Rational Solutions of High-Order Algebraic Ordinary Differential Equations

This paper considers algebraic ordinary differential equations(AODEs) and study their polynomial and rational solutions. The authors first prove a sufficient condition for the existence of a bound on the degree of the possible polynomial solutions to an AODE. An AODE satisfying this condition is called noncritical. Then the authors prove that some common classes of low-order AODEs are noncritical. For rational solutions, the authors determine a class of AODEs, which are called maximally comparable, such that the possible poles of any rational solutions are recognizable from their coefficients. This generalizes the well-known fact that any pole of rational solutions to a linear ODE is contained in the set of zeros of its leading coefficient. Finally, the authors develop an algorithm to compute all rational solutions of certain maximally comparable AODEs, which is applicable to 78.54% of the AODEs in Kamke's collection of standard differential equations.     

Determination and (re)parametrization of rational developable surfaces

The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors ?rst determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affrmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.     

Parametric equation solving and quantifier elimination in finite fields with the characteristic set method

For a parametric algebraic system in finite fields,this paper presents a method for computing the cover and the refined cover based on the characteristic set method.From the cover,the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains.By the refined cover,the author gives a complete classification of the number of solutions for this system,that is,the author divides the parameter space into several disjoint components,and on every component the system has a fix number of solutions.Moreover,the author develops a method of quantifier elimination for first order formulas in finite fields.     

A Survey of the Representations of Rational Ruled Surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design. A rational ruled surface may have different representations with respective advantages and disadvantages. In this paper, the authors revisit the representations of ruled surfaces including the parametric form, algebraic form, homogenous form and Plücker form. Moreover, the transformations between these representations are proposed such as parametrization for an algebraic form, implicitization for a parametric form, proper reparametrization of an improper one and standardized reparametrization for a general parametrization. Based on these transformation algorithms, one can give a complete interchange graph for the different representations of a rational ruled surface. For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces, the characterization methods are recalled to identify the ruled surfaces from them.

     

基于降阶观测器的双互质分解

基于降阶（非最小阶）观测器的设计,明确给出了一种新的线性时不变系统双互质分解的状态空间表示,并相应得到了真镇定控制器参数化结果的状态空间解释．与以往的结果相比,这种双互质分解更具一般性,更适用于稳定因式法.     

A proximal point algorithm for a system of generalized mixed variational inequalities

This paper introduces and considers a new system of generalized mixed variational inequalities in a Hilbert space,which includes many new and known systems of variational inequalities and generalized variational inequalities as special cases.By using the two concepts of 77-subdifferential andη-proximal mappings of a proper function,the authors try to demonstrate that the system of generalized mixed variational inequalities is equivalence with a fixed point problem.By applying the equivalence, a new and innovativeη-proximal point algorithm for finding approximate solutions of the system of generalized mixed variational inequalities will be suggested and analyzed.The authors also study the convergence analysis of the new iterative method under much weaker conditions.The results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

Computation of Invariant Curves and Identifying the Type of Critical Point

This paper considers two dimensional systems which have purely imaginary eigenvalues. In order to obtain more propositions of invariant curves, the authors transform the real systems into complex differential systems by using a suitable linear transformation. The authors also propose an algorithm to compute exponential factors. An improved method of constructing integrating factor by using all invariant curves is presented and can be used in determining the type of the equilibrium points.     

Implicitization using univariate resultants

<正> Among several implicitization methods,the method based on resultant computation is asimple and direct one,but it often brings extraneous factors which are difficult to remove.This paperstudies a class of rational space curves and rational surfaces by implicitization with univariate resultantcomputations.This method is more efficient than the other algorithms in finding implicit equationsfor this class of rational curves and surfaces.     

ON THE THEORY OF RESOLVENTS AND ITS APPLICATIONS

1.IntroductionSomefrequentlyusedalgebraicalgorithmssharethesamepropertythattheytransformasetofpolynomialequationstoasinglepolynomialequationsuchthatthezerosetofthepolynomialsetandthehypersurfacedefinedbythesinglepolynomialareequivalentincertainsense.FOralgorithmswiththisproperty,wemaymention'thealgorithmtofindaprimitiveelementforafinitelygeneratedalgebraicextensionfield[1],thealgorithmtofindaplanecurvewhichisbirationaltoaspacealgebraiccur.e[2]!etc.Inthispaper,wepresentageneralalgorithmwhichcan…     

A QUANTIFIER-ELIMINATION BASED HEURISTIC FOR AUTOMATICALLY GENERATING INDUCTIVE ASSERTIONS FOR PROGRAMS

A method using quantifier-elimination is proposed for automatically generating programinvariants/inductive assertions.Given a program,inductive assertions,hypothesized as parameterizedformulas in a theory,are associated with program locations.Parameters in inductive assertions arediscovered by generating constraints on parameters by ensuring that an inductive assertion is indeedpreserved by all execution paths leading to the associated location of the program.The method can beused to discover loop invariants-properties of variables that remain invariant at the entry of a loop.Theparameterized formula can be successively refined by considering execution paths one by one;heuristicscan be developed for determining the order in which the paths are considered.Initialization of programvariables as well as the precondition and postcondition,if available,can also be used to further refinethe hypothesized invariant.The method does not depend on the availability of the precondition andpostcondition of a program.Constraints on parameters generated in this way are solved for possiblevalues of parameters.If no solution is possible,this means that an invariant of the hypothesizedform is not likely to exist for the loop under the assumptions/approximations made to generate theassociated verification condition.Otherwise,if the parametric constraints are solvable,then undercertain conditions on methods for generating these constraints,the strongest possible invariant of thehypothesized form can be generated from most general solutions of the parametric constraints.Theapproach is illustrated using the logical languages of conjunction of polynomial equations as well asPresburger arithmetic for expressing assertions.     

New Results on Interval General Cohen-Grossberg BAM Neural Networks

This paper is concerned with an interval general Cohen-Grossberg bidirectional associative memory neural networks with mixed delays. Under proper conditions, the authors studied the existence,the uniqueness and the global exponential stability of almost automorphic solutions for the suggested system. The proposed method was mainly based on the exponential dichotomy of linear differential equation, the Banach's fixed point principle and the differential inequality techniques. The authors illustrate with an example to demonstrate the effectiveness of the proposed findings.     

基于均匀自组织映射遗传算法的梯级水库优化调度

针对遗传算法中初始解分布不均以及易早熟等问题,采用均匀设计方法来生成均匀分布的初始解以及自组织映射算法通过高低维空间映射来改变个体基因从而增强局部搜索能力,提出了均匀自组织映射遗传算法,弥补了传统遗传算法中初始解的生成过于随机以及进化过程中易陷入局部解的不足,并将此改进算法在梯级水库的长期优化调度中进行了应用.通过实例计算表明,与遗传算法以及标准粒子群算法相比,此方法拥有更好的全局寻优能力,与动态规划算法结果相近,并且有着较快的计算速度,从而验证了此方法用于处理梯级水库的长期优化调度问题的可行性与合理性.     

块对角可控规范型多输入线性系统及极点配置

基于向量空间的模结构分解和矩阵的有理标准形给出了定常多输入线性系统一类新的块对角可控规范型,其中的系统矩阵相似与一个块对角矩阵,该块对角矩阵类似于矩阵的有理标准形,与现在有的可控规范型比较,更容易反映系统的结构特征,证明步骤给出了求解方法。作为一个应用,讨论了定常多输入线性系统的极点配置问题,得到了反馈增益矩阵的一般表达式,此表达式中含有任意参数,此方法将多输入线性系统极点配置问题转化为个数为系统矩阵循环指数的单输入系统的极点配置问题,进而推导出确定一个反馈增益矩阵的最少元素个数即为系统的阶数。     

Invariant density,Lapunov exponent,and almost sure stability of Markovian-regime-switching linear systems

This paper is concerned with stability of a class of randomly switched systems of ordinary differential equations. The system under consideration can be viewed as a two-component process (X(t), α(t)), where the system is linear in X(t) and α(t) is a continuous-time Markov chain with a finite state space. Conditions for almost surely exponential stability and instability are obtained. The conditions are based on the Lyapunov exponent, which in turn, depends on the associate invariant density. Concentrating on the case that the continuous component is two dimensional, using transformation techniques, differential equations satisfied by the invariant density associated with the Lyapunov exponent are derived. Conditions for existence and uniqueness of solutions are derived. Then numerical solutions are developed to solve the associated differential equations.     

基于能力规划的三维概率选择矩阵优化算法

体系具有复杂性、巨大性和交互性等特征,体系需求的获取和分析则面临方案的不确定性和方案空间庞大等难题,利用启发式搜索算法可以求解规模较大的能力方案,但效率较低。在分析能力方案构成描述中定性、定量等要素特点的基础上,针对不同能力方案求解时算法执行效率差异较大的特征,提出面向智能Agent的三维概率选择矩阵算法,利用智能Agent自学习存储方式,将多种启发式优化算法求解不同类型的能力方案时的效率存储起来,建立三维选择矩阵,求解时动态选择效率高的算法,提高算法整体执行效率。在求解某体系能力规划方案时体现了此算法根据问题动态选择算法的优势。     

A Construction of 1-Resilient Boolean Functions with Good Cryptographic Properties

This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic immunity are also considered. A sufficient condition of the modified functions with optimal algebraic degree in terms of the Siegenthaler bound is proposed. The authors obtain a lower bound on the nonlinearity of the Tang-Carlet-Tang functions, which is slightly better than the known result. If the authors do not break the “continuity” of the support and zero sets, the functions constructed in this paper have suboptimal algebraic immunity. Finally, four specific classes of 1-resilient Boolean functions constructed from this construction and with the mentioned good cryptographic properties are proposed. Experimental results show that there are many 1-resilient Boolean functions have higher nonlinearities than known 1-resilient functions modified by Tu-Deng and Tang-Carlet-Tang functions.     

Solving the perspective-three-point problem using comprehensive Gröbner systems

A complete solution classification of the perspective-three-point (P3P) problem is given by using the Gröbner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be obtained by computing a comprehensive Gröbner system. Combining with properties of the generalized discriminant sequences, the authors give the explicit conditions to determine the number of distinct real positive solutions of the P3P problem. Several examples are provided to illustrate the effectiveness of the proposed conditions.     

ON THE ISOMORPHISMS OF SMOOTH ALGEBRAIC CURVES (Ⅰ)

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Application of Neurocomputing in Adaptive Control of Large-Scale Aerospace Systems

We are engaged in solving two difficult problems in adaptive control of the large-scale time-variant aerospace system. One is parameter identification of time-variant continuous-time state-space modei; the other is how to solve algebraic Riccati equation (ARE) of large order efficiently. In our approach, two neural networks are employed to independently solve both the system identification problem and the ARE associated with the optimal control problem. Thus the identification and the control computation are combined in closed-loop, adaptive, real-time control system . The advantage of this approach is that the neural networks converge to their solutions very quickly and simultaneously.     

Finding the topology of implicitly defined two algebraic plane curves

An algorithm is given for computing in a very efficient way the topology of two real algebraic plane curves defined implicitly.The authors preform a symbolic pre-processing that allows us later to execute all numerical computations in an accurate way.