The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistoo...The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.展开更多
The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength satisfying certain standard conditions. The con-ditions proposed in the early work of Armijo and Goldstein ar...The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength satisfying certain standard conditions. The con-ditions proposed in the early work of Armijo and Goldstein are sometimes replaced by those recommended by Wolf e because these latter conditions automatically allow positive definiteness of some popular quasi-Newton updates to be maintained.It is shown that a slightly modified form of quasi-Newton update allows positive definiteness to be maintained even if line searches based on the Armijo-Goldsteinconditions are used.展开更多
The class of symmetric definitely positive matrices is extremely important in the matrix theory. At present, positive definiteness of a symmetric matrix can be shown by determining the signs of its all ordered princip...The class of symmetric definitely positive matrices is extremely important in the matrix theory. At present, positive definiteness of a symmetric matrix can be shown by determining the signs of its all ordered principal minors or the signs展开更多
脑小血管病(small vessel disease,SVD)是一类以脑内小血管受损为主的临床影像综合征,可能会导致卒中、血管性认知障碍、神经心理疾病与其他功能障碍等。自2013年血管性神经病变的影像报告标准(standards for reporting vascular change...脑小血管病(small vessel disease,SVD)是一类以脑内小血管受损为主的临床影像综合征,可能会导致卒中、血管性认知障碍、神经心理疾病与其他功能障碍等。自2013年血管性神经病变的影像报告标准(standards for reporting vascular changes on neuroimaging,STRIVE)发布以来,SVD的神经影像学特征得到了初步的分类与标准化。然而,在临床实践与科学研究中,对SVD影像特征的认识和应用仍存在诸多不一致和不规范之处。随着对SVD病理生理机制的深入探索与影像技术的不断进步,新的SVD影像特征和定量标志物被相继发现,为SVD的诊断和评估提供了更为全面且精准的信息。在此基础上,STRIVE-2应运而生,以期能更全面地揭示SVD对脑功能与结构的影响。为了规范中国SVD的神经影像学评估和诊断,本共识将在STRIVE-2的基础上,结合中国具体国情,对SVD的神经影像学特征进行深入解读,旨在推动SVD影像学诊断术语的标准化,提高临床诊断的准确性,进一步促进相关领域的研究与进步。展开更多
基金Hong-Lin Liao was supported by National Natural Science Foundation of China(Grant No.12071216)Tao Tang was supported by Science Challenge Project(Grant No.TZ2018001)+3 种基金National Natural Science Foundation of China(Grants Nos.11731006 and K20911001)Tao Zhou was supported by National Natural Science Foundation of China(Grant No.12288201)Youth Innovation Promotion Association(CAS)Henan Academy of Sciences.
文摘The positive definiteness of real quadratic forms with convolution structures plays an important rolein stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysistool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators.More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, weshow that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Ourproof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolutionkernels and discrete complementary convolution kernels. To our knowledge, this is the first general result onsimple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using theunified theory, we obtain the stability for some simple nonuniform time-stepping schemes straightforwardly.
文摘The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength satisfying certain standard conditions. The con-ditions proposed in the early work of Armijo and Goldstein are sometimes replaced by those recommended by Wolf e because these latter conditions automatically allow positive definiteness of some popular quasi-Newton updates to be maintained.It is shown that a slightly modified form of quasi-Newton update allows positive definiteness to be maintained even if line searches based on the Armijo-Goldsteinconditions are used.
基金Project supported by the National Natural Science Foundation of China.
文摘The class of symmetric definitely positive matrices is extremely important in the matrix theory. At present, positive definiteness of a symmetric matrix can be shown by determining the signs of its all ordered principal minors or the signs
文摘脑小血管病(small vessel disease,SVD)是一类以脑内小血管受损为主的临床影像综合征,可能会导致卒中、血管性认知障碍、神经心理疾病与其他功能障碍等。自2013年血管性神经病变的影像报告标准(standards for reporting vascular changes on neuroimaging,STRIVE)发布以来,SVD的神经影像学特征得到了初步的分类与标准化。然而,在临床实践与科学研究中,对SVD影像特征的认识和应用仍存在诸多不一致和不规范之处。随着对SVD病理生理机制的深入探索与影像技术的不断进步,新的SVD影像特征和定量标志物被相继发现,为SVD的诊断和评估提供了更为全面且精准的信息。在此基础上,STRIVE-2应运而生,以期能更全面地揭示SVD对脑功能与结构的影响。为了规范中国SVD的神经影像学评估和诊断,本共识将在STRIVE-2的基础上,结合中国具体国情,对SVD的神经影像学特征进行深入解读,旨在推动SVD影像学诊断术语的标准化,提高临床诊断的准确性,进一步促进相关领域的研究与进步。
文摘We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.
