Exponential Function Research Articles

Non-singular control strategies for a class of strict-feedback nonlinear time-delay systems

This paper focuses on the non-singular control for strict-feedback nonlinear time-delay systems. The boundedness of all the signals of the closed-loop system is rigorously proved by introducing a class of Lyapunov-Krasovskii functionals, including exponential functions. Besides, novel auxiliary functions with continuous derivatives are proposed to construct the control law to avoid the discontinuity of the virtual controllers. Besides, the amount of computation is reduced by applying the new auxiliary functions when the other control methods require loads of calculations for handling singular problems. In the end, numerical examples are given to validate the effectiveness of the proposed control scheme.

Estimation of Land Deformation and Groundwater Storage Dynamics in Shijiazhuang–Baoding–Cangzhou–Hengshui Using Multi-Temporal Interferometric Synthetic Aperture Radar

Groundwater resources are crucial to socio-economic development and the ecosystem, and over-extraction can cause the groundwater level to drop, deplete reserves, and trigger geological hazards like land subsidence. The North China Plain (NCP) has experienced both subsidence and groundwater depletion due to over-extraction in the past 70 years. In this study, we used MT-InSAR technology and ascending C-band Sentinel-1 SAR data from 2017 to 2023 to study land deformation in the junction area of Shijiazhuang–Baoding–Cangzhou–Hengshui. We identified multiple subsidence funnels with a maximum rate exceeding −150 mm/year and a total deformation surpassing 600 mm. Seasonal decomposition methods accurately separated seasonal signals in the time-series deformation and groundwater level data. An exponential function model applied to long-term deformation showed no significant decrease in subsidence in severely affected areas. By modeling seasonal deformation and seasonal groundwater levels, we determined the elastic skeletal storage coefficients (Ske) to be in the range of 1.02 × 10−3~6.53 × 10−3 in subsidence areas. We obtained the spatiotemporal evolution of the total groundwater storage (TGWS), irreversible ground storage (IGWS), and recoverable ground storage (RGWS). The TGWS and IGWS decreased annually while the RGWS increased, which is attributable to the implementation of the South-to-North Water Diversion Project (SNWDP) and the issuance of groundwater withdrawal policies in the NCP.

Comparison of some Bayesian estimation methods for type-I generalized extreme value distribution with simulation

The Weibull distribution is considered one of the Type-I Generalized Extreme Value (GEV) distribution, and it plays a crucial role in modeling extreme events in various fields, such as hydrology, finance, and environmental sciences. Bayesian methods play a strong, decisive role in estimating the parameters of the GEV distribution due to their ability to incorporate prior knowledge and handle small sample sizes effectively. In this research, we compare several shrinkage Bayesian estimation methods based on the squared error and the linear exponential loss functions. They were adopted and compared by the Monte Carlo simulation method. The performance of these methods is assessed based on their accuracy and computational efficiency in estimating the scale parameter of the Weibull distribution. To evaluate their performance, we generate simulated datasets with different sample sizes and varying parameter values. A technique for pre-estimation shrinkage is suggested to enhance the precision of estimation. Simulation experiments proved that the Bayesian shrinkage estimator and shrinkage pre-estimation under the squared loss function method are better than the other methods because they give the least mean square error. Overall, our findings highlight the advantages of shrinkage Bayesian estimation methods for the proposed distribution. Researchers and practitioners in fields reliant on extreme value analysis can benefit from these findings when selecting appropriate Bayesian estimation techniques for modeling extreme events accurately and efficiently.

Vibration characteristics of a cylindrical sandwich shell with an FG auxetic honeycomb core and nanocomposite face layers subjected to supersonic fluid flow

In this paper, the flutter analysis is studied for a cylindrical sandwich shell subjected to external supersonic fluid flow. The sandwich shell consists of a re-entrant auxetic honeycomb (AH) core made of functionally graded materials (FGMs) which is covered with two polymeric face layers enriched with either carbon nanotubes (CNTs), graphene nanoplatelets (GNPs), or graphene oxide powders (GOPs). The volume fraction (percentage) of the ceramic phase in the functionally graded auxetic honeycomb (FGAH) core varies from zero at the inner surface to one at the outer one based on either power-law function (P-FGM), exponential function (E-FGM), or sigmoid function (S-FGM). It is assumed that the nanofillers are distributed uniformly inside the face layers. The first-order shear deformation theory (FSDT) and the piston theory are employed to provide the mathematical models of the shell and the aerodynamic pressure, respectively. The governing equations and boundary conditions are derived via Hamilton’s principle. An exact solution is performed in the circumferential direction via harmonic trigonometric functions (sine and cosine) and an approximate solution is performed in the axial direction utilizing the differential quadrature method (DQM). The natural frequencies and corresponding damping ratios are attained through the presented semi-analytical solution and the impacts of several factors on the natural frequencies and the critical aerodynamic pressure of the shell are examined. These factors the type of the FGM, the material index, and inclined angle of the cells in the FGAH core, the thickness of the FGAH core, the type and mass fraction of the nanofillers in the nanocomposite face layers, and boundary conditions.

Fluorescence lifetime of plant leaves with sub-nanosecond resolution

Abstract The study aimed to develop a measurement apparatus for in vivo chlorophyll-a (Chl-a) fluorescence decay measurements of plants by means of Time Correlated Single Photon Counting (TCSPC). In this approach, sub-nanosecond laser pulses with a repetition rate of 10 MHz are applied to excite the sample, followed by the analysis of arrival times of the emitted fluorescence photons. Photon statistics are generated by iteratively fitting the sum of 2 exponential functions. The tool was tested on both plastid and in vivo leaf samples of Savoy cabbage (Brassica oleracea var. sabauda) with 3-4 subsequent leaves giving a complete sample coverage starting from the outermost. The Chl-a fluorescence lifetime exhibited a gradual increase in both the isolated plastid suspensions and the in vivo leaf samples towards the innermost leaf layers explained by an increase of natural absence of light (etiolation syndrome). Furthermore, cadmium stress and iron deficiency were investigated on treated sugar beet (Beta vulgaris) samples in vivo using TCSPS measurements. The reduced fluorescence quenching resulted in an increased fluorescence lifetime. Finally, a long-term (10 week) testing of the setup was carried out on Chl-retaining resurrection Haberlea rhodopensis plants protecting themselves by an elevated non-photochemical quenching yielding a decrease of fluorescence lifetime during their desiccation.

Time distribution of stimulated transition probabilities

Abstract In recent years, an increasing number of experimental results have been published for a wide variety of systems in which high-precision, ultra-fast, and ultra-short laser pulses techniques have been used to study time-resolved dynamical processes. In the absence of a theoretical prediction of the time distribution of the transition probabilities, the lifetimes $\tau$ are generally extracted by fitting the decay time with an exponential, bi-exponential or three-exponential function. In fitting the data, the short-time behavior is generally neglected. The purpose of this study is to show that an explicit formula for the time distribution of the transition probability can be determined rigorously using time-dependent perturbation theory. A formula that perfectly fits an important subset of the experimental results from $t=0$ to $t=\infty$. We show that by following a route different from the usual procedure for deriving transition amplitudes and Einstein coefficients in time-dependent perturbation theory, one ends up with a time-dependent factor, that is, a temporal distribution function of the transition probabilities, and for the Einstein coefficients. The time distribution reported here looks in the time domain similar to the Planck distribution in the frequency domain. In fact, the behavior of the time distribution, with a maximum at $2\tau \ln 2$, and with $\tau$ playing the role of temperature, resembles the behavior of the Planck frequency distribution. We present several examples to demonstrate that the theoretical formula allows for easier adjustment of the experimental results reported in the literature. We also show that the time distribution explains the difference in resonance heights between the experimental and theoretical blue laser spectra.

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

Simple mathematical model for predicting COVID-19 outbreaks in Japan based on epidemic waves with a cyclical trend

BackgroundSeveral models have been used to predict outbreaks during the COVID-19 pandemic, with limited success. We developed a simple mathematical model to accurately predict future epidemic waves.MethodsWe used data from the Ministry of Health, Labour and Welfare of Japan for newly confirmed COVID-19 cases. COVID-19 case data were summarized as weekly data, and epidemic waves were visualized and identified. The periodicity of COVID-19 in each prefecture of Japan was confirmed using time-series analysis and the autocorrelation coefficient, which was used to investigate the longer-term pattern of COVID-19 cases. Outcomes using the autocorrelation coefficient were visualized via a correlogram to capture the periodicity of the data. An algorithm for a simple prediction model of the seventh COVID-19 wave in Japan comprised three steps. Step 1: machine learning techniques were used to depict the regression lines for each epidemic wave, denoting the “rising trend line”; Step 2: an exponential function with good fit was identified from data of rising straight lines up to the sixth wave, and the timing of the rise of the seventh wave and speed of its spread were calculated; Step 3: a logistic function was created using the values calculated in Step 2 as coefficients to predict the seventh wave. The accuracy of the model in predicting the seventh wave was confirmed using data up to the sixth wave.ResultsUp to March 31, 2023, the correlation coefficient value was approximately 0.5, indicating significant periodicity. The spread of COVID-19 in Japan was repeated in a cycle of approximately 140 days. Although there was a slight lag in the starting and peak times in our predicted seventh wave compared with the actual epidemic, our developed prediction model had a fairly high degree of accuracy.ConclusionOur newly developed prediction model based on the rising trend line could predict COVID-19 outbreaks up to a few months in advance with high accuracy. The findings of the present study warrant further investigation regarding application to emerging infectious diseases other than COVID-19 in which the epidemic wave has high periodicity.

Facial Recognition and Classification of Terracotta Warriors in the Mausoleum of the First Emperor Using Deep Learning

Abstract. The facial features of the Terracotta Warriors unearthed from the Mausoleum of the First Emperor of Qin are authentic depictions of the appearance of soldiers from the same period. Recognizing facial features to classify the Terracotta Warriors is one of the crucial aspects of archaeological research. Due to limitations in the collection of facial samples from the Terracotta Warriors, an enhanced SqueezeNet model is proposed for deep learning facial recognition. The FaceNet backbone feature extraction network has been improved by replacing the initial 7×7 convolution kernel with three 3×3 convolution kernels. The model's feature extraction layer is composed of alternating convolution layers, pooling layers, Fire modules, and pooling layers, with the introduction of an exponential function to smooth the shape of the loss function. Finally, facial classification of 295 Terracotta Warriors is accomplished using Agglomerative Clustering. The model demonstrates a facial recognition accuracy of 95.6%, showing a respective improvement of 4.1% and 2.8% compared to the classical SqueezeNet and Inception_ResNetV1 models. This approach better meets the requirements for facial recognition and classification of Terracotta Warriors, providing intelligent and efficient technical support for technological archaeology.

Effect of Micro-Cracks on Chloride Ion Diffusion in Concrete Based on Stochastic Aggregate Approach

For concrete structures in offshore areas, chloride ion erosion is one of the main factors affecting durability. It is crucial to evaluate the chloride ion permeability resistance of concrete structures. In this paper, a finite element simulation of the chloride ion diffusion process in concrete is conducted. A mass diffusion finite element model based on a random aggregate approach is established to investigate the influences of an aggregate, the interface transition zone, and micro-cracks on the chloride ion diffusion coefficients in concrete. The results show that the mass diffusion finite element analysis based on the exponential function model and the power function model can effectively simulate the chloride ion diffusion process in concrete. In addition, the data reveal that volume fraction and distribution aggregates considerably affect chloride ion diffusivity in concrete. Also, the interface transition zone significantly accelerates chloride ion diffusion in concrete. Moreover, this acceleration effect exceeds the barrier effect of an aggregate.

Geometric properties for a class of deformed trace functions

We investigate the geometric properties for a class of trace functions expressed in terms of the deformed logarithmic and exponential functions. We extend earlier results of Epstein, Hiai, Carlen and Lieb.

Dynamical and numerical analysis of classical multiple roots finding methods applied for different multiplicities

The aim of this paper is the study of the robustness of classical methods defined for finding multiple roots with single multiplicity m when they are used for approximating different roots of a function having different multiplicities. To do so, we first study the dynamical behaviour of such methods when they are applied for approximating multiple roots with different multiplicities of a polynomial. Secondly, based on the obtained dynamical results, we analyse numerically their behaviour and convergence when we carry out the approximation in the case of functions that combine polynomial, exponential and logarithmic functions.

Calibration of models predicting the load-bearing capacity of bonded anchors using a genetic algorithm

The paper deals with the topic of resistance of bonded headless post-installed anchors to uncracked concrete, in particular with the calibration of selected models predicting their ultimate tensile load. In particular, the optimization of two models for predicting the tensile capacity of an anchor is investigated in this paper. First model is based on the determination of the ultimate capacity based on separated failure modes (failure of the extraction of concrete cone and the bond failure). The second optimised model, proposed in this paper, summarizes the effect of both of these parameters into a single exponential function. A model combining the effect of concrete strength and adhesive strength has the potential to better capture the true nature of failure in which both materials are involved. In order to compare these models, an extensive database of experimental results was compiled (including own experiments and also results from various authors). The calibration consisted in finding the most appropriate values of the individual input parameters of the models to fit the experimental results as closely as possible. The models for predicting the tensile capacity of anchors are multiparametric. Therefore, a method using elements of genetic algorithms was used for optimization, suitable for this purpose. Several possible statistical evaluation criteria were used for the evaluation of the fit of the models. The optimization of the models showed that the proposed model combining the effect of concrete strength and bond strength can be optimized to better fit the experimental results.

Abundant soliton solutions on fractional kraenkel manna merle model (FKMM) via new extended of generalized exponential rational function approach (GERFA)

Abstract In this paper, we use the generalized exponential rational function approach (GERFA) for constructing new solitary wave solutions for the fractional Kraenkel Manna Merle (FKMM) model, which is saturated ferromagnetic materials (MF) with a field outside that has very little conductivity, reflects the nonlinear of an ultrashort pulse. The fractional performance of the proposed model is assessed using the beta-derivative. As a result, this investigation demonstrates the efficacy and simplicity of the offered strategy. Some of the revealed solutions' 3D graphs are also employed in the numerical simulations. The findings show that the model theoretically has extraordinarily rich soliton structures.

An Analytical Study on Growth Performance of Major Food Grain and Oilseed Crops in Context of Food Security: Rajasthan, India

Background: Productivity, as a source of growth has been a significant premise of analytical investigation in economics all along, however it gained an empirical momentum in the recent years only and has become a necessary index to growth. Present paper estimates growth rate of area, production and productivity of food grain and oilseed crops in Rajasthan. Methods: To analyse growth performance of food grain and oilseed crops Rajasthan time series, data on area, production and yield were collected from 1976-77 to 2018-19. The time-series data were collected on area, production and yield of selected crops from the Directorate of Economics and Statistics (DES), Directorate of Agriculture (DOA) and Rajasthan State Agriculture Department. The compound growth rates (CGRs) of area, production and productivity of selected crops in Rajasthan were calculated using the exponential function. Result: The study revealed that production, productivity and area of total food grains was recorded a positive and highly significant annual growth at the rate of 3.18%, 2.19% 0.63% per annum, respectively. The production, productivity and area of total oilseed was observed a positive and highly significant annual growth rate of 7.63%, 2.07% and 4.27% per annum, respectively. The area under barley, gram and sesamum reduced during the study period. Therefore, the technological support through the varietal improvement and extension support is required.

A comprehensive review of experimental studies on shear behavior of bolted rock joints with varying rock joint and bolt parameters and normal stress

AbstractThe shear characteristics of bolted rock joints are crucial for the stability of tunneling and mining, particularly in deep underground engineering, where rock bolt materials are exposed to high stress, water pressure, and engineering disturbance. However, due to the complex interaction between bolted rock joints and various geological contexts, many challenges and unsolved problems arise. Therefore, more investigation is needed to understand the shear performance of bolted joints in the field of deep underground engineering. This study presents a comprehensive review of research findings on the responses of bolted joints subjected to shearing under different conditions. As is revealed, the average shear strength of bolted rock joints increases linearly with the normal stress and increases with the compressive strength of rock until it reaches a stable value. The joint roughness coefficient (JRC) affects the contact area, friction force, shear strength, bending angle, and axial force of bolted rock joints. A mathematical function is proposed to model the relationship between JRC, normal load, and shear strength. The normal stress level also influences the deformation model, load‐carrying capacity, and energy absorption ratio of bolts within bolted rock joints, and can be effectively characterized by a two‐phase exponential equation. Additionally, the angle of the bolts affects the ratio of tensile and shear strength of the bolts, as well as the mechanical behavior of both bolted rock joints and surrounding rock, which favors smaller angles. This comprehensive review of experimental data on the shear behavior of bolted rock joints offers valuable theoretical insights for the development of advanced shear devices and further pertinent investigations.

Bottlebrush Block Copolymers at the Interface of Immiscible Liquids: Adsorption and Lateral Packing.

Amphiphilic bottlebrush block copolymers (BBCPs), having a hydrophilic bottlebrush polymer (BP) linked covalently to a hydrophobic BP, were found to segregate to liquid-liquid interfaces to minimize the free energy of the system. The key parameter influencing the outcome of the experiments is the ratio between the degree of polymerization of the backbone (NBB) and that of the side-chain brushes (NSC). Specifically, a spherical, star-like configuration results when NBB < NSC, while a cylindrical, bottlebrush-like shape is preferred when NBB > NSC. Dynamic interfacial tension (γ) and fluorescence recovery after photobleaching (FRAP) measurements show that the BBCP configuration influences the areal density and in-plane diffusion at the fluid interface. The characteristic relaxation times associated with BBCP adsorption (τA) and reorganization (τR) were determined by fitting time-dependent interfacial tension measurements to a sum of two exponential relaxation functions. Both τA and τR initially increased with NBB up to 92 repeat units, due to the larger hydrodynamic radius in solution and slower in-plane diffusivity, attributed to a shorter cross-sectional diameter of the side-chains near the block junction. This trend reversed at NBB = 190, with shorter τA and τR attributed to increased segregation strength and exposure of the bare water/toluene interface due to tilting and/or wiggling of the backbone chains, respectively. The adsorption energy barrier decreased with higher NBB, due to a reduced BBCP packing density at the fluid interface. This study provides fundamental insights into macromolecular assembly at fluid interfaces, as it pertains to unique bottlebrush block architectures.

Sugarcane leaf return and mixing significantly reduce slope runoff and soil loss

AbstractImproper management of sugarcane cultivation may cause severe soil erosion, especially in lateritic red soil hilly areas. Crop straw return to the field can effectively improve the soil structure, but there is a lack of evaluation studies of the soil and water conservation benefits of sugarcane leaf mixing and return to the field, and the underlying regulatory mechanism has rarely been described. Thus, we conducted a series of simulated rainfall experiments to investigate the impact of returning different mixed sugarcane leaf weights (0, 3, 6, 9, 12 and 15 t/ha) to the field on erodible lateritic red soil slopes that support sugarcane production under various slopes (5°, 10°, 15° and 20°) in different rainfall events (60 and 120 mm/h). The results indicated that returning sugarcane leaves into farmland could effectively regulate the process of runoff and sediment production on lateritic red soil slopes, and the optimal mixing amount was 9 t/ha. Under a low rainfall intensity, the erosion process was mainly affected by the slope, and the benefits of applying the sugarcane leaf mixing treatment to the slope to reduce runoff and sediment could reach 13.37–53.04% and 23.86–77.55%, respectively. However, under a high rainfall intensity, the sugarcane leaf mixing amount was the main influencing factor, the benefits for reducing runoff and sediment could reach 4.64–38.72% and 30.87–93.14%, respectively. The sugarcane leaf mixing amount exhibited a linear relationship with the Reynolds number (R2 &gt; 0.63), whereas the runoff velocity (R2 &gt; 0.96), Froude number (R2 &gt; 0.94) and drag coefficient (R2 &gt; 0.85) exhibited well‐fitting exponential function relationship with it. This study provides a reference for future endeavours regarding the inhibition mechanisms of sugarcane leaves mixed and returned to the field for decreasing slope erosion.

A new version of trial equation method for a complex nonlinear system arising in optical fibers

In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.