The exact solutions of the Riemann problems for the two different perturbed macroscopic production models are considered and constructed respectively for all the possible cases. It is found that the asymptotic limits of solutions to the Riemann problem for the first kind of perturbed macroscopic production model do not coverage to those of the pressureless gas dynamics model, because the delta shock wave in the limiting situation has different propagation speed and strength from those for the pressureless gas dynamics model. In order to remedy it, the second kind of perturbed macroscopic production model is proposed, whose asymptotic limits of Riemann solutions are identical with those of the pressureless gas dynamics model.

We present the mathematical analysis of the Isolation Random Forest Method (IRF Method) for anomaly detection, introduced in {\sc F.~T. Liu, K.~M. Ting, Z.-H. Zhou:}, {\it Isolation-based anomaly detection}, TKDD 6 (2012) 3:1–3:39. We prove that the IRF space can be endowed with a probability induced by the Isolation Tree algorithm (iTree). In this setting, the convergence of the IRF method is proved, using the Law of Large Numbers. A couple of counterexamples are presented to show that the method is inconclusive and no certificate of quality can be given, when using it as a means to detect anomalies. Hence, an alternative version of the method is proposed whose mathematical foundation is fully justified. Furthermore, a criterion for choosing the number of sampled trees needed to guarantee confidence intervals of the numerical results is presented. Finally, numerical experiments are presented to compare the performance of the classic method with the proposed one.

In wireless networked iterative learning control systems, the controller is separated from the plant, and additive noises, random delays and data dropouts arise in both sensor-to-controller and controller-to-actuator channels. In order to guarantee the convergence performance of such systems with the effect of these uncertainties, an input filter is designed based on a proportional iterative learning controller, so that updated inputs can be filtered at the actuator side. Specifically, two data transmission processes are first developed to describe the mix of those uncertainties in both channels by Bernoulli and Gaussian distributed variables with known distributions. Based on state augmentation, the two data transmission processes are further combined with the iterative learning process of controllers to build a unified filtering model. According to this unified model, an optimal filter is designed via the projection theory and implemented at the actuator side to filter the updated inputs in iteration domain. Moreover, the convergence performance of the filtering error covariance matrix is proved theoretically. Finally, some numerical results are given to illustrate the effectiveness of the proposed method.

In this paper, the anti-plane shear motion of an asymmetric three-layered inhomogeneous elastic plate has been examined. An asymptotic approach is employed for the present investigation. Both the generalized and unified dispersion relations within the long-wave low-frequency range have been determined. The obtained unified dispersion relation is investigated taking into account the recently analyzed material contrast for layered plate with mixed stiff-soft layers of different material properties. Finally, we make comparison with symmetric plate being a special case of the asymmetric plate under consideration in the end.

The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms: e.g., transforms appearing in the theory of integrable systems.

In this paper, we study an SIR epidemic model with ratio dependent incident rate function. We explore the impact of vaccination and treatment on the transmission dynamics of the disease. The treatment control strategies depend on the availability of maximal treatment capacity: treatment rate is constant when the number of infected individuals is greater than the maximal capacity of treatment and proportional to the number of infected individuals when the number of infected individuals is less than the maximal capacity of treatment. The existence and stability of the endemic equilibria are governed by the basic reproduction number and treatment control strategies. By carrying out rigorous mathematical analysis and numerical evaluations, it has been shown that (1) the sufficiently large value of the preventive vaccination rate can control the spread of disease, (2) a threshold level of the psychological (or inhibitory) effects in the incidence rate function is enough to decrease the infective population. Model system also undergoes a transcritical and a saddle-node bifurcation with respect to disease contact rate. In the presence of treatment strategies, system have multiple endemic equilibria and undergoes a backward bifurcation. The number of infected individuals decreases with respect to maximal treatment capacity and disease dies out from the system for large capacity of the treatment when constant treatment strategy is applied. Further, it is also found that the spread of disease can be suppressed by increasing treatment rate. Sensitivity analysis shows that the transmission and treatment rates are most sensitive parameters on the model system.

In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models are studied for their mathematical relevance and the numerical simulations which exhibit at most resemblance with turbulent flows. One of the mathematically well studied shell model of turbulence is called sabra shell model. This work concerns with two important issues related to shell model namely feedback stabilization and robust stabilization. We first address stabilization problem related to sabra shell model of turbulence and prove that the system can be stabilized via finite dimensional controller. Thus only finitely many modes of the shell model would suffice to stabilize the system. Later we study robust stabilization in the presence of the unknown disturbance and corresponding control problem by solving an infinite time horizon max-min control problem. We first prove the $H^ \infty$ stabilization of the associated linearized system and characterize the optimal control in terms of a feedback operator by solving an algebraic riccati equation. Using the same riccati operator we establish asymptotic stability of the nonlinear system.

During earthquakes, the liquid-filled storage tank generates hydrodynamic pressures, in addition to hydrostatic pressure, on the solid domain of the tank. The theoretical background of hydrodynamic pressure analysis, as well as the numerical simulation of the liquid-filled cylindrical concrete tank, is the focus of this paper. The Finite Element Method (FEM) modeling, along with Arbitrary Lagrangian-Eulerian and Fluid-Structure Interactions formulation, are used for simulating the seismic response of cylindrical concrete liquid-filled tank, fixed to the rigid foundation. The Loma Prieta accelerogram is utilized for recording the seismic ground motion. In the numerical study, two states are observed: 1) static condition where only hydrostatic pressure acts, and 2) seismic excitation where hydrodynamic pressure occurs. When exposed to an earthquake situation, the tank liquid gives the total pressure of the liquid domain. The dynamic analysis considers the pressure response of the liquid domain, as well as the stress response of the solid domain of the coupled system, i.e., liquid-filled cylindrical concrete tank.

Polynomial interpolation with equidistant nodes is notoriously unreliable due to the Runge phenomenon, and is also numerically ill-conditioned. By taking advantage of the optimality of the interpolation processes on Chebyshev nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev points, which are selected from a satisfactory uniform grid, for polynomial interpolation. Yet, little literature exists on the computation of these points. In this study, we investigate the properties of the mock-Chebyshev nodes and propose a subsetting method for constructing mock-Chebyshev grids. Moreover, we provide a precise formula for the cardinality of a satisfactory uniform grid. Some numerical experiments using the points obtained by the method are given to show the effectiveness of the proposed method and numerical results are also provided.

We present the method for detection of particle groups involved in collective motion based on network analysis. Knowing the positions and velocities of individual particles, a “velocity similarity graph’‘ is built, where the graph vertices represent the particles. The vertex pairs are connected by the edge if the distance between the respective particles is small enough. The edge weight is calculated to be inversely proportional to the difference in the respective particle velocities, i.e., the vertex pairs representing nearby particles having similar velocities are connected by edges of larger weight. If a group of particles moves in a coordinated matter, the particles in this group will have similar velocities, therefore, the corresponding vertices in the graph will be connected by edges of larger weight in the representing graph. Having produced the velocity similarity graph, identification of particle groups becomes equivalent to the problem of “community detection” in graph analysis. The algorithms and techniques developed for community detection in graphs can be thereby applied for identification of particle groups involved in coordinated motion in granular matter. We illustrate this approach by an example of granular media filled in a rotating cylinder.

In this paper, we further Meirong Zhang, et al.’s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied.

In this research study, fuzzy fractional differential equations in presence of the Atangana-Baleanu-Caputo differential operators are analytically and numerically treated using extended reproducing Kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions; analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three-dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

In this paper, we consider two dimensional inverse problem for a fractional diffusion equation. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. Also the stability estimate is obtained.

Nickel alloys are widely used in the production of gas turbine parts. The alloys show resistance to mechanical and chemical degradation under severe long-term stress and high temperatures. One of the major mechanical properties of the alloys is the high-temperature rupture strength, which is measured after a specimen is heated to a certain temperature and held for a certain time considering deformation. Determining the influence of certain elements on the properties of an alloy is a complex scientific and engineering problem that affects the time and cost of developing new materials. Simulation is a great chance to cut costs. In this paper, we predict a high-temperature strength based on the composition of refractory elements in alloys using a deep learning artificial neural network. We build the model based on prior knowledge of the composition of the alloys, information on the role of alloying elements, type of crystallization, test temperature and time, and the tensile strength. Successful simulation results show the applicability of this method in practice.

In this paper, we consider the artificial neural networks for solving the differential equation with boundary layer, in which the gradient of the solution changes sharply near the boundary layer. The solution of the boundary layer problems poses a huge challenge to both traditional numerical methods and artificial neural network methods. By theoretical analyzing the changing rate of the weights of first hidden layer near the boundary layer, a mapping strategy is added in traditional neural network to improve the convergence of the loss function. Numerical examples are carried out for the 1D and 2D convection-diffusion equation with boundary layer. The results demonstrate that the modified neural networks significantly improve the ability in approximating the solutions with sharp gradient.

Changing and moving toward online shopping has made it necessary to customize customers' needs and provide them more selective options. The buyers search the products' features before deciding to purchase items. The recommender systems facilitate the searching task for customers via narrowing down the search space within the specific products that align the customer needs. Clustering, as a typical machine learning approach, is applied in recommender systems. As an information filtering method, a recommender system clusters user's data to indicate the required factors for more accurate predictions by calculating the similarity between members of a cluster. In this study, using the Gaussian mixture model clustering and considering the scores distance and the value of scores in the Pearson correlation coefficient, a new method is introduced for predicting scores in machine learning recommender systems. To study the proposed method's performance, a Movie Lens data set is evaluated, and the results are compared to some other recommender systems, including the Pearson correlation coefficients similarity criteria, K-means, and fuzzy C-means algorithms. The simulation results indicate that our method has less error than others by increasing the number of neighbors. The results also illustrate that when the number of users increases, the proposed method's accuracy will increase. The reason is that the Gaussian mixture clustering chooses similar users and considers the scores distance in choosing similar neighbors to the active user.

The aim of this work is to prove analytically the existence of symmetric periodic solutions of the family of Hamiltonian systems with Hamiltonian function H(q_1,q_2,p_1,p_2)= 1/2(q_1^2+p_1^2)+1/2(q_2^2+p_2^2)+ a q_1^4+b q_1^2q_2^2+c \q_2^4 with three real parameters a, b and c. Moreover, we characterize the stability of these periodic solutions as function of the parameters. Also, we find a first-order analytical approach of these symmetric periodic solutions. We emphasize that these families of periodic solutions are different from those that exist in the literature.