Least squares X=±Xη* solutions to split quaternion matrix equation AXAη*=B

Liu, Xin; Zhang, Yang

doi:10.1002/mma.6033

Cited by 11 publications

(2 citation statements)

References 22 publications

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“…According to the definitions of the Frobenius norm of the split quaternion matrix in the papers [22,24,28,29], we obtain the following results:…”

Section: Proposition 23 ([17]) Suppose That a B ∈ H N×nmentioning

confidence: 99%

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Jiang,

Wang,

Guo

et al. 2024

Math Methods in App Sciences

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With the breakthroughs made by physicists in high‐dimensional mathematics, it has become possible to represent and solve a number of classical mathematical physics problems using the split quaternion algebra. In this paper, we study the least squares approximation of a class of Schrödinger equations in split quaternionic mechanics and propose two algebraic algorithms to the generalized right eigen‐problem for an i‐Hermitian split quaternion matrix pencil by using two isomorphic mappings. Numerical examples show the effectiveness of the proposed theories and algorithms.

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“…According to the definitions of the Frobenius norm of the split quaternion matrix in the papers [22,24,28,29], we obtain the following results:…”

Section: Proposition 23 ([17]) Suppose That a B ∈ H N×nmentioning

confidence: 99%

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Jiang,

Wang,

Guo

et al. 2024

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…In addition, the character of highly localized two‐photon excitation at the focal point facilitates a variety of biological applications requiring high spatial and temporal resolution [7, 8] . As a consequence, great efforts have been made to develop new materials for TPEF imaging [9] . It is worth noting that a suitable TPEF material usually requires a large two‐photon action cross section ( δ TPA Φ em ), which defines the high two‐photon brightness.…”

Section: Introductionmentioning

confidence: 99%

Multipolar Porphyrin‐Triazatruxene Arrays for Two‐Photon Fluorescence Cell Imaging

Zhang

Gong

Zhang

et al. 2020

Chemistry A European J

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Two-photon excitedf luorescent (TPEF) materials are highly desirable for bioimaging applications owing to their unique characteristics of deep-tissue penetration and high spatiotemporal resolution. Herein, by connectingo ne, two, or three electron-deficient zinc porphyrin units to an electron-rich triazatruxene core via ethynyl p-bridges, conjugated multipolar molecules TAT-(ZnP) n (n = 1-3) were developed as TPEF materialsf or cell imaging. The three new dyes presenth igh fluorescence quantum yields (0.40-0.47) and rationally improved two-photon absorption (TPA) properties. In particular, the peak TPAc ross sectiono fTAT-ZnP (436 GM) is significantly largert han that of the ZnP reference (59 GM). The d TPA values of TAT-(ZnP) 2 and TAT-(ZnP) 3 further increase to 1031 and up to 1496 GM, respectively,i ndicatingt he effect of incorporated ZnP units on the TPA properties. The substantial improvemento ft he TPEF properties is attributed to the formation of p-conjugated quadrapole/octupole molecules and the extensiono fD-p-AD systems,w hich has been rationalized by density function theory(DFT) calculations. Moreover, all of the three new dyes displayg ood biocompatibility and preferential targeting ability towardc ytomembrane, thusc an be superior candidates for TPEF imaging of living cells. Overall, this work demonstratedapromisings trategy for the development of porphyrin-based TPEF materials by the construction and extensiono fD-p-AD multipolar array.

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The generalized quaternion matrix equation AXB+CX^⋆D=E

Liu

Zhang

2020

Math Methods in App Sciences

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In this paper, we discuss the generalized quaternion matrix equation AXB+CX⋆D=E, where X⋆ is one of X, X*, the η‐conjugate or the η‐conjugate transpose of X with η∈{i,j,k}. Two new real representations of a generalized quaternion matrix are proposed. By using this method, the criteria for the existence and uniqueness of solutions to the mentioned matrix equation as well as the existence of X=± X⋆ solutions to the generalized quaternion matrix equation AXB+CXD=E are derived in a unified way.

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Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

Cited by 11 publications

References 22 publications

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Multipolar Porphyrin‐Triazatruxene Arrays for Two‐Photon Fluorescence Cell Imaging

The generalized quaternion matrix equation AXB+CX^⋆D=E

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Least squares X=±Xη* solutions to split quaternion matrix equation AXAη*=B

Cited by 11 publications

References 22 publications

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Algebraic algorithms for a class of Schrödinger equations in split quaternionic mechanics

Multipolar Porphyrin‐Triazatruxene Arrays for Two‐Photon Fluorescence Cell Imaging

The generalized quaternion matrix equation AXB+CX⋆D=E

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Least squares X=±X^η* solutions to split quaternion matrix equation AXA^η*=B

The generalized quaternion matrix equation AXB+CX^⋆D=E