In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native sp...In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.展开更多
In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This spac...In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L2(Ω) is also solved by the author.展开更多
As the rate-determining step in native chemical ligation reactions, the thiol–thioester exchange step is important in determining the efficiency of the ligations of peptides. In the present study, systematic theoreti...As the rate-determining step in native chemical ligation reactions, the thiol–thioester exchange step is important in determining the efficiency of the ligations of peptides. In the present study, systematic theoretical calculations were carried out on the relationships between the structure of different thioesters and the free energy barriers of the thiol–thioester exchange step. According to the calculation results, the thiol–thioester exchange step is disfavored by the steric hindrance around the carbonyl center, while the electronic effect（i.e. conjugation and hyper-conjugation effects） becomes important when the steric hindrance is insignificant.展开更多
文摘In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.
文摘In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L2(Ω) is also solved by the author.
基金NSFC (No. 21202006)FRFCU (No. FRF-TP14-015A2) for financial supports and Super-computer Center of Shanghai and Shenzhen for technical supports
文摘As the rate-determining step in native chemical ligation reactions, the thiol–thioester exchange step is important in determining the efficiency of the ligations of peptides. In the present study, systematic theoretical calculations were carried out on the relationships between the structure of different thioesters and the free energy barriers of the thiol–thioester exchange step. According to the calculation results, the thiol–thioester exchange step is disfavored by the steric hindrance around the carbonyl center, while the electronic effect（i.e. conjugation and hyper-conjugation effects） becomes important when the steric hindrance is insignificant.
