Exponential Function Research Articles

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.

Artificial Neural Network Accuracy Optimization Using Transfer Function Methods on Various Human Gait Walking Environments

A bionic leg with ergonomic functionality can increase the user’s independence. However, an ergonomic bionic leg can be challenged to be developed. One of its challenges is related to functionality, where the bionic leg motor can be rugged to adapt to the user. One of the solutions for the bionic leg challenge is the application of a motor driver controlled by the user’s muscle signal. EMG signal can be utilized as the user’s signal source. The EMG signal is then fed back into the motor device. EMG signals obtained during a natural walking environment can result in smooth and natural movement. This study classifies EMG signals into 8 classes: a controlled walking environment (treadmill walking with various speeds) and a natural walking environment (ground walking, upstairs and downstairs walking). This research aims to optimize the ANN method using transfer function variations. The best method will be used to train EMG-driven motors for future studies related to bionic legs. The best ANN parameter in this research using Levenberg-Marquardt backpropagation as a training algorithm with transfer function pairing of the exponential function: Hyperbolic tangent sigmoid transfer function and SoftMax transfer function with 98.8% accuracy and 0.036 MSE value. The best method from the experiment and ANN classification can be used as a training method for a bionic leg in future research.

The Cobb-Douglas Production Function and the Old Bowley's Law

Bowley's law, also referred to as the law of the constant wage share, was a noteworthy empirical finding in economics, suggesting that a nation's wage share tended to remain stable over time, as observed through most of the 20th century. The wage share represents the proportion of a country's economic output that is distributed to employees as compensation for their labor, usually in the form of wages. The term ''Bowley's law'' was coined in 1964 by Paul Samuelson, the first American laureate of the Nobel memorial prize in economic sciences. He attributed this principle to Sir Arthur Bowley, an English economist, mathematician, and statistician. In this paper, we introduce a mathematical model derived from data for the American economy, originally employed by Cobb and Douglas in 1928 to validate the renowned Cobb-Douglas production function. We utilize symmetry methods, particularly those developed by Peter Olver, to elucidate the validity of Bowley's law within our model's framework. By employing these advanced mathematical techniques, our objective is to elucidate the factors contributing to the stability of the wage share over time. We demonstrate that the validity of both Bowley's law and the Cobb-Douglas production function arises from the robust growth of an economy, characterized by expansion in capital, labor, and production, which can be approximated by an exponential function. Through our analysis, we aim to offer valuable insights into the underlying mechanisms that support Bowley's law and its implications for comprehending income distribution patterns in economies.

A Hydration-Based Integrated Model to Evaluate Properties Development and Sustainability of Oyster Shell Powder–Cement Binary Composites

Currently, oyster shell powder (OSP) is becoming more widely used in the production of cement-based materials. The purpose of this study is to propose a predictive model that can predict the properties of concrete materials incorporating oyster shell powder. The methods of this prediction model are given as follows. First, based on the measurement results of the heat of hydration in the first 7 days, the prediction parameters of the hydration model are obtained. Secondly, based on the hydration model, the measured results of the heat of hydration were extrapolated, and the heat of hydration from the start of stirring to day 28 was calculated. From the calculation results, the developments of compressive strength, ultrasonic velocity, and surface electrical resistivity were estimated. Finally, we evaluated the CO2 emissions of concrete incorporating oyster shell powder. The CO2 emissions corresponding to unit compressive strength and unit surface electrical resistivity were calculated. The important conclusions of the prediction model are given as follows. First, for different substitution amounts of oyster shell powder, the model result shows that the ultimate value of the heat of hydration corresponding to the unit cement mass is the same, i.e., 454.27 J/g. While the substitution amount of oyster shell powder increases from 0% to 30%, the model result shows that the cumulative 28-day hydration heat for 1 g cement increases the powder amount from 405.7 J/g to 419.3 J/g. Secondly, as the amount of substituted oyster shell powder increases from 0% to 30%, the model result shows that the cumulative 28-day heat of hydration per gram of cementitious material decreases this amount from 405.7 J/g to 293.4 J/g. Compressive strength, ultrasonic pulse velocity, and surface electrical resistivity can all be expressed as exponential functions of the heat of hydration. For compressive strength, ultrasonic pulse velocity, and surface electrical resistivity, the coefficients of determination for the simulation results and experimental results are 0.8396, 0.7195, and 0.9408, respectively. Finally, as the amount of substituted oyster shell powder increases from 0% to 30%, the model result shows that the CO2 emission per unit of compressive strength increases from 10.18 kg/MPa to 16.51 kg/MPa. As the amount increases from 0% to 30%, the model result shows that the CO2 emission corresponding to the unit surface electrical resistivity does not change significantly. In summary, the importance of this model is that it can predict various properties of concrete mixed with oyster shell powder, reduce the number of experiments, and promote the engineering application of oyster shell powder concrete.

A thermodynamically consistent anisotropic continuum damage mechanics model with damage-dependent equivalent strain

Abstract This paper proposes a new thermodynamically consistent anisotropic continuum damage mechanics model that the degradation of material is described by a set of damage variables. We adopt a set of novel equivalent strains which are functions of both the strain and the damage state variables as well. A viscous stabilization method is considered to improve the convergence during the material softening. Thus, the evolution of damage state variables are updated through a Newton-Raphson iterative process. The expression of the consistent tangent stiffness matrix is also derived. A smoothed exponential damage evolution shape function with rescaling regularization method is proposed to describe the material softening process and the mesh dependency phenomenon is relieved. Several sets of numerical examples including three-point-bending of a notched-beam, uniaxial tension of unnotched and notched plate of composite material with multiple layups are presented to verify the anisotropic damage models.

HJ Semi Symmetric Cryptographic Algorithm

This paper presents an algorithm for performing encryption and decryption using modular exponentiation mainly based on Euler’s Theorem. The algorithm uses the properties of modular arithmetic to ensure secure communication. The gcd function is employed to find coprime numbers within a given range, which are then utilized as bases for encryption and decryption. The modular Exponentiation function efficiently computes the exponentiation of a base to a power modulo a given modulus. Through a step-by-step process, the algorithm encrypts a numerical message using the selected base and then decrypts it, ensuring the original message is recovered accurately. Experimental results demonstrate the effectiveness and reliability of the algorithm for secure data transmission. In the conclusion we discussed the effectiveness of the algorithm compared to the existing ones such as Elgamal algorithm and RSA Cryptography. The algorithm presented in this paper publishes only the modulus (integer) number and protocols of key exchanges kept secret between sender and receiver; therefore, the algorithm is not fully asymmetric but semi symmetric.

The [formula omitted]-logistic growth model. Qualitative and quantitative dynamics

The κ-exponential function, representing a generalization of the exponential function, has been firstly introduced in physics, and, then, it has been considered in a noteworthy number of fields because of its ability to take rare events into account. Among the possible applications of this function, one of particular interest is in economics in which rare events may consist in natural disasters, such as earthquakes that reduce the supply of capital, or epidemics or other external shocks influencing the supply of intermediate inputs, human or physical capital. Starting from the κ-exponential function, the κ-logistic function, which is a generalization of the sigmoidal function, can be obtained and used to describe production functions in a unique setting to take into account (1) several shapes usually considered in economics (i.e. concave and non-concave production functions), (2) economies at different development levels, and, (3) the possible occurrence of rare events. In this paper, we investigate the economic growth model as proposed by Böhm and Kaas (2000), wherein the production function utilizes the κ-logistic function. We provide theoretical results confirmed by extensive computational experiments and in line with economic literature showing that a poverty trap may emerge together with fluctuations, multistability and complex dynamics.