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This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.

Statistics are mathematical tools applying scientific investigations, such as engineering and medical and biological analyses. However, statistical methods are often improved. Nowadays, statisticians try to find an accurate way to solve a problem. One of these problems is estimation parameters, which can be expressed as an inverse problem when independent variables are highly correlated. This paper’s significant goal is to interpret the parameter estimates of double generalized Rayleigh distribution in a regression model using a wavelet basis. It is difficult to use the standard version of the regression methods in practical terms, which is obtained using the likelihood. Since a noise level usually makes the result of estimation unstable, multicollinearity leads to various estimates. This kind of problem estimates that features of the truth are complicated. So it is reasonable to use a mixed method that combines a fully Bayesian approach and a wavelet basis. The usual rule for wavelet approaches is to choose a wavelet basis, where it helps to compute the wavelet coefficients, and then, these coefficients are used to remove Gaussian noise. Recovering data is typically calculated by inverting the wavelet coefficients. Some wavelet bases have been considered, which provide a shift-invariant wavelet transform, simultaneously providing improvements in smoothness, in recovering, and in squared-error performance. The proposed method uses combining a penalized maximum likelihood approach, a penalty term, and wavelet tools. In this paper, real data are involved and modeled using double generalized Rayleigh distributions, as they are used to estimate the wavelet coefficients of the sample using numerical tools. In practical applications, wavelet approaches are recommended. They reduce noise levels. This process may be useful since the noise level is often corrupted in real data, as a significant cause of most numerical estimation problems. A simulation investigation is studied using the MCMC tool to estimate the underlying features as an essential task statistics.

As more long-span bridges continue to be completed and opened to traffic, the safety of cars driving across the bridge has attracted more and more attention, especially when the car is suddenly affected by the crosswind, the car is likely to have direction deviation or even a rollover accident. In this paper, the large eddy simulation method is used to study the flow field characteristics and safety of the car on the bridge under the turbulent crosswind. The numerical simulation model is established by referring to the Donghai Bridge, and the correctness of the car model is validated by combining with the data of wind tunnel test. The influence of factors such as the porosity and height of the bridge guardrail and the Reynolds number of airflow on the flow field characteristics is analyzed. The study shows that, in order to ensure the safety of cars on the bridge, the bridge guardrail porosity should be small, 35.8% is more suitable, the guardrail height should be more suitable within the range of 1.5–1.625 meters, and the Reynolds number should not be 3.51e + 5. The research results of this paper will provide reference for the optimal design of bridge guardrail.

This paper studies the effects of Southwest Airlines, the largest low-cost carrier (LCC) in the U.S., on other carriers’ payoff functions and entry probabilities. A static entry game model is developed and estimated by viewing entry as an indicator of underlying profitability and making use of Nash Equilibrium. Results indicate that Southwest has a remarkable and negative impact on the payoffs of other carriers. This impact is firm-specific, with LCCs being more affected than full-service carriers (FSCs). Comparing the two service types, the results show that Southwest’s nonstop presence apparently imposes more downward pressure on opponents’ profits than its connecting presence. A counterfactual experiment is then conducted. Once Southwest is counterfactually removed, the probability of each carrier entering a market significantly changes. This paper examines Southwest’s impacts from a new perspective and extends literature on entry game estimation.

In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.

This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that G is edge- k -group choosable if its line graph is k -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).

This paper proposes a novel predictive strategy based on a model predictive control (MPC) for the interior permanent magnet synchronous motors (IPMSMs) driven by a three-level simplified neutral-point clamped inverter (3L-SNPC) for electric vehicle applications (EVAs). Based on the prediction of the future behavior of the controlled variables, a predefined multiobjective cost function incorporates the control objectives which are evaluated for every sampling period to generate the optimal switching state applied directly to the inverter without the modulation stage. The control objectives in this paper are tracking current capacity, neutral-point voltage balancing, common-mode voltage control, and switching frequency reduction. The principal concepts of the novel scheme are summarized as follows: first, the delay compensation based on the long horizon of prediction is adopted by a multilevel power converter structure. Second, based on the modified Lyapunov candidate function, both stability and recursive feasibility are ensured of the proposed predictive scheme. Third, the practicability of the real-time implementation is improved by the proposed “static voltage vector” (SVV) and “single state variation” (SSV) principles. Finally, the proposed concepts are implemented in the novel predictive control formulation as additional constraints without compromising the complexity and the good performances of the predictive controller. Therefore, only the switching states that guarantee the stability and the reduction of calculation burden criteria are considered in the evaluation of cost function. The proposed predictive scheme based on the “SVV” principle has demonstrated superior performance in simulation compared with the proposed scheme with the “SSV” principle. The computational burden and switching frequency rates are reduced by 35% and 56.22%, respectively.

Linearized model of the system is often used in control design. It is generally believed that we can obtain the linearized model as long as the Taylor expansion method is used for the nonlinear model. This paper points out that the Taylor expansion method is only applicable to the linearization of the original nonlinear function. If the Taylor expansion is used for the derived nonlinear equation, wrong results are often obtained. Taking the linearization model of the maglev system as an example, it is shown that the linearization should be carried out with the process of equation derivation. The model is verified by nonlinear system simulation in Simulink. The method in this paper is helpful to write the linearized equation of the control system correctly.

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

In this paper, by employing the Zhang neural network (ZNN) method, an effective continuous-time LU decomposition (CTLUD) model is firstly proposed, analyzed, and investigated for solving the time-varying LU decomposition problem. Then, for the convenience of digital hardware realization, this paper proposes three discrete-time models by using Euler, 4-instant Zhang et al. discretization (ZeaD), and 8-instant ZeaD formulas to discretize the proposed CTLUD model, respectively. Furthermore, the proposed models are used to perform the LU decomposition of three time-varying matrices with different dimensions. Results indicate that the proposed models are effective for solving the time-varying LU decomposition problem, and the 8-instant ZeaD LU decomposition model has the highest precision among the three discrete-time models.

The excessive vertical vibration of structures induced by walking pedestrians has attracted considerable attention in the past decades. The bipedal walking models proposed previously, however, merely focus on the effects generated by legs and ignore the effects of the dynamics of body parts on pedestrian-structure interactions. The contribution of this paper is proposing a novel pedestrian-structure interaction system by introducing the concept of the continuum and a different variable stiffness strategy. The dynamic model of pedestrian-structure coupling system is established using the Lagrange method. The classical mode superposition method is utilized to calculate the response of the structure. The state-space method is employed to determine natural frequencies and damping ratio of the coupled system. Based on the proposed model, numerical simulations and parametric analysis are conducted. Numerical simulations have shown that the continuum enables the pedestrian-structure system to achieve the stable state more efficiently than the classic model does, which idealizes the body as a concentrated or lumped mass. The parametric study reveals that the presence of pedestrians is proved to significantly decrease the frequency of human-structure interaction system and improve its damping ratio. Moreover, the parameters of the bipedal model have a noticeable influence on the dynamic properties and response of the pedestrian-structure system. The bipedal walking model proposed in this paper depicts a pattern of pedestrian-structure interactions with different parameter settings and has a great potential for a wide range of practical applications.

In this paper, a switched finite-time source-seeking strategy is proposed for underactuated ships with a signal sensor. In practical application, for sensors in some complex environment, due to the lack of the ability to perceive the location of the source or its own position, we use the nonmodel gradient estimation of extremum seeking algorithm. This paper presents a three-step switching control method for the first time. In the first step, we prove that the closed-loop system of the underactuated ship is stable in a finite time with a constant yaw angular acceleration. In the second step, under the yaw angle acceleration controller, one of the position coordinates of the underactuated ship sensor is realized to be close to the center of the signal source through the average method. The third step is to drive the other position coordinates of the underactuated ship sensor to the center of the signal source under the action of the surge line speed controller. Simulation results show the effectiveness of the proposed strategy.

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

Every graph G = V , E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e . A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k -vertex-coloring of a graph G = V , E is a mapping c : V ⟶ S ( S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple O n 2 time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.

In this paper, we focus on the safety supervision of inland vessels. This paper especially aims at studying the vessel target detection and dynamic tracking algorithm based on computer vision and the target fusion algorithm based on multisensor. For the vessel video target detection and tracking, this paper analyzes the current widely used methods and theories. Additionally, facing the application scenarios and characteristics of inland vessels, a comprehensive vessel video target detection algorithm is proposed in this paper. It is combined with a three-frame difference method based on Canny edge detection and a background subtraction method based on mixed Gaussian background modeling. Besides, for the multisensor target fusion, the processing method of laser point cloud data and automatic identification system (AIS) data is analyzed in this paper. Based on the idea of fuzzy mathematics, this paper proposes a method for calculating the fuzzy correlation matrix with normal membership function, which realizes the fusion of vessel track features of laser point cloud data and AIS data under dynamic video correction. Finally, through this method, a set of vessel situation active intelligent perception systems based on multisensor fusion was developed. Experiments show that this method has better environmental applicability and detection accuracy than traditional manual detection and any single monitoring method.

To achieve numerical simulation of large deformation evolution processes in underground engineering, the barycentric interpolation test function is established in this paper based on the manifold cover idea. A large-deformation numerical simulation method is proposed by the double discrete method with the fixed Euler background mesh and moving material points, with discontinuous damage processes implemented by continuous simulation. The material particles are also the integration points. This method is called the manifold cover Lagrangian integral point method based on barycentric interpolation. The method uses the Euler mesh as the background integral mesh and describes the deformation behavior of macroscopic objects through the motion of particles between meshes. Therefore, this method can avoid the problem of computation termination caused by the distortion of the mesh in the calculation process. In addition, this method can keep material particles moving without limits in the set region, which makes it suitable for simulating large deformation and collapse problems in geotechnical engineering. Taking a typical slope as an example, the results of a slope slip surface obtained using the manifold cover Lagrangian integral point method based on barycentric interpolation proposed in this paper were basically consistent with the theoretical analytical method. Hence, the correctness of the method was verified. The method was then applied for simulating the collapse process of the side slope, thereby confirming the feasibility of the method for computing large deformations.

The purpose of this paper is to introduce and analyze the shrinking projection algorithm with errors for a finite set of costerro bounded linear mappings in the setting of uniformly convex smooth Banach spaces. Here, under finite dimensional or compactness restriction or the error term being zero, the strong limit point of the sequence stated in the iterative scheme for these mappings in uniformly convex smooth Banach spaces was studied. This paper extends Ezearn and Prempeh’s result for nonexpansive mappings in real Hilbert spaces.

This paper is intended to study the limit theorem of Markov chain function in the environment of single infinite Markovian systems. Moreover, the problem of the strong law of large numbers in the infinite environment is presented by means of constructing martingale differential sequence for the measurement under some different sufficient conditions. If the sequence of even functions gnx,n≥0 satisfies different conditions when the value ranges of x are different, we have obtained SLLN for function of Markov chain in the environment of single infinite Markovian systems. In addition, the paper studies the strong convergence of the weighted sums of function for finite state Markov Chains in single infinitely Markovian environments. Although the similar conclusions have been carried out, the difference results performed by previous scholars are that we give weaker different sufficient conditions of the strong convergence of weighted sums compared with the previous conclusions.

Relevance vector machine (RVM) is a Bayesian sparse kernel method for regression in statistical learning theory, which can avoid principal limitations of the support vector machine (SVM) and result in faster performance on test data while maintaining comparable generalization error. In this paper, we develop a logic framework of the evidence function approximation associated with RVM based on Taylor expansion instead of traditional technology called “completing the square.” While constructing the term of completing the square, we have to find the term of completing the square by making use of some skill, which in practice increases the difficulty in dealing with the evidence function approximation associated with RVM. The logical framework in this paper based on Taylor expansion shows some advantages compared with the conventional method of completing the square, which is easier to be enforced due to the clear logical framework and avoids the difficulty in looking for the term of completing the square intentionally. From symmetry of covariance in a multivariate Gaussian distribution and algebraic knowledge, we derive approximation and maximization of the evidence function associated with RVM, which is consistent with the previous result using the novel logical framework. Finally, we derive the EM algorithm for RVM, which is also consistent with the previous result except that we use the precision matrix as the covariance.

The traditional spectral analysis method is used to study the characteristics of bearing fault signals in frequency domain, which is reasonable and effective in general cases. However, it is proved that the fault signals have heavy tails in this paper, which are α stable distribution, and 1<α<2, and even the noises belong to α stable distribution. Then the conventional spectral analysis methods degenerate and even fail under α stable distribution environment. Several improved frequency spectral analysis methods are proposed employing fractional lower order covariation or fractional lower order covariance in this paper, including fractional lower order Blackman-Tukey covariation spectrum (FLOBTCS), fractional lower order periodogram covariation spectrum (FLOPCS), and fractional lower order welch covariation spectrum (FLOWCS). In order to suppress side lobe and improve resolution, we present novel fractional lower order autoregression (FLO-AR) and fractional lower order autoregressive moving average (FLO-ARMA) parameter model frequency spectrum methods, and the calculation steps are summarized. The proposed spectrum methods are compared with the existing methods based on second-order statistics under Gaussian and SαS distribution environments, and the results show that the new algorithms have better performance than the traditional methods. Finally, the improved methods are applied to estimate frequency spectrums of the normal and outer race fault signals, and it is demonstrated that they are effective for fault diagnosis.

In manifold learning, the aim of alignment is to derive the global coordinate of manifold from the local coordinates of manifold’s patches. At present, most of manifold learning algorithms assume that the relation between the global and local coordinates is locally linear and based on this linear relation align the local coordinates of manifold’s patches into the global coordinate of manifold. There are two contributions in this paper. First, the nonlinear relation between the manifold’s global and local coordinates is deduced by making use of the differentiation of local pullback functions defined on the differential manifold. Second, the method of local linear iterative alignment is used to align the manifold’s local coordinates into the manifold’s global coordinate. The experimental results presented in this paper show that the errors of noniterative alignment are considerably large and can be reduced to almost zero within the first two iterations. The large errors of noniterative/linear alignment verify the nonlinear nature of alignment and justify the necessity of iterative alignment.

First of all, we establish a three-dimension open Kaldorian business cycle model under the condition of the fixed exchange rate. Secondly, with regard to the model, we discuss the existence of equilibrium point and the stability of the system near it with a time delay in currency supply as the bifurcating parameters of the system. Thirdly, we discuss the existence of Hopf bifurcation and investigate the stability of periodic solution generated by the Hopf bifurcation; then the direction of the Hopf bifurcation is given. Finally, a numerical simulation is given to confirm the theoretical results. This paper plays an important role in theoretical researching on the model of business cycle, and it is crucial for decision-maker to formulate the macroeconomic policies with the conclusions of this paper.

In this paper, a kind of BAM neural networks with leakage delays in the negative feedback terms and time-varying delays in activation functions was considered. By constructing a suitable Lyapunov function and using inequality techniques, some sufficient conditions to ensure the existence and exponential stability of antiperiodic solutions of these neural networks were derived. These conditions extend some results recently appearing in recent papers. Lastly, an example is given to show the feasibility of these conditions.

This paper deals with the classic problem of the synthesis of planar linkages for path generation. Based on the Fourier theory, the task curve and the synthesized four-bar coupler curve are regarded as the same curve if their Fourier descriptors match. Using Fourier analysis, a curve must be given as a function of time, termed a parametrization. In practical applications, different parametrizations can be associated with the same task and coupler curve, respectively; however, these parametrizations are Fourier analyzed to different Fourier descriptors, thus resulting in the mismatch of the task and coupler curve. In this paper, we present a parametrization-invariant method to eliminate the influence of parametrization on the values of Fourier descriptors by unifying given parametrizations to the arc length parametrization; meanwhile, a new design space decoupling scheme is introduced to separate the shape, size, orientation, and location matching of the task and four-bar curve, which leads naturally to an efficient synthesis approach.

The aim of this paper is to extend the modified Fourier-Ritz approach to evaluate the free vibration of four-parameter functionally graded moderately thick cylindrical, conical, spherical panels and shells of revolution with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. In the modified Fourier-Ritz approach, the admissible functions of the structure elements are expanded into the improved Fourier series which consist of two-dimensional (2D) Fourier cosine series and auxiliary functions to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions and then solve the natural frequencies by means of the Ritz method. As one merit of this paper, the functionally graded cylindrical, conical, spherical shells are, respectively, regarded as a special functionally graded cylindrical, conical, spherical panels, and the coupling spring technology is introduced to ensure the kinematic and physical compatibility at the common meridian. The excellent accuracy and reliability of the unified computational model are compared with the results found in the literatures.

This paper investigates the exponential synchronization problem of stochastic complex dynamical networks with impulsive perturbation and Markovian switching. The complex dynamical networks consist ofκmodes, and the networks switch from one mode to another according to a Markovian chain with known transition probability. Based on the Lyapunov function method and stochastic analysis, by employingM-matrix approach, some sufficient conditions are presented to ensure the exponential synchronization of stochastic complex dynamical networks with impulsive perturbation and Markovian switching, and the upper bound of impulsive gain is evaluated. At the end of this paper, two numerical examples are included to show the effectiveness of our results.

This technical paper deals with the problem of stochastic stability and stabilization for a class of linear Markovian jumping systems with discrete time-varying delay. A novel delay-dependent stochastic stability criterion for Markovian delay systems is established based on new augmented Lyapunov-Krasovskii functional and delay fractioning techniques. Then a state feedback controller is designed to guarantee the stochastic stability of the resulting closed-loop system. Numerical examples are provided to illustrate the effectiveness of the proposed design approach in this paper.

This paper presents a fault diagnosis algorithm to estimate the fault for a class of linear discrete systems based on an adaptive fault estimation observer. And observer gain matrix and adaptive adjusting rule of the fault estimator are designed. Furthermore, the adaptive regulating algorithm can guarantee the first-order difference of a Lyapunov discrete function to be negative, so that the observer is ensured to be stable and fault estimation errors are convergent. Finally, simulation results of an aircraft F-16 illustrate the advantages of the theoretic results that are obtained in this paper.

This paper addresses the short-term scheduling problem of hydrothermal power systems, which results in a large-scale mixed-integer nonlinear programming problem. The objective consists in minimizing the operation cost over a two-day horizon with a one-hour time resolution. To solve this difficult problem, a Lagrangian Relaxation (LR) based on variable splitting is designed where the resulting dual problem is solved by a Bundle method. Given that the LR usually fails to find a feasible solution, we use an inexact Augmented Lagrangian method to improve the quality of the solution supplied by the LR. We assess our approach by using a real-life hydrothermal configuration extracted from the Brazilian power system, proving the conceptual and practical feasibility of the proposed algorithm. In summary, the main contributions of this paper are (i) a detailed and compatible modelling for this problem is presented; (ii) in order to solve efficiently the entire problem, a suitable decomposition strategy is presented. As a result of these contributions, the proposed model is able to find practical solutions with moderate computational burden, which is absolutely necessary in the modern power industry.

A shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.

An instrumental matrix approach to design output feedback passive controller for switched singular systems is proposed in this paper. The nonlinear inequality condition including Lyapunov inverse matrix and controller gain matrix is decoupled by introducing additional instrumental matrix variable. Combined with multiple Lyapunov function method, the nonlinear inequality is transformed into linear matrix inequality (LMI). An LMI condition is presented for switched singular system to be stable and passive via static output feedback under designed switching signal. Moreover, the conditions proposed do not require the decomposition of Lyapunov matrix and its inverse matrix or fixing to a special structure. The theoretical results are verified by means of an example. The method introduced in the paper can be effectively extended to a single singular system and normal switched system.

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

This paper analyses the dynamics of dual-channel energy supply chain model with heterogeneous retailers (as regards the type of expectations’ formation). On the basis of analyzing the stabilities of four fixed points in the three-dimensional dynamic system, local stable regions of Nash equilibrium are obtained. Effects ofSon the stable regions and profit are studied. Simulation results show that the adjustment of price speed has an obvious impact on the complexity of competition. The performances of the model in different period are measured by using the index of average profit. The results show that unstable behavior in economic system is often an unfavorable outcome. So this paper discusses the application of parameters control method when the model is in chaos and then allows the oligarchs to eliminate the negative effects.

The paper focuses on studying the Noether theorem for nonholonomic systems with time delay. Firstly, the differential equations of motion for nonholonomic systems with time delay are established, which is based on the Hamilton principle with time delay and the Lagrange multiplier rules. Secondly, based upon the generalized quasi-symmetric transformations for nonconservative systems with time delay, the Noether theorem for corresponding holonomic systems is given. Finally, we obtain the Noether theorem for the nonholonomic nonconservative systems with time delay. At the end of the paper, an example is given to illustrate the application of the results.

Traditional and typical iterative learning control algorithm shows that the convergence rate of error is very low for a class of regular linear systems. A fast iterative learning control algorithm is designed to deal with this problem in this paper. The algorithm is based on the traditional P-type iterative learning control law, which increases the composition of adjacent two overlapping quantities, the tracking error of previous cycle difference signals, and the current error difference. Using convolution to promote Young inequalities proved strictly that, in terms of Lebesgue-p norm, when the number of iterations tends to infinity, the tracking error converges to zero in the system and presents the convergence condition of the algorithm. Compared with the traditional P-type iterative learning control algorithm, the proposed algorithm improves convergence speed and evades the defect using the norm metric’s tracking error. Finally, the validation of the effectiveness of the proposed algorithm is further proved by simulation results.

In this paper, we investigate the monotone variational inequality in Hilbert spaces. Based on Censor’s subgradient extragradient method, we propose two modified subgradient extragradient algorithms with self-adaptive and inertial techniques for finding the solution of the monotone variational inequality in real Hilbert spaces. Strong convergence analysis of the proposed algorithms have been obtained under some mild conditions.

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law A B C † = C † B † A † . Moreover, several equivalent statements of ℛ A A ∗ A B C = ℛ A B C and ℛ C ∗ C A B C ∗ = ℛ A B C ∗ are also deducted by the theory of operators.

The major purpose of this paper is to use the fractional integral operator in terms of extended generalized Bessel function to estimate new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions. We prove a set of inequalities for the fractional integral operator in terms of extended generalized Bessel function integrals with one and two parameters. Also, we discussed some special cases of the obtained result.

The most frequent method for modeling count responses in numerous investigations is the Poisson regression model. Under simple random sampling, this paper offers utilizing Poisson regression-based mean estimator and discovers its associated formula of the mean square error (MSE). The MSE of the proposed estimator is compared to the MSE of traditional ratio estimators in theory. As a result of these evaluations, the proposed estimator has been proven to be more efficient than traditional estimators. Furthermore, the practical results corroborated the theoretical findings.

The aim of this paper is to unify the extended Mittag-Leffler function and generalized Q function and define a unified Mittag-Leffler function. Both the extended Mittag-Leffler function and generalized Q function can be obtained from the unified Mittag-Leffler function. The Laplace, Euler beta, and Whittaker transformations are applied for this function, and generalized formulas are obtained. These formulas reproduce integral transformations of various deduced Mittag-Leffler functions and Q function. Also, the convergence of this unified Mittag-Leffler function is proved, and an associated fractional integral operator is constructed.

The generalized log-logistic distribution is especially useful for modelling survival data with variable hazard rate shapes because it extends the log-logistic distribution by adding an extra parameter to the classical distribution, resulting in greater flexibility in analyzing and modelling various data types. We derive the fundamental mathematical and statistical properties of the proposed distribution in this paper. Many well-known lifetime special submodels are included in the proposed distribution, including the Weibull, log-logistic, exponential, and Burr XII distributions. The maximum likelihood method was used to estimate the unknown parameters of the proposed distribution, and a Monte Carlo simulation study was run to assess the estimators’ performance. This distribution is significant because it can model both monotone and nonmonotone hazard rate functions, which are quite common in survival and reliability data analysis. Furthermore, the proposed distribution’s flexibility and usefulness are demonstrated in a real-world data set and compared to its submodels, the Weibull, log-logistic, and Burr XII distributions, as well as other three-parameter parametric survival distributions, such as the exponentiated Weibull distribution, the three-parameter log-normal distribution, the three-parameter (or the shifted) log-logistic distribution, the three-parameter gamma distribution, and an exponentiated Weibull distribution. The proposed distribution is plausible, according to the goodness-of-fit, log-likelihood, and information criterion values. Finally, for the data set, Bayesian inference and Gibb’s sampling performance are used to compute the approximate Bayes estimates as well as the highest posterior density credible intervals, and the convergence diagnostic techniques based on Markov chain Monte Carlo techniques were used.

The unsteady flow of Jeffrey fluid along with a vertical plate is studied in this paper. The equations of momentum, energy, and generalized Fourier’s law of thermal flux are transformed to non-dimensional form for the proper dimensionless parameters. The Prabhakar fractional operator is applied to acquire the fractional model using the constitutive equations. To obtain the generalized results for velocity and temperature distribution, Laplace transform is performed. The influences of fractional parameters α , β , γ , thermal Grashof number Gr , and non-dimensional Prandtl number Pr upon velocity and temperature distribution are presented graphically. The results are improved in the form of decay of energy and momentum equations, respectively. The new fractional parameter contains the Mittag-Leffler kernel with three fractional parameters which are responsible for better memory of the fluid properties rather than the exponential kernel appearing in the Caputo–Fabrizio fractional operator. The Prabhakar fractional operator has advantage over Caputo–Fabrizio in the real data fitting where needed.

The integration of smart grid and Internet of Things (IoT) has been facilitated with the proliferation of electric vehicles (EVs). However, due to EVs’ random mobility and different interests of energy demand, there exists a significant challenge to optimally schedule energy supply in IoT. In this paper, we propose a secure game theoretic scheme for charging EVs supplied by mobile charging stations (MCSs) in IoT, considering the dynamic renewable energy source. Firstly, the charging system composed of MCSs is developed to implement the charging service. Secondly, when the secure charging scheme of EV users is designed, the utility function of each entity in the charging system is formulated to express the trading relationship between EV users and MCSs. Moreover, with consideration of the competition and cooperation, we propose a Stackelberg game framework with sub-noncooperative optimization. Thirdly, the existence and uniqueness of both Stackelberg equilibrium (SE) and Nash equilibrium (NE) are theoretically analyzed and proved. Through the presented distributed energy scheduling algorithm, we can achieve the optimal solution. Finally, numerical results demonstrate the effectiveness and efficiency of our proposal through comparison with other existing schemes.

This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.

In this paper, we prove three supercongruences on sums of binomial coefficients conjectured by Z.-W. Sun. Let p be an odd prime and let h ∈ ℤ with 2 h − 1 ≡ 0 mod p . For a ∈ ℤ + and p a > 3 , we show that ∑ k = 0 p a − 1 h p a − 1 k 2 k k − h / 2 k ≡ 0 mod p a + 1 . Also, for any n ∈ ℤ + , we have ν p ∑ k = 0 n − 1 h n − 1 k 2 k k − h / 2 k ≥ ν p n , where ν p n denotes the p -adic order of n . For any integer m ≡ 0 mod p and positive integer n , we have 1 / p n ∑ k = 0 p n − 1 p n − 1 k 2 k k / − m k − m m − 4 / p ∑ k = 0 n − 1 n − 1 k 2 k k / − m k ∈ ℤ p , where − ˙ is the Legendre symbol and ℤ p is the ring of p -adic integers.

In this paper, a numerical study is presented for the 3D mathematical model of bioconvective boundary layer flow having nanoparticles and motile microorganisms on a curved sheet under isothermal conditions. Using an appropriate choice of similarity transformations, the problem reduces to coupled ordinary nonlinear equations, and this system is then treated with bvp4c (a MATLAB-based solver) to get the desired solution with good accuracy. The repercussion of distinct important dimensionless numbers such as thermophoresis, buoyancy ratio, Lewis number, and Brownian number on the velocity, temperature, and volume fraction of nanoparticles is presented graphically and is discussed in context with their importance on flow dynamics. Moreover, the physical impact of various parameters on motile microorganism density, the local Sherwood number, the local Nusselt number, and the local skin friction coefficients is analyzed and presented in tables. Qualitative analysis also reveals that the Brownian motion parameter, Peclet number, and Schmidt number have an inverse impact on the density microorganisms.

In this paper, a PD + controller combined with the sliding mode surface is proposed to improve system convergence rate and efficiency on control torque for satellite attitude tracking control. A sliding mode surface with the maneuver stage is constructed by the Euler axis; hence, the constant angular velocity is achieved. The PD + controller with the variable structure and auxiliary term is constructed to track the desired sliding mode surface. The Lyapunov function in PD control analysis is modified to simplify stability analysis. Model uncertainty, external disturbance, control torque constraint, and angular velocity constraint are taken into consideration, and a novel method to reduce the overshoot of angular velocity is proposed. The performance and superiority of the proposed method are demonstrated by numerical simulation results.

In the compressed sensing (CS) reconstruction algorithms, the problems of overestimation and large redundancy of candidate atoms will affect the reconstruction accuracy and probability of the algorithm when using Sparsity Adaptive Matching Pursuit (SAMP) algorithm. In this paper, we propose an improved SAMP algorithm based on a double threshold, candidate set reduction, and adaptive backtracking methods. The algorithm uses the double threshold variable step-size method to improve the accuracy of sparsity judgment and reduces the undetermined atomic candidate set in the small step stage to enhance the stability. At the same time, the sparsity estimation accuracy can be improved by combining with the backtracking method. We use a Gaussian sparse signal and a measured shock wave signal of the 15psi range sensor to verify the algorithm performance. The experimental results show that, compared with other iterative greedy algorithms, the overall stability of the DBCSAMP algorithm is the strongest. Compared with the SAMP algorithm, the estimated sparsity of the DBCSAMP algorithm is more accurate, and the reconstruction accuracy and operational efficiency of the DBCSAMP algorithm are greatly improved.

In order to resolute the micro-Doppler frequency ambiguity caused by radar pulse repetition frequency not high enough (i.e., pulse dimension does not satisfy the requirement of Nyquist sampling theorem), this paper presents a micro-Doppler frequency ambiguity resolution method based on complex-valued U-net. The echo sequence is interpolated by zeros in the pulse dimension to increase the equivalent pulse repetition frequency, so that the echo sequence after zero interpolation contains the real micro-Doppler frequency; at the same time, some new frequency components are generated. The variation law of the echo sequence frequency after zero interpolation is analyzed. Then, the echo sequence in time domain after zero interpolation is transformed to the time-frequency domain by short-time Fourier transform (STFT). Finally, the time-frequency results can be segmented by the model, which is trained by complex-valued U-net to eliminate the redundant frequencies generated by zero interpolation; thus, the reconstruction of real micro-Doppler frequency is realized. Theoretical analysis and simulation results show that the proposed method can solve the problem of micro-Doppler frequency ambiguity. Compared with fully convolution network (FCN) and fully convolution residual network (FCRN), the proposed method has better performance and robustness.