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Due to the hasty growth of digital technology, securing medical images for transmission and storage is a significant issue. Information on the texture of medical images is essential for diagnosis. Hence, protecting these content has become a great challenge in the e-Healthcare system. So, we propose a novel dynamic circular bit shifting and bit-flipping process in the diffusion phase of the image cryptosystem to provide high level protection for the medical image. In the developed framework, at first, the hyperbolic chaotic map is executed for generating three hyper chaotic series and these chaotic series are utilized for encryption process. In encryption process, primarily the secret image is confused by utilizing any two chaotic series, then the confused image is divided into blocks and each block is diffused by executing the proposed dynamic circular bit shifting and bit-flipping process which greatly increases the probability and complexity level of the developed cryptosystem. Evaluation using different medical images represents that the proposed model has robustness, high efficiency, key sensitivity and also has the ability to withstand different attacks. This has been represented using the results of security measures and statistical analysis, such as Shannon entropy (7.997), CC (< 0), NPCR (99.63 %), UACI (37.14 %), SSIM (< 0), FSIM (< 0.5), PSNR (6.88). These results prove that the developed medical crypt can efficiently provide better security for medical images utilized in real time medical application like e-Healthcare and telemedicine.

Due to the hasty growth of digital technology, securing medical images for transmission and storage is a significant issue. Information on the texture of medical images is essential for diagnosis. Hence, protecting these content has become a great challenge in the e-Healthcare system. So, we propose a novel dynamic circular bit shifting and bit-flipping process in the diffusion phase of the image cryptosystem to provide high level protection for the medical image. In the developed framework, at first, the hyperbolic chaotic map is executed for generating three hyper chaotic series and these chaotic series are utilized for encryption process. In encryption process, primarily the secret image is confused by utilizing any two chaotic series, then the confused image is divided into blocks and each block is diffused by executing the proposed dynamic circular bit shifting and bit-flipping process which greatly increases the probability and complexity level of the developed cryptosystem. Evaluation using different medical images represents that the proposed model has robustness, high efficiency, key sensitivity and also has the ability to withstand different attacks. This has been represented using the results of security measures and statistical analysis, such as Shannon entropy (7.997), CC (< 0), NPCR (99.63 %), UACI (37.14 %), SSIM (< 0), FSIM (< 0.5), PSNR (6.88). These results prove that the developed medical crypt can efficiently provide better security for medical images utilized in real time medical application like e-Healthcare and telemedicine. [ABSTRACT FROM AUTHOR]

In this article, we study the complex coupled Kuralay system (CKS), which have many applications in different field such as ferromagnetic material and fiber optics. We implement an analytical method, namely, the tanh–coth method to considered system to formulate a system of algebraic equations. Solving the obtained system, we acquire different traveling wave solutions that are expressed in term of hyperbolic tangent functions and hyperbolic cotangent functions. Also, we employ the energy balance technique to the selected model to sort out the behavior of traveling wave solutions. By this method, we obtain periodic wave solution that are expressed in term of cosine function. In order to show the graphical presentation of the acquired solitary waves solutions, 3D and 2D curve are created by selecting well-suited parametric values via Mathematica.

The work presented in this article has been motivated by the recent research going on the Hankel determinant bounds and their related consequences, as well as the techniques used previously by many different authors. We aim to establish a new subfamily of holomorphic functions connected with the hyperbolic tangent function with bounded boundary rotation. We investigate the sharp estimate of the third Hankel determinant for this newly defined family of functions. Moreover, for the defined functions family, the Krushkal inequality, the first four initial sharp bounds of the logarithmic coefficients and the sharp second Hankel determinant of the logarithmic coefficients are given.

A system with distributed parameters in its mechanical part is considered. Examples of such a system are given. The reason for taking into account the distribution of parameters in such systems is indicated. Existing methods of research and calculation of control systems are considered. The boundary value problem for a system with distributed parameters is presented. The use of a surveillance device as one of the ways to implement a control system for a system with distributed parameters is proposed. The necessity of using a closed control system and the complexity of its implementation are demonstrated. A block diagram of the control method being developed is presented. Also, transfer functions describing a system with distributed parameters are given; the mathematical calculation of the surveillance device is given as well. A method for implementing a closed control system along an intermediate coordinate for an electromechanical system with distributed parameters in the mechanical part using a surveillance device has been obtained. The surveillance device is in feedback and restores the output speed without measuring it directly. A general view of the transfer function is determined for the surveillance device. The advantages and disadvantages of the transfer function of the surveillance device and a graphical view of a simple implementation of the surveillance device with a dedicated auxiliary device in feedback are presented. An approximation that is used for systems with distributed parameters is described. Approximation conclusions are obtained. According to the parameters of the experimental installation, approximated transfer functions of the mechanical part of the system with distributed parameters without a surveillance device and with a surveillance device are obtained. The LACH of the mechanical part of the system with distributed parameters with a surveillance device and without a surveillance device for an experimental installation for the simplest case is presented, as well as a method for implementing a closed control system along an intermediate coordinate for an electromechanical system with distributed parameters in the mechanical part using a surveillance device.

In the present article, we define and investigate a new subfamily of holomorphic functions connected with the cosine hyperbolic function with bounded turning. Further some interesting results like sharp coefficients bounds, sharp Fekete-Szegö estimate, sharp $ 2^{nd} $ Hankel determinant and non-sharp $ 3^{rd} $ order Hankel determinant. Moreover, the same estimates have been investigated for 2-fold, 3-fold symmetric functions, the first four initial sharp bounds of logarithmic coefficient and sharp second Hankel determinant of logarithmic coefficients fort his defined function family.

This study is concerned with an observer-based independent joint control scheme for a coupled two-link rigid-flexible robotic manipulator considering the effects of actuator saturation and external disturbances. A distributed parameter model of the robotic manipulator system described by ordinary and partial differential equation (ODE-PDE) is derived by using Hamilton's Principle. An anti-saturation controller is developed to cope with the input saturation problem by using hyperbolic tangent and hyperbolic secant function. Considering the external disturbance, a disturbance observer is designed to estimate the external disturbance. The observer-based independent joint controller can adjust the joint angle and suppress the vibration simultaneously without the need for end-mounted actuators. The extended LaSalle's Invariance Principle is adopted to strictly prove the asymptotic stability of the robotic manipulator system. The simulation comparisons validate the effectivity of the proposed controllers and observers. [ABSTRACT FROM AUTHOR]

The work presented in this article has been motivated by the recent research going on the Hankel determinant bounds and their related consequences, as well as the techniques used previously by many different authors. We aim to establish a new subfamily of holomorphic functions connected with the hyperbolic tangent function with bounded boundary rotation. We investigate the sharp estimate of the third Hankel determinant for this newly defined family of functions. Moreover, for the defined functions family, the Krushkal inequality, the first four initial sharp bounds of the logarithmic coefficients and the sharp second Hankel determinant of the logarithmic coefficients are given. [ABSTRACT FROM AUTHOR]

In the present article, we define and investigate a new subfamily of holomorphic functions connected with the cosine hyperbolic function with bounded turning. Further some interesting results like sharp coefficients bounds, sharp Fekete-Szego estimate, sharp 2nd Hankel determinant and nonsharp 3rd order Hankel determinant. Moreover, the same estimates have been investigated for 2-fold, 3-fold symmetric functions, the first four initial sharp bounds of logarithmic coefficient and sharp second Hankel determinant of logarithmic coefficients fort his defined function family. [ABSTRACT FROM AUTHOR]

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models. [ABSTRACT FROM AUTHOR]

In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

The aim was to evaluate the performance of the latest quantitative marker for intrathecal IgG synthesis and to compare it with other established markers used for the same purpose. We retrospectively applied Auer's and Reiber's intrathecal IgG synthesis formulae in a cohort of 372 patients under investigation for central nervous system demyelination who had undergone lumbar puncture and oligoclonal bands (OCBs) detection for demonstrating intrathecal IgG synthesis. A ROC analysis revealed Auer's formula had lower sensitivity (68%) compared to Reiber's formula (83%) and IgG index (89%), in our cohort of patients that exhibited normal to mildly elevated albumin quotients (4.48 ± 3.93). By excluding possible sources of errors, we assume that Auer's formula is less sensitive than other established tools for the "prediction" of the detection of OCBs in routine cerebrospinal fluid (CSF) analyses due to the mathematical model used. Given the ability of Reiber's hyperbolic formula to describe the blood–CSF IgG distribution across a wide range of blood–brain barrier functionality, its use and the use of similar formulae are recommended for the discrimination between CNS-derived and blood-derived molecules in clinical laboratories. [ABSTRACT FROM AUTHOR]

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models. [ABSTRACT FROM AUTHOR]

Beam models are widely applied in civil engineering, transport, and industry because the beams are basic structural elements. When dealing with the high-order modes of beam in the context of applying the modal analysis method, the numerical instability issue affects the numeric simulation accuracy in many boundary conditions. There are two solutions in literature to overcome this shortcoming, namely refinement of the asymptotic form for the high order modes and reshaping the terms within the equation of the modes to eliminate the source of the numerical instability. In this paper, the numerical instability issue is signalled when the standard form of Green's function, which includes hyperbolic functions, is applied to a free-free Timoshenko length-long beam. A new way is proposed based on new set of eigenfunctions, including an exponential function, to construct a new form of Green's function. To this end, it starts from a new general form of Green's function and the characteristic equation is obtained; then, based on the boundary condition, the Green's function associated to the differential operator of the free-free Timoshenko beam is distilled. The numerical stability of the new form of the Green's function is verified in a numerical application and the results are compared with those obtained by using the standard form of the Green's function. [ABSTRACT FROM AUTHOR]

• A dynamic control algorithm based on state transition is proposed for vehicle following control. • Safe and efficient steady-following state is regarded as the optimization objective. • The absolute and relative braking modes can be flexibly chosen by the follower for safety control. • The following vehicle can adjust its own behavior in safety, efficiency, and smoothness. • The synchronous control of velocity, acceleration, and inter-vehicle distance is well achieved. This study aims to improve the safety and efficiency of vehicle-following behavior through a smooth vehicle behavior adjustment process. A dynamic control algorithm based on state transition is proposed for the following vehicle to enable it to incorporate better behavioral adjustments. This algorithm organically integrates a new generalized velocity difference control with the velocity control based on hyperbolic function to exploit its advantages and avoid weaknesses. The new generalized mathematical model based on fitting function is built for vehicle-following control within the neighborhood of a safe and efficient steady-following state, outside which, the velocity control method based on hyperbolic function is used by the following vehicle to adjust its behavior. In this algorithm, the absolute and relative braking modes can be flexibly selected by the following vehicle for safety control. Then, we establish a state transition diagram and a state optimization model with the safe and efficient steady-following state as the optimization objective of the vehicle-following behavior. The simulation shows that the current or a new safe and efficient steady-following state can be maintained, recovered, or reestablished by implementing the dynamic control algorithm based on state transition, and the synchronous control of the following vehicle's velocity, acceleration, and inter-vehicle distance can be realized by the following vehicle adjusting its own behavior safely, efficiently, and smoothly. [ABSTRACT FROM AUTHOR]

In this article, we consider the existence and uniqueness of solutions for a class of initial value problems of fuzzy Caputo-Katugampola fractional differential equations and the stability of the corresponding fuzzy fractional differential equations. The discussions are based on the hyperbolic function, the Banach fixed point theorem and an inequality property. Two examples are given to illustrate the feasibility of our theoretical results. [ABSTRACT FROM AUTHOR]

Fourteen main tree species from typical natural broad-leaved stands in the mid-subtropical zone of the Jian’ou Wanmulin Provincial Nature Reserve were studied to reveal the characteristics and provide a typical reference for the tree height diameter ratio of main broad-leaved tree species for the sustainable management of artificial, semi-natural, and natural broad-leaved forests in mid-subtropical zones. The mean tree height diameter ratio, standard deviation, variation coefficient, Mann-Whitney U test, Spearman rank correlation coefficient, exponential function, and hyperbola equation were used to analyze the characteristics of the tree height diameter ratio of the main tree species in each storey (including three strata and the whole stand). The results showed that the mean and maximum values of the tree height diameter ratio of the main tree species of typical natural broad-leaved stands in the mid-subtropical zone decreased with increasing stratum height. The mean values were significantly (P＜0.05) or extremely significantly different (P＜0.01) among each stratum. The correlation between tree height to diameter ratio and diameter at breast-height (DBH) of the main tree species in each storey showed an extremely significant negative correlation (P＜0.01). The correlation coefficients of strata Ⅰ and Ⅱ were significantly higher than those of the whole stand and stratum Ⅲ. The absolute values of the correlation coefficients were 0.806 1-0.895 3 for stratum Ⅰ, 0.788 2-0.964 3 for stratum Ⅱ, 0.419 0-0.809 1 for stratum Ⅲ, and 0.419 0-0.848 9 in the whole stand. The effects of the exponential and hyperbolic functions on the relationship between tree height diameter ratio and DBH of strata Ⅰ and Ⅱ of the main tree species were better than those of the whole stand and stratum Ⅲ. The R², Erms, and Ema of the main tree species in the whole stand and stratum Ⅲ were relatively high in the exponential function fitting, while those in strata Ⅰ and Ⅱ were relatively high in the hyperbolic fitting. The tree height to diameter ratio of the main tree species was not constant but varied with stratum and DBH. The characteristics of the tree height diameter ratio of each tree species can be expressed by the relationship curve between the tree height diameter ratio and the DBH of each stratum of the main tree species. [ABSTRACT FROM AUTHOR]

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models.

Mathematical modeling of many physical process leads to nonlinear evolution equations. The investigation of traveling wave solutions of nonlinear partial differential equations plays significant role in the study of nonlinear physical phenomena. In this article, the generalized (G' =G) - expansion method using generalized wave transformation is applied to find exact traveling wave solutions of the loaded modified Zakharov-Kuznetsov(ZK) equation. The performance of the method is reliable, useful and gives more new general exact solutions than the existing methods. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The results emphasize the power of proposed methods in providing distinct solutions of different physical structures in nonlinear science. The graphical presentations are demonstrated for some of newly obtained solutions. It is shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details. [ABSTRACT FROM AUTHOR]

This brief introduces a novel cost function framework for developing robust algorithms for adaptive filtering by embedding the standard cost function into the arctangent framework. This proposed framework is called the arctangent cost function framework. Based on this, we propose an arctangent family of robust algorithms for adaptive filtering. The performance of the proposed family of algorithms is tested through simulation studies in system identification scenarios that confirm the enhanced performance achieved by the arctangent family of algorithms over standard algorithms. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove that, for x>0 $x>0$, 1−exp(−x2x2+1)

A chaotic system with a hyperbolic function flux-controlled memristor is designed, which exhibits conditional symmetry and attractor growing. The newly introduced cosine function keeps the polarity balance when some of the variables get polarity inversed and correspondingly conditional symmetric coexisting chaotic attractors are coined. Due to the periodicity of the cosine function, the memristive system with infinitely many coexisting attractors shows attractor growing in some special circumstances. Analog circuit experiment proves the theoretical and numerical analysis.

The main contribution of this paper is to formulate a robust-adaptive and stable state-space speed control strategy for DC motors. The linear-quadratic-integral (LQI) controller is utilized as the baseline controller for optimal speed-regulation, accurate reference-tracking and elimination of steady-state fluctuations in the motor’s response. To reject the influence of modelling errors, the LQI controller is augmented with a Lyapunov-based model reference adaptation system (MRAS) that adaptively modulates the controller gains while maintaining the asymptotic stability of the controller. To further enhance the system’s robustness against parametric uncertainties, the adaptation gains of MRAS online gain-adjustment law are dynamically adjusted, after every sampling interval, using smooth hyperbolic functions of motor’s speed-error. This modification significantly improves the system’s response-speed and damping against oscillations, while ensuring its stability under all operating conditions. It dynamically re-configures the control-input trajectory to enhance the system’s immunity against the detrimental effects of random faults occurring in practical motorized systems such as bounded impulsive-disturbances, modelling errors, and abrupt load–torque variations. The efficacy of the proposed control strategy is validated by conducting credible hardware-in-the-loop experiments on QNET 2.0 DC Motor Board. The experimental results successfully validate the superior tracking accuracy and disturbance-rejection capability of the proposed control strategy as compared to other controller variants benchmarked in this article.

The aim was to evaluate the performance of the latest quantitative marker for intrathecal IgG synthesis and to compare it with other established markers used for the same purpose. We retrospectively applied Auer’s and Reiber’s intrathecal IgG synthesis formulae in a cohort of 372 patients under investigation for central nervous system demyelination who had undergone lumbar puncture and oligoclonal bands (OCBs) detection for demonstrating intrathecal IgG synthesis. A ROC analysis revealed Auer’s formula had lower sensitivity (68%) compared to Reiber’s formula (83%) and IgG index (89%), in our cohort of patients that exhibited normal to mildly elevated albumin quotients (4.48 ± 3.93). By excluding possible sources of errors, we assume that Auer’s formula is less sensitive than other established tools for the “prediction” of the detection of OCBs in routine cerebrospinal fluid (CSF) analyses due to the mathematical model used. Given the ability of Reiber’s hyperbolic formula to describe the blood–CSF IgG distribution across a wide range of blood–brain barrier functionality, its use and the use of similar formulae are recommended for the discrimination between CNS-derived and blood-derived molecules in clinical laboratories.

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models.

Beam models are widely applied in civil engineering, transport, and industry because the beams are basic structural elements. When dealing with the high-order modes of beam in the context of applying the modal analysis method, the numerical instability issue affects the numeric simulation accuracy in many boundary conditions. There are two solutions in literature to overcome this shortcoming, namely refinement of the asymptotic form for the high order modes and reshaping the terms within the equation of the modes to eliminate the source of the numerical instability. In this paper, the numerical instability issue is signalled when the standard form of Green’s function, which includes hyperbolic functions, is applied to a free-free Timoshenko length-long beam. A new way is proposed based on new set of eigenfunctions, including an exponential function, to construct a new form of Green’s function. To this end, it starts from a new general form of Green’s function and the characteristic equation is obtained; then, based on the boundary condition, the Green’s function associated to the differential operator of the free-free Timoshenko beam is distilled. The numerical stability of the new form of the Green’s function is verified in a numerical application and the results are compared with those obtained by using the standard form of the Green’s function.

In applied sciences it is always useful to improve the catalogue of definite integrals available in tables. This present paper is a compendium of definite integrals involving a hyperbolic quotient function expressed in terms of the Lerch function. A substantial portion of the results are new. [ABSTRACT FROM AUTHOR]

In this paper, we establish an interesting chain of sharp inequalities involving Toader-Qi mean, exponential mean, logarithmic mean, arithmetic mean and geometric mean. This greatly improves some existing results. [ABSTRACT FROM AUTHOR]

Global orbits connect the saddle points in an infinite period through the homoclinic and heteroclinic types of manifolds. Different from the periodic movement analysis, it requires special strategies to obtain expression of the orbit and detect the associated profound dynamic behaviors, such as chaos. In this paper, a global dynamic frequency method is applied to detect the homoclinic and heteroclinic bifurcation of the complicated nonlinear systems. The so-called dynamic frequency refers to the newly introduced frequency that varies with time t, unlike the usual static variable. This new method obtains the critical bifurcation value as well as the analytic expression of the orbit by using a standard five-step hyperbolic function-balancing procedure, which represents the influence of the higher harmonic terms on the global orbit and leads to a significant reduction of calculation workload. Moreover, a new homoclinic manifold analysis maps the periodic excitation onto the target global manifold that transfers the chaos discussion of non-autonomous systems into the orbit computation of the general autonomous system. That strategy unifies the global bifurcation analysis into a standard orbit approximation procedure. The numerical simulation results are shown to compare with the predictions. [ABSTRACT FROM AUTHOR]

In this article, we study the complex coupled Kuralay system (CKS), which have many applications in different field such as ferromagnetic material and fiber optics. We implement an analytical method, namely, the tanh – coth method to considered system to formulate a system of algebraic equations. Solving the obtained system, we acquire different traveling wave solutions that are expressed in term of hyperbolic tangent functions and hyperbolic cotangent functions. Also, we employ the energy balance technique to the selected model to sort out the behavior of traveling wave solutions. By this method, we obtain periodic wave solution that are expressed in term of cosine function. In order to show the graphical presentation of the acquired solitary waves solutions, 3D and 2D curve are created by selecting well-suited parametric values via Mathematica. • Nonlinear Kuralay System is considered. • Implementation of tanh-coth and energy balance method. • Bright, periodic and singular solitons are studied. [ABSTRACT FROM AUTHOR]

The hyperbolic cosine function with high-order errors can be utilized to improve the accuracy of adaptive filters. However, when initial weight errors are large, the hyperbolic cosine-based adaptive filter (HCAF) may be unstable. In this paper, a novel normalization based on the logarithmic hyperbolic cosine function is proposed to achieve the stabilization for the case of large initial weight errors, which generates a logarithmic HCAF (LHCAF). Actually, the cost function of LHCAF is the logarithmic hyperbolic cosine function that is robust to large errors and smooth to small errors. The transient and steady-state analyses of LHCAF in terms of the mean-square deviation (MSD) are performed for a stationary white input with an even probability density function in a stationary zero-mean white noise. The convergence and stability of LHCAF can be therefore guaranteed as long as the filtering parameters satisfy certain conditions. The theoretical results based on the MSD are supported by the simulations. In addition, a variable scaling factor and step-size LHCAF (VSS-LHCAF) is proposed to improve the filtering accuracy of LHCAF further. The proposed LHCAF and VSS-LHCAF are superior to HCAF and other robust adaptive filters in terms of filtering accuracy and stability. [ABSTRACT FROM AUTHOR]

This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details.

The main contribution of this paper is to formulate a robust-adaptive and stable state-space speed control strategy for DC motors. The linear-quadratic-integral (LQI) controller is utilized as the baseline controller for optimal speed-regulation, accurate reference-tracking and elimination of steady-state fluctuations in the motor's response. To reject the influence of modelling errors, the LQI controller is augmented with a Lyapunov-based model reference adaptation system (MRAS) that adaptively modulates the controller gains while maintaining the asymptotic stability of the controller. To further enhance the system's robustness against parametric uncertainties, the adaptation gains of MRAS online gain-adjustment law are dynamically adjusted, after every sampling interval, using smooth hyperbolic functions of motor's speed-error. This modification significantly improves the system's response-speed and damping against oscillations, while ensuring its stability under all operating conditions. It dynamically re-configures the control-input trajectory to enhance the system's immunity against the detrimental effects of random faults occurring in practical motorized systems such as bounded impulsive-disturbances, modelling errors, and abrupt load–torque variations. The efficacy of the proposed control strategy is validated by conducting credible hardware-in-the-loop experiments on QNET 2.0 DC Motor Board. The experimental results successfully validate the superior tracking accuracy and disturbance-rejection capability of the proposed control strategy as compared to other controller variants benchmarked in this article. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, for x > 0 , 1 − exp (− x 2 x 2 + 1 ) < tanh x < 1 − exp (− x 3 x 3 + 1 ) 3. This solves an open problem proposed by Ivády. [ABSTRACT FROM AUTHOR]

Let Ipx be the modified Bessel function of the first kind of order p. The upper and lower bounds in the form of simple rational functions about cosht and (sinht)/t for the function I0x are obtained. The corresponding inequalities for the Toader-Qi mean do not match those in the existing literature.

The aims of this paper are to introduce and validate a new distribution, which is related to the hyperbolic function. The differential equation is applied to obtain the survival and probabilistic functions. The moment generating function is provided by using the integral transform. The proposed distribution is applied to real data sets. It is demonstrated that it can be used as an alternative model in various disciplines such as electronics, financial, weather, and arrival times.

The Smith's approximation formula for the gamma function is given by Γ (x + 1 2) = 2 π (x e) x (2 x tanh ⁡ 1 2 x) x / 2 (1 + O (1 x 5 )) , as x → ∞. In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions: Γ (x + 1 / 2) 2 π (x / e) x ∼ (2 x tanh ⁡ 1 2 x) x / 2 exp ⁡ (∑ n = 3 ∞ [ (2 n) ! − (2 n − 1) 2 2 n − 1 ] (1 − 2 1 − 2 n) 2 n (2 n − 1) (2 n) ! B 2 n x 2 n − 1 ) , as x → ∞ , and give an estimate of remainder in the above asymptotic series, where B 2 n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ (x + 1 / 2) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established. [ABSTRACT FROM AUTHOR]

A brief account of our recent works dealing with the problem of characterizing hyperbolic functions through the use of functional equations with side conditions is reported. Specifically, the fact that each of the n functions having a zero sum over n points lying in a hyperplane is necessarily a linear function, is generalized and the results so obtained are applied to characterize hyperbolic functions. [ABSTRACT FROM AUTHOR]

Snake robots could be utilized in many fields because of their hyper-redundant properties, although there are still control problems when they are operated in complex environments. For example, a helical rolling motion has been used to climb a pipe. By using this kind of motion, a snake robot can move along the inside or outside of a pipe. However, this motion has limitations when the robot moves along a pipe containing a high gap or a branch point. In this study, we propose a type of motion for snake robots that involves wrapping around the outside of a pipe to overcome a branch point on it. This new motion uses a hyperbolic function to make a helical wave curve, which is then propagated by shifting the shape of the hyperbolic function along the body of the snake robot. The joint angles of the snake robot are derived by calculating the curvature and torsion of the curve on the basis of the formula of the continuous curve model. Finally, the results of simulations performed using the Robot Operating System and Gazebo programs are shown to validate the effectiveness of the new motion. [ABSTRACT FROM AUTHOR]