Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable coup...Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.展开更多
Matrix-valued data have found extensive applications in various fields,such as modern biomedical imaging,chemometrics,and economics.In this paper,we address the problem of generalized trace regression involving matrix...Matrix-valued data have found extensive applications in various fields,such as modern biomedical imaging,chemometrics,and economics.In this paper,we address the problem of generalized trace regression involving matrix-valued covariates.To estimate the unknown parameters,we propose a penalty that combines the MCP nuclear norm and two-dimensional spline lasso.This penalty accounts for the potential low-rank and row/column order structures in the matrix-valued covariates.We establish the general theory and explicit statistical convergence rate of the resulting estimator.Through simulations,we demonstrate the advantages of our proposed method compared to other competing methods.Finally,we apply our approach to analyze the brain-image datasets related to Alzheimer’s disease,identifying several efficient regions that illustrate the mechanism of Alzheimer.展开更多
The two gate formula for a diffusion X_t in R^n is considered.The formula gives an expressionof the density function of X_t in terms of the path integral of the Brownian bridge starting from xand ending on y at time t.
基金supported by the National Natural Science Foundation of China(1127100861072147+1 种基金11071159)the First-Class Discipline of Universities in Shanghai and the Shanghai University Leading Academic Discipline Project(A13-0101-12-004)
文摘Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.
基金supported by the NSF of China(Grant No.12301377)Jiangxi Provincial NSF(Grant No.20232BAB211014)+4 种基金Ling Peng’s research was supported by the NSF of China(Grant No.12201259)the NSF of Jiangxi Province(Grant Nos.20224BAB211008,20252BAC240162)the China Postdoctoral Science Foundation(Grant No.2024M751232)supported by the NSF of China(Grant No.12201260),Jiangxi Provincial NSF(Grant No.20212BAB211010)China Postdoctoral Science Foundation(Grant No.2022M711425)。
文摘Matrix-valued data have found extensive applications in various fields,such as modern biomedical imaging,chemometrics,and economics.In this paper,we address the problem of generalized trace regression involving matrix-valued covariates.To estimate the unknown parameters,we propose a penalty that combines the MCP nuclear norm and two-dimensional spline lasso.This penalty accounts for the potential low-rank and row/column order structures in the matrix-valued covariates.We establish the general theory and explicit statistical convergence rate of the resulting estimator.Through simulations,we demonstrate the advantages of our proposed method compared to other competing methods.Finally,we apply our approach to analyze the brain-image datasets related to Alzheimer’s disease,identifying several efficient regions that illustrate the mechanism of Alzheimer.
基金This project is supported by the National Natural Science Foundation of China
文摘The two gate formula for a diffusion X_t in R^n is considered.The formula gives an expressionof the density function of X_t in terms of the path integral of the Brownian bridge starting from xand ending on y at time t.
