Exponential Function Research Articles

Analysis of low temperature flexural creep properties of mineral powder reinforced asphalt MPRA through combined experimentation and finite element simulation

The low temperature flexural creep properties of Mineral Powder Reinforced Asphalts (MPRAs) are systematically studied using both experimentations and finite element (FE) simulations. First, a new method to obtain the viscoelastic parameters of the pure asphalt is introduced. A digital twin for MPRA is then generated using Matlab, and a reliable simplification of the geometric dimension is adopted and verified. Second, a novel concept of equivalent particle size (EPS) is introduced in the FE models to represent the agglomeration effect of mineral powder in an asphalt. Four different basic EPSs are adopted to simulate the MPRAs with different powder-cement ratios. Third, the exponential function between EPS and simulated creep stiffness is fitted, and a reliable EPS for a specific sample is deduced using the fitted exponential function and experimental result, and it is verified using FE simulations. Finally, the digital twin for the MPRA proposed in this study is further verified by predicting the flexural creep properties at another temperature. The results show that the novel method to obtain the pure asphalt viscoelastic parameters and the digital twin for MPRA are reliable. EPS proposed in this paper could suitably characterize the agglomeration effect of mineral powder in FE simulations. The stress and strain inside the asphalt matrix decrease with smaller EPS. An accurate EPS for a specific MPRA can be deduced using experimental results and the exponential function fitted using different EPSs and simulated creep stiffness. Creep stiffness is larger under bigger powder-cement ratios, whereas the creep rate shows an opposite trend. Mineral powder tends to agglomerate more under bigger powder-cement ratios because the accurate EPS is bigger for MPRA having a bigger powder-cement ratio. The reliable digital twin for MPRA so obtained could be used to consistently predict the flexural creep properties at a different temperature.

Traveling wave solutions of Fordy–Gibbons equation

The Fordy–Gibbons equation is a nonlinear differential equation. Physically, the motion of a damped oscillator with a more complex potential than in basic harmonic motion is described by the Fordy–Gibbons equation. For the equation under consideration, numerous novel families of precise analytical solutions are being successfully found. The soliton solutions are represented as rational and exponential functions. To further illustrate the potential and physical behavior of the equation, the findings are also stated visually. Three approaches are suggested in this paper for solving the Fordy–Gibbons equation. These solutions are new solutions.

Analysis of the nonlinear Fitzhugh–Nagumo equation and its derivative based on the Rabotnov fractional exponential function

The Fitzhugh–Nagumo equation is an essential governing equation that explains the process of impulse transmission from the nerves. This work suggests a novel generalized arbitrary-order Fitzhugh–Nagumo equation that is based on the recently developed nonsingular fractional operator. The modified Fitzhugh–Nagumo equation has been generalized using fractional Yang–Abdel–Cattani operator to provide a more accurate representation of the equation with memory effects and non-local behavior. Further, we discussed some interesting and new properties of the proposed fractional operator and the Rabotnov exponential fractional kernel with a modified Sumudu integral transform. In order to solve the proposed nonlinear differential equation, the homotopy perturbation method coupled with the modified Sumudu integral transform technique is implemented to examine the result of a newly proposed Yang–Abdel–Cattani fractional Fitzhugh–Nagumo equation, which converges to the exact solution. The nonlinear terms of this equation are handled in terms of He’s polynomials. The existence and uniqueness of the solution have been demonstrated using the Banach space fixed point theorem. The convergence and error analysis of the suggested technique are also discussed. The findings are expressed in terms of generalized Mittag-Leffler functions. The obtained results are plotted with the help of MATLAB R2016a mathematical software. The results in this work are more accurate, and it is proposed that the new Yang–Abdel–Cattani fractional derivative operator is an effective tool for finding the results of any other nonlinear problems arising in science and engineering. All three problems have been computed for different orders to confirm the significance of the fractionalizing concept in the proposed equation. The increase of the fractional order value ℘ to 1 confirms that our proposed time-fractional Fitzhugh–Nagumo equation in the Yang–Abdel–Cattani sense gets close to the analytic solution.

Soliton waves with optical solutions to the three-component coupled nonlinear Schrödinger equation

This study uses the modified Sardar sub-equation method to find novel soliton solutions to the nonlinear three-component coupled nonlinear Schrödinger equation (NLSE), which is used for pulse propagation in nonlinear optical fibers. Multi-component NLSE equations are widely used because they can represent a wide range of complex observable systems and more dynamic patterns of localized wave solutions. The optical solutions proposed in this study are novel and can be described using hyperbolic, trigonometric, and exponential functions. These solutions are categorized as bright, dark, singular, combo bright-singular, and periodic solutions. Some solutions’ dynamic behaviors are demonstrated by selecting appropriate physical parameter values. The results and computational analysis indicate that the techniques provided are simple, effective, and adaptable. They can be applied to a variety of nonlinear evolution equations, whether stable or unstable, and can be used in fields such as mathematics, mathematical physics, and applied sciences.

An Examination of Learning Using Fourier Analysis of Mathematical Models of Consciousness

The concept of consciousness remains quite possibly the ultimate mystery for humanity. It is somehow linked to the ability to learn and experience the external environment. The process of learning is of utmost importance to any species and yet remains largely not understood. While many experts from varying fields throughout history have attempted to quantify and measure learning, all failed to capture the process with a mathematical explanation. The purpose of this paper is to present three different models of consciousness and their related Fourier transform and evaluate the ability of each to capture some of the behavior that is understood about the conscious experience. There is some evidence that the experience of consciousness as a function of time can be measured in an expression of phase. With this hypothesis, the technique of Fourier transform becomes a useful tool to examine a mathematical model of consciousness that can be transformed from the phase plane to the linearization of time. This paper summarizes some current research on consciousness and learning and perception of time. This paper then presents three different models of consciousness represented as a pulse function, Dirac Delta function, and exponential decay function and examines each model utilizing Fourier transform. Finally, this paper concludes that the intersection between psychology, neuroscience and cognitive science can be made by utilizing analysis tools in the field of applied mathematics.

Pengaruh Penggunaan Abu Cangkang Terhadap Kuat Tekan Beton

The background of this research is that there is still a lot of oyster shell material that has not been utilized, meanwhile the shell has good mechanical properties as well as concrete. Up to this moment, the shell is not used properly so it needs to studied the use as concrete mixture. By the utilization of this waste is expected to reduce the environmental impact. This study was aims to obtain a regression equation of the compressive strength of concrete using shells, the shells used in this study were oyster shells (ostreidae), tested at 28 days of age with 3 variations in the percentage of the use of shell ash material and with 1 variation of normal concrete. The percentage variations used are 0% variation, 5% variation, 10% variation, and 15% variation. In addition, based on the regression equation, the strong relationship between variables will be obtained which is indicated by the value of r square. The implementation of this research was carried out by carrying out a compressive test of concrete cylinder samples with 3 variations of the water cement rasio (W/C), is W/C 0.4, W/C 0.5, and W/C 0.6. The activity was continued by recording and analyzing data by conducting a regression analysis between the relationship between the percentage of shell use and the compressive strength of concrete. The regression equation used is the exponential regression equation, linear regression, polynomial regression order 2. From the test data, it is found that the regression equation is the best for W/C variation 0.4, obtained polynomial regression equation order 2, y = -0.0351x2 + 0, 3279x + 25.075 with r² = 0.9999, for the W/C variation of 0.5, the second order polynomial regression equation is obtained, y = -0.0702x2 + 0.8062x + 23.406 with r² = 0.8983, and for the W/C variation of 0.6 , the second order polynomial regression equation is obtained, y = -0.0404x2 + 0.4008x + 19.074 with a value of r² = 0.9167..

Modeling soil moisture from in situ portable X-ray spectrometer measurements: A novel approach for correcting geochemical data across different environments and climatic conditions

The portable X-ray fluorescence (pXRF) spectrometer is widely employed for in situ analysis of both contaminated and uncontaminated soils. However, the accuracy of the measurements can be significantly affected by soil moisture, resulting in unreliable soil pollution monitoring. This effect has already been studied and quantified, but this is ineffective if the soil moisture in the field is unknown. Given the considerable variability of soil moisture conditions across time and space, significant bias during in situ investigations remains a main issue. This study introduces a novel method to estimate soil moisture directly from pXRF field measurements, enabling its reliable use in almost any field condition. The study was conducted using soil samples and in situ pXRF soil surface measurements in Estarreja (Portugal) and Vicdessos (France). In the first experiment, the innovative approach involved modeling soil moisture directly from the raw XRF measurement errors obtained in moist soils using multiple regression. In the second experiment, metal concentrations were modeled as an exponential function of the moisture content. The final model integrates both approaches to correct field data from geochemical mapping in diverse environments, including a coastal region in Portugal and a mountainous region in France. Our findings demonstrate that this simple, efficient and cost-effective method accurately predicts soil moisture (U) using pXRF, as shown by the equation Umeasured = 1.0028 x Uestimated (r2 = 0.9715). The model effectively corrected up to 70% of moisture-induced errors in metal concentrations in the wettest soils and produced more reliable soil Fe, Pb, and Zn maps. Specifically, the accuracy improvement was at least 32% in drier soils (Portugal) and at least 55% in wetter soils (France). This study offers a cost-effective, efficient solution for employing pXRF in geochemical mapping across different climatic conditions and soil environments.

Modulation instability and extraction of fractional optical solitons in the presence of generalized Kudryashov’s law and dual form of non-local nonlinearity

This research delves into the study of optical solitons governed by Kudryashov’s law of nonlinear refractive index, incorporating a conformable fractional derivative, perturbed terms, and a quadrupled-power law in optical fibers. Employing a novel approach known as the generalized exp(− S(ζ) ) expansion method, we identify novel optical soliton solutions characterized by exponential, rational, and hyperbolic functions featuring a conformable fractional derivative. These solutions manifest as unified bright-dark, singular, singular periodic, kink, anti-kink, and novel solitary waves. We employ 3D, contour, and 2D plots to analyze the behavior of these solutions across different temporal and fractional-order derivative parameters, offering an understanding of their physical interpretations. Furthermore, we perform modulation instability (MI) analysis that depends on standard linear stability analysis to derive the MI gain spectrum, demonstrating how well our methodology works for nonlinear problems in the natural sciences and engineering domains. This approach provides an effective way to investigate optical solutions in various integer and fractional Schrödinger equations.

The Relevance of the New Exponential Equation of State for Semiconductors

The Relevance of the New Exponential Equation of State for Semiconductors

Experimental study on dynamic characteristics of frozen saline silty clay under cyclic loading

In order to investigate the dynamic characteristics of frozen saline silty clay (studied saline soil) under reciprocating loading, a series of cyclic tri-axial experiments have been conducted for studied saline soil with 0.0, 0.5, 1.5, and 2.5 % of Na2SO4 contents under confining pressures 0 MPa–18 MPa at −6 °C, respectively. The test results show that as the increase of number of cycle N, the accumulated axial strain εaN continuously increases, and the relationship between εaN and N can be described by an exponential function: εaN=aNb; As the number of cycle N increasing, dynamic shear modulus GdN and dynamic bulk modulus KdN increase firstly and then decrease slightly; Salt content and confining pressure have great influences on the εaN, GdN and KdN. Within the same loading cycle, GdN and KdN decrease until reaching their minimums when the salt content is 0.5 %, and then GdN and KdN increase as the increase of salt content. With the change of salt content, εaN presents an opposite change law to GdN and KdN. For the studied saline soil with same salt content, during the same loading cycle, εaN and GdN increase until reaching their minimums, and then decrease as the increase of confining pressure, KdN increases with the increase of confining pressure. This study can be applied to evaluate the elasto-plastic properties of frozen soils and saline soils subjected to cyclic dynamic loading.

Investigating the dynamics of water–sand mixing inrush in viscous sand layers: insights from laboratory experiments

The geological hazard of water–sand inrush is a matter of concern for infrastructure construction and resource exploration activities in China, due to the complex interplay between groundwater dynamics and the stability properties of sand particles. This phenomenon is characterized by its intensity, hazardous nature, and unpredictable behavior. Following comprehensive analysis, this study identifies the critical factors influencing water–sand inrush processes as fissure width, water stress (waterhead height), in-situ sand ground stress within the sand stratum, and clay content. To investigate these factors experimentally, a custom-designed hydraulically coupled water–sand inrush test apparatus was used. The apparatus was equipped with a cylinder to apply ground stress, a pneumatic diaphragm pump to regulate water stress, and a bottom opening in the sand layer. Tests were conducted to investigate the dynamic response of water–sand inrush events under various combinations of factor levels. The findings revealed that the critical value for inrush is only present in the fissure width, which was observed to be 3 mm for the tested sand material. Unlike fissure width, the other factors do not have definitive critical values but instead modulate the intensity of the inrush process without determining its occurrence. The ‘inrush rate’ serves as a measure of the severity of water–sand inrush disasters and shows a linear increase with both increasing groundwater stress and fissure width, a negative exponential function relationship between the inrush rate and the clay content. Notably, ground stress does not exert a significant influence on the intensity of the inrush process itself. Under constant conditions, the inrush rate remains relatively constant across different levels of sand ground stress, for instance, in the experiments, the inrush rate was measured at 1.606 kg/s when the water stress was 0.1 MPa and the fissure width was 5 mm. Grey relation analysis was used to examine the sensitivity of each factor’s influence on the inrush rate. The results showed that water stress has the greatest impact on the intensity of water–sand inrush, followed by ground or soil stress, clay content, and the width of the fissures in the sand layer.

Development of the GeoGebra’s Guidebook in Creating Mathematics Learning Media Based on Ethnomathematics

This research aims to develop a guidebook for the use of GeoGebra in creating mathematics learning media based on ethnomathematics for university students in mathematics education study program at Pattimura University PSDKU Maluku Barat Daya. The method used in this research is Research and Development (R&D) with 4D model developed by Thiagarajan et al. (1974) which consists of 4 stages, namely Define, Design, Develop and Disseminate. This method and model are very feasible and systematically supports the development of the guidebook. By involving 2 validators, namely material experts and linguists, the draft of the guidebook was then revised and tested on a small limited group. Furthermore, based on descriptive statistical analysis and qualitative descriptive analysis, it can be concluded that the reference book using GeoGebra has been successfully prepared and meets the criteria of valid and effective. In addition, the use of this guidebook can also increase students' knowledge in understanding material related to cartesian coordinates, especially in drawing graphs of linear functions, quadratic functions, polynomials, trigonometric functions, exponent functions, and logarithms.

Spectrality of infinite convolutions and random convolutions

In this paper, we explore spectral measures whose square integrable spaces admit a family of exponential functions as an orthonormal basis. Our approach involves utilizing the integral periodic zeros set of Fourier transform to characterize spectrality of infinite convolutions generated by a sequence of admissible pairs. Then we delve into the analysis of the integral periodic zeros set. Finally, we show that given finitely many admissible pairs, almost all random convolutions are spectral measures. Moreover, we give a complete characterization of spectrality of random convolutions in some special cases.

Identifying and estimating solar cell parameters using an enhanced slime mould algorithm

This study proposed an enhanced slime mould algorithm (ESMA) for identifying the solar cells’ parameters for five photovoltaic (PV) models, making two modifications to the original slime mould algorithm (SMA). The first modification was an arbitrary average position among the new individual position of slime and the best individual position of the slime mound currently found to resolve the issue of local optimum. Second, the tangent hyperbolical function of the formula p in the original SMA was replaced with an exponential function to create more variations for selecting the updated equation. The proposed ESMA was used to resolve the problem of estimating PV parameters based on the empirical current-voltage (I-V) data. Specifically, ESMA was evaluated on the parameter estimation in several photovoltaic models, i.e., the one-diode model (ODM), dual-diode model (DDM), and PV-module model (PMM). In general, ESMA outperformed the original SMA and other recent algorithms. Also, in order to provide a close approximation of the empirical I-V data of the real PV modules and cells, ESMA was able to determine the optimal parameter values for photovoltaic models.

Cell size induced bias of current density in hypertrophic cardiomyocytes

ABSTRACT Alterations in ion channel expression and function known as “electrical remodeling” contribute to the development of hypertrophy and to the emergence of arrhythmias and sudden cardiac death. However, comparing current density values – an electrophysiological parameter commonly utilized to assess ion channel function – between normal and hypertrophied cells may be flawed when current amplitude does not scale with cell size. Even more, common routines to study equally sized cells or to discard measurements when large currents do not allow proper voltage-clamp control may introduce a selection bias and thereby confound direct comparison. To test a possible dependence of current density on cell size and shape, we employed whole-cell patch-clamp recording of voltage-gated sodium and calcium currents in Langendorff-isolated ventricular cardiomyocytes and Purkinje myocytes, as well as in cardiomyocytes derived from trans-aortic constriction operated mice. Here, we describe a distinct inverse relationship between voltage-gated sodium and calcium current densities and cell capacitance both in normal and hypertrophied cells. This inverse relationship was well fit by an exponential function and may be due to physiological adaptations that do not scale proportionally with cell size or may be explained by a selection bias. Our study emphasizes the need to consider cell size bias when comparing current densities in cardiomyocytes of different sizes, particularly in hypertrophic cells. Conventional comparisons based solely on mean current density may be inadequate for groups with unequal cell size or non-proportional current amplitude and cell size scaling.

An Efficient Algorithm for Basic Elementary Matrix Functions with Specified Accuracy and Application

If the matrix function f(At) posses the properties of f(At)=gf(tkA, then the recurrence formula fi−1=gfi,i=N,N−1,⋯,1,f(tA)=f0, can be established. Here, fN=f(AN)=∑j=0majANj,AN=tkNA. This provides an algorithm for computing the matrix function f(At). By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f(At). It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f(AN) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations.

Analytical and Numerical Investigation for the Inhomogeneous Pantograph Equation

This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions.

Predictive Pattern of Undrained Shear Strength in Stabilized Sulfur Rich Silty Soil Based on Binder and Initial Mixing Water Content

A laboratory investigation was conducted to identify principal variables-initial mixing water content, porosity, and binder content- impacting undrained shear strength (qu) of stabilized sulfur-rich silty soil. An equation for predicting qu of stabilized soil was established based on the experimental data. Initially, samples were prepared with soils sample with different initial water and binder contents. Multicem, a binder consisting of a mix of cement and cement kiln dust, was added to the samples. Three different percentages of Multicem were mixed at five different soil water contents to measure qu of stabilized mixtures to understand how water content and porosity levels in the samples affect the performance of the binder and their combined impact on the strength of the samples. The soil-binder mixtures were compacted and subsequently cured in laboratory-controlled environment. The prepared samples were tested in uniaxial compression test apparatus. The results evidenced that binder content and corresponding porosity affect the strength of specimens at an equal water content. The results showed that at equal initial mixing water content, the qu of a sample increased by increasing binder content. Furthermore, it was observed that increase of binder content has a reverse effect on porosity. It was appeared lowering the soil water content, initially increased the strength until an optimum water content. Further lowering water content increased the porosity and consequently decreased qu of samples. Moreover, a ratio of porosity/volumetric binder content was chosen to evaluate the impact of these two variables on strength of samples. This study showed that qu is an exponential function of porosity/binder volumetric content ratio which depends on initial mixing water content of mixtures. It was shown at water content lower than the optimum, results of stabilization are more effective than in soil at higher water contents. Therefore, reducing the water content and thereby porosity has more significant effect on improving qu than increasing the binder content.

NTCE-KD: Non-Target-Class-Enhanced Knowledge Distillation.

Most logit-based knowledge distillation methods transfer soft labels from the teacher model to the student model via Kullback-Leibler divergence based on softmax, an exponential normalization function. However, this exponential nature of softmax tends to prioritize the largest class (target class) while neglecting smaller ones (non-target classes), leading to an oversight of the non-target classes's significance. To address this issue, we propose Non-Target-Class-Enhanced Knowledge Distillation (NTCE-KD) to amplify the role of non-target classes both in terms of magnitude and diversity. Specifically, we present a magnitude-enhanced Kullback-Leibler (MKL) divergence multi-shrinking the target class to enhance the impact of non-target classes in terms of magnitude. Additionally, to enrich the diversity of non-target classes, we introduce a diversity-based data augmentation strategy (DDA), further enhancing overall performance. Extensive experimental results on the CIFAR-100 and ImageNet-1k datasets demonstrate that non-target classes are of great significance and that our method achieves state-of-the-art performance across a wide range of teacher-student pairs.

The role of chlorine promoters in mediating particle size effects in silver-catalyzed ethylene epoxidation

Measured chlorine coverages over Ag/α-Al2O3 catalysts with varying Ag weight loadings (1.3–35 wt%) and different Ag particle size distributions when parsed in terms of Cl coverages over particles of varying size constituted in the distribution reveal that Ag particles of size below ∼30 nm are covered in more than 1 monolayer (ML) of chlorine in presence of 3.5 ppm C2H5Cl. These assessments of Cl coverages were made possible by correlating the cumulative Cl coverage to the lognormal surface area distribution of 22 Ag/α-Al2O3 samples using a single exponential decay function. Fully chlorided small Ag particles exhibit low rates of epoxidation and only large Ag particles which exhibit sub-monolayer Cl coverages catalyze ethylene epoxidation with high ethylene oxide (EO) rates and selectivity. The Ag particle size dependence of Cl coverages plausibly explains why ≥ 100 nm Ag particles are typical in promoted Ag/α-Al2O3 ethylene epoxidation catalysts.