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Tensor convolutions and Hankel tensors
Changqing XU, Yiran XU
Front. Math. China. 2017, 12 (6): 1357-1373.
https://doi.org/10.1007/s11464-017-0666-8
Let be an mth order n-dimensional tensor, where m, nare some positive integers and N:= m(n−1).Then is called a Hankel tensor associated with a vector if for each k= 0, 1, …,Nwhenever σ= (i1, …,im) satisfies i1 +…+im = m+k.We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k= 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.
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Approximation algorith ms for nonnegative polynomial optimization problems over unit spheres
Xinzhen ZHANG, Guanglu ZHOU, Louis CACCETTA, Mohammed ALQAHTANI
Front. Math. China. 2017, 12 (6): 1409-1426.
https://doi.org/10.1007/s11464-017-0644-1
We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.
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Diophantine inequality involving binary forms
Quanwu MU
Front. Math. China. 2017, 12 (6): 1457-1468.
https://doi.org/10.1007/s11464-017-0602-y
Let be an integer, and set and respectively. Suppose that are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2, . . . , λr are nonzero real numbers with λ1/λ2 irrational, and λ1Φ1 (x1, y1) + λ2Φ2 (x2, y2) + · · · + λrΦr (xr, yr) is indefinite. Then for any given real η and σ with 0<σ<22−d, it is proved that the inequality has infinitely many solutions in integers x1, x2, . . . , xr, y1, y2, . . . , yr. This result constitutes an improvement upon that of B. Q. Xue.
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Robust inference in linear mixed model with skew normal-symmetric error
Mixia WU, Ye TIAN, Aiyi LIU
Front. Math. China. 2017, 12 (6): 1483-1500.
https://doi.org/10.1007/s11464-017-0660-1
Linear mixed effects models with general skew normal-symmetric (SNS) error are considered and several properties of the SNS distributions are obtained. Under the SNS settings, ANOVA-type estimates of variance components in the model are unbiased, the ANOVA-type F-tests are exact F-tests in SNS setting, and the exact confidence intervals for fixed effects are constructed. Also the power of ANOVA-type F-tests for components are free of the skewing function if the random effects normally distributed. For illustration of the main results, simulation studies on the robustness of the models are given by comparisons of multivariate skew-normal, multivariate skew normal-Laplace, multivariate skew normal-uniform, multivariate skew normal-symmetric, and multivariate normal distributed errors. A real example is provided for the illustration of the proposed method.
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Solvability of finite groups
Jia ZHANG, Baijun GAO, Long MIAO
Front. Math. China. 2017, 12 (6): 1501-1514.
https://doi.org/10.1007/s11464-017-0643-2
H is called an -embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that Hp ∈ Sylp (B) and B is -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that and d divides |P|. If every subgroup H of P with is -embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5'-group, (3)
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