Solving Linear Differential Problems with Parameters

机译：用参数求解线性微分问题

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We present algorithms for parametric problems in differential algebra that can be formulated in a suitable first-order language L. The atomic L-formulas are linear ODEs of arbitrary order with parametric coefficients of arbitrary degrees. Using rather weak axioms on differential fields or differential algebras that are realized in natural function domains, we establish explicit quantifier elimination algorithms for L providing also parametric sample solutions for purely existential problems. These sample solutions are "generic" solutions of univariate parametric linear ODEs that can be realized by concrete functions in the natural function domains mentioned above. We establish upper complexity bounds for the elimination algorithms that are elementary recursive for formulas of bounded quantifier alternation, in particular doubly exponential for existential formulas. Our results are in contrast to Seidenberg's model theoretic elimination theory for non-linear problems that is non elementary recursive, requires very strong axioms that are not realizable in natural function domains, and does not provide sample solutions.

机译：我们提出了微分代数中参数问题的算法，可以用合适的一阶语言L表示。原子L公式是任意阶的线性ODE，其参数系数为任意度。使用在自然函数域中实现的微分场或微分代数上的相当弱的公理，我们为L建立了明确的量词消除算法，还提供了纯粹存在问题的参数样本解。这些样本解决方案是单变量参数线性ODE的“通用”解决方案，可以通过上述自然函数域中的具体函数来实现。我们为消除算法建立了复杂度上限，消除算法对于有界量词交替的公式是基本递归的，对于存在性公式而言尤其是双指数。我们的结果与赛登伯格针对非基本递归的非线性问题的模型理论消除理论形成了鲜明对比，该理论是非基本递归的，需要非常强大的公理，这些公理不能在自然函数域中实现，并且不提供样本解决方案。

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来源
《International Workshop on Computer Algebra in Scientific Computing(CASC 2005); 20050912-16; Kalamata(GR)》|2005年|P.469-488|共20页
会议地点 Kalamata(GR)
作者
Volker Weispfenning;
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作者单位

University of Passau, Germany;

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会议组织
原文格式 PDF
正文语种 eng
中图分类计算技术、计算机技术;代数、数论、组合理论;
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Solving Linear Differential Problems with Parameters

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