An epidemic model that describes the dynamics of the spread of infectious diseases is proposed. Two different types of infectious diseases that spread through both horizontal and vertical transmission in the host population are considered. The basic reproduction number $R_0$ is determined. The local and the global stability of all possible equilibrium points are achieved. The local bifurcation analysis and Hopf bifurcation analysis for the four-dimensional epidemic model are studied. Numerical simulations are used to confirm our obtained analytical results.

Interval type 2 fuzzy numbers are a special kind of type 2 fuzzy numbers. These numbers can be described by triangular and trapezoidal shapes. In this paper, first, perfectly normal interval type 2 trapezoidal fuzzy numbers with their left-hand and right-hand spreads and their core have been introduced, which are normal and convex; then a new type of fuzzy arithmetic operations for perfectly normal interval type 2 trapezoidal fuzzy numbers has been proposed based on the extension principle of normal type 1 trapezoidal fuzzy numbers. Moreover, in this proposal, linear programming problems with resources and technology coefficients are perfectly normal interval type 2 fuzzy numbers. To solve this kind of fuzzy linear programming problems, a method based on the degree of satisfaction (or possibility degree) of the constraints has been introduced. In this method the fulfillment of the constraints can be measured with the help of ranking method of fuzzy numbers. Optimal solution is obtained at different degree of satisfaction by using Barnes algorithm with the help of MATLAB. Finally, the optimal solution procedure is illustrated with numerical example.

Chemical entropy generation and magnetohydrodynamic effects on the unsteady heat and fluid flow through a porous medium have been numerically investigated. The entropy generation due to the use of a magnetic field and porous medium effects on heat transfer, fluid friction, and mass transfer have been analyzed numerically. Using a similarity transformation, the governing equations of continuity, momentum, and energy and concentration equations, of nonlinear system, were reduced to a set of ordinary differential equations and solved numerically. The effects of unsteadiness parameter, magnetic field parameter, porosity parameter, heat generation/absorption parameter, Lewis number, chemical reaction parameter, and Brinkman number parameter on the velocity, the temperature, the concentration, and the entropy generation rates profiles were investigated and the results were presented graphically.

Steady flow of a couple-stress fluid in constricted tapered artery has been studied under the effects of transverse magnetic field, moving catheter, and slip velocity. With the help of Bessel’s functions, analytic expressions for axial velocity, flow rate, impedance, and wall shear stress have been obtained. It is of interest to note that these solutions can be used for different types of fluid flow in tubes and not only the case of blood. The effects of various geometric parameters, the parameters arising out of the fluid considered and the magnetic field, are discussed by considering the slip velocity, the catheter velocity, and tapering angle. The study of the above model is very important as it has direct applications in the treatment of cardiovascular diseases.

The main purpose of this paper is to study the convergence properties of Generalized Fibonacci Sequences and the series of partial sums associated with them. When the proper values of an $s\times s$ real matrix $A$ are real and different, we give a necessary and sufficient condition for the convergence of the matrix sequence $A,A^{\mathrm{2}},A^{\mathrm{3}},\dots$ to a matrix $B$.

This paper presents a control problem for the optimization of the production and setup activities of an industrial system operating in an uncertain environment. This system is subject to random disturbances (breakdowns and repairs). These disturbances can engender stock shortages. The considered industrial system represents a well-known production context in industry and consists of a machine producing two types of products. In order to switch production from one product type to another, a time factor and a reconfiguration cost for the machine are associated with the setup activities. The parts production rates and the setup strategies are the decision variables which influence the inventory and the capacity of the system. The objective of the study is to find the production and setup policies which minimize the setup and inventory costs, as well as those associated with shortages. A modeling approach based on stochastic optimal control theory and a numerical algorithm used to solve the obtained optimality conditions are presented. The contribution of the paper, for industrial systems not studied in the literature, is illustrated through a numerical example and a comparative study.

Designing the jet ejector optimally is a challenging task and has a great impact on industrial applications. Three different sets of nozzles (namely, 1, 3, and 5) inside the jet ejector are compared in this study by using numerical simulations. More precisely, dynamics of bubble coalescence and breakup in the multinozzle jet ejectors are studied by means of Computational Fluid Dynamics (CFD). The population balance approach is used for the gas phase such that different bubble size groups are included in CFD and the number densities of each of them are predicted in CFD simulations. Here, commercial CFD software ANSYS Fluent 14.0 is used. The realizable $k$-$\epsilon$ turbulence model is used in CFD code in three-dimensional computational domains. It is clear that Reynolds-Averaged Navier-Stokes (RANS) models have their limitations, but on the other hand, turbulence modeling is not the key issue in this study and we can assume that the RANS models can predict turbulence of the carrying phase accurately enough. In order to validate our numerical predictions, results of one, three, and five nozzles are compared to laboratory experiments data for Cl₂-NaOH system. Predicted gas volume fractions, bubble size distributions, and resulting number densities of the different bubble size groups as well as the interfacial area concentrations are in good agreement with experimental results.

We perturb the differential system ${\dot{x}}_{\mathrm{1}}=-x_{\mathrm{2}}(\mathrm{1}+x_{\mathrm{1}})$, ${\dot{x}}_{\mathrm{2}}=x_{\mathrm{1}}(\mathrm{1}+x_{\mathrm{1}})$, and ${\dot{x}}_k=\mathrm{0}$ for $k=\mathrm{3},\dots ,d$ inside the class of all polynomial differential systems of degree $n$ in ${\mathbb{R}}^d$, and we prove that at most $n^{d-\mathrm{1}}$ limit cycles can be obtained for the perturbed system using the first-order averaging theory.

Chaotic convection in a viscoelastic fluid saturated porous layer, heated from below, is studied by using Oldroyd’s type constituting relation and in the presence of an internal heat source. A modified Darcy law is used in the momentum equation, and a heat source term has been considered in energy equation. An autonomous system of fourth-order differential equations has been deduced by using a truncated Fourier series. Effect of internal heat generation on chaotic convection has been investigated. The asymptotic behavior can be stationary, periodic, or chaotic, depending upon the flow parameters. Construction of four-scroll, or “two-butterfly,” and chaotic attractor has been examined.

Factor analysis models with continuous and ordinal responses are a useful tool for assessing relations between the latent variables and mixed observed responses. These models have been successfully applied to many different fields, including behavioral, educational, and social-psychological sciences. However, within the Bayesian analysis framework, most developments are constrained within parametric families, of which the particular distributions are specified for the parameters of interest. This leads to difficulty in dealing with outliers and/or distribution deviations. In this paper, we propose a Bayesian semiparametric modeling for factor analysis model with continuous and ordinal variables. A truncated stick-breaking prior is used to model the distributions of the intercept and/or covariance structural parameters. Bayesian posterior analysis is carried out through the simulation-based method. Blocked Gibbs sampler is implemented to draw observations from the complicated posterior. For model selection, the logarithm of pseudomarginal likelihood is developed to compare the competing models. Empirical results are presented to illustrate the application of the methodology.

This paper first presents a new generally perturbed compressed sensing (CS) model $y=(A+E)(x+u)+e$, which incorporated a general nonzero perturbation $E$ into sensing matrix $A$ and a noise $u$ into signal $x$ simultaneously based on the standard CS model $y=Ax+e$ and is called noise folding in completely perturbed CS model. Our construction mainly will whiten the new proposed CS model and explore in restricted isometry property ($\mathrm{R}\mathrm{I}\mathrm{P}$) and coherence of the new CS model under some conditions. Finally, we use OMP to give a numerical simulation which shows that our model is feasible although the recovered value of signal is not exact compared with original signal because of measurement noise $e$, signal noise $u$, and perturbation $E$ involved.

Our main endeavor is to investigate the effect of the uncertain market size in the second period on the pricing strategies in the remanufacturing market. Observing the previous research, we find that the market size in the second period is always supposed to be certain. However, there is substantial empirical and experimental evidence that the market size in reality deviates from this assumption. In fact, though the market size in the first period is definitized, it is difficult to confirm the change of the market scale in the second period, since this change is affected by all kinds of elements, such as the awareness of environmental protection, some consumers’ psychological factors, and the related governments’ policies. Hence, we pay attention to the case in which the change rate of the market scale in the second period is random variable. We suppose that this rate satisfies uniform distribution on $[\mathrm{0,2}]$. Underlying this assumption, we further provide an insight into the game-playing relationship between original equipment manufacturers (OEMs) and remanufacturers. Moreover, we delicately and subtly incorporate the game theory, stochastic analysis, adversarial risk analysis (ARA), and optimization methods into the pricing strategies in the remanufacturing market. Last but not least, considerable efforts and attempts have been made to subtly test the sensitivity of an optimal solution to the different parameters.

The variational homotopy perturbation method VHPM is used for solving $n$-dimensional Burgers’ system. Some examples are examined to validate that the method reduced the calculation size, treating the difficulty of nonlinear term and the accuracy.

A networked control system (NCS) is a control system where components such as plants and controllers are connected through communication networks. Self-triggered control is well known as one of the control methods in NCSs and is a control method that for sampled-data control systems both the control input and the aperiodic sampling interval (i.e., the transmission interval) are computed simultaneously. In this paper, a self-triggered model predictive control (MPC) method for discrete-time linear systems with disturbances is proposed. In the conventional MPC method, the first one of the control input sequence obtained by solving the finite-time optimal control problem is sent and applied to the plant. In the proposed method, the first some elements of the control input sequence obtained are sent to the plant, and each element is sequentially applied to the plant. The number of elements is decided according to the effect of disturbances. In other words, transmission intervals can be controlled. Finally, the effectiveness of the proposed method is shown by numerical simulations.

We study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings. We obtain some important properties of our class of mappings including the demiclosedness property and the closedness and convexity of the fixed point set. We then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP for certain class of these mappings.

We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value function $V$ often does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such cases $V$ can be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.

We study the inextensible flows of curves in 3-dimensional Euclidean space ${\mathbb{R}}^{\mathrm{3}}$. The main purpose of this paper is constructing and plotting the surfaces that are generated from the motion of inextensible curves in ${\mathbb{R}}^{\mathrm{3}}$. Also, we study some geometric properties of those surfaces. We give some examples about the inextensible flows of curves in ${\mathbb{R}}^{\mathrm{3}}$ and we determine the curves from their intrinsic equations (curvature and torsion). Finally, we determine and plot the surfaces that are generated by the motion of those curves by using Mathematica 7.

Creep process of linear viscoelastic materials is described by the integral equation of Boltzmann-Volterra in which creep kernel is approximated by Rabotnov’s fractional exponential function. The creep equation contains four unknown parameters: $\alpha$, singularity parameter; $\beta$, fading parameter; $\lambda$, rheological parameter; and ${\epsilon }_0$, conditionally instantaneous strain. Two-stage determination method of creep parameters is offered. At the first stage, taking into account weak singularity properties of Abel’s function at the initial moment of loading, parameters ${\epsilon }_0$ and $\alpha$ are determined. At the second stage, using already known parameters ${\epsilon }_0$ and $\alpha$, parameters $\beta$ and $\lambda$ are determined. Analytical expressions for calculating these parameters are obtained. An accuracy evaluation of the offered method with using experimentally determined creep strains of material Nylon 6 and asphalt concrete showed its high accuracy.

We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We have managed to obtain a particular solution of this equation in an explicit form.

This paper discusses the construction of new $C^{\mathrm{2}}$ rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parameters ${\mathrm{\alpha }}_i$, ${\mathrm{\beta }}_i$, and ${\gamma }_i$. The sufficient conditions for the positivity are derived on one parameter ${\mathrm{\gamma }}_i$ while the other two parameters ${\mathrm{\alpha }}_i$ and ${\mathrm{\beta }}_i$ are free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation with $C^{\mathrm{2}}$ continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion and $C^{\mathrm{2}}$ continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivatives $d_i$, $i=\mathrm{1},\dots ,n-\mathrm{1}$. Comparisons with existing schemes also have been done in detail. From all presented numerical results the new $C^{\mathrm{2}}$ rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated is $f\left(t\right)\in C^{\mathrm{3}}\left[t_{\mathrm{0}},t_n\right]$ is also investigated in detail.

Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex body $K$ is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on $K$. These regions are bounded by arcs of involutes of $\partial K$ and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.

A numerical investigation is made to determine the occurrence of the multiple solutions of MHD Casson fluid in a porous channel. Governing partial differential equation of the proposed problem converted into nonlinear ordinary differential equations by using similarity transformation. Numerical technique known as shooting method is used to investigate the existence of the multiple solutions for the variations of different parameters. Effects of physical parameters on velocity profile, temperature, concentration, and skin friction are presented in pictorial and tabulation representation.

A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms and $L^2$-norm), are investigated. Numerical test is provided.

We derive closed formulas for the energy of circulant graphs generated by $\mathrm{1}$ and $\gamma$, where $\gamma ⩾\mathrm{2}$ is an integer. We also find a formula for the energy of the complete graph without a Hamilton cycle.

We propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponential-power moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort.

This paper addresses a single-machine scheduling problem with sequence-dependent family setup times. In this problem the jobs are classified into families according to their similarity characteristics. Setup times are required on each occasion when the machine switches from processing jobs in one family to jobs in another family. The performance measure to be minimized is the total tardiness with respect to the given due dates of the jobs. The problem is classified as $\mathcal{N}\mathcal{P}$-hard in the ordinary sense. Since the computational complexity associated with the mathematical formulation of the problem makes it difficult for optimization solvers to deal with large-sized instances in reasonable solution time, efficient heuristic algorithms are needed to obtain near-optimal solutions. In this work we propose three heuristics based on the Iterated Local Search (ILS) metaheuristic. The first heuristic is a basic ILS, the second uses a dynamic perturbation size, and the third uses a Path Relinking (PR) technique as an intensification strategy. We carry out comprehensive computational and statistical experiments in order to analyze the performance of the proposed heuristics. The computational experiments show that the ILS heuristics outperform a genetic algorithm proposed in the literature. The ILS heuristic with dynamic perturbation size and PR intensification has a superior performance compared to other heuristics.

Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and low-rank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as statistic analysis and system control. This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints. We apply the widely used alternating direction method of multipliers to solve the model and get a novel algorithm. The applicability and efficiency of the new algorithm are demonstrated in numerical experiments. Recovery results show that our algorithm is helpful.

This paper shows how to value multiasset options analytically in a modeling framework that combines both continuous and discontinuous variations in the underlying equity or foreign exchange processes and a stochastic, two-factor yield curve. All correlations are taken into account, between the factors driving the yield curve, between fixed income and equity as asset classes, and between the individual equity assets themselves. The valuation method is applied to three of the most popular two-asset options.

Several authors have examined the long swings hypothesis in exchange rates using a two-state Markov switching model. This study developed a model to investigate long swings hypothesis in currencies which may exhibit a $k$-state $(k\ge \mathrm{2})$ pattern. The proposed model was then applied to euros, British pounds, Japanese yen, and Nigerian naira. Specification measures such as AIC, BIC, and HIC favoured a three-state pattern in Nigerian naira but a two-state one in the other three currencies. For the period January 2004 to May 2016, empirical results suggested the presence of asymmetric swings in naira and yen and long swings in euros and pounds. In addition, taking $\mathrm{0.5}$ as the benchmark for smoothing probabilities, choice models provided a clear reading of the cycle in a manner that is consistent with the realities of the movements in corresponding exchange rate series.

We obtain the sharp bounds for the spectral radius of a nonnegative matrix and then obtain some known results or new results by applying these bounds to a graph or a digraph and revise and improve two known results.

In order to classify a finite group using its elements orders, the order classes are defined. This partition determines the number of elements for each order. The aim of this paper is to find the order classes of 2-generator $p$-groups of class 2. The results obtained here are supported by Groups, Algorithm and Programming (GAP).

The vec operator transforms a matrix to a column vector by stacking each column on top of the next. It is useful to write the vec of a block-structured matrix in terms of the vec operator applied to each of its component blocks. We derive a simple formula for doing so, which applies regardless of whether the blocks are of the same or of different sizes.

A graceful labeling of a tree $T$ with $n$ edges is a bijection $f:V(T)\to \{\mathrm{0,1},\mathrm{2},\dots ,n\}$ such that $\{|f(u)-f(v)|:uv\in E(T)\}$ equal to $\{\mathrm{1,2},\dots ,n\}$. A spider graph is a tree with at most one vertex of degree greater than $\mathrm{2}$. We show that all spider graphs with at most four legs of lengths greater than one admit graceful labeling.

We propose a new image watermarking scheme based on the real SVD and Arnold scrambling to embed a color watermarking image into a color host image. Before embedding watermark, the color watermark image $W$ with size of $M\times M$ is scrambled by Arnold transformation to obtain a meaningless image $\tilde{W}$. Then, the color host image $A$ with size of $N\times N$ is divided into nonoverlapping $N/M\times N/M$ pixel blocks. In each $(i,j)$ pixel block $A_{i,j}$, we form a real matrix $C_{i,j}$ with the red, green, and blue components of $A_{i,j}$ and perform the SVD of $C_{i,j}$. We then replace the three smallest singular values of $C_{i,j}$ by the red, green, and blue values of ${\tilde{W}}_{ij}$ with scaling factor, to form a new watermarked host image ${\tilde{A}}_{ij}$. With the reserve procedure, we can extract the watermark from the watermarked host image. In the process of the algorithm, we only need to perform real number algebra operations, which have very low computational complexity and are more effective than the one using the quaternion SVD of color image.

This paper presents a theoretical analysis and derives the amplifier output noise power spectral density result in a closed form when the input to the amplifier is a band limited Gaussian noise. From the computed power spectral density the NPR is evaluated by a simple subtraction. The method can be applied to any amplifier with known input-output characteristics. The method may be applied to analyze various other important characteristics of the nonlinear amplifier such as spectral regrowth that refers to the spreading of the signal bandwidth when a band limited signal is inputted to the nonlinear amplifier. The paper presents numerical results on the NPR as a function of the noise bandwidth, depth level of the notch, and the output power back-off obtained from the analysis presented in the paper.

Through an Alexandrov-Fenchel inequality, we establish the general Brunn-Minkowski inequality. Then we obtain the uniqueness of solutions to a nonlinear elliptic Hessian equation on ${\mathbb{S}}^n$.

The $N$ person games in which each player maximizes his payoff function are considered. We have studied an interesting question for the cooperative game theory about the usefulness of uniting the $N$ players in a union. The aim of such cooperation is for each player to get a positive increase to his guaranteed payoff. We have obtained some effective sufficient conditions under which the joining of the players in union is useful for each player. The linear case, specially, is being considered. In the second part of the paper, we have studied the question about the usefulness of cooperation of the $N$ players in the presence of the $(N+\mathrm{1})$th player, an ill-intentioned destructive player, whose whole aim is not to win but to harm each player individually, and also the union of these players, for example, global terrorism. It should be noted that the considered situation in the second part is related to A. V. Kryazhimskiy’s talk delivered in the summer of 2014. We obtain constructive conditions under which the union of the players is beneficial in this situation as well.