An infinite family of bounds on zeros of analytic functions and relationship to Smale's bound

Smale's analysis of Newton's iteration function induce a lower bound on the gap between two distinct zeros of a given complex-valued analytic function . In this paper we make use of a fundamental family of iteration functions , , to derive an infinite family of lower bounds on the above gap. However, even for , where coincides with Newton's, our lower bound is more than twice as good as Smale's bound or its improved version given by Blum, Cucker, Shub, and Smale. When is a complex polynomial of degree , for small the corresponding bound is computable in arithmetic operations. For quadratic polynomials, as increases the lower bounds converge to the actual gap. We show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of . In particular, using the latter result, we show that, given a complex polynomial , , for each we can compute upper and lower bounds and such that the roots of lie in the annulus . In particular, , ; and , , where . An application of the latter bounds is within Weyl's classical quad-tree algorithm for computing all roots of a given complex polynomial.

     

机载可见光侦察系统最大探测距离地面等效实验方法研究

分析了地面定点验证法和飞行实验法测量机载可见光侦察系统最大探测距离的缺点和局限性,在地面定点验证方法的基础上,设计了机载可见光探测距离地面等效替代实验方法,避免了飞行实验带来的实验费用高、风险大的缺点,并通过飞行实验对探测距离实验结果进行比较和验证。结果表明该等效替代实验方法合理可行,实验效费比高、可操作性强,实验结果具有较高的准确度,对可见光侦察设备性能评估有一定的借鉴意义。     

ON PROPERTIES OF MEROMORPHIC SOLUTIONS FOR COMPLEX DIFFERENCE EQUATION OF MALMQUIST TYPE

In this paper, we present the properties on zeros, fixed points, poles, Borel exceptional value of finite order transcendental meromorphic solutions of complex difference equation of Malmquist typewhere n（∈ N） 〉 2, P（f（z）） and Q（f（z）） are relatively prime polynomials in f（z） with rational coefficients a8 （s = 0, 1,…,p） and bt （t = 0, 1,… ,q） such that aoapbq 7≠ O, and also consider the existence and the forms on rational solutions of this type of difference equations. Some examples are also listed to show that the assumptions of theorems, in certain senses, are the best possible.