An alternative proof of Kharitonov's theorem

An alternative proof is presented of Kharitonov's theorem for real polynomials. The proof shows that if an unstable root exists in the interval family, then another unstable root must also show up in what is called the Kharitonov plane, which is delimited by the four Kharitonov polynomials. This fact is proved by using a simple lemma dealing with convex combinations of polynomials. Then a well-known result is utilized to prove that when the four Kharitonov polynomials are stable, the Kharitonov plane must also be stable, and this contradiction proves the theorem     

A note on the stability of delta-operator-induced systems

Points out that stability of delta-operator-induced real polynomials is equivalent to a shifted Hurwitz stability of their reciprocal counterparts. This fact reveals a simple and clear link between the delta-domain stability and Hurwitz stability for real polynomials.     

Characterization of robust root loci of polytopes of polynomials

This paper deals with the robust root locus problem of a polytope of real polynomials. First, a simple and efficient algorithm is presented for testing if the value set of a polytopic family of polynomials includes the origin of the complex plane. This zero-inclusion test algorithm is then applied along with a pivoting procedure to construct the smallest set of regions in the complex plane which characterizes the robust root loci of a polytope of polynomials.     

On the sixty-four polynomials of Djaferis to stabilize an intervalplant

This paper shows that 32 of the 64 polynomials developed by Djaferis to stabilize an interval plant with a controller are superfluous. Thus it is sufficient to stabilize 32 polynomials simultaneously. Moreover the construction developed in this note allows the implementation of these polynomials in a simple way     

Hardwired polynomial evaluation

This paper is devoted to the evaluation of polynomials and elementary functions by special-purpose circuits. First we recall the basic results concerning the approximation of mathematical functions by polynomials (these results enable us to compute every continuous function if we are able to compute polynomials); then we describe a simple operator, suitable for VLSI implementation, which evaluates a polynomial in the range [0, 1]. Finally, we give some complexity results about the evaluation of the most usual elementary functions with our operator.     

An elementary proof of Kharitonov's stability theorem withextensions

Gives an elementary proof of Kharitonov's theorem using simple complex plane geometry without invoking the Hermite-Bieler theorem. Kharitonov's theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval. The result states that the whole class is Hurwitz if and only if four special, well-defined polynomials are Hurwitz. The paper also gives elementary proofs of two previously known extensions: for polynomials of degree less than six, the requirement is reduced to fewer than four polynomials; and the theorem is generalized to polynomials with complex coefficients     

基于代数神经网络的置换多项式判定网络学习算法

从神经元的运算特性入手,对神经元的激发函数,网络结构,学习目标三方面进行了推广,设计出了一类用于有限域上置换多项式判定的多项式神经网络模型,它们是单输入单输出的3层神经网络。给出了两类置换多项式判定的离散网络模型学习算法,该算法简单可行,易于实现。     

Using symbolic computation to find algebraic invariants

Implicit polynomials have proved themselves as having excellent representation power for complicated objects, and there is growing use of them in computer vision, graphics, and CAD. A must for every system that tries to recognize objects based on their representation by implicit polynomials are invariants, which are quantities assigned to polynomials that do not change under coordinate transformations. In the recognition system developed at the Laboratory for Engineering Man-Machine Studies in Brown University (LEMS), it became necessary to use invariants which are explicit and simple functions of the polynomial coefficients. A method to find such invariants is described and the new invariants presented. This work addresses only the problem of finding the invariants; their stability is studied in another paper     

Real and complex polynomial stability and stability domain construction via network realizability theory

Network realzability theory provides the basis for a unified approach to the stability of a polynomial or a family of polynomials. In this paper conditions are given, in terms of certain decompositions of a given polynomial, that are necessary and sufficient for the given polynomial to be Hurwitz. These conditions facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. This approach provides a simple interpretation of recent results for polynomials with real coefficients and also leads to the formulation of corresponding results for the case of polynomials with complex coefficients.     

Improving multivariate Horner schemes with Monte Carlo tree search

Optimizing the cost of evaluating a polynomial is a classic problem in computer science. For polynomials in one variable, Horner’s method provides a scheme for producing a computationally efficient form. For multivariate polynomials it is possible to generalize Horner’s method, but this leaves freedom in the order of the variables. Traditionally, greedy schemes like most-occurring variable first are used. This simple textbook algorithm has given remarkably efficient results. Finding better algorithms has proved difficult. In trying to improve upon the greedy scheme we have implemented Monte Carlo tree search, a recent search method from the field of artificial intelligence. This results in better Horner schemes and reduces the cost of evaluating polynomials, sometimes by factors up to two.     

Fitting curves and surfaces with constrained implicit polynomials

A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. The paper suggests a different approach-to design parameterized families of polynomials whose zero-sets are guaranteed to satisfy certain topological properties. Namely, we construct families of polynomials with star-shaped zero-sets, as well as polynomials whose zero-sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse. This is more rigorous than using heuristics which may fail and result in pathological zero-sets. The ability to parameterize these families depends heavily on the ability to parameterize positive polynomials. To achieve this, we use some powerful results from real algebraic geometry     

Improving semi-empirical equations of ultimate bearing capacity of shallow foundations using soft computing polynomials

This study presents the ultimate bearing capacity of shallow foundations in meaningful ways and improves its semi-empirical equations accordingly. Approaches including weighted genetic programming (WGP) and soft computing polynomials (SCP) are utilized to provide accurate prediction and visible formulas/polynomials for the ultimate bearing capacity. Visible formulas facilitate parameter studies, sensitivity analysis, and applications of pruning techniques. Analytical results demonstrate that the proposed SCP is outstanding in both prediction accuracy and provides simple polynomials as well. Notably, the SCP identifies that the shearing resistance angle and foundation geometry impact on improving the Vesic's semi-empirical equations.     

Extreme-point robust stability results for discrete-timepolynomials

This paper provides some new extreme-point robust-stability results for discrete-time polynomials with special uncertainties in the coefficient space. The proofs, obtained using the bilinear transformation, are simple, and the results specialize to existing robust-stability results     

基于两变量Hahn矩的图像分析

提出了一种新的、以两变量离散正交Hahn多项式为核函数的图像矩,推导了正则化后,两变量离散正交Hahn多项式的简单的计算方法。对二值图像、灰度图像以及噪声图像的重建实验表明：相对于同系数的单变量的Hahn矩,两变量Hahn矩的重建误差更小。因此,它们能够更好地提取图像的特征。     

Improving analytical models of circular concrete columns with genetic programming polynomials

This study improves weighted genetic programming and uses proposed novel genetic programming polynomials (GPP) for accurate prediction and visible formulas/polynomials. Representing confined compressive strength and strain of circular concrete columns in meaningful representations makes parameter studies, sensitivity analysis, and application of pruning techniques easy. Furthermore, the proposed GPP is utilized to improve existing analytical models of circular concrete columns. Analytical results demonstrate that the GPP performs well in prediction accuracy and provides simple polynomials as well. Three identified parameters improve the analytical models—the lateral steel ratio improves both compressive strength and strain of the target models of circular concrete columns; compressive strength of unconfined concrete specimen improves the strength equation; and tie spacing improves the strain equation.     

Endpaper: FRPOLY: A benchmark revisited

The FRPOLY Lisp performance benchmark [3] was based on a circa-1968 piece of code for computing powers of polynomials. We address two questions: (a) What algorithm would you use if you really wanted to compute powers of polynomials fast? and (b) Given that Common Lisp supports many types of data structures other than the simple lists used for the benchmark, what more efficient representations might be appropriate to use for polynomials?     

利用矩阵变换求n个多项式的最大公因式及判断复常系数线性系统的稳定性

本文给出通过矩阵变换的方法求n个多项式最大公因式以及判断复常系数线性系统的稳定性判据,这些算法是很简单的,改进了传统的欧几里德算法和胡维茨算法。     

Robust stability manifolds for multilinear interval systems

The stability of a class of multilinearly perturbed families of systems is considered. It is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved. The extremal property of these subsets is established. The results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces     

区间多项式族左扇区稳定鲁棒性及不变惯性定理

本文考虑区间多项式关于左扇区的稳定鲁棒性,用值集排零方法推导出强哈氏定理和不变惯性定理,并给出了构造强哈氏多项式及棱边族的简明算法。     

Robust stability analysis of polynomials with linearly dependentcoefficient perturbations

A computational tractable procedure for robust pole location analysis of uncertain linear time-invariant dynamical systems, whose characteristic polynomial coefficients depend linearly on parameter perturbations, is proposed. It is shown that, in the case of linearly dependent coefficient perturbations, the stability test with respect to any unconnected domain of the complex plane can be carried out, and the largest stability domain in parameter space can be computed by using only a quick test on a particular set of polynomials named vertex polynomials. The procedure requires only one sweeping function and simple geometrical considerations at each sweeping step. This leads to a very short execution time, as is shown in an example. A unification with Kharitonov's theory and edge theorem is also provided