Novel method based on ant colony opti mization for solving ill-conditioned linear systems of equations

A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations.

Interval Criteria for Oscillation of Forced Second-order Half-linear Differential Equations

By employing a generalized Riccati technique and an integral averaging tech-nique, new interval oscillation criteria are established for the forced second-order half-lineardifferential equation [r(t)|x′ (t)|α-1x′ (t)]′ + q(t)|x(t)|α-1x(t) = e(t).

A Mathod to Solve Systems of Fuzzy Linear Equations

Many systems of fuzzy linear equations do not have solutions when the solution concept is based on α cuts and interval arithmetic. In this paper,we establish the relations between the systems of fuzzy linear equations and the possibilistic linear programming problems and present an alternative method of solving the systems of fuzzy linear equations.

Numerical solutions of linear quadratic control for time-varying systems via symplectic conservative perturbation

Optimal control system of state space is a conservative system, whose approximate method should be symplectic conservation. Based on the precise integration method, an algorithm of symplectic conservative perturbation is presented. It gives a uniform way to solve the linear quadratic control （LQ control） problems for linear timevarying systems accurately and efficiently, whose key points are solutions of differential Riccati equation （DRE） with variable coefficients and the state feedback equation. The method is symplectic conservative and has a good numerical stability and high precision. Numerical examples demonstrate the effectiveness of the proposed method.

Perturbation Analysis of Continuous-Time Linear Time-Invariant Systems

In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-time linear time-invariant systems. It solves the perturbed linear time-invariant systems via Riccati differential equations and continuous-time algebraic Riccati equations in finite and infinite time horizons. We derive the explicit expressions of measuring the perturbation bounds of condition numbers with respect to the solution of the linear time-invariant systems. Furthermore, condition numbers and their upper bounds of Riccati differential equations and continuous-time algebraic Riccati equations are also discussed. Numerical simulations show the sharpness of the perturbation bounds computed via the proposed methods.

A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach

This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.

Equivalent method for accurate solution to linear interval equations

Based on linear interval equations, an accurate interval finite element method for solving structural static problems with uncertain parameters in terms of optimization is discussed. On the premise of ensuring the consistency of solution sets, the original interval equations are equivalently transformed into some deterministic inequations. On this basis, calculating the structural displacement response with interval parameters is predigested to a number of deterministic linear optimization problems. The results are proved to be accurate to the interval governing equations. Finally, a numerical example is given to demonstrate the feasibility and efficiency of the proposed method.

LINEAR EQUATIONS OVER CONES,COLLATZ-WIELANDT NUMBERS AND ALTERNATING SEQUENCES

Let K be a proper cone in R^x,let A be an n×n real matrix that satisfies AK(?)K,letb be a given vector of K,and let λbe a given positive real number.The following two lin-ear equations are considered in this paper:(i)(λⅠ_n-A)x=b,x∈K,and(ii)(A-λⅠ_n)x=b,x∈K.We obtain several equivalent conditions for the solvability of the first equation.

Optimal Solution of Fuzzy Transportation Problem Using Octagonal Fuzzy Numbers

In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics to solve a system of linear equations in many applications in science.Systems of concurrent linear equations play a vital major role in operational research.The main perspective of this research paper is to find out the minimum amount of transportation cost of some supplies through a capacitated network formerly the availability and the demand notes are octagonal fuzzy numbers.Octagonal fuzzy numbers are used and showed a membership function.To illustrate this method,a fuzzy transportation problem is solved by using octagonal fuzzy numbers using the ranking technique.It is shown that it is the best optimal solution and it is demonstrated with a numerical example.

Solution and Application of the Matrix Equation Ax=b

In this paper,we first give the solution concept of the fuzzy matrix equation =. Secondly,we discuss the property of the solution and give the method of solving the fuzzy matrix equation A=. Finally,we present an application of solving fuzzy matrix equation A= to the fuzzy linear regression analysis,establish a new model of fuzzy linear regression,and introduce a new method of estimating parameters.

A TWO-STEP EXPLICIT METHOD FOR A CLASS OF LINEAR PERIODIC INITIAL VALUE PROBLEMS

This paper presents a two-step explicit method of order four for solving aclass of linear periodic initial value problems. At each computational step, only tworight function evaluations and one derivative evaluation are employed. Basing on aspecial vector operation, the method can be extended to the vector-applicable in multi-dimensional space.

New Interval Oscillation Criteria for Forced Second-order Half-linear Differential Equations

这份报纸讨论强迫的秒顺序的一个班一半线性的微分方程。由使用概括 Riccati 技术和平均技术，一些新间隔摆动标准被获得。

Type Synthesis of Walking Robot Legs

Walking robots use leg structures to overcome obstacles or move on complicated terrains. Most robots of current researches are equipped with legs of simple structure. The specific design method of walking robot legs is seldom studied. Based on the generalized-function(GF) set theory, a systematic type synthesis process of designing robot legs is introduced. The specific mobility of robot legs is analyzed to obtain two main leg types as the goal of design.Number synthesis problem is decomposed into two stages, actuation and constraint synthesis by name,corresponding to the combinatorics results of linear Diophantine equations. Additional restrictions are discussed to narrow the search range to propose practical limb expressions and kinematic-pair designs. Finally, all the fifty-one leg structures of four subtypes are carried out, some of which are chosen to make up robot prototypes, demonstrating the validity of the method. This paper proposed a novel type synthesis methodology, which could be used to systematically design various practical robot legs and the derived robots.

An Interval Matrix Based Generalized Newton Method for Linear Complementarity Problems

The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.

An O(k²+kh²+h²) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

设计参数不确定条件下滚动直线导轨运动性能分析

为了研究几何参数不确定性对直线导轨动态运动性能的影响,用区间数表示几何参数的不确定性。基于赫兹接触理论,分析滚动直线导轨实际受载情况,建立了二自由度动力学分析模型,求解得到滚动直线导轨水平方向和垂直方向动态位移;根据直线导轨精度等级,确定水平和垂直方向位移公差范围。通过对比实际动态位移与位移公差范围,得到了直线导轨水平方向和垂直方向上动态位移位于运动公差合理区间内的时间占比。通过对不同精度等级的SHS-45R直线导轨的分析可以得到,随着导轨精度等级的提高,水平和垂直方向上动态位移位于运动公差合理区间内部的比例逐渐变小。

一类6p^(2)阶群的自同态与自同构数量

结合一次同余方程的相关知识,研究了一类三元生成的6p^(2)阶非交换群的元素性质及自同态和自同构,计算了其自同态和自同构数量,验证其自同态数量是满足T.Asai和T.Yoshida的猜想的.

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill’s equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.

PERTURBATIONAL SOLUTIONS FOR FUZZY-STOCHASTIC FINITE ELEMENT EQUILIBRIUM EQUATIONS (FSFEEE)

In this paper, the random interval equilibrium equations (RIEE) is obtained by lambda-level cutting the fuzzy-stochastic finite element equilibrium equations (FSFEEE). Based on the relations between the variables of equilibrium equations, solving RIEE is transformed into solving two kinds of general random equilibrium equations (GREE). Then the recursive equations of evaluating the random interval displacement is derived from the small-parameter perturbation theory. The computational formulae of statistical characteristic of the fuzzy random displacements, the fuzzy random strains and the fuzzy random stresses are also deduced in detail.

A NEW SOLUTION TO STRUCTURAL FUZZY FINITE ELEMENT EQUILIBRIUM EQUATIONS (SFFEEE)

A speedy accurate solution to structural fuzzy finite element equilibrium equations (SFFEEE), by combining the definition of the solution of interval equations with the mechanical meaning of the structural finite element equilibrium equations (SFEEE), was put forward. The fuzzification of the SFFEEE, which is discussed in this paper, originates from that of material property, structural boundary conditions and external loading. The computing quantity of this solution is almost equal to that of the general finite element method (GFEM).