Search

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.

Brassinosteroids (BRs) are essential plant hormones. In angiosperms, brassinolide and castasterone, the first and second most active BRs, respectively, are synthesised by CYP85A2 and CYP85A/A1, respectively. BRs in angiosperms function through an essential receptor, BR Insensitive 1 (BRI1). In addition, some angiosperms also have non-essential BRI1-like 1/3 (BRL1/3). In conifers, BRs promote seed germination under drought stress; however, how BRs function in gymnosperms is unknown. In this study, we performed functional complementation of BR biosynthesis and receptor genes from Picea abies with respective Arabidopsis mutants. We found that P. abies possessed functional PaCYP85A and PaBRL1 but not PaCYP85A2 or PaBRI1, and this results in weak BR signaling, and both PaCYP85A and PaBRL1 were abundantly expressed. However, neither BR treatment of P. abies seedlings nor expression of PaBRL1 in the Arabidopsis Atbri1 mutant promoted plant height, despite the fact that BR-responsive genes were activated. Importantly, chimeric AtBRI1 replaced with the BR-binding domain of PaBRL1 complemented the Atbri1 phenotypes. Furthermore, PaBRL1 had less kinase activity than BRI1 in vitro. Overall, P. abies had weak but still active BR signaling, explaining aspects of its slow growth and high stress tolerance. Our study sheds light on the functional and evolutionary significance of distinct BR signaling that is independent of BRI1 and brassinolide.

This paper discusses how to compute all real solutions of the second-order cone tensor complementarity problem when there are finitely many ones. For this goal, we first formulate the second-order cone tensor complementarity problem as two polynomial optimization problems. Based on the reformulation, a semidefinite relaxation method is proposed by solving a finite number of semidefinite relaxations with some assumptions. Numerical experiments are given to show the efficiency of the method.

$$\mathbf {P}$$-tensor and $$\mathbf {P}_0$$-tensor are introduced in tensor complementarity problem, which have wide applications in game theory. In this paper, we establish SDP relaxation algorithms for detecting $$\mathbf {P}(\mathbf {P}_0)$$-tensor. We first reformulate $$\mathbf {P}(\mathbf {P}_0)$$-tensor detection problem as polynomial optimization problems. Then we propose the SDP relaxation algorithms for solving the reformulated polynomial optimization problems. Numerical examples are reported to show the efficiency of the proposed algorithms.

A real symmetric matrix (resp., tensor) is said to be copositive if the associated quadratic (resp., homogeneous) form is greater than or equal to zero over the nonnegative orthant. The problem of detecting their copositivity is NP-hard. This paper proposes a complete semidefinite relaxation algorithm for detecting the copositivity of a matrix or tensor. If it is copositive, the algorithm can get a certificate for the copositivity. If it is not, the algorithm can get a point that refutes the copositivity. We show that the detection can be done by solving a finite number of semidefinite relaxations, for all matrices and tensors., Comment: 16 pages

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the contrast, every nonsymmetric tensor has finitely many H-eigenvalues. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every nonsymmetric tensor that has finitely many real Z-eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every nonsymmetric tensor, we can compute all its real H-eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations. Various examples are demonstrated., Comment: 19 pages

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.

A straightforward synthetic strategy toward [1,2,5]thiadiazole-fused heteroacences via the Pd-catalyzed carbonylative (PdCC) annulation utilizing the building blocks of 5,6-substituted benzo[2,1,3- b ]thiadidazole (BT) and 1 atm CO has been successfully developed. Preliminarily, solution-processed Field-effect transistor (FET) devices based on FTC8 exhibited a promising mobility up to 0.177 cm 2 /(Vs) with a current on/off ratio of nearly 1.2 × 10 5 under an ambient atmosphere. This chemistry opens new perspectives toward high performance heteroacenes.

The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint, and then convert the problem of computing the largest unitary eigenvalue to a real equality constrained polynomial optimization problem, which can be solved by the Jacobian semidefinite programming relaxation method. Numerical examples are presented to show the availability of this approach.

A facile synthetic strategy toward thieno[3,2-b]indole (TI) derivatives was developed by the Cadogan annulation method. Three donor–acceptor (D–A) conjugated polymers P1, P2 and P3 containing N-alkyl-TI derivatives and 4,7-dithien-5-yl-2,1,3-benzodiathiazole (DTBT) derivatives were successfully synthesized and applied to bulk heterojunction (BHJ) polymer solar cells (PSCs). Different side chains were introduced to TI units (for P2) or DTBT units (for P3), the results indicate that the bandgaps, energy levels and photovoltaic performance were finely tuned by the side chains in the TI-DTBT copolymer. Power conversion efficiencies (PCEs) based on the device structure of ITO/PEDOT:PSS/polymer:PC71BM/Ca/Al exhibit a large distinction (1.61% for P1, 0.53% for P2 and 2.73% for P3) under optimal device fabrication conditions. The optimized devices based on P3:PC71BM blends with a relatively higher mobility (2.84 × 10−5 cm2 V−1 s−1) show the best PCE under air mass 1.5 global (AM 1.5 G) irradiation of 100 mW cm−2, which is in good agreement with its high current density and light absorption property. Accordingly, the TI unit can be used as the efficient donor units for D–A conjugated donor materials for PSCs application. In addition, this work suggests that the side chains on low bandgap polymers significantly impact their molecular energy levels and the observed PCEs of the corresponding BHJ solar cells.

A universal synthetic strategy toward thieno[3,2-b]indole (TI) derivatives was developed. Three conjugated polymers (PTIBT, PTITBT and PTIDTBT) containing N-alkyl-TI as the donor units, 2,1,3-benzodiathiazole (BT) as the acceptor units and thiophene as the spacers were synthesized. The thiophene spacers have a dramatic impact on the physical and electrochemical properties of these copolymers. These polymer donors were used for the fabrication of bulk heterojunction polymer solar cells (PSCs). Preliminarily, power conversion efficiencies (PCEs) based on the device structure of ITO/PEDOT:PSS/polymer:PC71BM/Ca/Al exhibit a large distinction (1.61% for PTIBT, 5.83% for PTITBT and 1.79% for PTIDTBT) at optimal device fabrication conditions. The device based on PTITBT : PC71BM (1 : 3, w/w) shows the best PCE of 5.83% (Voc = 0.69 V, Jsc = 13.92 mA cm−2, FF = 61.8%), which represents one of the best performances among PCDTBT analogues. In addition, the Jsc of 13.92 mA cm−2 is also among the highest Jsc values of all PCDTBT analogues. On the basis of our results, one can conclude that incorporating TI and its derivatives into donor–acceptor conjugated polymers is a feasible and effective way to develop novel donor materials for high efficiency PSC applications.

Comon's Conjecture claims that for a symmetric tensor, its rank and its symmetric rank coincide. We show that this conjecture is true under an additional assumption that the rank of that tensor is not larger than its order. Moreover, if its rank is less than its order, then all rank decompositions are necessarily symmetric rank decompositions.