Exponential Form Research Articles

Analytical solution for rainwater infiltration in monolithic soil covers under heavy rainfall and its implications for practice

This study established an analytical solution for rainwater infiltration in MSCs during heavy rainfall assuming that the soil hydraulic properties follow exponential forms. The analytical solution was used to calculate the factor of safety (FOS) of MSCs after the verification with numerical simulations. It is found that the FOS for the potential slip surface at the cover bottom remined the lowest during and after heavy rainfall. The percolation and FOS might present pronounced “lag effects”, meaning that the maximum percolation rate and minimum FOS occurred after heavy rainfall. A parametric study was conducted to reveal the influencing factors on the hydraulic response and slope stability of MSCs based on the analytical solution. Relevant results demonstrate that the hydraulic performance and slope stability could be improved by decreasing soil saturated hydraulic conductivity, increasing soil desaturation coefficient and lowering water level in landfills. The results also reveal the existence of a threshold of soil saturated hydraulic conductivity (1×10<sup>-8</sup> m/s for this study) for controlling the hydraulic performance and slope stability of MSCs. Furthermore, the analytical solution was applied to determine the rainfall intensity-duration threshold curves of MSCs. The results indicate that the obtained curves of MSCs satisfied exponential forms.

Delineating Trajectories of Social-Emotional Competence in Infants and Toddlers.

The acquisition of social-emotional competence (SEC) in early childhood has implications for critical child and adult outcomes, such as school readiness, educational and occupational attainment, and mental health. To elucidate this developmental process, normative trajectories of social-emotional competence in infants and toddlers were modeled using longitudinal mixed effects modeling, including the evaluation of child and family characteristics as moderators. The SEC of 12-36-month-old children (N=256, 83% White, 51% female) was assessed in a cohort-sequential design using the Infant Toddler Social-Emotional Assessment Competence scale. Trajectories were modeled using linear, quadratic, exponential, and logistic mean forms. Following base model selection, child sex, maternal education, parental occupation, family income, and number of siblings were separately added to the model to assess their effect on trajectories. Results show that infants and toddlers SEC follows a quadratic pattern of growth. Additionally, girls had higher scores than boys at 12months with similar slopes. Number of siblings was also significant at 12months such that children with fewer siblings had higher scores than those with more with similar slopes. This suggests a female advantage in early SEC acquisition exists even before 12months and that sibling number may moderate SEC in infancy and toddlerhood.

Compact Analysis of the Necessity of Padé Approximation for Delayed Continuous-time Models in LQR, H-Infinity and Root Locus Control Strategies

This paper presents a comprehensive analysis of the need for the Padé approximation for continuous-time models with delays, focusing on its critical role in addressing the control challenges posed by time delays. Time delays, often referred to as dead times, transport delays or time lags, are inherent in a wide range of industrial and engineering processes. These delays introduce phase shifts that degrade control performance by reducing control bandwidth and threatening the stability of closed-loop systems. Accurate modelling and compensation of these delays is essential to maintain system stability and ensure effective control. This paper highlights the difficulties that arise when using advanced control techniques such as root locus (RL), linear quadratic regulator (LQR) and H-infinity (H_∞) control in systems with delays. Representing delays in exponential form leads to an infinite number of state problems, complicating the design and analysis of controllers in such systems. To address these challenges, the Padé approximation is proposed as an effective method for approximating time delays with rational polynomials of appropriate order. This approach allows for more accurate simulation, system analysis and controller design, thereby mitigating the problems caused by delays. The paper also provides a detailed comparative analysis between the Padé approximation and Taylor polynomials, demonstrating the superiority of the former in achieving accurate delay modelling and control performance. The results show that the use of Padé approximation not only improves the accuracy of system models, but also improves the robustness and stability of control strategies such as RL, LQR, and H_∞. These results highlight the importance of the Padé approximation as a valuable tool in the design of delay-affected control systems, offering significant advantages for both theoretical and practical applications.

Isospin violation effect and three-body decays of the Tcc+ state

In this work, we make a study of the Tcc+ state observed by the LHCb collaboration in 2021. In obtaining the effective potentials using the one-boson-exchange potential model we use an exponential form factor and find that, in the short and medium range, the contributions of the π, ρ, and ω exchanges are comparable while in the long range the pion-exchange contribution is dominant. Based on the assumption that Tcc+ is a loosely bound state of D*D, we focus on its three-body decay using the meson-exchange method. Considering that the difference between the thresholds of D*+D0 and D*0D+ is even larger than the binding energy of Tcc+, the isospin-breaking effect is amplified by the small binding energy of Tcc+. Explicitly including such an isospin-breaking effect we obtain, by solving the Schrödinger equation, that the probability of the isoscalar component is about 91% while that of the isovector component is around 9% for Tcc+. Using the experimental value of the mass of Tcc+ as an input, we obtain the wave function of Tcc+ and further obtain its width via the three-body hadronic as well as the radiative decays. The total width we obtain is in agreement with the experimental value of the LHCb measurement with a unitarized Breit-Wigner profile. Conversely, the current results support the conclusion that Tcc+ is a hadronic molecule of D*D. Published by the American Physical Society 2024

Exploring Kink Solitons in the Context of Klein–Gordon Equations via the Extended Direct Algebraic Method

This work employs the Extended Direct Algebraic Method (EDAM) to solve quadratic and cubic nonlinear Klein–Gordon Equations (KGEs), which are standard models in particle and quantum physics that describe the dynamics of scaler particles with spin zero in the framework of Einstein’s theory of relativity. By applying variables-based wave transformations, the targeted KGEs are converted into Nonlinear Ordinary Differential Equations (NODEs). The resultant NODEs are subsequently reduced to a set of nonlinear algebraic equations through the assumption of series-based solutions for them. New families of soliton solutions are obtained in the form of hyperbolic, trigonometric, exponential and rational functions when these systems are solved using Maple. A few soliton solutions are considered for certain values of the given parameters with the help of contour and 3D plots, which indicate that the solitons exist in the form of dark kink, hump kink, lump-like kink, bright kink and cuspon kink solitons. These soliton solutions are relevant to actual physics, for instance, in the context of particle physics and theories of quantum fields. These solutions are useful also for the enhancement of our understanding of the basic particle interactions and wave dynamics at all levels of physics, including but not limited to cosmology, compact matter physics and nonlinear optics.

Stability and qualitative dynamics of a class of three‐dimensional difference equation system of exponential type

In this paper, we study the boundedness, persistence, and global stability of the equilibrium point and rate of convergence of positive solutions of the exponential form of a system of difference equations where are constants and are initial values. Furthermore, some numerical examples are also given, which support our theoretical results.

Exploring the Diversity of Kink Solitons in (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves.

Chiral-even twist-3 GPDs for the proton in a spectator diquark model

We investigate the chiral-even twist-3 generalized parton distributions (GPDs) of valence quarks in the proton at nonzero skewness ξ, using a spectator model with scalar and axial-vector diquarks. We consider the exponential form factor for the nucleon-quark-diquark vertex and the axial-vector diquark with light-cone transverse polarization. We analyze the dependence of GPDs on the longitudinal momentum fraction x at different ξ, and on the square of the transverse momentum transfer ΔT2 at different x. Our numerical results reveal distinct discontinuities in all twist-3 GPDs except G1 and G˜1. By taking the forward limit, we obtain the twist-3 parton distribution function gT, which encodes the transverse spin distribution of quarks. We also compare the kinetic orbital angular momentum and the spin-orbit correlations of quarks defined by the twist-2 and twist-3 GPDs, respectively. Published by the American Physical Society 2024

Longitudinal wave propagation in a practical metamaterial lattice

A practical metamaterial lattice is constructed by integrating curved beams, four-link mechanisms, and cantilever beams with lumped masses. It can generate two complete low-frequency bandgaps due to the lateral local resonance, inertia mass, and the main chain. The effective mass density and stiffness are obtained using different effective models, which show negative within the bandgaps. The analysis of the energy distribution and the space wave attenuation reveals that the metamaterial can attenuate the elastic waves in an exponential form within the bandgaps along the lattice. The finite element model is established to show the dynamic behaviour of the elastic wave propagation in the frequency domain and transient domain. Both results show that waves can be efficiently blocked within the bandgaps, while outside the bandgaps, waves can propagate without any attenuation. Finally, the experimental model of practical metamaterial is constructed, and the test piece is excited by a force hammer. Experimental results verify that the practical metamaterial can efficiently suppress the vibration within the bandgap frequency and validate the accuracy of the theoretical prediction.

Quantum geometrical properties of topological materials

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models.

On the existence and avoidance of singularity in [formula omitted] gravity using Raychaudhuri equation

The paper deals with the modified Raychaudhuri equation in the framework of homogeneous and isotropic f(Q) gravity model subject to the parametrization f(Q)=Q+F(Q). Focusing of a congruence of time-like geodesics has been studied via the convergence condition for two choices of F(Q) namely, a polynomial in Q and an exponential form of Q. Finally, possibilities for the avoidance of initial big-bang singularity in both the models have been examined via the signature of Raychaudhuri scalar.

Micro–Macro-Analysis and Model Derivation of Electrical Resistivity of Ningxia Cement–Loess

In recent years, highway infrastructure in the Ningxia region of China has rapidly advanced. Cement–loess is extensively utilized in the roadbed and foundation reinforcement. It is necessary to conduct micro–macro-analysis and model derivation of the electrical resistivity on Ningxia cement–loess, which are beneficial for both the practical application of electrical resistivity and the evaluation of the geotechnical properties of cement–loess. Therefore, a series of electrical resistivity measurements, microstructural observations (scanning electron microscopy), mineral analyses (thermogravimetric analysis), and theoretical analyses were adopted on the cement–loess. The following conclusions can be drawn: The electrical resistivity is negatively related to dry density and water content, while it is positively related to cement dosage and curing age. A cement dosage of 6% exhibits a lower hydration reaction potential compared to 12%, causing a slower increase in electrical resistivity. The formation of calcium silicate gel around particles results in particle clustering and pore filling, reducing the pore area and increasing electrical resistivity. Increased hydration also decreases microscopic orientation, contributing to a higher electrical resistivity of cement–loess. Finally, a new three-dimensional electrical resistivity model was created, finding that the electrical resistivity of Ningxia cement–loess was determined by the dry density, water content (ρd·w), cement dosage, and curing age (aw·T) in an exponential function form. The new three-dimensional electrical resistivity model could be used in the high-efficiency evaluation of the cement–loess geotechnical parameter, offering valuable insights for the monitoring and maintenance of road infrastructure.

Entropy optimization of MHD second-grade nanofluid thermal transmission along stretched sheet with variable density and thermal-concentration slip effects

The goal of present investigation is to explore the influence of exponential variable density and entropy optimization on second-grade nanofluid heating efficiency and mass-concentration transmission along extended surface using external magnetic-field and temperature-concentration slip effects. To enhance the motion of nanoparticles and thermal efficiency, the influence of exponential form of temperature-based density on magnetically charged second-grade nanomaterial is main novelty of this research. For higher temperature difference, the entropy optimization is used. The defined formulation of stream functions and similarities are used to convert leading second-grade nanofluid model into ordinary differential form. The efficient Keller box method and Newton Raphson technique are applied to compute numerical results. The final algebraic equations are solved through global matrix for unknown physical quantities. The consequence of all physical constraints on velocity/U profile, temperature/θ field, concentration/ϕ shapes, skin friction coefficient, Nusselt and Sherwood number are analyzed pictorially and numerically. The following range of parameters 0.1 ≤ ξ ≤ 2.0, 0.0 ≤ n ≤ 1.2, 0.1 ≤ Ec ≤ 2.0, 0.07 ≤ Pr ≤ 7.0, 0.01 ≤ Nt ≤ 0.8, 0.01 ≤ Nb ≤ 0.9 is used. It is found that velocity field increases with maximum amplitude as variable density, magnetic force and temperature-slip constraint. It is noted that the slip behavior in temperature field and concentration field are increased with convective boundary conditions. It is depicted that local Nusselt quantity and local Sherwood quantity increases as buoyancy force and Prandtl coefficient increases.

Long-lived recurrent fluxes of energetic ions from solar coronal holes

The results of studying the relative abundances and energy spectra of 4He, C, O, and Fe suprathermal ions in particle fluxes from long-lived near-equatorial coronal holes are presented. It was found that the ion energy spectra had a power or exponential form and, in a number of events, increased particle fluxes were observed in the region of ion energies (~0.2–1.5 MeV/nucleon), creating a fracture in the spectra, which may be due to additional ions accelerated beyond 1 au.

Bond-slip model of corroded plain round bars in low-strength concrete under cyclic and monotonic loading

The effect of corrosion on the overall behavior of beam-type bond-slip samples constructed from low-strength concrete and plain round bars was examined in this study. First, a set of nominally identical specimens underwent testing under both monotonic and cyclic loading, and subsequently, the bond-slip interaction was assessed for each individual sample. The observed failure mode for plain round bars was direct pullout without concrete splitting apart, characterized by the loss of cohesion between the rebar and the adjacent concrete surface. Then, an analytical relationship was established by fitting a curve to the average experimental data using the method of least squares. Corrosion-degradation in the bond stress was considered by incorporating an exponential component into the equation of the reference bond-slip curve (i.e., null corrosion). Besides, the degradation in bond strength was predicted as a function of corrosion level, which is in the form of an exponential curve. The maximum surface crack width, an easily quantified variable on-site, was correlated with the bond strength of corroded bars. The regression analysis successfully established the optimal relationship between the maximum surface crack width and the corrosion level in an exponential form as well. Notably, the majority of the data in all cases fell within the 95% confidence interval.

Nonminimal coupling holographic superconductors

We explore a holographic superconductor model in which a real scalar field is nonminimally coupled to a gauge field. We consider several types of the nonminimal coupling function h(ψ) including exponential, hyperbolic (cosh), power-law, and fractional forms. We investigate the influences of the nonminimal coupling parameter α on condensation, critical temperature, and conductivity. We can categorize our results in two groups. In the first group, conductor/superconductor phase transition is easier to occur for larger values of α, while in the second group stronger effects of the nonminimal coupling makes the formation of scalar hair harder. Although the real and imaginary parts of conductivity are impressed by different forms of h(ψ), they follow some universal behaviors such as connecting with each other through Kramers–Kronig relation in ω → 0 limit or the appearance of gap frequency at low temperatures around ωg ∼ 8 Tc, which shifts to larger values by increasing the strength of α. Among all forms of h(ψ) we observe that h(ψ) = 1 + αψ2 gives us better information in wide range of nonminimal coupling constant and temperature. Choosing the best form of h(ψ), we construct a family of solutions for holographic conductor/superconductor phase transitions to discover the effect of the hyperscaling violation when the gauge and scalar fields are nonminimally coupled. we find that the critical temperature increases for higher effects of hyperscaling violation θ and nonminimal coupling constant α. By increasing these two parameters, we obtain lower values of condensation which means that conductor/superconductor phase transition will acquire easier. Furthermore, we understand that the hyperscaling violation affects the conductivity σ of the holographic superconductors and changes the expected relation in the gap frequency. Some universal behaviors like infinite DC conductivity are observed. In addition, we consider a five-dimensional Gauss–Bonnet (GB) black hole with a flat horizon. We find out that the critical temperature decreases for larger values of GB coupling constant, λ, or smaller values of nonminimal coupling constant, α, which means that the condensation is harder to form. Moreover, we study the electrical conductivity in the holographic setup. We observe that the gap frequency shifts to larger values for stronger λ, and becomes flat by increasing α.

On the study of analytical soliton solutions and interaction aspects to the Estevez-Mansfield-Clarkson equation arising in diversity of fields

Abstract The beta fractional form of the Estevez-Mansfield-Clarkson equation is under consideration and this study is done with the assistance of methods such as modified F-expansion method and the logarithmic transformation. A variety of analytical solutions like bright, dark, mixed, singular, bright-dark, and combined solitons are extracted. Moreover, multi waves structures, interaction with double exponential form, breather waves, mixed type solutions as well as periodic cross kink solutions have been analyzed. The governing equation is converted into an ordinary differential equation by employing an appropriate wave transformation with the β-derivative in order to achieve the desired solutions. The applied approaches have substantial computational capability, enabling them to efficiently address exact solutions with high accuracy in these systems. The results indicate that the equation under investigation theoretically contains a substantial number of soliton solution structures. Additionally, in order to examine the behaviors of the solutions at various parameter values, we plot a variety of graphs that incorporate pertinent parameters. The results of this study have the potential to improve understanding of the nonlinear dynamic characteristics displayed by the specified system and to confirm the effectiveness of the techniques that have been implemented.

Comparison and Optimization of Light Use Efficiency-Based Gross Primary Productivity Models in an Agroforestry Orchard

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.

Failure mechanism of loess landslide induced by water stagnation on the combined surface

In order to reveal the destructive mechanism of loess landslide induced by stagnant water on the combined surface, and to clarify the influence of the main control factors, this paper takes a typical loess landslide in northern Shaanxi as the research object, analyzes the structure of the rock and soil body, and the excavation and filling construction through the geohazard survey, and analyzes the process of traction sliding caused by the stagnant water on the combined surface at the different stages of the project by combining with the calculation of the stability of the slope body. Further the article analyses the process of traction sliding caused by water on the combined area due to construction by means of a discrete element model, and delves into the mechanism of strength reduction of saturated loess. The results show that: 1) the combined surface stagnant water type loess landslide has the characteristics of sudden sliding and rapid evolution, which is highly hazardous and difficult to prevent and control; 2) the slope destabilization is controlled by the engineering geological conditions, and the slope excavation changes the original mechanical equilibrium conditions of the slope, which provides the dynamic conditions for the traction sliding of the slope; 3) the change of the hydrogeological environment results in the obstruction of the natural drainage channel, which leads to the formation of continuous sliding surface due to stagnant water on the combined surface, and the formation of a continuous sliding surface due to stagnant water on the combined surface. Surface stagnant water to form a continuous slippery surface, inducing the overall destabilization of the slope damage; 4) loess strength index with the increase of saturation and the exponential function form of reduction, and when the saturation degree reaches more than 80%, the strength index of the soil body to reach the basic stability. The article expanding the ideas of landslide control and analysis, and the research results will provide a theoretical basis for the design of junction landslide management in the loess areas of northern Shaanxi.

Static, cylindrically symmetric spacetime in the coincident f(Q) gravity

In this paper we consider a static, cylindrically symmetric spacetime with coincident f(Q) gravity. Since the field equation of this spacetime in symmetric teleparallel gravity is suitable for choosing the function of f(Q) in the form of power series and exponential forms which are in coherent with cosmological observations, anisotropic perfect fluid solutions of these forms are discussed for this spacetime. Energy densities, directional pressures and energy conditions are plotted and analysed for a few different metric potentials. Although corresponding cosmic strings violate all energy conditions in both f(Q) functions, the corresponding Levi-Civita solution violates all energy conditions in the power law function of f(Q), but for exponential f(Q) gravity they are satisfied in small regions.