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Proper resolutions and Gorensteinness in extriangulated categories
Jiangsheng HU, Dondong ZHANG, Panyue ZHOU
Front. Math. China. 2021, 16 (1): 95-117.
https://doi.org/10.1007/s11464-021-0887-8
Let be an extriangulated category with a proper class of -triangles, and an additive full subcategory of . We provide a method for constructing a proper -resolution (resp., coproper - coresolution) of one term in an -triangle in from that of the other two terms. By using this way, we establish the stability of the Gorenstein category in extriangulated categories. These results generalize the work of Z. Y. Huang [J. Algebra, 2013, 393: 142{169] and X. Y. Yang and Z. C. Wang [Rocky Mountain J. Math., 2017, 47: 1013{1053], but the proof is not too far from their case. Finally, we give some applications about our main results.
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Fourier transform of anisotropic mixed-norm Hardy spaces
Long HUANG, Der-Chen CHANG, Dachun YANG
Front. Math. China. 2021, 16 (1): 119-139.
https://doi.org/10.1007/s11464-021-0906-9
Let be the anisotropic mixed-norm Hardy space associated with defined via the radial maximal function, and let f belong to the Hardy space . In this article, we show that the Fourier transform coincides with a continuous function g on in the sense of tempered distributions and, moreover, this continuous function g; multiplied by a step function associated with ; can be pointwisely controlled by a constant multiple of the Hardy space norm of f: These proofs are achieved via the known atomic characterization of and the establishment of two uniform estimates on anisotropic mixed-norm atoms. As applications, we also conclude a higher order convergence of the continuous function g at the origin. Finally, a variant of the Hardy{Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained. All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces with , and are even new for isotropic mixed-norm Hardy spaces on .
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Bi-block positive semidefiniteness of bi-block symmetric tensors
Zheng-Hai HUANG, Xia LI, Yong WANG
Front. Math. China. 2021, 16 (1): 141-169.
https://doi.org/10.1007/s11464-021-0874-0
The positive definiteness of elasticity tensors plays an important role in the elasticity theory. In this paper, we consider the bi-block symmetric tensors, which contain elasticity tensors as a subclass. First, we define the bi-block M-eigenvalue of a bi-block symmetric tensor, and show that a bi-block symmetric tensor is bi-block positive (semi)definite if and only if its smallest bi-block M-eigenvalue is (nonnegative) positive. Then, we discuss the distribution of bi-block M-eigenvalues, by which we get a sufficient condition for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor involved. Particularly, we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite, including bi-block (strictly) diagonally dominant symmetric tensors and bi-block symmetric (B)B0-tensors. These give easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of a bi-block symmetric tensor. As a byproduct, we also obtain two easily checkable suffcient conditions for the strong ellipticity of elasticity tensors.
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Biquadratic tensors, biquadratic decompositions, and norms of biquadratic tensors
Liqun QI, Shenglong HU, Xinzhen ZHANG, Yanwei XU
Front. Math. China. 2021, 16 (1): 171-185.
https://doi.org/10.1007/s11464-021-0895-8
Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.
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12 articles
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