The cross-dimensional dynamical systems have received increasing research attention in recent years.This paper characterizes the structure features of the cross-dimensional vector space.Specifically,it is proved that ...The cross-dimensional dynamical systems have received increasing research attention in recent years.This paper characterizes the structure features of the cross-dimensional vector space.Specifically,it is proved that the completion of cross-dimensional vector space is an infinite-dimensional separable Hilbert space.Hence,it means that one can isometrically and linearly embed the crossdimensional vector space into theℓ^(2),which is known as the space of square summable sequences.This result will be helpful in the modeling and analyzing the dynamics of cross-dimensional dynamical systems.展开更多
In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B＇ in separable complex Hilbert spaces H and H＇, respectively. It is found that if the ...In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B＇ in separable complex Hilbert spaces H and H＇, respectively. It is found that if the mapping f ∈ C^1＋α at z0 ∈ B with f（zo） = wo ∈ OB＇, then the Fr＆het derivative operator Df（z0） maps the tangent space Tz0（ B^n） to Tw0（ B＇）, the holomorphic tangent space Tz0^（1,0） to Tw0（1,0）（ B＇）,respectively.展开更多
The necessary and sufficient conditions are given so that a non-anticipative transformation in Hilbert space is isometric. In terms of second order Wiener process, these conditions assure that a non-anticipative trans...The necessary and sufficient conditions are given so that a non-anticipative transformation in Hilbert space is isometric. In terms of second order Wiener process, these conditions assure that a non-anticipative transformation of Wiener process is a Wiener process, too.展开更多
基金supported by the National Natural Science Foundation of China under Grant No.61673129the Key Programs in Shaanxi Province of China under Grant No.2021JZ-12Science and the Technology Bureau Project of Yulin under Grant Nos.2019-89-2 and 2019-89-4。
文摘The cross-dimensional dynamical systems have received increasing research attention in recent years.This paper characterizes the structure features of the cross-dimensional vector space.Specifically,it is proved that the completion of cross-dimensional vector space is an infinite-dimensional separable Hilbert space.Hence,it means that one can isometrically and linearly embed the crossdimensional vector space into theℓ^(2),which is known as the space of square summable sequences.This result will be helpful in the modeling and analyzing the dynamics of cross-dimensional dynamical systems.
基金supported by the National Natural Science Foundation of China(Nos.11671361,11571256)the Zhejiang Provincial Natural Science Foundation of China(No.LY14A010008)
文摘In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B＇ in separable complex Hilbert spaces H and H＇, respectively. It is found that if the mapping f ∈ C^1＋α at z0 ∈ B with f（zo） = wo ∈ OB＇, then the Fr＆het derivative operator Df（z0） maps the tangent space Tz0（ B^n） to Tw0（ B＇）, the holomorphic tangent space Tz0^（1,0） to Tw0（1,0）（ B＇）,respectively.
文摘The necessary and sufficient conditions are given so that a non-anticipative transformation in Hilbert space is isometric. In terms of second order Wiener process, these conditions assure that a non-anticipative transformation of Wiener process is a Wiener process, too.
