Hyperbolic Function Research Articles

OAM Mode Propagation and Supercontinuum Generation in a Nested Photonic Crystal Fiber

Abstract The proposed work presents a nested Photonic Crystal Fiber(PCF) with
a hexagonal lattice of air holes, capable of supporting Orbital Angular Momentum
modes(OAM) and generating a wide-ranging supercontinuum spectrum, further
enhancing its utility in various optical applications. The Photonic Crystal Fiber
consists of a dense flint SF6 and a very light flint LLF1, fabricated from Schott. Finite
Element Method is used for the analysis of Nested Photonic Crystal Fiber characteris-
tics using COMSOL Multiphysics. A total of 120 Orbital Angular Momentum modes
modes have been obtained in the fiber, with 86 in the outer core and 34 in the inner
core. The generalised Nonlinear Schrodinger Equation provides a numerical description
of the pulse that is propagating through the PCF. In this article, the proposed nested
PCF can be used to generate supercontinuum by pumping with a hyperbolic secant
pulse, with red an average power of 71 mW. The resulting supercontinuum spans an
impressive bandwidth, extending from approximately 1122 to 2072 nm.

Deep interval type-2 generalized fuzzy hyperbolic tangent system for nonlinear regression prediction

Recently, due to the rapid rise of artificial intelligence (AI), considerable progress has been made in the field of nonlinear regression prediction. However, many existing methods suffer from the issues of rule and parameter explosion and poor accuracy, particularly for high-dimensional data with uncertainty. To address these limitations, this paper proposes a deep interval type-2 generalized fuzzy hyperbolic tangent system (DIT2GFHS). First, a novel neural network-based implementation of the interval type-2 fuzzy generalized fuzzy hyperbolic tangent system (IT2GFHS) is introduced to improve the efficiency of system parameter updates and optimization. Then, using a hierarchical and block-based framework, multiple IT2GFHSs are stacked layer by layer from bottom to top to construct the DIT2GFHS, with each layer’s fuzzy subsystems being independent of the others. Additionally, DIT2GFHS incorporates optimization algorithms and the Adam optimizer for training, thereby avoiding the tedious manual parameter tuning process. The detailed analysis of the construction manner and internal mechanisms for DIT2GFHS indicates that it features a reduced number of parameters, a transparent and clear structure, strong capability in handling uncertainty, and favorable accuracy. Notably, the small number of parameters and the explicit structure reduce computational and hardware burdens while maintaining interpretability. Finally, extensive experimental studies on both relatively low-dimensional and high-dimensional datasets are conducted. The results demonstrate that DIT2GFHS achieves excellent performance with fewer parameters compared to many deep-structured models, including deep fuzzy systems and deep learning models. This highlights its potential impact in addressing practical nonlinear regression problems.

Microstructure evolution modeling of Ti6Al4V alloy during cutting using the Particle Finite Element Method and homogeneous field distributions

This paper aims to explore the evolution of microstructural parameters induced during the cutting of Ti6Al4V alloy (TC4). The microstructure characteristics of the workpiece material is directly tied to its mechanical response during machining. During TC4 cutting, microstructure evolution is observed as a result of severe plastic deformations. The characterization of this phenomenon has significant interest from both academia and industry. In this work we present a developed modeling technique which combines the Particle Finite Element Method (PFEM) with incremental homogeneous field distributions. First, the PFEM is extended to perform a thermo-mechanical analysis capable of capturing the material responses of TC4 during orthogonal cutting. To generate serrated chips, an appropriate strain softening-based constitutive plasticity model i.e., TANH (Hyperbolic TANgent) is utilized. The PFEM’s validity is checked through comparison with available experimental results in terms of chip shapes and cutting forces. Second, the evolution of microstructural parameters such as dislocation density, vacancy concentration, dynamic recrystallization (DRx) grain size, and hardness is incrementally developed and incorporated into the PFEM using internal state variables as homogeneous field distributions. The Johnson–Mehl–Avrami–Kolmogorov (JMAK) model and Hall-Petch equation are applied for predicting grain size and hardness, respectively. The parameters of the corresponding models are modified for TC4 to accurately capture their alterations. Lastly, the predicted results of the microstructure evolution in serrated chips and machined surfaces, including average grain size and hardness, are compared with experiments, demonstrating good agreement. This implies that the PFEM combined with microscale schemes can reliably simulate the machining process of the TC4.

Study on a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in nonlinear physics: Multiple soliton solutions, lump solutions, and breather wave solutions

In this work, we study an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equations that appear in many nonlinear physics applications. We show that this extended equation retains its complete integrability via Painlevé analysis. We explore multiple soliton solutions by using the Hirota bilinear method. Moreover, we derive lump solutions where two numerical examples are tested. Breather wave solutions were also explored by using a variety of distinct schemes. We also determine other traveling wave solutions, rational solutions, periodic solutions, exponential solutions, ratio of trigonometric or hyperbolic functions, and others.

Damage Identification Using Meta-Lens and Dual-Enhanced Backscattered Waves: Numerical and Experimental Design

This paper introduces a Meta-lens designed for the detection of minute damage in Structural Health Monitoring (SHM) by amplifying backscattered waves originating from anomalous areas. Our approach starts with the development of an easily installed Meta-lens, optimizing wave focusing on an aluminium plate through strategic planning of the A0 wave trajectory, resembling a hyperbolic secant. Subsequently, we detail the configuration of the actuator, lens, and sensor in both healthy and damaged scenarios, the latter involving a through-hole defect in the plate. A comparative analysis of signals, accounting for damage size and location variations, is conducted against a baseline. The results demonstrate that the Meta-lens induces a unique wave behaviour, resulting in double intensification of backscattered signals, significantly increasing sensitivity to damage features compared to systems without the lens. The focusing effect of the Meta-lens is further validated experimentally, which shows good agreement with the simulation results. The findings underscore the significant potential of employing the Meta-lens for damage detection, thereby eliminating the reliance on intricate sensor networks and complex signal post-processing.

Multivariate Smooth Symmetrized and Perturbed Hyperbolic Tangent Neural Network Approximation over Infinite Domains

In this article, we study the multivariate quantitative smooth approximation under differentiation of functions. The approximators here are multivariate neural network operators activated by the symmetrized and perturbed hyperbolic tangent activation function. All domains used here are infinite. The multivariate neural network operators are of quasi-interpolation type: the basic type, the Kantorovich type, and the quadrature type. We give pointwise and uniform multivariate approximations with rates. We finish with illustrations.

Manufacture of a 4-Degree-of-Freedom Robot to Support an IRB 120 Robot

In this work, we present the construction and control of a four-degrees-of-freedom (DOF) manipulator aimed at addressing one of the key challenges faced by the Academy-Industry Cooperation Center (CCAI): the need for mechatronic equipment to support and facilitate the development of advanced robotic cells. We begin by designing the robot’s structure and components using SolidWorks software for computer-aided design (CAD) modeling. This ensures that all the links and parts fit together properly without collisions. The robot links are then manufactured using 3D printing. Additionally, we performed kinematic modeling, dynamic analysis, and PI-V control, along with control using a trigonometric function (hyperbolic tangent). To evaluate the robot’s movement, we simulate these processes using Matlab R2019a/Simulink software, focusing on key parameters such as position, velocity, and acceleration, which inform the design of PI-V control for each link. We also present the electrical and electronic designs, followed by system implementation. The kinematics of the robot play a crucial role in the dynamics and controller design. We validate the kinematics using Peter Corke’s libraries based on the Denavit–Hartenberg parameters. The results show that the controller based on the trigonometric function improves the response time, particularly enhancing the performance of axes 2 and 3.

The hot deformation behavior of a L12 precipitation strengthened face centered cubic high entropy alloy

In this work, the hot compressive deformation behavior of a homogenized Ni45Co15Cr15Fe15Al8Ti2 high entropy alloy was investigated at deformation temperatures of 900–1200°C and strain rates of 10−3-1 s−1. Initially, combined the true stress-strain curves and hyperbolic sine function model, the constitutive equation for the flow stress of the alloy during the hot deformation process was established. Additionally, the hot processing map of the alloy at a true strain of ε = 0.5 was plotted based on the DMM model. Coupled with microstructural observations, the optimal hot processing parameters for the alloy were determined to be T = 1070–1200 °C and ε̇= 0.001–0.05 s−1. Finally, the deformed microstructure of the alloy under different temperatures and strain rates was characterized using EBSD, and the dynamic recrystallization mechanisms was revealed. The flow stress decreases with the increasing deformation temperature and decreasing strain rate. As the deformation temperature increases from 900°C to 1200°C, the dynamic recrystallization mechanism transitions from continuous dynamic recrystallization (CDRX) to discontinuous dynamic recrystallization (DDRX). The findings of this study will provide valuable guidance for the development of hot forging parameters of the L12 strengthened FCC high-entropy alloys.

Elimination of oscillation causing Hopf bifurcations in engineering problems

Bifurcation analysis was performed on various engineering process problems that exhibit undesirable oscillation causing Hopf bifurcations. Hopf bifurcations result in oscillatory behavior which is problematic for optimization and control tasks. Additionally, the presence of oscillations causes a reduction in product quality and in some cases causes equipment damage. The hyperbolic tangent function activation factor is normally used in neural networks and optimal control problems to eliminate spikes in optimum profiles. Spikes are similar to oscillatory profiles and this is the motivation to investigate whether the hyperbolic tangent function activation factor can eliminate the oscillation causing Hopf bifurcations. The results of this paper show that the hyperbolic tangent function activation factor eliminates the Hopf bifurcations. Bifurcation analysis is performed using The MATLAB software MATCONT. Five examples involving problems that exhibit Hopf bifurcations are presented.

Machine learning assisted prediction of the nitric oxide (NO) solubility in various deep eutectic solvents

Deep eutectic solvents (DESs) are recently proposed as green materials to remove nitric oxide (NO) from released streams into the atmosphere. The mathematical aspect of this process attracted less attention than it deserved. A straightforward approach in this field will help engineer DES chemistry and optimize the equilibrium conditions to maximize the amount of removed NO. This study covers this gap by constructing a reliable artificial neural network (ANN) to correlate the NO removal capacity of DES with equilibrium pressure/temperature and solvent chemistry. So, firstly, the physical meaningful features are selected to make the DES chemistry quantitative. It was found that the density is the best representative for the hydrogen-bound acceptor and hydrogen-bound donor. Also, the density and viscosity of the DESs exhibit the highest correlation with the NO solubility. Then, the hyperparameters of three famous ANN types (feedforward, recurrent, and cascade) are determined by combining trial-and-error and sensitivity analyzes. Finally, the ranking test distinguishes the ANN type with the lowest uncertainty toward estimating NO dissolution in DESs. The cascade neural network (CNN) with twelve and one neurons in the hidden and output layers equipped with the tangent hyperbolic and radial basis transfer functions is identified as the best ANN type for the given purpose. This model predicts 292 DES-NO equilibrium records collected from the literature with mean absolute errors = 0.033, relative absolute errors = 1.49 %, mean squared errors = 0.002, and coefficient of determination = 0.9998. Also, the present study helps understand the role of DES chemistry and operating conditions on the amount of removable NO by DESs. 1,3-dimethylthioureaP4444Cl (3:1) is recognized as the best DES to separate NO molecules from gaseous streams, respectively. The simulation results show that the unit mass of the best DES is capable of absorbing up to ∼27 mol of NO.

Individualized muscle architecture and contractile properties of ankle plantarflexors and dorsiflexors in post-stroke individuals

ObjectiveThis study was to investigate alterations in contractile properties of the ankle plantar- and dorsiflexors in post-stroke individuals. The correlation between muscle architecture parameters and contractile properties was also evaluated.MethodsEight post-stroke individuals and eight age-matched healthy subjects participated in the study. Participants were instructed to perform maximal isometric contraction (MVC) of ankle plantar- and dorsiflexors at four ankle angles, and isokinetic concentric contraction at two angular velocities. B-mode ultrasound images of gastrocnemius medialis (GM) and tibialis anterior (TA) were collected simultaneously during the MVC and isokinetic measurements. Individualized torque-angle and torque-angular velocity relations were established by fitting the experimental data using a second-order polynomial and a rectangular hyperbola function, respectively. Muscle structure parameters, such as fascicle length, muscle thickness and pennation angle of the GM and TA muscles were quantified.ResultsPost-stroke subjects had significantly smaller ankle plantarflexor and dorsiflexor torques. The muscle structure parameters also showed a significant change in the stroke group, but no significant difference was observed in the TA muscle. A narrowed parabolic shape of the ankle PF torque-fiber length profile with a lower width span was also found in the stroke group.ConclusionThis study showed that the contractile properties and architecture of ankle muscles in post-stroke individuals undergo considerable changes that may directly contribute to muscle weakness, decreased range of motion, and impaired motion function in individuals after stroke.

Optimized Fault Classification in Electric Vehicle Drive Motors Using Advanced Machine Learning and Data Transformation Techniques

The increasing use of electric vehicles has made fault diagnosis in electric drive motors, particularly in variable speed drives (VSDs) using three-phase induction motors, a critical area of research. This article presents a fault classification model based on machine learning (ML) algorithms to identify various faults under six operating conditions: normal operating mode (NOM), phase-to-phase fault (PTPF), phase-to-ground fault (PTGF), overloading fault (OLF), over-voltage fault (OVF), and under-voltage fault (UVF). A dataset simulating real-world operating conditions, consisting of 39,034 instances and nine key motor features, was analyzed. Comprehensive data preprocessing steps, including missing value removal, duplicate detection, and data transformation, were applied to enhance the dataset’s suitability for ML models. Yeo–Johnson and Hyperbolic Sine transformations were used to reduce skewness and improve the normality of the features. Multiple ML algorithms, including CatBoost, Random Forest (RF) Classifier, AdaBoost, and quadratic discriminant analysis (QDA), were trained and evaluated using Bayesian optimization with cross-validation. The CatBoost model achieved the best performance, with an accuracy of 94.1%, making it the most suitable model for fault classification in electric vehicle drive motors.

On Bergman-Orlicz and Campanato-Orlicz spaces of hyperbolic harmonic functions on the real unit ball

Let Φ be a growth function. In this paper, we characterize Bergman-Orlicz spaces B α Φ of hyperbolic harmonic functions on the real unit ball B in terms of invariant gradients and some integral functions. As an application, several characterizations of h-harmonic Campanato-Orlicz spaces are established. This is a continuous work of [X. Fu and Q. Shi. Complex Var. Elliptic Equ. 2023].

A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques

Abstract Soliton solutions play a crucial role in modeling stable phenomena across optical communications, fluid dynamics, and plasma physics, owing to their stability and persistence in solving nonlinear equations. This study centers on the extended Sakovich equation, emphasizing the importance of soliton solutions in predicting and controlling localized wave behaviors, which advances nonlinear dynamics and its various applications due to its integrable properties and flexible soliton characteristics. This equation is applicable across diverse fields such as fluid dynamics, nonlinear optics, and plasma physics, where it effectively models nonlinear wave phenomena, including solitons and shock waves. Additionally, it provides crucial insights into wave propagation in biological systems and acoustics, making it a valuable tool for analyzing complex wave dynamics. Additionally, we investigate bifurcation and modulation instability within this equation, employing the improved Sardar subequation method and the ℛ ′ ℛ , 1 ℛ \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{{ {\mathcal R} }^{^{\prime} }}{ {\mathcal R} },\frac{1}{ {\mathcal R} }\right) method to derive solitary wave solutions. These methods yield a diverse range of waveforms – hyperbolic, trigonometric, and rational functions – validated rigorously using Mathematica software for accuracy. Graphical representations vividly display various soliton patterns, such as singular, multi-singular, periodic singular, kink, anti-kink, bell-shaped, Kuznetsov–Ma Breather, and parabolic-shaped, highlighting their effectiveness in revealing innovative solutions. Furthermore, a comparative analysis verified the novelty of our derived soliton solutions. This research significantly contributes to advancing soliton solutions for the Sakovich equation, promising diverse applications across scientific disciplines.

The design of an eco-tax on tourism waste for sustainable destinations, based on environmental efficiency and spatial dependence

This paper analyzes environmental efficiency in the tourism sector, considering the joint production of desirable (overnight stays) and undesirable (tourism waste) outputs while addressing spatial dependence among destinations. Employing the enhanced hyperbolic distance function methodology, we integrate stochastic frontier analyses and spatial econometric techniques. The study investigates the 41 tourist areas of Spain between 2000 and 2019, and the results obtained reveal that they operate with a high average efficiency level and exhibit diminishing returns to scale. A significant contribution of this study, besides providing a useful reference for the design of policies to support destinations with efficiency gaps, is the estimation of shadow prices for waste in terms of overnight stays. This enables the calculation of a different ecotax for each tourist zone, providing valuable insights into the formulation of effective tourism policies that can guide policymakers in designing sustainable development strategies. The results obtained indicate that the most popular Spanish destinations, such as the Canary Islands and several coastal areas around the Mediterranean, should implement higher ecotaxes ranging from 3.3 to 6 euros per overnight stay. Additionally, calculating the shadow prices of tourism waste generated by each visitor can contribute to raising their environmental awareness.

VC dimension of Graph Neural Networks with Pfaffian activation functions

Graph Neural Networks (GNNs) have emerged in recent years as a powerful tool to learn tasks across a wide range of graph domains in a data-driven fashion. Based on a message passing mechanism, GNNs have gained increasing popularity due to their intuitive formulation, closely linked to the Weisfeiler–Lehman (WL) test for graph isomorphism, to which they were demonstrated to be equivalent (Morris et al., 2019 and Xu et al., 2019). From a theoretical point of view, GNNs have been shown to be universal approximators, and their generalization capability — related to the Vapnik Chervonekis (VC) dimension (Scarselli et al., 2018) — has recently been investigated for GNNs with piecewise polynomial activation functions (Morris et al., 2023). The aim of our work is to extend this analysis on the VC dimension of GNNs to other commonly used activation functions, such as the sigmoid and hyperbolic tangent, using the framework of Pfaffian function theory. Bounds are provided with respect to the architecture parameters (depth, number of neurons, input size) as well as with respect to the number of colors resulting from the 1–WL test applied on the graph domain. The theoretical analysis is supported by a preliminary experimental study.

Thermal energy analysis using artificial neural network and particle swarm optimization approach in partially ionized hyperbolic tangent material with ternary hybrid nanomaterials

This investigation holds significant pragmatic implications for endeavors aimed at curbing energy losses stemming from diverse factors. A neural network propelled by artificial intelligence, employing the Levenberg-Marquardt technique (ANN-LMM) has been devised for integrating ternary hybrid nanoparticles into a partially ionized hyperbolic tangent liquid flowing over an extended melting surface (PIHTL-SMS). The substance motion equivalence is delineated, considering the rotational outcome. The heat energy is formulated by amalgamating viscous intemperance and Joule heat contributions. To streamline complexity, the resulting PDEs are transmuted into a series of ordinary differential equations (ODEs) through resemblance transformations. A reference dataset for ANN-LMM is produced encompassing diverse significant model permutations and pretending situations utilizing the Lobatto III-A statistical technique. This reference data undergoes verification, evaluation and training procedures to refine the estimated explanation towards achieving anticipated outcomes. The precision, constancy, capability and resilience of ANN-LMM are substantiated concluded mean squared error (MSE)-based fitness curves, error histograms, regression plots and absolute error evaluations. A relative examination elucidates the correctness of the suggested solver, exhibiting entire errors within the array of 10-10 to 10-06 for all significant constraints. Resulting differential equations are also solved using particle swarm optimization (PSO) approach. In PSO several parameters are optimized to enhance the performance of the algorithm. Optimizing these parameters help to improve the effectiveness and efficiency of the PSO algorithm for given problem. PSO converges quickly to optimal or near-optimal solutions, making it efficient for problems with large domain. Several pivotal graphs are constructed to illustrate the impact of emergent constraints on fluid temperature and velocity profiles. The outcomes underscore the numerical technique as a potent instrument for tackling the intricate conjoined ODEs system prevalent in fluid mechanism and allied intemperance presentations in technology. Additionally, improvements in the Forchheimer constraint and the Weissenberg number are deemed imperative for regulating fluid velocity. Unlike prior research that mostly concentrated on single or binary nanofluids, this work presents the integration of ternary hybrid nanoparticles into a partly ionized hyperbolic tangent liquid, a unique technique. Improved accuracy and processing efficiency are also provided by using an ANN-LMM neural network to solve the complicated transformed ODEs. Comparison of ANN, PSO results and existing results are done which shows validity of the current analysis. This work is unique in that it offers a deeper understanding of fluid behavior at advanced thermal settings by including emergent restrictions, viscous dissipation, and Joule heating.

Distributed Adaptive Multi‐Lane Fusion Control for 2‐D Plane Vehicle Platoon With Distance Constraints and Angle Sensor Faults

ABSTRACTThis article investigates the distributed adaptive multi‐lane fusion control for two‐dimensional (2‐D) plane vehicle platoons with variable distance constraints, input hysteresis, and angle sensor faults. Variable distance constraints occur within a limited time interval (VDCOLT), which limits vehicle spacing within a variable range during preset time intervals while keeping the vehicle spacing unconstrained at other times. Combined with a transformed error function, a new shifting function is introduced to convert the constrained spacing error variables into the unconstrained variables while keeping the unconstrained variable unchanged. Designing the position controller ensures that the distance satisfies VDCOLT and reduces the adverse effects caused by unknown signs of input hysteresis by using a Nussbaum function. Moreover, a new angle controller based on the hyperbolic tangent function is designed to solve the problem of angle sensor faults. Furthermore, the stability of the vehicle platoon is achieved through the backstepping control and sliding‐mode control methods. Finally, a numerical simulation is provided to verify the proposed techniques.

Optimizing EPR pulses for broadband excitation and refocusing

In this paper, we numerically optimize broadband pulse shapes that maximize Hahn echo amplitudes. Pulses are parameterized as neural networks (NN), nonlinear amplitude limited Fourier series (FS), and discrete time series (DT). These are compared to an optimized choice of the conventional hyperbolic secant (HS) pulse shape. A power constraint is included, as are realistic shape distortions due to power amplifier nonlinearity and the transfer function of the microwave resonator. We find that the NN, FS, and DT parameterizations perform equivalently, offer improvements over the best HS pulses, and contain a large number of equivalent optimal maxima, implying the flexibility to include further constraints or optimization goals in future designs.

Non-Fourier thermal focusing by gradient thermal metamaterials based on the Cattaneo–Vernotte model

Direct methods to manipulate internal heat conduction in solids remain nearly elusive. In this work, we propose a two-dimensional (2D) gradient thermal metamaterial lens designed to control the internal non-Fourier heat conduction. This gradient thermal metamaterial comprises cylindrical structures made of dermis immersed in a stratum background with an effective refractive index adhering to the hyperbolic secant profile rule. Subsequently, the effective refractive index of the thermal metamaterial is defined and numerically assessed using the extended plane wave expansion method. Thereafter, the non-Fourier heat conduction is analyzed numerically via the finite element method in the time-domain. Our findings indicate that the proposed gradient thermal metamaterial can focus the heat effectively and is akin to a flat thermal lens. The proposed flat thermal lens offers promising avenues for enhancing the material properties by laser-based technologies.