Formalization of implication based fuzzy reasoning method

Fuzzy reasoning includes a number of important inference methods for addressing uncertainty. This line of fuzzy reasoning forms a common logical foundation in various fields, such as fuzzy logic control and artificial intelligence. The full implication triple I method (a method only based on implication, TI method for short) for fuzzy reasoning is proposed in 1999 to improve the popular CRI method (a hybrid method based on implication and composition). The current paper delves further into the TI method, and a sound logical foundation is set for the TI method based on the monoidal t-norm based logical system MTL.     

等腰三角形Mindlin板的自由振动分析

提出了一种新方法来对基于 Mindlin剪切变形理论的等腰三角形板进行自由振动分析 .此方法采用了一种新的基函数并利用 pb-2 Rayleigh-Ritz边界函数得到了一种新型的 Ritz方法 .这种方法的有效性通过收敛性和对比性分析得到了证实 .数值结果表明此方法相当精确有效 .     

Finding roots of a real polynomial simultaneously by means of Bairstow's method

Aberth's method for finding the roots of a polynomial was shown to be robust. However, complex arithmetic is needed in this method even if the polynomial is real, because it starts with complex initial approximations. A novel method is proposed for real polynomials that does not require any complex arithmetic within iterations. It is based on the observation that Aberth's method is a systematic use of Newton's method. The analogous technique is then applied to Bairstow's procedure in the proposed method. As a result, the method needs half the computations per iteration than Aberth's method. Numerical experiments showed that the new method exhibited a competitive overall performance for the test polynomials.     

A coding method for a sequent calculus of propositional logic

The paper deals with a coding method for a sequent calculus of the propositional logic. The method is based on the sequent calculus. It allows us to determine if a formula is derivable in the calculus without constructing a derivation tree. The main advantage of the coding method is its compactness in comparison with derivation trees of the sequent calculus. The coding method can be used as a decision procedure for the propositional logic.     

Amplitude–shape approximation as an extension of separation of variables

Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd.     

On the final steps of Newton and higher order methods

The Newton method is one of the most used methods for solving nonlinear system of equations when the Jacobian matrix is nonsingular. The method converges to a solution with Q-order two for initial points sufficiently close to the solution. The method of Halley and the method of Chebyshev are among methods that have local and cubic rate of convergence. Combining these methods with a backtracking and curvilinear strategy for unconstrained optimization problems these methods have been shown to be globally convergent. The backtracking forces a strict decrease of the function of the unconstrained optimization problem. It is shown that no damping of the step in the backtracking routine is needed close to a strict local minimizer and the global method behaves as a local method. The local behavior for the unconstrained optimization problem is investigated by considering problems with two unknowns and it is shown that there are no significant differences in the region where the global method turn into a local method for second and third order methods. Further, the final steps to reach a predefined tolerance are investigated. It is shown that the region where the higher order methods terminate in one or two iteration is significantly larger than the corresponding region for Newton’s method.     

The method of alternating projections and the method of subspace corrections in Hilbert space

A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating projections is an iterative algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces by alternatively computing the best approximations from the individual subspaces which make up the intersection. The method of subspace corrections is an iterative algorithm for finding the solution of a linear equation in a Hilbert space by approximately solving equations restricted on a number of closed subspaces which make up the entire space. The new identity given in the paper provides a sharpest possible estimate for the rate of convergence of these algorithms. It is also proved in the paper that the method of alternating projections is essentially equivalent to the method of subspace corrections. Some simple examples of multigrid and domain decomposition methods are given to illustrate the application of the new identity.

     

扰动的Couette－Poiseuille流的特征值的计算

本文考虑的问题是二维粘性渠流．对０到２０００之间的雷诺数，计算了平稳扰动的Ｃｏｕｅｔｔｅ－Ｐｏｉｓｅｕｉｌｌｅ流的下游特征值，其特征方程类似于Ｏｒｒ－Ｓｏｍｍｅｒｆｅｌｄ方程。所用的方法是谱方法和初值方法（复合矩阵方法）．就几种有趣的流量，给出了相应的特征值的计算结果。这些特征值确定了扰动的衰减率．     

Reconstruction of boundary regimes in the inverse problem of thermal convection of a high-viscosity fluid

A problem of reconstruction of boundary regimes in a model for free convection of a high-viscosity fluid is considered. A variational method and a quasi-inversion method are suggested for solving the problem in question. The variational method is based on the reduction of the original inverse problem to some equivalent variational minimum problem for an appropriate objective functional and solving this problem by a gradient method. When realizing the gradient method for finding a minimizing element of the objective functional, an iterative process actually reducing the original problem to a series of direct well-posed problems is organized. For the quasi-inversion method, the original differential model is modified by means of introducing special additional differential terms of higher order with small parameters as coefficients. The new perturbed problem is well-posed; this allows one to solve this problem by standard methods. An appropriate choice of small parameters gives an opportunity to obtain acceptable qualitative and quantitative results in solving the inverse problem. A comparison of the methods suggested for solving the inverse problem is made with the use of model examples.     

Solution of the stochastic boundary-value problem of elasticity theory for composites with disordered structures in the correlative approximation of the method of quasi-periodic components

The method of quasi-periodic components, a new method of statistical mechanics of composites, is presented. In correlative approximation, this method makes it possible to reduce the problem of solving the stochastic boundary-value problem of elasticity theory for composites with disordered structures to a simpler problem for an individual cell with one inclusion in a homogeneous elastic medium. The generalized volumetric forces on the cell boundary take into account the random distribution of inclusions in the composite fragment studied. The problem for one inclusion cell can be solved, for example, by the boundary element method. The numerical solution in the correlative approximation of the method of quasi-periodic components for inhomogeneous interphase stress fields in the matrix of an epoxy composite containing randomly distributed unidirectional fibers is given. A comparison with the known analytical solutions obtained by other authors confirms the high accuracy of the correlative approximation.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 465–478, July–August, 1999.