This paper establishes the geometric framework of manifold learning.After summarizing the requirements of the classical manifold learning methods,we construct the smooth homeomorphism between the manifold and its tang...This paper establishes the geometric framework of manifold learning.After summarizing the requirements of the classical manifold learning methods,we construct the smooth homeomorphism between the manifold and its tangent space.Then we propose a new algorithm via homeomorphic tangent space(LHTS).We also present another algorithm via compactness(CSLI)by analyzing the topological properties of manifolds.We illustrate our algorithm on the completed manifold and non-completed manifold.We also address several theoretical issues for further research and improvements.展开更多
It is shown that K 4(i,j,l,k,m,n) is chromatically unique if three numbers among i,j,l,k,m,n have the same value and the other three numbers are not equal but larger than that value.
In this note, we reported some results of homeomorphic classification of graphlikemanifolds of which contractions are edges of pyramids, and the number of vertexes of thebase of a pyramid, equal to 6 or 7 or 8.
This paper provides a regularity theorem for certain 2nth-order differential operator Bλthat arises in considering some moving behaviors of certain objects, which includes (n + 2)homeomorphisms and extended homeomorp...This paper provides a regularity theorem for certain 2nth-order differential operator Bλthat arises in considering some moving behaviors of certain objects, which includes (n + 2)homeomorphisms and extended homeomorphisms in 'real nonlinear case' or (n + 3) linearhomeomorphisms and extened linear homeomorphisms in 'complex linear case' for this operator. It is useful to illustrate some stabilities in both directions of certain objects in theirmoving processes.展开更多
In this paper we give a new and elementary proof to the following fact:each closed orientable surface of positive genus admits a both chaotic and expansive homeomorphism.Further more,we show that the homeomorphisms gi...In this paper we give a new and elementary proof to the following fact:each closed orientable surface of positive genus admits a both chaotic and expansive homeomorphism.Further more,we show that the homeomorphisms given are also weakly mixing.展开更多
Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold BM, and fπ ： π1 （M）→π1 （M） be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the...Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold BM, and fπ ： π1 （M）→π1 （M） be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of fπ, namely, rankFix（fπ） 〈 2rankπ1（M）, which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.展开更多
Suppose that D is a domain in (n 2) and y= f(x): D → R^n is a homeomorphism. We prove that if the modulus dilatation K(x, f) satisfies the condition A then f(x) is ACL.
Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, pro...Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, provided that the exceptional set E has σ-finite H^1-measure.Moreover, f has finite distortion, differentiable a.e. and the Jacobian Jf 〉 0 a.e.展开更多
In this paper,a distortion theorem and some equicontinuity and compactness theorems are obtained for the homeomorphism f with the sphere dilatation H（x,f）∈L<sub>loc</sub>（D）.
Kerékjártó[1, pp. 224—226] showed that every periodic self-homeomorphism of a disk is topologically conjugate to either a rotation or a reflection, which is restated by Brechner in Ref. [2]. According ...Kerékjártó[1, pp. 224—226] showed that every periodic self-homeomorphism of a disk is topologically conjugate to either a rotation or a reflection, which is restated by Brechner in Ref. [2]. According to this conclusion, every orientation-preserving periodic selfhomeomorphism of a disk must be topologically conjugate to some rotation with a turning range 2kπ/m radian (where m and k are relatively prime positive integers, m≥k), and every orientation-reversing periodic self-homeomorphism of a disk must be topologically con-展开更多
We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover. we apply t...We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover. we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is-exceptional if and only if μ(B)=0 for any measure μ of finite energy integral.展开更多
We introduce the concept of asymptotic pseudo orbit tracing property (APOTP) and obtain a new condition by the APOTP for which a homeomor-phism is a non-wandering homeomorphism.
Let X be a Jordan domain satisfying certain hyperbolic growth conditions.Assume that φ is a homeomorphism from the boundary ?X of X onto the unit circle.Denote by h the harmonic diffeomorphic extension of φ from X o...Let X be a Jordan domain satisfying certain hyperbolic growth conditions.Assume that φ is a homeomorphism from the boundary ?X of X onto the unit circle.Denote by h the harmonic diffeomorphic extension of φ from X onto the unit disk.We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of h.These generalize the Sobolev regularity of h in [A.Koski,J.Onninen,Sobolev homeomorphic extensions,J.Eur.Math.Soc.23(2021) 4065-4089,Theorem 3.1].展开更多
Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary ...Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary condition for a quasisymmetric homeomorphism to belong to QS *(S 1) from the aspect of cross-ratio distortion.展开更多
文摘This paper establishes the geometric framework of manifold learning.After summarizing the requirements of the classical manifold learning methods,we construct the smooth homeomorphism between the manifold and its tangent space.Then we propose a new algorithm via homeomorphic tangent space(LHTS).We also present another algorithm via compactness(CSLI)by analyzing the topological properties of manifolds.We illustrate our algorithm on the completed manifold and non-completed manifold.We also address several theoretical issues for further research and improvements.
文摘It is shown that K 4(i,j,l,k,m,n) is chromatically unique if three numbers among i,j,l,k,m,n have the same value and the other three numbers are not equal but larger than that value.
基金Supported by the State Ethnic Affairs Commission of PRC in 2000 year
文摘In this note, we reported some results of homeomorphic classification of graphlikemanifolds of which contractions are edges of pyramids, and the number of vertexes of thebase of a pyramid, equal to 6 or 7 or 8.
文摘This paper provides a regularity theorem for certain 2nth-order differential operator Bλthat arises in considering some moving behaviors of certain objects, which includes (n + 2)homeomorphisms and extended homeomorphisms in 'real nonlinear case' or (n + 3) linearhomeomorphisms and extened linear homeomorphisms in 'complex linear case' for this operator. It is useful to illustrate some stabilities in both directions of certain objects in theirmoving processes.
基金The first author is supported by the Special Foundation of National Prior Basis Research of China(Grant No.G1999075108)the second author is supported by National Natural Science Foundation of China(11171320 and 11431012).
文摘In this paper we give a new and elementary proof to the following fact:each closed orientable surface of positive genus admits a both chaotic and expansive homeomorphism.Further more,we show that the homeomorphisms given are also weakly mixing.
基金Supported by NSFC(Grant No.11201364)"the Fundamental Research Funds for the Central Universities"
文摘Let M be a compact connected orientable Seifert manifold with hyperbolic orbifold BM, and fπ ： π1 （M）→π1 （M） be an automorphism induced by an orientation-reversing homeomorphism f of M. We give a bound on the rank of the fixed subgroup of fπ, namely, rankFix（fπ） 〈 2rankπ1（M）, which is an analogue of inequalities on surface groups and hyperbolic 3-manifold groups.
基金Project supported by the National Natural Science Foundation of China and JTU
文摘Suppose that D is a domain in (n 2) and y= f(x): D → R^n is a homeomorphism. We prove that if the modulus dilatation K(x, f) satisfies the condition A then f(x) is ACL.
基金Supported partially by the Academy of Finland(Grant No.131477)the Magnus Ehrnrooth foundation
文摘Let f ： Ω→ f（Ω） belong to R^n be a W^1,1-homeomorphism with L^1-inegrable inner We show that finiteness of min{lipf（x）, kf（x））, for every x∈ Ω／E, implies that f^-1 ∈ W^1,n and has finite distortion, provided that the exceptional set E has σ-finite H^1-measure.Moreover, f has finite distortion, differentiable a.e. and the Jacobian Jf 〉 0 a.e.
基金Supported by the National Natural Science Foundation of China the Dctoral Foundation of the Education Commission of China
文摘In this paper,a distortion theorem and some equicontinuity and compactness theorems are obtained for the homeomorphism f with the sphere dilatation H（x,f）∈L<sub>loc</sub>（D）.
文摘Kerékjártó[1, pp. 224—226] showed that every periodic self-homeomorphism of a disk is topologically conjugate to either a rotation or a reflection, which is restated by Brechner in Ref. [2]. According to this conclusion, every orientation-preserving periodic selfhomeomorphism of a disk must be topologically conjugate to some rotation with a turning range 2kπ/m radian (where m and k are relatively prime positive integers, m≥k), and every orientation-reversing periodic self-homeomorphism of a disk must be topologically con-
文摘We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover. we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is-exceptional if and only if μ(B)=0 for any measure μ of finite energy integral.
基金Project supported by the National Natural Science Foundation of China(10361001)the Natural Science Foundation of the Committee of Education of Jiangshu Province (02KJB110008).
文摘We introduce the concept of asymptotic pseudo orbit tracing property (APOTP) and obtain a new condition by the APOTP for which a homeomor-phism is a non-wandering homeomorphism.
基金partially supported by the Young Scientist Program of the Ministry of Science and Technology of China(2021YFA1002200)supported by National Natural Science Foundation of China(12101226)+1 种基金partially supported by the National Natural Science Foundation of China(12101362)supported by Shandong Provincial Natural Science Foundation(ZR2021QA032)。
文摘Let X be a Jordan domain satisfying certain hyperbolic growth conditions.Assume that φ is a homeomorphism from the boundary ?X of X onto the unit circle.Denote by h the harmonic diffeomorphic extension of φ from X onto the unit disk.We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of h.These generalize the Sobolev regularity of h in [A.Koski,J.Onninen,Sobolev homeomorphic extensions,J.Eur.Math.Soc.23(2021) 4065-4089,Theorem 3.1].
文摘Let QS* (S 1) be the space of quasisymmetric homeomorphisms of the unit circle such that the corresponding subspace of the universal Teichmu¨ller space has Weil-Petersson metric.In this paper we give a necessary condition for a quasisymmetric homeomorphism to belong to QS *(S 1) from the aspect of cross-ratio distortion.
