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The (b; c)-inverse in semigroups and rings with involution
Xiaofeng CHEN, Jianlong CHEN
Front. Math. China. 2020, 15 (6): 1089-1104.
https://doi.org/10.1007/s11464-020-0880-7
We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a2)* = a2; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible.
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On intervals and sets of hypermatrices (tensors)
Saeed RAHMATI, Mohamed A. TAWHID
Front. Math. China. 2020, 15 (6): 1175-1188.
https://doi.org/10.1007/s11464-020-0884-3
Interval hypermatrices (tensors) are introduced and interval -hypermatrices are uniformly characterized using a finite set of 'extreme' hypermatrices, where can be strong P, semi-positive, or positive definite, among many others. It is shown that a symmetric interval is an interval (strictly) copositive-hypermatrix if and only if it is an interval (E) E0-hypermatrix. It is also shown that an even-order, symmetric interval is an interval positive (semi-) definite-hypermatrix if and only if it is an interval P (P0)-hypermatrix. Interval hypermatrices are generalized to sets of hyper-matrices, several slice-properties of a set of hypermatrices are introduced and sets of hypermatrices with various slice-properties are uniformly characterized. As a consequence, several slice-properties of a compact, convex set of hyper-matrices are characterized by its extreme points.
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Uniform supertrees with extremal spectral radii
Wen-Huan WANG, Ling YUAN
Front. Math. China. 2020, 15 (6): 1211-1229.
https://doi.org/10.1007/s11464-020-0873-6
A supertree is a connected and acyclic hypergraph. We investigate the supertrees with the extremal spectral radii among several kinds of r-uniform supertrees. First, by using the matching polynomials of supertrees, a new and useful grafting operation is proposed for comparing the spectral radii of supertrees, and its applications are shown to obtain the supertrees with the extremal spectral radii among some kinds of r-uniform supertrees. Second, the supertree with the third smallest spectral radius among the r-uniform supertrees is deduced. Third, among the r-uniform supertrees with a given maximum degree, the supertree with the smallest spectral radius is derived. At last, among the r-uniform starlike supertrees, the supertrees with the smallest and the largest spectral radii are characterized.
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Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games
Qingfeng ZHU, Lijiao SU, Fuguo LIU, Yufeng SHI, Yong’ao SHEN, Shuyang WANG
Front. Math. China. 2020, 15 (6): 1307-1326.
https://doi.org/10.1007/s11464-020-0889-y
We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.
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14 articles
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