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Note on the root locus method
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作者 吴晓林 夏莉 王志珍 《Journal of Shanghai University(English Edition)》 CAS 2009年第2期105-107,共3页
A new approach to the calculation of the points at which the root locus crosses the imaginary axis is proposed and the corresponding parameters are given. Further, this method to analyze polynomial convexity is used. ... A new approach to the calculation of the points at which the root locus crosses the imaginary axis is proposed and the corresponding parameters are given. Further, this method to analyze polynomial convexity is used. Examples are given for illustration. It is shown that this approach is simple and useful to determine the Hurwitz stable polynomial. 展开更多
关键词 singular frequency root locus stable polynomial CONVEXITY
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Context-free Grammars for Triangular Arrays
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作者 Robert X.J.HAO Larry X.W.WANG Harold R.L.YANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2015年第3期445-455,共11页
We consider context-free grammars of the form G = {f → f^b1+b2+1g^a1+a2, g → f^b1 g^a1+1},where ai and bi are integers sub ject to certain positivity conditions. Such a grammar G gives rise to triangular arrays... We consider context-free grammars of the form G = {f → f^b1+b2+1g^a1+a2, g → f^b1 g^a1+1},where ai and bi are integers sub ject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T(n, k)}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn (x) =∑k=0^n T(n, k)x^k. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn+1(x) = Pn(Tn(x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Br?ndén, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh.Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers. 展开更多
关键词 Context-free grammar stable polynomials real-rootedness Bessel numbers Whitneynumbers
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