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A general version of the Morse-Sard theorem.
1Department of Mathematics, Zhejiang University, Hangzhou 310027, China. hyjiang@math.zju.edu.cn
Journal of Zhejiang University. Science
|October 21, 2004
Summary
This study solves Arthur Sard's 1965 conjecture for k>=2, proving that for a function f, the image of a set A defined by the rank of its derivative is d-null, where d is a specific dimension.
Area of Science:
- Differential Geometry
- Real Analysis
- Geometric Measure Theory
Background:
- Arthur Sard's 1965 statement concerns the measure of the image of a set of critical points under a smooth map.
- Understanding the properties of sets defined by the rank of the derivative is crucial in geometric analysis.
Purpose of the Study:
- To completely solve Arthur Sard's 1965 statement for the case where k is greater than or equal to 2.
- To determine the d-null property of the image of a set defined by the rank of the derivative of a function.
Main Methods:
- The study defines a set A based on the rank of the derivative of a function f: R^m -> R^n.
- It utilizes concepts from differential geometry and measure theory to analyze the properties of f(A).
- A specific dimension 'd' is calculated as d=r+(m-r)/(k+alpha).
Main Results:
- The core finding is that for functions f in C^k(R^m, R^n) with k>=2 and alpha in (0,1], the set f(A) is d-null.
- This result provides a complete solution to Sard's statement under the specified conditions.
Conclusions:
- The study successfully resolves a significant problem posed by Arthur Sard.
- The findings contribute to the understanding of geometric measure theory and the behavior of functions on sets defined by derivative ranks.