The proof of the Jordan Curve Theorem (JCT) in this paper is focused on a graphic illustration and analysis ways so as to make the topological proof more understandable, and is based on the Tverberg’s method, which i...The proof of the Jordan Curve Theorem (JCT) in this paper is focused on a graphic illustration and analysis ways so as to make the topological proof more understandable, and is based on the Tverberg’s method, which is acknowledged as being quite esoteric with no graphic explanations. The preliminary constructs a parametrisation model for Jordan Polygons. It takes quite a length to introduce four lemmas since the proof by Jordan Polygon is the approach we want to concern about. Lemmas show that JCT holds for Jordan polygon and Jordan curve could be approximated uniformly by a sequence of Jordan polygons. Also, lemmas provide a certain metric description of Jordan polygons to help evaluate the limit. The final part is the proof of the theorem on the premise of introduced preliminary and lemmas.展开更多
New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations a...New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.展开更多
文摘The proof of the Jordan Curve Theorem (JCT) in this paper is focused on a graphic illustration and analysis ways so as to make the topological proof more understandable, and is based on the Tverberg’s method, which is acknowledged as being quite esoteric with no graphic explanations. The preliminary constructs a parametrisation model for Jordan Polygons. It takes quite a length to introduce four lemmas since the proof by Jordan Polygon is the approach we want to concern about. Lemmas show that JCT holds for Jordan polygon and Jordan curve could be approximated uniformly by a sequence of Jordan polygons. Also, lemmas provide a certain metric description of Jordan polygons to help evaluate the limit. The final part is the proof of the theorem on the premise of introduced preliminary and lemmas.
文摘New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.
