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This paper applies mechanism design to the supply chain enterprise’s pollution abatement problem with carbon tax. To maximize the government’s expected utility, an uncertain contract model is presented in the framework of principal-agent theory, where the government’s assessment of the supply chain enterprise’s carbon emission level is described as an uncertain variable. Afterwards, the equivalent model is provided to obtain the optimal contract for the uncertain pollution abatement problem. The results demonstrate that the supply chain enterprise’s optimal output decreases with the carbon emission level. Furthermore, the government’s optimal transfer payment decreases with the carbon emission level if the carbon tax is low. In contrast, if the carbon tax is high, the optimal transfer payment increases with the carbon emission level. In addition, an increase in the carbon emission level decreases the optimal utilities of both the government and the supply chain enterprise and also leads to the supply chain enterprise’s incremental marginal utility. Finally, we provide a numerical example, which illustrates the effectiveness and practicability of the proposed model.

There are many uncertainties with respect to the assessment of slope stability, and those associated with soil properties should be given particular attention. The uncertainty theory provides an alternative to treat these uncertainties using parochial cognitive sources. A novel methodology is proposed to evaluate the stability of slopes based on an uncertain set. The soil properties involved in the deterministic methods, i.e., shear strength parameters and unit weight, are expressed as uncertain sets, and their membership functions can be assumed to be triangular or trapezoidal for a homogeneous or two-layered slope, respectively. The parameter values of membership functions are designed according to the means and variations of the soil properties, and then the expected safety factor can be calculated through the operational laws. Two numerical examples including a homogeneous slope and a two-layered slope illustrate the suitability of the proposed methodology. The relationship between the variation in the safety factor and the changes in the soil properties is investigated; moreover, the determination of the parameter values of membership is also discussed.

Precision deep-hole parts are widely used in various fields of industrial production and their machining quality has a great impact on fatigue limit, geometric accuracy, and stability of products. Since roundness and straightness errors are essential technical indexes to evaluate the machining quality of deep-hole parts, accurate measurement and effective evaluation of them are of great significance to ensure the performance of related products. A multi-sensor integrated device that can measure two kinds of shape errors simultaneously was developed based on laser displacement sensor, two-dimensional position-sensitive detector, angle sensor, and laser distance sensor. Aiming at the problem of roundness error evaluation, the solution process of the control points of the minimum zone circle was optimized by calculating the distance between points and searching according to the polygon removal rule. Besides, the rotating projection method was used to evaluate the straightness error effectively. Eventually, the effectiveness of the measuring device and the shape error evaluation method was verified by experimental research.

Robustness refers to the ability of a system to maintain its original state under a continuous disturbance conditions. The deviation argument (DA) and stochastic disturbances (SDs) are enough to disrupt a system and keep it off course. Therefore, it is of great significance to explore the interval length of the deviation function and the intensity of noise to make a system remain exponentially stable. In this paper, the robust stability of Hopfield neural network (VPHNN) models based on differential algebraic systems (DAS) is studied for the first time. By using integral inequalities, expectation inequalities, and the basic control theory method, the upper bound of the interval of the deviation function and the noise intensity are found, and the system is guaranteed to remain exponentially stable under these disturbances. It is shown that as long as the deviation and disturbance of a system are within a certain range, there will be no unstable consequences. Finally, several simulation examples are used to verify the effectiveness of the approach and are described below.

Chatter is undesirable self-excited vibrations, which always lead to adverse effects during milling process. Selecting a reasonable combination of cutting parameters is an effective way to avoid chatter. Based on the mathematical model of milling process and the Floquet theory, the stable cutting area can be determined. The stability lobe diagrams (SLD) could be obtained by different interpolation methods. To study the effect of higher order interpolation methods on the accuracy and efficiency of milling stability prediction, the state item, the time-delayed item, and the periodic-coefficient item of the state-space equation are approximated by different higher order interpolation methods, respectively. The calculations show that when the state item is approximated by the third-order Hermite interpolation polynomial, third-order Newton interpolation of the time-delayed item can improve the accuracy of SLD, while higher order interpolation of periodic-coefficient item has negative effect on improving effectiveness and efficiency compared to high-order interpolation of the state item and the time-delayed item. In order to obtain the SLD of milling process more accurately, an updated full-discretization milling stability prediction method which based on the third-order Hermite-Newton interpolation polynomial approximation is proposed in this paper. By dividing the tooth passing period equally into a finite set of time intervals, the third-order Hermite interpolation polynomial and the third-order Newton interpolation polynomial are utilized in each time interval to estimate the state item and the time-delayed item, respectively. The comparison of convergence rate of the critical eigenvalues and the SLD of the proposed method between the existing methods is carried out. The results indicate that the proposed method show a faster convergence rate than that of other methods, and its SLD is more close to the ideal ones with small number of time intervals.

Slope stability assessment is a difficult geotechnical problem because of the many uncertainties involved. The uncertainties associated with soil parameters are significant, originating from limited and imprecise information, which allows for the soil parameters to be expressed as uncertain variables. The distributions of these uncertain variables are built on the basis of the structure of the soil layers of a slope, and the safety factor of slope stability is defined as a ratio of resisting and acting moments under uncertain environments. The expected value of the safety factor is therefore calculated by utilizing the operational laws of the uncertain variables. The concept of the reliability index of slope stability is also introduced in order to evaluate the risk of slope instability. Three numerical examples, including a homogeneous slope and two non-homogeneous slopes, are used to verify the proposed model; meanwhile, the results are compared with those of the Fellenius method. The numerical results indicate that the proposed model can not only improve the reliability of the calculated safety factor but also reduce the computational complexity compared to deterministic methods. Fifteen project instances taken from the literature are used to investigate the validity of reliability index, and the results illustrate that the reliability index is another useful indicator to assess the stability of a slope.

Based on third-order Newton interpolation polynomial and direct integration scheme (DIS), this paper proposes a method to generate stability lobe diagram in milling process. The dynamic model of milling process with consideration of regeneration effect is described by time periodic delay-differential equation (DDE). Then, the DDE is rewritten as state space equation by a transformation. After equally discretizing the time delay into a series of small time intervals, the state space equation of milling system is integrated on the small time interval. Both the state term and delayed term are interpolated by third-order Newton interpolation polynomial, and the periodic-coefficient matrix is interpolated by first-order Newton interpolation polynomial. The state transition matrix which reflects the discrete mapping relation of dynamic responses for current tooth pass period and immediate previous tooth pass period is obtained directly. The accuracy of the proposed method is evaluated by comparing with benchmark methods in terms of the rate of convergence. The efficiency of the proposed method is verified through the comparison of computational time with existing methods. The proposed method is proved to be an accurate and efficient method by the comparison results. The distinction between up-milling and down-milling operations is also analyzed by comparing the stability lobe diagrams for these two operations. Besides, according to the analysis of rate of convergence, the number of substitutions, which are used to convert the variables located out of the required range into the required range, may affect the results of stability lobe diagrams. Moreover, the stability lobe diagram cannot be generated by using fourth-order updated full-discretization method.

Based on orthogonal polynomial approximation scheme, this paper presents several stability prediction methods using different kinds of orthogonal polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). Firstly, this work employs the classical Legendre and Chebyshev polynomials to approximate the state term, delayed term, and periodic-coefficient matrix. With the help of direct integration scheme (DIS), the state transition matrixes which indicate the mapping relations of the dynamic response between the current tooth pass and the previous tooth pass are obtained. The stability lobe diagrams for single degree of freedom (DOF) and two DOF milling models are generated by using the Legendre and Chebyshev polynomial approximation-based methods. The rate of convergence of the Legendre and Chebyshev polynomial-based methods is compared with that of the benchmark first-order semi-discretization method (1stSDM). The comparison results indicate that the rate of convergence and the numerical stability of the Legendre and Chebyshev polynomial-based methods are both need to be improved. In order to develop new methods with high rate of convergence and numerical stability base on DIS, the monic orthogonal polynomial sequences are constructed by using Gram-Schmidt orthogonalization to approximate the state term, delayed term, and periodic-coefficient matrix. The rate of convergence and the computational efficiency of the monic orthogonal polynomial-based methods are evaluated by comparing with those of the benchmark 1stSDM. The results turn out that the monic orthogonal polynomial-based methods are advantageous in terms of the rate of convergence and numerical stability. The stability lobe diagrams for single DOF and two DOF milling models obtained by the monic orthogonal polynomial-based methods are compared with those obtained by the 1stSDM. Finally, the monic orthogonal polynomial-based methods are proved to be the effective and efficient methods to predict the milling stability.

In this paper, we consider the following inverse eigenvalue problem: to construct a real symmetric doubly arrow matrix A from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. We also give a necessary and sufficient condition in order that the constructed matrices can be nonnegative. Our results are constructive and they generate algorithmic procedures to construct such matrices.

In the paper, based on the the minimal projective bimodule resolutions of the system quiver algebras of minimal wild representation type, we calculate explicitly the dimensions of all Hochschild homology groups of the system quiver algebras of minimal wild representation type by means of combinatorics.

National Natural Science Foundation of China [10961010]; Natural Science Foundation of Jiangxi, China [2007GZS1760]; Jiangxi education office, China [GJJ08432]

In this paper, we develop a modified gradient based algorithm for solving matrix equationsAXB+CXTD=F. Different from the gradient based method introduced by Xie et al., 2010, the information generated in the first half-iterative step is fully exploited and used to construct the approximate solution. Theoretical analysis shows that the new method converges under certain assumptions. Numerical results are given to verify the efficiency of the new method.

We consider the following inverse eigenvalue problem: to construct a special kind of matrix (real symmetric doubly arrow matrix) from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. Our results are constructive and they generate algorithmic procedures to construct such matrices.

A real symmetric unipotent matrix P is said to be n×n generalized reflection matrix. A real matrix A is said to be a generalized reflexive matrix with respect to generalized reflection matrix dual (P, Q) if A = PAQ. This paper involves related inverse eigenvalue problems of generalized reflexive matrices and their optimal approximation. Necessary and sufficient conditions for the solvability of the problem are derived, the general expression of the solution is given. The optimal approximate solution is also provided.

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers ¦O to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list except for one (the Perron eigenvalue) are pure image numbers. Lets. Let ( ,bi, bi,,b i, b i)be a list of complex numbers with ¦N,bj > 0 for j =1,2,,k . A simple necessary and sufficient conditions for the existence of an entry wise nonnegative 2k +1 order matrix A with spectrum ¦O are presented , and the proof is elementary.

A real symmetric unipotent matrix P is said to be generalized anti-reflection matrix. A real matrix A is said to be a generalized anti-reflexive matrix with respect to generalized reflection matrix dual (P, Q) if A =-PAQ. In this paper, the left and right inverse eigenpairs problems for generalized anti-reflexive matrices are considered. We obtain the necessary and sufficient conditions for the solvability of the problem and we present the general expression of the solution. The related optimal approximation problem to a given matrix over the solution set is solved. In addition, a numerical algorithm and examples to solve the problem are given.

In the paper, we put forward the linear weighting method for solving a type of non-differentiable multiobjective programming problem, and use it to solve an example.

In this paper, we consider the generalized reflexive solutions of the matrix equation AX = B. The necessary and sufficient conditions for the solvability of the problem are given, moreover, a simple and eigenvector-free formula of the general solution to the matrix equation is presented using Moore-Penrose generalized inverses of the coefficient matrix A. In addition, in corresponding solution set of the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been provided.

In this paper we study a kind of inverse eigen value problem for a special kind of real symmetric matrices: the real symmetric Arrow-plus-Jacobi matrices. That is, matrices which look like arrow matrices forward and Jacobi backward, from the (p,p)station, 1 ? p ? n. We give a necessary and sufficient condition for the existence of such two matrices. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix.

In this paper we study a kind of inverse eigenvalue problem for a special kind of real symmetric matrices: the real symmetric Arrow-plus-Jacobi matrices. That is, matrices which look like arrow matrices forward and Jacobi backward, from the station, . We give a necessary and sufficient condition for the existence of such two matrices. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix.

A class of inverse eigenvalue problem is proposed for real symmetric positive definite Arrow matrices. Necessary and sufficient conditions for the existence of a unique solution of this problem, as well as the analytic formula of this solution are derived, Our results are on structive, in the sense that they generate an algorithmic procedure to construct the matrix.

A new Kantorovich type inequality for Hermite matrices is proposed in this paper. It holds for the invertible Hermite matrices and provides refinements of the classical results. Elementary methods suffice to prove the inequality.

A class of inverse eigenvalue problem is proposed for real symmetric positive definite Arrow matrices. Necessary and sufficient conditions for the existence of a unique solution of this problem, as well as the analytic formula of this solution are derived.

Efficient methods based on Galerkin projections and the Wiener-Askey polynomial chaos are constructed to solve the steady/unsteady state diffusion problems with uncertainty. Some numerical examples are used to confirm the efficiency of the methods. The exponential convergence of the methods is demonstrated for each problem.

One purpose of a well test is to get reservoir parameters. The problem is how to analyze the transient well test parameters and find relationship between them. We present an effective method to analyze the correlation between the well test interpretation parameters, which is a combination of grey correlation analysis and fuzzy clustering. In particular, we use grey correlation analysis to describe the correlation between parameters, and use fuzzy clustering method to divide them into different categories according to the degree of the correlation between all parameters. A simple example shows the algorithmic strategies to implementing them. In the example, we consider six parameters and each of them contains 92 data points. The example presented in this paper shows that it's a flexible and cost-effective quantitative analysis method for transient test parameters analysis, it is particularly suitable for analyze relevance of time series data.

We present several matrix Kantorovich-type inequalities, which improve the results obtained in Liu and Neudecker (1996). Elementary methods suffice to prove the inequalities.