The Quotient Space Theory of Problem Solving

‘The Schoenflies Theorem after Mazur, Morse, and Brown’ provides two proofs of the Schoenflies theorem. The Schoenflies theorem states that every bicollared embedding of an (n – 1)-sphere in the n-sphere splits the n-sphere into two balls. This chapter provides two proofs. The first is due to Mazur and Morse; it utilizes an infinite ‘swindle’ and a classical technique called push-pull. The second proof, due to Brown, serves as an introduction to shrinking, or decomposition space theory. The latter is a beautiful, but outmoded, branch of topology that can be used to produce non-differentiable homeomorphisms between manifolds, especially from a manifold to a quotient space. Techniques from decomposition space theory are essential in the proof of the disc embedding theorem.

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Target detection based on granularity computing of quotient space theory using SAR image

2010 IEEE International Conference on Image Processing ◽

10.1109/icip.2010.5651698 ◽

2010 ◽

Cited By ~ 3

Author(s):

Bin Zou ◽

Qingchao Jia ◽

Lamei Zhang ◽

Ye Zhang

Keyword(s):

Target Detection ◽

Quotient Space ◽

Sar Image ◽

Space Theory

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Quotient Space Based Problem Solving

10.1016/c2012-0-06873-1 ◽

2014 ◽

Keyword(s):

Problem Solving ◽

Quotient Space

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Product evolutionary design driven by environmental performance

Concurrent Engineering ◽

10.1177/1063293x18805200 ◽

2018 ◽

Vol 27 (1) ◽

pp. 40-56

Author(s):

Lei Zhang ◽

Chenxing Zheng ◽

Yu Zheng ◽

Haihong Huang ◽

Qingdi Ke

Keyword(s):

Environmental Performance ◽

Quotient Space ◽

Evolutionary Process ◽

Evolutionary Design ◽

Space Theory ◽

Design Constraints ◽

Evolutionary Method ◽

Operation Modes ◽

Better Than ◽

Evolutionary Product

This article is in terms of product environmental performance demand and proposes four structure evolutionary operation modes which include combined evolutionary method, decomposition evolutionary method, replacement evolutionary method, and material-changing evolutionary method to express the structure evolutionary process of products. Through the quotient space theory and proposed method combined with probability statistics, probability mapping from environmental performance to product structure is established and the evolutionary individuals with outstanding environmental performance are listed. Through the analysis to the specific conditions of the evolutionary individuals, the design constraints are extracted, and the objective function of environmental performance is established. This article presents an interactive genetic algorithm as evolutionary algorithm and combines it with four structure evolutionary operation modes to conduct corresponding gene manipulation and generates evolutionary product. Finally, the proposed methodology is successfully applied to engine gear chamber and the environmental impact is found to be better than before evolution.

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