Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,su...Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.展开更多
Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of t...Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).展开更多
With the development of digitalization in healthcare,more and more information is delivered and stored in digital form,facilitating people’s lives significantly.In the meanwhile,privacy leakage and security issues co...With the development of digitalization in healthcare,more and more information is delivered and stored in digital form,facilitating people’s lives significantly.In the meanwhile,privacy leakage and security issues come along with it.Zero watermarking can solve this problem well.To protect the security of medical information and improve the algorithm’s robustness,this paper proposes a robust watermarking algorithm for medical images based on Non-Subsampled Shearlet Transform(NSST)and Schur decomposition.Firstly,the low-frequency subband image of the original medical image is obtained by NSST and chunked.Secondly,the Schur decomposition of low-frequency blocks to get stable values,extracting the maximum absolute value of the diagonal elements of the upper triangle matrix after the Schur decom-position of each low-frequency block and constructing the transition matrix from it.Then,the mean of the matrix is compared to each element’s value,creating a feature matrix by combining perceptual hashing,and selecting 32 bits as the feature sequence.Finally,the feature vector is exclusive OR(XOR)operated with the encrypted watermark information to get the zero watermark and complete registration with a third-party copyright certification center.Experimental data show that the Normalized Correlation(NC)values of watermarks extracted in random carrier medical images are above 0.5,with higher robustness than traditional algorithms,especially against geometric attacks and achieve watermark information invisibility without altering the carrier medical image.展开更多
文摘Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently,especially related to vortices and turbulence.Several decompositions of the velocity gradient tensor,such as the triple decomposition of motion(TDM)and normal-nilpotent decomposition(NND),have been proposed to analyze the local motions of fluid elements.However,due to the existence of different types and non-uniqueness of Schur forms,as well as various possible definitions of NNDs,confusion has spread widely and is harming the research.This work aims to clean up this confusion.To this end,the complex and real Schur forms are derived constructively from the very basics,with special consideration for their non-uniqueness.Conditions of uniqueness are proposed.After a general discussion of normality and nilpotency,a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed,with a brief comparison of complex and real decompositions.Based on that,several confusing points are clarified,such as the distinction between NND and TDM,and the intrinsic gap between complex and real NNDs.Besides,the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case.But their justification is left to further investigations.
文摘Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).
基金supported in part by the Natural Science Foundation of China under Grants 62063004the Key Research Project of Hainan Province under Grant ZDYF2021SHFZ093+1 种基金the Hainan Provincial Natural Science Foundation of China under Grants 2019RC018 and 619QN246the postdoctoral research from Zhejiang Province under Grant ZJ2021028.
文摘With the development of digitalization in healthcare,more and more information is delivered and stored in digital form,facilitating people’s lives significantly.In the meanwhile,privacy leakage and security issues come along with it.Zero watermarking can solve this problem well.To protect the security of medical information and improve the algorithm’s robustness,this paper proposes a robust watermarking algorithm for medical images based on Non-Subsampled Shearlet Transform(NSST)and Schur decomposition.Firstly,the low-frequency subband image of the original medical image is obtained by NSST and chunked.Secondly,the Schur decomposition of low-frequency blocks to get stable values,extracting the maximum absolute value of the diagonal elements of the upper triangle matrix after the Schur decom-position of each low-frequency block and constructing the transition matrix from it.Then,the mean of the matrix is compared to each element’s value,creating a feature matrix by combining perceptual hashing,and selecting 32 bits as the feature sequence.Finally,the feature vector is exclusive OR(XOR)operated with the encrypted watermark information to get the zero watermark and complete registration with a third-party copyright certification center.Experimental data show that the Normalized Correlation(NC)values of watermarks extracted in random carrier medical images are above 0.5,with higher robustness than traditional algorithms,especially against geometric attacks and achieve watermark information invisibility without altering the carrier medical image.
