Computing constraint sets for differential fields

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.

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On separating transcendency bases for differential fields

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1955-0071420-5 ◽

1955 ◽

Vol 6 (5) ◽

pp. 726-726 ◽

Cited By ~ 1

Author(s):

A. Seidenberg

Keyword(s):

Differential Fields

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A Proposed Method to Group the Solutions From Dimensional Synthesis: Planar Triads for Six Precision Position

22nd Biennial Mechanisms Conference: Mechanism Design and Synthesis ◽

10.1115/detc1992-0332 ◽

1992 ◽

Author(s):

Xiancheng Lu ◽

Chuen-Sen Lin

Keyword(s):

Infinite Number ◽

Numerical Solutions ◽

Angular Displacement ◽

Dimensional Synthesis ◽

Number Of Solutions ◽

Jacobian Matrices ◽

Design Task ◽

Numerical Examples ◽

Computer Automation ◽

Constraint Sets

Abstract In this paper, a method has been proposed to group into six sets the infinite number of solutions from dimensional synthesis of planar triads for six precision positions. The proposed method reveals the relationships between the different configurations of the compatibility linkage and the sets of numerical solutions from dimensional synthesis. By checking the determinant signs and the contunities of values of the sub-Jacobian matrices and their derivatives with respect to the independent angular displacement for all constraint sets in the compatibility linkage, it enables the computer to identify and group the synthesized solutions. Numerical examples have been given to verify the applicability of this method. Six sets of the partial triad Burmester curves have been plotted based on grouped solutions. Suitable solutions can be easily found from the partial triad Burmester curves and utilized for the prescribed design task. This method provides a useful tool to group the dimensional synthesis solutions and enhances the computer automation in the design of linkage mechanisms.

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Adapting Rabin’s Theorem for Differential Fields

Models of Computation in Context - Lecture Notes in Computer Science ◽

10.1007/978-3-642-21875-0_22 ◽

2011 ◽

pp. 211-220

Author(s):

Russell Miller ◽

Alexey Ovchinnikov

Keyword(s):

Differential Fields

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