Exponential Function Research Articles

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.

ADAPTIVE CONTROL OF A SUBMARINE FOR VARIOUS DIVING DEPTHS USING AN EXPONENTIAL

Due to the progress of deep submergence capabilities, a submarine is extensively employed on many sides of marine science. For that reason, it has become necessary to design an effective submarine position controller while achieving a fast and stable mission. An adaptive analytical scheme for the underwater submarine was developed. The main purpose of the adaptive analytical scheme is to provide a powerful analytical controller that can successfully steer the submarine from the current state under the water to the surface of water in a short time. To achieve that, a seventh-term exponential function was proposed to reshape the reference diving depth while maintaining the control variable limitation and satisfying initial and final boundary conditions. The direct search method with two cascade loops was employed to achieve a minimum cost function by determining an appropriate constant of the function and minimum final time. However, using the direct search method can be time-consuming since a number of algebraic operations are executed inside these two loops. Therefore, the curve fitting method was used to fit the set of direct search method data using power functions. Then, at a certain diving depth, the function coefficients were computed based on the final settling time and function constant using the Gaussian elimination method. A nominal numerical simulation of the submarine’s model was implemented using both adaptive analytical and linear controllers.

Improved weighted nuclear norm with total variation for removing multiplicative noise

This paper introduces an improved weighted nuclear norm with a total variation model tailored for removing multiplicative noise. The model incorporates a weight matrix to regularize the residual matrix, effectively leveraging image redundancy to differentiate various statistical properties of the noise. Since there is no guarantee of a unique solution, the model is reformulated as a linear equality constraint problem and decomposed into two subproblems. These are addressed by using the alternating direction method of multipliers and the split Bregman method, respectively. In addition, each alternative update step has a closed-form and convergent solution. After obtaining the denoised image in the log-domain, the recovered image is given by using the exponential function and bias correction. Experimental evaluations demonstrate the efficacy of our algorithms in enhancing image restoration quality.

Security Analysis of Distributed Consensus Filtering Under Replay Attacks.

This work studies the security of consensus-based distributed filtering under the replay attack, which can freely select a part of sensors and modify their measurements into previously recorded ones. We analyze the performance degradation of distributed estimation caused by the replay attack, and utilize the Kullback-Leibler (K-L) divergence to quantify the attack stealthiness. Specifically, for a stable system, we prove that under any replay attack, the estimation error is not only bounded, but also can re-enter the steady state. In that case, we prove that the replay attack is ϵ -stealthy, where ϵ can be calculated based on two Lyapunov equations. On the other hand, for an unstable system, we prove that the trace of estimation error covariance is lower bounded by an exponential function, which indicates that the estimation error may diverge due to the attack. In view of this, we provide a sufficient condition to ensure that any replay attack is detectable. Furthermore, we analyze the case that the adversary starts to attack only if the current measurement is close to a previously recorded one. Finally, we verify the theoretical results via several numerical simulations.

Reduced Geostatistical Approach With a Fourier Neural Operator Surrogate Model for Inverse Modeling of Hydraulic Tomography

AbstractIn this study, we propose a new approach for inverse modeling of hydraulic tomography (HT) using the Fourier neural operator (FNO) as a surrogate forward model. FNO is a deep learning model that directly parameterizes the integral kernel in Fourier space to learn solution operators for parametric partial differential equations (PDEs). We trained a highly accurate FNO to learn the solution operators of PDEs in HT, which can efficiently generate multiple channels of hydraulic head fields corresponding to specific pumping tests in new parameter fields through a single forward pass. The trained FNO surrogate model is integrated with the reduced geostatistical approach (RGA) to provide a powerful tool, named RGA‐FNO, for inverse modeling of Gaussian random fields. RGA utilizes principal components to effectively reduce problem dimensions and encode geostatistical information. FNO benefits from deep learning automatic differentiation, enabling efficient backpropagation of gradients during inverse modeling. Our results show that RGA‐FNO can efficiently produce satisfactory estimations for random transmissivity fields with an exponential covariance function, which is non‐differentiable and visually non‐smooth. The RGA‐FNO application on steady‐state multi‐well HT is better than the single‐well transient pumping test for inverse modeling. The data required for inverse modeling applications of RGA‐FNO increases with the variance of the random transmissivity fields. FNO and other neural operator models have the potential to play an important role in groundwater modeling, especially in inverse modeling and experimental design, which often requires a large number of forward simulations.

Intensification of grape marc (Vitis vinifera) exploitation by subcritical water-ethanol extraction: Effect on polyphenol bioactivities and kinetic modelling

Subcritical solvent extraction (SSE) is an efficient and versatile technology for the recovery of bioactive compounds from food by-products. The present work aims to study the SSE efficiency and kinetics for the extraction of grape marc polyphenols. Water, ethanol, and a 50% water-ethanol mixture (EtOH 50%) were used as green solvents. The higher polyphenol content (4.05 ± 0.23 gGAE/100 gd. m.) and antioxidant activity (42.30 ± 0.68 mg α-toc/mL) were obtained with subcritical EtOH 50% at 120 °C and 10.3 MPa, respectively, 1.5- and 3-fold higher than the conventional extraction. According to kinetic modeling, SSE was four-fold faster and a degradation phenomenon of polyphenols was highlighted for times longer than 10 min. The two-site kinetic model and a second order exponential decay function suitably described the extraction kinetics and degradation phase, respectively (R2 > 0.97). A remarkable effect was highlighted on the polymerization degree and galloylation percentage of proanthocyanidins by SSE, and an increase in their antioxidant activity was observed.

Investigation on optimization-oriented EPC method in analyzing the non-linear oscillations under multiple excitations

The optimization-oriented exponential-polynomial-closure (OEPC) method is extended and investigated to analyze stochastic nonlinear oscillators under multiple excitations, with the purpose of obtaining probabilistic solutions for the corresponding system. The presented method extends the original EPC projection procedure for solving the FPK equation by minimizing an objective function, which is defined as the spatial integration of the weighted square residual error. Using the exponential polynomial function, the parameters within it can be located by seeking the minimum of the objective function. In this paper, the novel method has been tested with four examples of strong nonlinearities raised by polynomial nonlinear terms and parametric excitations. The results provide adequate evidence that the OEPC approach delivers notably improved accuracy compared to the Gaussian closure method and demonstrates superior efficiency compared to Monte Carlo simulation. The OEPC method presents a viable alternative for investigating stochastic nonlinear oscillators.

A quasi-exponential distribution of interfacial voids and its effect on the interlayer strength of 3D printed concrete

In this work, a novel theoretical model of the void length probability distribution in 3D printed concrete is established based on a zigzag analog of the layer interface. A quasi-exponential distribution of void length is predicted and subsequently validated on both the zigzag analog and the actual 3D printed concrete, with different void ratios that determine the descending rate and node intervals that decide the horizontal scaling of the distribution. Moreover, the relationships between the interlayer strength, void ratio, and quantity of voids are also studied based on the theoretical model. It is found that the quantity of voids is symmetric about a void ratio of 0.5, and the decrease in the interlayer strength against the void ratio is non-linear which is approximated well by combined exponential and linear functions. This work is believed to reveal the nature of the interfacial void distributions and significantly advance the understanding of the layer interface in 3D printed concrete. The code for the zigzag analog with the computation of its interlayer strength is publicly available at: https://github.com/Human-HLW/Layer-interface.

Nonlinear two-component system of time-fractional PDEs in [formula omitted]-dimensions: Invariant subspace method combined with variable transformation

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite-dimensional invariant linear product spaces for the given system using the invariant subspace method combined with variable transformation. Additionally, we present how to use the obtained invariant linear product spaces to derive the generalized separable exact solutions of the discussed system. We also note that the discussed method will help to reduce the nonlinear two-component system of TFPDEs in (2+1)-dimensions into the nonlinear two-component system of TFPDEs in (1+1)-dimensions, which again reduces to a system of time-fractional ODEs through the obtained invariant linear product spaces. More specifically, the significance and efficacy of the systematic investigation of the discussed method have been investigated through the initial and boundary value problems of the generalized nonlinear two-component system of time-fractional reaction–diffusion equations (TFRDEs) in (2+1)-dimensions for finding the generalized separable exact solutions, which can be expressed in terms of the exponential, trigonometric, polynomial, Euler-Gamma and Mittag-Leffler functions. Also, 2D and 3D graphical representations of some of the obtained solutions are presented for different values of fractional orders.

Study and Analysis of Some Impact on GDP of Nepal by Increase the Exports and Decreasing Import, on Economic Growth of Nepal with the Help of Exponential Function

Study and Analysis of Some Impact on GDP of Nepal by Increase the Exports and Decreasing Import, on Economic Growth of Nepal with the Help of Exponential Function

Critical radius prediction for small radius curved section of two-lane secondary highways based on speed characteristics

Small radius curved sections of two-lane secondary highways have low technical indicators, complex alignments, and frequent accidents. In order to improve the safety of traffic operation, the article investigates the curved sections of two-lane secondary highways with different radii and obtains the travelling speeds of different vehicle types at different section locations. By studying the trajectory characteristics and speed characteristics of vehicles travelling in curves, the velocity difference ratio, which responds to the continuity of driving speed, was defined. Then, based on this, this paper investigates the vehicle motion law of small radius curved road sections. The results show that the speed change of large cars in small radius curves is similar to a “U" shape, while the speed change of small cars is similar to a “V" shape. The speed adjustments of the larger cars occur mainly within the range of gentle curves, whereas the speed changes of the smaller cars are large throughout the range of curves. There is a significant exponential function relationship between curve radius and 85 % percentile speed, from which the curve radius threshold can be predicted. This provides a basis for engineering construction of curved road sections and optimization of curve design indicators, which greatly improves the traffic safety level of curved road sections.

An event-based distributed least square linear equation solver employing network flow

In this article, we study a distributed, second-order, event-triggered network flow for solving linear algebraic equations described by the form z=Hy in the sense of least squares. An undirected graph is utilized to describe the distributed computation network. In the network, each node broadcasts its dynamic states to its neighbors in an asynchronous fashion, and updates the flow at the isolated event times. We construct a specific trigger function to determine the event times, in which an exponential decay function is introduced as a comparison term. We show that if the linear algebraic equation has a unique least squares solution, all nodes states converge to this unique solution exponentially fast, when the applied network is connected and the decay rate of the comparison term satisfies certain bounds determined jointly by the topology and the linear equations information. We obtain a strictly positive lower bound on the time interval between consecutive events for time t∈[0,∞), and therefore the event-triggered system does not exhibit Zeno behavior. We provide numerical examples to verify the effectiveness of the proposed event-based network flow. In addition, we illustrate the importance of the strictly positive inter-event time interval, in comparison with the case that the inter-event time interval may converge to zero when time t goes to infinity.

A novel term-structure-based Heston model for implied volatility surface

It remains a challenge for existing financial models to accurately capture the shapes of implied volatility (IV) of options with all maturities simultaneously. Inspired by Chen et al.’s empirical finding, ‘the shorter the option expiry, the higher the IV is’, we propose a novel and simple approach to solve this challenge, by introducing a term-structure-based correction (i.e. an exponential increase function of the options expiry) to the volatility of volatility (vol–vol) term of the classical Heston stochastic volatility model. We derive an approximate formula for the IV under the corrected model with the perturbation method and further apply the formula to predict the IVs of options written on the Shanghai Stock Exchange 50 ETF. Numerical experiments and empirical results show that the introduction of a term-structure-based correction function surely overcomes the deficiency of the classical Heston model in capturing the short-term IVs, thus improving notably its performance of IV forecasting.