Hyperbolic Function Research Articles

Predicting and analysis of urban wastewater quantity to be treated by using Neural Network and ODE model: A case study

Wastewater data, including Population of the Municipality Served by the Wastewater Treatment Plants (WWTPs), Capacity, Number of WWTPs, Amount of Daily Wastewater Discharged per Person and Wastewater Treated in WWTPs (y(t)) obtained from Turkish Statistical Institute (TUIK) for Kayseri province for 2003-2020 were mathematically modelled and analysed with Artificial Neural Network (ANN) and Differential equations. Firstly, the data were augmented through ARIMA since TUIK data numbers are insufficient to ANN training and later the data were normalized. The augmented normalized data was trained with ANN twice, thus the effect of other variables on the y(t) variable was shown and mathematical ANN activation functions in the form of a tangent hyperbolic function was proposed for this variable. Then, arbitrary parameters used for a linear system consisting of differential equations representing the five variables mentioned above were estimated using normalized original data and thus the ODE (ordinary differential equation) model was proposed. Two ANN models and ODE model were evaluated on normalized real TUIK data and the performances of these three models were compared. Among these mathematical models, the model that gave the minimum MSE (mean squared error) has been determined as the ODE model. Finally, future predictions were made for the y(t) variable with the ODE model.

Satellite observations and modeling of the plasmapause structure and dynamics

We investigated the structure and dynamics of the plasmapause using electron density data from the Arase (ERG) satellite between 2017 and 2019. By fitting the electron density profile to a hyperbolic function using the least-squares method, 1891 plasmapause events were identified. The hyperbolic function’s fitting parameters were used to estimate the plasmapause’s location and thickness. The plasmapause location and thickness were mapped onto the geomagnetic equator plane. We examined the dependence of the plasmapause location and thickness on the geomagnetic disturbance. The median plasmapause location on the geomagnetic equator plane was 32,089 km from Earth’s center. Plasmapause locations depend on geomagnetic disturbances that vary between four and six radii of the Earth.In contrast, the median plasmapause thickness obtained by fitting a hyperbolic function on the geomagnetic equator plane was 2472 km. The thickness changed from ∼1500 km at night to ∼ 3000 km during the daytime. The thickness was several times larger than the ring current H+ Larmor radius, suggesting an association with plasma instability in the plasmapause due to the penetration of magnetospheric electric fields.Additionally, we developed a three-dimensional empirical electron density model with historical effects associated with refilling the plasmasphere, plasmapause, and magnetosphere. The International Reference Ionosphere (IRI) model can be enhanced to incorporate the electron density model of the plasmasphere and estimate the total electron content (TECp) in the plasmasphere using the empirical model. Our analysis provides valuable insights into plasmapause and contributes to a better understanding of the inner magnetosphere.

Distributed Two-Channel Dynamic Event-Triggered Adaptive Finite-Time Fault-Tolerant Containment Control for Multi-Leader UAV Formations

This paper addresses the distributed adaptive finite-time containment control problem in multi-leader UAV formations with actuator failures, limited communication, and external disturbances. A two-channel dynamic event-triggered strategy based on adaptive and sliding mode control is proposed as a containment control scheme, which solves the contradiction between the need to include containment error in the continuous term of the sliding mode surface in the adaptive law and the discontinuity of the communication between neighboring subsystems (follower UAVs) by constructing intermediate variables. Unlike the traditional event triggering condition that can only be applied to asymptotically convergent systems, the dynamic triggering condition designed in this paper, based on auxiliary variables, hyperbolic tangent function, and adaptive techniques, realizes finite-time convergence during the adjustment of the system and sets a more reasonable lower limit for the triggering thresholds, thus further reducing the communication frequency and ensuring the finite-time convergence of the system. The stability of the closed-loop system can be proved by the Lyapunov theorem. Zeno behavior can be avoided. The simulation results verify the effectiveness of the developed control algorithm.

Topology‐preserving phase‐field modeling of elastic bending energy

AbstractIn vesicle membranes, the minimum of the elastic bending energy determines the equilibrium shape. We can reformulate the energy using a phase‐field function and optimize it using the standard gradient flow approach. The method, in its typical formulation, allows topological changes in the membrane; however, in certain events, like in the simulation of blood cells, maintaining the initial topology of the membrane is crucial. In this study, we add a constraint on the phase‐field method to preserve the topology. We note that even if the phase‐field method is formulated using a hyperbolic tangent function, during a change in topology, the membrane's profile deviates from the tanh function. Owing to this idea, the constraint imposes an additional requirement on the profile to maintain the tanh shape, thus preserving the topology. We perform extensive experiments in two‐dimensional and three‐dimensional scenarios to demonstrate that our method preserves the topology during the simulation.

Onset of Darcy–Brinkman convection with thermal anisotropy in an inclined porous layer

In this article, we study the linear instability and the nonlinear stability (through energy functional) analyses of thermal convection in an inclined Darcy–Brinkman porous layer considering uniformly heated horizontal rigid, impermeable walls from below and above. The effects of a uniform internal heat source and anisotropy in effective thermal diffusivity on heat transfer are also considered. Heating the porous layer from below yields the temperature gradient, influencing the buoyancy and making the convection happen. This temperature gradient also impacts the base state. The basic solution for velocity incorporates both hyperbolic and polynomial functions, significantly increasing the complexity of linear and nonlinear analyses. The Chebyshev-tau method, together with the QZ algorithm, is used to solve the linear and nonlinear perturbed system of equations numerically. The region of subcritical instability is obtained by comparing the linear and nonlinear thresholds for the longitudinal and transverse rolls, respectively. We found that perturbations for longitudinal and transverse rolls do not grow after inclination is more than 30.3° and 31.3°, respectively. It has been noted that in transverse roll scenarios, the flow becomes stabilized when the inclination angle, ϕ, is equal to or exceeds 60°, where ϕ plays a leading role in surpassing the impact of internal heating. However, when the inclination angle is ϕ<60°, then internal heating dominates and destabilizes the flow. For the longitudinal rolls, the internal heating dominates the whole range of ϕ, destabilizing the system. Furthermore, it can be seen that the Darcy number (Da) and the anisotropic thermal diffusivity (ξ) delay the onset of convection.

Preferable price to buy an electric two-wheeled vehicle for college students

This research delves into the determinants affecting Taiwanese college students’ purchasing behavior towards two-wheeled electric vehicles (ETWVs) and the financial allocation they are willing to commit. A choice experiment design was utilized, with a sample of approximately 900 college students, regular users of two-wheeled gasoline vehicles (TWGs). The students engaged in one of two choice scenarios, presented with general characteristics, operational efficiency, vehicle cost, and subsidy strategies. Subsequent to the scenarios, respondents indicated their willingness to pay (WTP) for a ETWV. ETWVs, apart from being an emerging transportation alternative, also present potential environmental and health benefits due to reduced air pollution. The dependent inverse hyperbolic sine (IHS) double-hurdle model was employed, facilitating the accommodation of heteroscedasticity and correlated normality. The outcomes suggest that purchasing behavior for ETWVs among college students is shaped by factors such as purchase subsidy, tax breaks, and ETWVs’ capability in supporting students’ external travel requirements. The average WTP for ETWVs among this demographic stands at US$ 1,410. Importantly, the dependent IHS double-hurdle model, considering heteroscedasticity, proved statistically superior to its counterparts. Insights from this research offer crucial information on the price range students are likely to accommodate for ETWVs and provide a foundation for framing incentive-driven policies to amplify ETWV adoption

On a geometric version of the Grothendieck conjecture for configuration spaces of hyperbolic curves

On a geometric version of the Grothendieck conjecture for configuration spaces of hyperbolic curves

An analog of the pendulum/rotor system with elementary solutions

Abstract We explore a novel one-dimensional model potential, V(x), which mimics many aspects of the classic pendulum/rotor system, including low- and high-energy limits which are exactly solvable, as well as a divergent period separating the two. Using energy methods, the classical period, T(E), is evaluated in closed form for all values of the energy (0 < E < ∞) in terms of simple logarithmic functions, in contrast to the pendulum/rotor system where the results require elliptic integrals. We find an unexpected duality between the exact expressions for the period on either side of the divergence. Using both energy methods, and a Newton’s law differential equation approach, we present closed form expressions for trajectory solutions, x(t) and p(t) = mv(t) (or the linear momentum), in terms of hyperbolic functions, which explicitly give the same values for T(E).

Chemical shrinkage characteristics and prediction model of desulfurization electrolytic manganese residue-cement composite slurries

To reveal the chemical shrinkage (CS) characteristics of desulphurized electrolytic manganese residue (D-EMR) cement composite slurries, the CS was measured for slurries with different water-cement ratios (0.3 and 0.4) and varying D-EMR content (0 wt%, 5 wt%, 10 %, 15 wt%, 20 wt%, 25 wt%, and 30 wt%) using the expansion method. Fourier transform infrared spectroscopy (FTIR), X-ray diffraction (XRD), and backscattered electron (BSE) image analysis were used to investigate the hydration products and pore porosity at different stages. In addition, the correlation between CS and hydration products was simulated using GEMS thermodynamics. The results indicate that as the water-cement ratio increases, the CS of D-EMR cement composite slurries increases significantly. At a water-cement ratio of 0.4, the CS decreases with increasing D-EMR replacement levels before 1829 h of hydration, but this trend reverses after 1829 h. At 8 days, the CS values for replacement levels of 5 wt%, 10 wt%, 15 wt%, 20 wt%, 25 wt%, and 30 wt% were 97.2 %, 94.9 %, 87.3 %, 84.4 %, 81.5 %, and 78.6 % of those of the pure cement pastes, respectively. The corresponding limiting chemical shrinkage values were 0.0712 mL/g, 0.0725 mL/g, 0.0742 mL/g, 0.0756 mL/g, 0.0769 mL/g, and 0.0780 mL/g. Finally, the prediction of long-term CS of D-EMR cement composite slurries is well described by both hyperbolic function prediction and thermodynamic simulations. These findings provide valuable theoretical support for the use of D-EMR as a supplementary cementitious material in construction.

A robust approach to satellite image encryption using chaotic map and circulant matrices

AbstractIn the modern era, where satellite imagery is vital for applications like ecological monitoring and national security, ensuring the safety and integrity of these data repositories is crucial. This study presents an improved satellite image encryption technique that combines the cryptographic strength of the circulant matrix in the Hill cipher with the dynamic characteristics of the hyperbolic tangent tent map, further enhanced by the Kronecker XOR product. The algorithm initiates with computing alterations by a shift amount. After preserving the leftmost pixel in each row, it executes XOR operations between alternating rows, combining the value of the current even or odd row with corresponding pixels in the adjacent rows followed by encryption using the Hill cipher. The resulting image undergoes a diffusion process utilizing a hyperbolic tangent tent map. The Kronecker XOR product operation is then applied to individual pixels to produce a secure image followed by additional diffusion with keys from the hyperbolic tangent tent map to achieve the final encrypted image. We conducted simulations using MATLAB to assess the efficiency of the proposed satellite image encryption from theoretical and statistical perspectives. The results exhibit robust encryption performance as demonstrated by metrics such as an entropy value of 7.9982, a UACI of 33.5333%, and an NPCR of 99.6038%. The experiment results demonstrate the proposed image encryption scheme's reliability, practicability, and efficiency in securing satellite images during data storage and transmission. Comprehensive testing against various attacks including correlation, histogram, chi‐square, NPCR, PSNR, UACI, SSIM, key space and key sensitivity analysis confirms the scheme's robustness, efficiency and speed. These findings verify the scheme's ability to come across the most stringent encryption and decryption standards, making it an effective solution for securing sensitive satellite image data.

Detecting edges of geologic sources from gravity or magnetic anomalies through a novel algorithm based on hyperbolic tangent function

Detecting edges of geologic sources from gravity or magnetic anomalies through a novel algorithm based on hyperbolic tangent function

A novel deep learning‐based technique for driver drowsiness detection

AbstractEvery year, many people lose their lives because of road accidents. It is evident from statistics that drowsiness is one of the main causes of a large number of car accidents. In our research, we wish to solve this major problem by measuring the drowsiness level of the human brain while driving. The study aims to develop a novel technique to detect different alertness levels (i.e., awake, moderately drowsy, and maximally drowsy) of a person while driving. A hybrid model using a stacked autoencoder and hyperbolic tangent Long Short‐Term Memory (TLSTM) network with attention mechanism is designed for this purpose. The designed model uses different biopotential signals, such as electroencephalography (EEG), facial electromyography (EMG), and different biomarkers, such as pulse rate, respiration rate galvanic skin response, and head movement to detect a person's alertness level. Here, the stacked autoencoder model is used for automated feature extraction. TLSTM is used to predict a person's alertness level using stacked autoencoder network‐extracted features. The proposed model can classify awake, moderately drowsy, and maximally drowsy states of a person with accuracies of 99%, 98.3%, and 98.6%, respectively. The novel contributions of the paper includes (i) incorporation of an attention mechanism into the TLSTM network of the proposed hybrid model to focus on the emphatic states to enhance classification accuracy, and (ii) utilization of EEG, facial EMG, pulse rate, respiration rate, galvanic skin reaction, and head movement pattern to assess a person's alertness level.

Hot Torsion Tests of AA6082 Alloy

Materials characterization and the knowledge of their elastic-plastic behavior are of fundamental importance for the design of industrial manufacturing processes. Nowadays, FEM simulation is the main tool used to optimize product quality and minimize scraps, and the numerical codes have evolved over the years to obtain accurate solutions with reduced computational times. Nevertheless, in order to perform reliable simulations, it is necessary to include accurate modeling of the material flow stress. Hot torsion is a powerful method for the characterization of the material flow stress because, tests can be carried out at constant speeds and temperatures, reaching large strain values, and thus getting over the limits of compression and tensile tests. In this paper the hot torsion characterization applied to AA6082 alloy is presented: tests were performed with equivalent strain rates of 0.01, 0.1, 1, and 10 s-1, in the temperature range from 440 to 550 °C (from 713.15 to 823.15 K). The results are presented in terms of equivalent stress vs equivalent strain. Finally, the material flow stress curve was predicted by the Hyperbolic sine model and Hensel-Spittel law, and the material parameters A, m1-9 are provided for the temperature expressed in °C and K.

Validation of a novel effort-discounting assessment and evaluation of the effort-delay confound on effort discounting.

A vast literature highlights the prevalence of impulsive decision making in maladaptive outcomes. Most research has focused on one form-delay discounting. Less research has focused on effort discounting, possibly because of a lack of a standardized task for assessment. In published effort-discounting tasks, effort is conceptualized in many ways, making it difficult to compare findings across studies. Additionally, most effort-discounting tasks do not control for the time inherent in completing the effortful task, which makes it difficult to disentangle effort discounting from delay discounting. The current study evaluated the validity of a novel hypothetical effort-discounting task. The novel task was used to evaluate the influence of the effort-delay confound on rates of effort discounting in humans. Participants were randomly assigned to complete a confounded or a controlled version of the novel effort-discounting task. The effort-discounting data were well described by hyperbolic and exponential functions. When effort and delay were confounded, effort-discounting rates were significantly higher than when effort alone influenced discounting. The results suggest that data that are produced by effort-discounting tasks that do not control the effort-delay confound should be interpreted cautiously because they are also influenced by delay discounting. Task limitations and future directions are discussed.

Reciprocal Hyperbolic Series of Ramanujan Type

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

Multi-phased solutions of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion and the general rule of swapping

Based on the conventional solution of Prandtl's stress function for an orthotropic rectangular bar under Saint-Venant's torsion, one can derive displacements, shear strains, shear stresses and torsional rigidity. The conventional solution of Prandtl's stress function has a hyperbolic function for the coordinate in the bar's thickness direction, and a trigonometric function for the coordinate in the width direction. This paper raises questions about the solution. Why is the solution not arranged in the opposite way? Why is the hyperbolic function not for the coordinate in the width direction and the trigonometric function for the coordinate in the thickness direction? How is it that these obvious questions have never been addressed? This study rearranges the solution of the conventional Prandtl's stress function using the TSAI technique and finds that the solution is multi-phased, indicating that the coordinates in the width and thickness directions and their corresponding parameters are swappable, a phenomenon proposed as the general rule of swapping.

Numerical illustration using finite difference method for the transient flow through porous microchannel and statistical interpretation of entropy using response surface methodology

The current article discloses the influence of the hyperbolic tangent nanofluid on time dependent flow through a microchannel when a magnetic field is applied. The porous medium was incorporated using the Darcy–Forchheimer model. The chemical reaction is explained by Arrhenius activation energy. Temperature is determined by convective boundary conditions. The irreversibility occurring in the flow is analyzed. The modeled problem gives rise to partial differential equations, which are computed by finite difference method. Response surface methodology, an optimization technique, is used to attain the optimal conditions for entropy generated for the flow of fluid. Results of the analysis reveal that concentration decreases with the rise in reaction rate parameter and increases with activation energy parameter. Prandtl and Eckert numbers, with their increase, enhance entropy, and fluid friction irreversibility is at its highest. Perfect co-relation is attained for the model by the response surface methodology, with a co-relation coefficient of 100 %. The Weissenberg number is highly sensitive to change in the present modeling, followed by Darcy and Reynolds numbers. The Reynolds number and Darcy number show positive sensitivity, while the Weissenberg number shows negative sensitivity to the entropy generated.

Millimeter accuracy depth estimation in concrete using smartphone-equipped handheld GPR and hyperbolic curve fitting with antenna distance consideration

In the management and maintenance of structural facilities, knowing the cover thickness of rebar and the depth of buried objects is of paramount importance. Currently, Smartphone-Equipped Handheld Ground Penetrating Radars (GPR) are primarily used for detecting rebar, yet the Electromagnetic Induction (EMI) method remains the sole available technique for determining the cover thickness of rebar. This necessitates the use of two separate devices to understand the subsurface condition of rebar. In this study, we introduce a new algorithm that utilizes Hyperbolic Curve Fitting to determine depth and relative permittivity from reflection images derived from a single GPR measurement, using metal depth estimation as an exemplar. Our contributions include the automatic detection of the hyperbola's centerline before curve fitting, separation of the hyperbola into left and right for enhanced fitting accuracy by removing points near the vertex, and incorporating a rigorous theoretical model that took into account the distance between the transmitting and receiving antennas, known as the inter-antenna distance. This algorithm had enabled us to meet the performance requirements for the EMI method as stipulated by the Japanese Society for Non-Destructive Inspection (JSNDI). Notably, our method retained its accuracy even at depths exceeding the 100 mm limitation of the EMI method.

Research on temperature performance prediction of vortex tubes based on artificial neural networks

Abstract This study constructs a hybrid neural network model by integrating the physical constraints of the Bernoulli equation and Nikolaev's formula. The model is designed to explore and predict the variation pattern of the cold end temperature in a vortex tube. The input parameters include inlet pressure, inlet temperature, and cold mass fraction, with the cold end temperature as the output parameter. The network employs a multilayer feedforward model and the Levenberg-Marquardt learning algorithm, using a hyperbolic tangent function as the activation function. To evaluate the statistical validity of the developed model, the coefficient of determination (R²) and root mean square error (RMSE) are utilized, along with an analysis of the model's uncertainty and robustness. The hybrid model achieves an R² of 0.9936 and an RMSE of 0.3392, demonstrating strong performance in terms of uncertainty and robustness. These results indicate that the model accurately predicts the cold end temperature variation in the vortex tube. Furthermore, the findings reveal an optimal pressure range (0.49 MPa to 0.76 MPa) and cold mass fraction range (0.1 to 0.2) that minimize the cold end temperature.

Enhanced Cyclically Stable Plasticity Model for Multiaxial Behaviour of Magnesium Alloy AZ31 under Low-Cycle Fatigue Conditions.

Magnesium alloys, particularly AZ31, are promising materials for the modern automotive industry, offering significant weight savings and environmental benefits. This research focuses on the challenges associated with accurate modelling of multiaxial cyclic plasticity at small strains of AZ31 under low-cycle fatigue conditions. Current modelling approaches, including crystal plasticity and phenomenological plasticity, have been extensively explored. However, the existing models reach their limits when it comes to capturing the complexity of cyclic plasticity in magnesium alloys, especially under multiaxial loading conditions. To address this gap, a cyclically stable elastoplastic model is proposed that integrates elements from existing models with an enhanced algorithm for updating stresses and hardening parameters, using the hyperbolic tangent function to describe hardening and ensure a stabilised response with closed hysteresis loops for both uniaxial and multiaxial loading. The model is based on a von Mises yield surface and includes a kinematic hardening rule that promises a stable simulation of the response of AZ31 sheets under cyclic loading. Using experimental data from previous studies on AZ31 sheets, the proposed model is optimised and validated. The model shows promising capabilities in simulating the response of AZ31 sheet metal under different loading conditions. It has significant potential to improve the accuracy of fatigue simulations, especially in the context of automotive applications.