanwhile, the effective reproduction number for two epidemicswere investigated by surveillance data, and the results showed that Jinzhou wave 1 reached the peak on November 1 and was controlled 7 days later, and that Jinzhou wave 2 reached the peak on November 28 and was controlled 5 days later. Moreover, the impacts of non-pharmaceutical interventions (awareness delay, peak delay, control intensity)were discussed extensively, the variations of infection scales for Omicron variant and EG.5 variant were also discussed. Furthermore, the investigations on peaks and infection scales for two epidemics in dynamic zero-COVID policy were operated by the SVEIR model with control. The investigations on public medical requirements of Jinzhou City and Liaoning Province were analyzed by using SVEIR model without control, which provided a possible perspective on variant evolution in the future. [ABSTRACT FROM AUTHOR]

The basic reproduction number, R0, is a well-known quantifier of epidemic spread. However, a class of existing methods for estimating R0 from incidence data early in the epidemic can lead to an over-estimation of this quantity. In particular, when fitting deterministic models to estimate the rate of spread, we do not account for the stochastic nature of epidemics and that, given the same system, some outbreaks may lead to epidemics and some may not. Typically, an observed epidemic that we wish to control is a major outbreak. This amounts to implicit selection for major outbreaks which leads to the over-estimation problem. We formally characterised the split between major and minor outbreaks by using Otsu's method which provides us with a working definition. We show that by conditioning a 'deterministic' model on major outbreaks, we can more reliably estimate the basic reproduction number from an observed epidemic trajectory. [ABSTRACT FROM AUTHOR]

The determinant formula for tensor Pade-type approximation (TPTA) is proposed to compute the tensor exponential function eAt, where A represents an order-p (p ≥ 3) tensor in n-dimensional Euclidean space. Furthermore, an error formula for TPTA is also derived. It is demonstrated that the determinant formula can be utilized to approximate any arbitrary tensor function defined by a power series. Numerical illustrations are presented to showcase the efficiency of the determinant formula. [ABSTRACT FROM AUTHOR]

Some Cauchy-type mean-value theorems for Chebychev's inequality, Steffensen's inequality, midpoint rule, and Simpson's rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials, and some trigonometric functions. Further, we obtain some exponential, logarithmic, and trigonometric equations and give two inequalities for midpoint and Simpson's rules. [ABSTRACT FROM AUTHOR]

Distance decay of urban park visitation (e.g., visitation number and visitation frequency) has been widely acknowledged and is increasingly integrated into urban park planning and management considering spatial accessibility and service equity. However, thorough understandings especially concerning the variations among visitors with different personal characteristics and visitation patterns are still scarce. Taking Changsha, China as an example, we collected data on visitation distance (i.e., the distance between urban parks and visitor's homes) and visitation frequency of 2535 urban park visitors, modeled the distance decay of visitation density and visitation frequency, and investigated their variations among visitors with different personal characteristics and visitation patterns. The results show that: (1) The median visitation distance was 1.3 km and the median visitation frequency was 24 times per season. (2) Both visitation density and visitation frequency showed clear spatial patterns of distance decay and can be effectively modeled using common distance decay functions (e.g., power function, exponential function, and logarithmic function). (3) Visitors' characteristics (e.g., gender and age) and visitation patterns (e.g., duration time, transportation modes, and visitation purposes) significantly impact visitation distance, visitation frequency, and the characteristics of distance decay (i.e., the rate of distance decay). These findings extend our understanding of the distance decay of urban park visitation which can help better urban park planning and management. [ABSTRACT FROM AUTHOR]

This paper presents robust input shaping commands with first-order actuators utilizing a classical robust input shaper for practical applications in input shaping technology. An ideal input shaping command can deviate due to actuator dynamics so that the modified command has a detrimental effect on the performance of oscillation reduction in feedforward control applications. A zero-vibration-derivative (ZVDF) shaper with first-order actuators is analytically proposed using a phasor–vector approach, an exponential function for the approximation of the dynamic response of first-order actuators and the usage of the ZVD shaper. In addition, an equivalent transformation is utilized based on the superposition principle for the convenient inclusion of first-order actuator dynamics and is applied to the individual segment input command. The residual deflection and robustness of the proposed robust input shaping commands are numerically evaluated and compared with those of a conventional ZVD shaper with respect to the parameter uncertainties of flexible systems and actuators. The robust input shaping commands that are possible with first-order actuators are experimentally validated, presenting a better robustness and residual deflection reduction performance than the classical ZVD shaper on a mini bridge crane. [ABSTRACT FROM AUTHOR]

Shallow mixing layers developed in partially-distributed canopy flows (PCFs) are important for water conveyance, sediment transport and solute dispersion. In this paper, we address the response of shallow mixing layers in PCFs to several influential factors associated with bed friction, water shallowness and canopy denseness using depth-averaged Large Eddy Simulation (DA-LES). The results show that under certain canopy denseness, shallow mixing layers in PCFs display asymptotic behaviour for the variation of bed friction and water shallowness, consistent with classic shallow mixing layers. The canopy drag encourages the asymmetric growth of the inner and outer parts of the mixing layer, and this effect decreases when bed friction and water shallowness become pronounced. Characteristic velocity and length scales show similarity well delineated by exponential decay function regarding a stabilization efficiency parameter. The exception happens for the inner length scale of the mixing layer when only changing canopy density. We conclude that changing the canopy denseness can encourage the overall shift of mixing layer scaling regarding a broad range of bed friction and water shallowness. Furthermore, the bed friction parameter thresholding triggering flow instability of mixing layers in PCFs is found about 0.03, lower than 0.09 for splitter-plate induced mixing layers. [ABSTRACT FROM AUTHOR]

In this work, we consider the degenerate Frobenius-Euler-Genocchi polynomials utilizing the degenerate exponential function and the degenerate Changhee-Frobenius-Euler-Genocchi polynomials utilizing the degenerate logarithm function. Then, we analyze some summation and addition formulas for these polynomials. In addition, we derive some correlations with degenerate Stirling numbers of both kinds and degenerate Frobenius-Euler polynomials. Moreover, we present difference and derivative operator rules for the generalized degenerate Frobenius-Euler-Genocchi polynomials. Lastly, we show certain zeros of both the degenerate Frobenius-Euler-Genocchi polynomials and the degenerate Changhee-Frobenius-Euler-Genocchi polynomials and provide their beautifully graphical representations. [ABSTRACT FROM AUTHOR]

Let , , be the class of entire functions of exponential type bounded on the real line. For a number and a sequence of complex numbers satisfying the condition , the operator on defined by is considered. Obviously, The main purpose of the paper is to describe all extremal functions for this inequality. Theorem 1 proved in the paper asserts that if (1) for some and (2) there exists an with such that for all , then the set of all extremal functions for the above inequality coincides with the set of functions of the form , . The proof of Theorem 1 essentially uses Theorem 2, which says that if and there exists a point for which and , then , . Theorem 3 gives general examples of operators satisfying both conditions of Theorem 1. In particular, such is the fractional derivative operator for and . [ABSTRACT FROM AUTHOR]

Knowledge of the spatial and temporal variations in soil nutrients is crucial for designing efficient site-specific nutrient management plans, which can improve crop yields and maximize nutrient use efficiency. The present study was conducted to evaluate and compare the status and spatio-temporal distribution pattern of available nitrogen (AHN), phosphorus (AP) and potassium (AK) and some selected soil properties [soil pH and soil organic matter (SOM)] in cultivated soils of Southwestern China over a 15-year period (2007–2022). We visualized the correlations among soil properties, AHN, AP and AK and analyzed the spatial structures of these parameters. A total of 3845 topsoil (0–20 cm) samples (3331 in 2007 and 514 in 2022) were collected from the cultivated areas of Jiangjin District, Chongqing, Southwestern China. Soil pH, SOM, AHN and AK showed moderate variability with coefficient of variation (CV) values varying between 10 and 100%, except AP (CV > 100%, high variability). The mean soil pH, SOM, AP and AK in 2022 were significantly higher than those in 2007. AHN was significantly positively correlated with SOM (r = 0.531, p < 0.01 in 2007, r = 0.768, p < 0.01 in 2022) and significantly negatively correlated with soil pH (r = −0.186, p < 0.01 in 2007, r = −0.102, p < 0.05 in 2022). AP was significantly negatively correlated with soil pH (r = −0.075, p < 0.01 in 2007, r = −0.126, p < 0.01 in 2022). AK was significantly positively correlated with SOM (r = 0.164, p < 0.01 in 2007, r = 0.229, p < 0.01 in 2022), ANH (r = 0.131, p < 0.01 in 2007, r = 0.251, p < 0.01 in 2022) and AP (r = 0.145, p < 0.01 in 2007, r = 0.52, p < 0.01 in 2022). The exponential function performed best for the soil properties and all three nutrients, with higher R2 values (0.203 to 0.93 in 2007 and 0.316 to 0.796 in 2022) in both years. The nugget/sill ratios (which varied from 31.58% to 72% in 2007 and from 29.31% to 47.02% in 2022) indicated the moderate spatial dependence of all soil parameters, except AK in 2022 (nugget/sill ratio = 23.81%, strong spatial dependence). During the study period, soil pH, AP and AK increased in the central and northern areas; SOM increased in the northern and southwestern parts; AHN decreased in the central areas. The current study highlighted the change in spatial variability of soil pH, AHN, AP and AK in the study area over 15 years. [ABSTRACT FROM AUTHOR]

In this work, we introduce a new definition for the fractional differential operator that generalizes several well-known fractional differential operators. In fact, we introduce the notion of the p -proportional ω -weighted κ -Hilfer derivative includes an exponential function, D a , λ σ , ρ , p , κ , ω , and then we consider a non-instantaneous impulse differential inclusion containing D a , λ σ , ρ , p , κ , ω with order σ ∈ (1 , 2) and of kind ρ ∈ [ 0 , 1 ] in Banach spaces. We deduce the relevant relationship between any solution to the studied problem and the integral equation that corresponds to it, and then, by using an appropriate fixed-point theorem for multi-valued functions, we give two results for the existence of these solutions. In the first result, we show the compactness of the solution set. Next, we introduce the concept of the (p , ω , κ) -generalized Ulam-Hyeres stability of solutions, and, using the properties of the multi-valued weakly Picard operator, we present a result regarding the (p , ω , κ) -generalized Ulam-Rassias stability of the objective problem. Since many fractional differential operators are particular cases of the operator D a , λ σ , ρ , p , κ , ω , our work generalizes a number of recent findings. In addition, there are no past works on this kind of fractional differential inclusion, so this work is original and enjoyable. In the last section, we present examples to support our findings. [ABSTRACT FROM AUTHOR]

High-modulus asphalt mixtures (HMAMs) are potential materials for achieving long-life pavement performance, and understanding their long-term performance degradation mechanism is crucial for analyzing pavement performance degradation under long-term exposure to traffic and environmental coupling conditions. This paper aims to examine the long-term evolution of high- and low-temperature performance and water stability in HMAMs derived from modified nature asphalt, prepared using microparticle suspension technology, under heavy load and humid heat conditions. Four asphalt mixtures (AC-13, BBME-13, SMA-13, SBSAC-13) were prepared, and then their dynamic moduli were tested. A test plan for the long-term performance of the mixtures was designed, and the performances of HMAMs with different gradations and time were analyzed. Furthermore, corresponding long-term performance degradation models of HMAMs were established. The results show that the HMAMs of modified natural asphalt have better high-temperature performance than HMAMs of high-modulus agent/SBS-modified asphalt. However, following prolonged thermal aging and freeze-thaw cycles, the low-temperature performance shows the opposite trend. In addition, thermal aging had the same effect on the long-term low-temperature performance of both types of HMAM, with the HMAMs of modified natural asphalt being less affected by long-term freeze-thaw cycling. The long-term bending tensile strain attenuation of HMAMs can be simulated using an exponential function. Both types of HMAMs display similar water stability under dry-wet cycles and dynamic water erosion condition. The variations in residual stability during dry-wet cycles can be accurately represented using a power function. [ABSTRACT FROM AUTHOR]

In this paper, the Bayesian estimators for the Inverse Weibull Distribution (IWD) scale parameter are derived when the shape parameter of distribution is known. The Bayesian estimators for the parameter are obtained by using the Gamma prior under the different types of loss functions such as square error loss function (Self), Entropy loss function (Elf), Precautionary loss function (Plf), Linear exponential loss function (Linexlf) and nonlinear exponential loss function (Nlinexlf). A classical maximum likelihood estimator (mle) for the parameter is also derived. To compare the efficiency of the parameter estimation methods, a simulation study is carried out. The comparison is based on mean square error. [ABSTRACT FROM AUTHOR]

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev's integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard's numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. [ABSTRACT FROM AUTHOR]

This paper is concerned with applying the Bayesian and E-Bayesian approaches to estimating the unknown parameters of the modified Topp–Leone–Chen distribution under a progressive Type-II censored sample plan. The paper explores the complexities of different estimating methods and investigates the behavior of the estimates through some computations. The Bayes and E-Bayes estimators are obtained under two distinct loss functions, the balanced squared error loss function, as a symmetric loss function, and the balanced linear exponential loss function, as an asymmetric loss function. The estimators are derived using gamma prior and uniform hyperprior distributions. A numerical illustration is given to examine the theoretical results through using the Metropolis–Hastings algorithm of the Markov chain Monte Carlo method of simulation by the R programming language. Finally, real-life data sets are applied to prove the flexibility and applicability of the model. [ABSTRACT FROM AUTHOR]

It is well known that the first stage of mine-to-mill optimization is rock fragmentation by blasting. The degree of rock fragmentation can be expressed in terms of average grain (X50) size and size distribution. There are approaches in which exponential functions are used to estimate the size distribution of the pile that will be formed before blasting. The most common of these exponential functions used to estimate the average grain size is the Kuz–Ram and KCO functions. The exponential functions provide a curve from 0% to 100% using the mean grain size (X50), characteristic size (XC), and uniformity index (n) parameters. This distribution curve can make predictions in the range of fine grains and coarse grains outside the acceptable error limits in some cases. In this article, the usability of the hyperbolic tangent function, which is symmetrical at origin, in the estimation of the size distribution as an alternative to the exponential distribution functions used in almost all estimation models is investigated. As with exponential functions, the hyperbolic tangent function can express the aggregated size distribution as a percentage with reference to the variables X50 and XC. It has been shown that the hyperbolic tangent function provides 99% accuracy to the distribution of fine grains and coarse grains of the pile formed as a result of blasting data for the characteristic size (XC) parameter and the uniformity index (n). [ABSTRACT FROM AUTHOR]