A test for robust Hurwitz stability of convex combinations of complex polynomials

In this paper we present a method for testing the Hurwitz property of a segment of polynomials (1−λ)p₀(s)+λp₁(s), where λ∈[0,1] and p₀(s) and p₁(s) are nth-degree polynomials with complex coefficients. The method consists in constructing a parametric Routh-like array with polynomial entries and generating Sturm sequences for checking the absence of zeros of two real λ-polynomials of degrees 2 and 2n in the interval (0,1). The presented method is easy to implement. Moreover, it accomplishes the test in a finite number of arithmetic operations because it does not invoke any numerical root-finding procedure.     

On orthogonal polynomials

A study is made of the orthogonal polynomials on certain curves in the complex plane. Necessary and sufficient conditions for a set of polynomials to be orthogonal on the curves are obtained in terms of symmetric matrices. The relations of the symmetric matrices to Toeplitz matrices and innerwise matrices are shown.     

Design of 2-D stable analog and recursive digital filters using properties of the derivative of even or odd Parts of Hurwitz polynomials

In this paper, a method for the design of 2-D analog and recursive digital filters is presented. Starting from a structure in the analog domain, suitable even or odd parts of two-variable Hurwitz polynomials are generated. This enables 2-variable very strictly Hurwitz polynomials (VSHP) to be obtained,² thus avoiding non-essential singularities of the second kind. Thus it will ensure a stable 2-D recursive digital filter obtained by the use of bilinear transformations. Examples are given to illustrate the method.     

凸分析的点化

本文给出了相对于凸分析中函数与集合整体性质的局部化,即定义了集合与函数的局部凸点,并且描述了该对象的性质,对已有结论进行了推广。     

Operations on power series and polynomials

When an operation is carried out on polynomials or power series the result is usually a new power series. The general term of the series is the product of a multiplier which depends on the operation being performed and another factor which is common to the power series being operated on. In this paper equations are developed for the general term of the resultant series when certain operations are carried out on a series or on several series. Convenient equations are given for obtaining those factors common to a given series, and the lower order factors are tabulated. Several useful examples are given.     

函数的凸包及其模拟方法

给出了一种简单实用的构造函数凸包的构造方法,介绍了该算法在偏微分方程近似解模拟中的应用。     

Knowledge base combinations and firm growth

The link between knowledge and firm growth has been a core topic in economics of innovation for a long time. However, despite strong theoretical arguments, empirical evidence remains inconclusive. One important reason for this conundrum may be the failure of standard indicators to capture firm innovation activities comprehensively. We contribute to overcoming this limitation by looking in the knowledge processes that drive variegated forms of innovation and aim thereby to establish a solid relationship with firm growth in more detail. Our arguments draw on the differentiated knowledge base approach, distinguishing between analytical, synthetic, and symbolic knowledge. We measure the three types of knowledge bases with detailed longitudinal linked-employer-employee micro-data from Sweden. Econometric findings based on a very large sample of small and medium-sized firms indicate significantly positive effects of the three knowledge types, and in particular combinations thereof, on firm growth. In addition, we show that not only high-growth but also slow-growth firms benefit immensely from the use of combinatory knowledge bases. We find evidence on a curvilinear relation between knowledge bases and growth of firms. Beyond certain thresholds increasing the knowledge bases further results in decreasing firm growth. Our results remain robust in a wide range of specifications and econometric models.     

A mechanical method for graphical solution of polynomials

A mechanical synthesizer with thirty harmonic elements (fifteen sine components and fifteen cosine components) may be used to graph a polynomial in a complex plane. The sum of the sine components is recorded by a tracing point which moves vertically. The sum of the cosine components is recorded by horizontal motion of a drawing board.     

Coefficient relationship between rook and chromatic polynomials

It is shown that a p+q node graph G_c representation can be drawn for a chessboard C(p,q) i.e. an array of p rows and q columns. It is shown further that the coefficients of the rook polynomial for C(p,q) correspond 1:1 inversely with the coefficients of the chromatic polynomial for G_c written in factorial form.     

根平方平均的最优凸组合不等式

本文得到最大值α和最小值β,使得对所有的a,b>0,a≠b双向不等式αC(a,b)+(1-α)A(a,b)相似文献   

Analysis of linear time-varying systems by shifted legendre polynomials

This paper analyses the linear time-varying system by the shifted Legendre polynomials expansion. Using the operational matrix for integrating the shifted Legendre polynomials, the dynamic equation of a linear time-varying system is reduced to a set of simultaneous linear algebraic equations. The coefficients of the shifted Legendre polynomials expansion can be determined by using the least-squares method. An example is given to demonstrate the accuracy of shifted Legendre polynomials expansion of finite terms and it is compared with the results of the Laguerre method.     

Evaluating combinations of ranked lists and visualizations of inter-document similarity

We are interested in how ideas from document clustering can be used to improve the retrieval accuracy of ranked lists in interactive systems. In particular, we are interested in ways to evaluate the effectiveness of such systems to decide how they might best be constructed. In this study, we construct and evaluate systems that present the user with ranked lists and a visualization of inter-document similarities. We first carry out a user study to evaluate the clustering/ranked list combination on instance-oriented retrieval, the task of the TREC-6 Interactive Track. We find that although users generally prefer the combination, they are not able to use it to improve effectiveness. In the second half of this study, we develop and evaluate an approach that more directly combines the ranked list with information from inter-document similarities. Using the TREC collections and relevance judgments, we show that it is possible to realize substantial improvements in effectiveness by doing so, and that although users can use the combined information effectively, the system can provide hints that substantially improve on the user's solo effort. The resulting approach shares much in common with an interactive application of incremental relevance feedback. Throughout this study, we illustrate our work using two prototype systems constructed for these evaluations. The first, AspInQuery, is a classic information retrieval system augmented with a specialized tool for recording information about instances of relevance. The other system, Lighthouse, is a Web-based application that combines a ranked list with a portrayal of inter-document similarity. Lighthouse can work with collections such as TREC, as well as the results of Web search engines.     

Analysis and parameter estimation of bilinear systems via Chebyshev polynomials

The operational properties of the integration and product of Chebyshev polynomials are used in the analysis of bilinear systems by the approximation of time functions by truncated Chebyshev series. The operational properties are also applied to determine the unknown parameters of a general bilinear system from the input-output data. Examples with excellent results are given.     

Analysis of time-varying discrete systems using discrete legendre orthogonal polynomials

A finite series expansion method using discrete Legendre orthogonal polynomials (DLOPs) is applied to analyze linear time-varying discrete systems. An effective algorithm is derived to establish a representation which relates the DLOP coefficient vector of a product function to those of its two-component functions. By using this representation, along with the time-shift operational matrix of DLOP, a linear time-varying difference equation is converted into a set of linear algebraic equations. It is therefore convenient for computer computation.     

Matching polynomials: A matrix approach and its applications

A new approach is formulated for the matching polynomial m(G) of a graph G. A matrix A(G) is associated with G. A certain function defined on A(G) yields the matching polynomial of G. This approach leads to a simple characterization of m(G). It also facilitates a technique for constructing graphs with a given matching polynomial.     

一种变量核凸空间内期望收益均衡控制模型

通过对贝叶斯博弈经济增长指数的数学模型构建和预测,在变量核凸空间内实现对区域经济的期望收益均衡控制,传统的期望收益均衡控制模型采用非线性模型下的非线性步进寻优方法,寻优速度慢,且最终收敛误差较大。通过引入期望收益函数和期望均衡的方法,给出理性交换协议的公平性描述,构建一种变量核凸空间内期望收益均衡控制模型,提高数学模型的控制准确性,给出变量核凸空间内期望收益均衡控制条件,最后进行期望均衡点线性数列的收敛性和全局稳定性证明,实现了变量核凸空间内期望收益均衡控制,且控制模型是全局稳定的。     

A new type of transitional filter based on summation of polynomials

A new transitional filter based on the sum of filter polynomials is proposed and discussed. It shall be shown that the approach allows for greater flexibility in adjusting passband and stopband gain shapes. The sensitivity of the filter transfer function is analyzed and related to the approximation problem. Some new modified classical filters are also derived including maximally flat (Butterworth) and equiripple (Chebyshev) types. Several examples of combining Butterworth, Chebyshev and sync-tuned filters are presented and compared to those obtained using the conventional product method. The new transitional filter provides an alternative to other forms and provides a means for isolating the effects of non-uniformly distributed pole angles.     

随机矩阵特征多项式的相关函数的行列式表示

利用正交多项式的性质给出了高斯辛系综中酉辛群上的随机矩阵特征多项式的相关函数和矩的简洁的行列式表示,且行列式的元为正交多项式。     

Synthesis of singly-terminated ladder networks by multivariable array using homogeneous polynomials

This paper discusses a multivariable array whose elements are multivariable homogeneous polynomials which may have missing terms in any of the variables. This array permits one to obtain conditions for the realizability of a driving-point function or a transfer function as a resistively terminated low-pass ladder network. By reac- tance transformations, other forms of ladder structures can be obtained. The case of cascade-separable ladder networks is discussed in detail.