Chebyshev polynomials are used as a reservoir for generating intricate classes of symmetrical and chaotic pattems, and have been used in a vast anaount of applications. Using extended Chebyshev polynomial over finite ...Chebyshev polynomials are used as a reservoir for generating intricate classes of symmetrical and chaotic pattems, and have been used in a vast anaount of applications. Using extended Chebyshev polynomial over finite field Ze, Algehawi and Samsudin presented recently an Identity Based Encryption （IBE） scheme. In this paper, we showed their proposal is not as secure as they chimed. More specifically, we presented a concrete attack on the scheme of Algehawi and Samsudin, which indicated the scheme cannot be consolidated as a real altemative of IBE schemes since one can exploit the semi group property （bilinearity） of extended Chebyshev polynomials over Zp to implement the attack without any difficulty.展开更多
For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiene...For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space.By this result,it is known that in the sense of information-based complexity,if permissible information functionals are Hermite data,then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.展开更多
We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmen...We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction-diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally,we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.展开更多
基金Acknowledgements The authors would like to thank the reviewers for their detailed reviews and constructive comments, which have helped improve the quality of this paper. This work was partically supported by National Natural Science Foundation of China under Crants No. 61172085, No. 61103221, No. 61133014, No. 11061130539 and No. 61021004.
文摘Chebyshev polynomials are used as a reservoir for generating intricate classes of symmetrical and chaotic pattems, and have been used in a vast anaount of applications. Using extended Chebyshev polynomial over finite field Ze, Algehawi and Samsudin presented recently an Identity Based Encryption （IBE） scheme. In this paper, we showed their proposal is not as secure as they chimed. More specifically, we presented a concrete attack on the scheme of Algehawi and Samsudin, which indicated the scheme cannot be consolidated as a real altemative of IBE schemes since one can exploit the semi group property （bilinearity） of extended Chebyshev polynomials over Zp to implement the attack without any difficulty.
文摘For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space.By this result,it is known that in the sense of information-based complexity,if permissible information functionals are Hermite data,then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.
文摘We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction-diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally,we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.
