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All biological functions evolve by fixing beneficial mutations and removing deleterious ones. Therefore, continuously fixing and removing the same essential function to separately diverge monophyletic gene families sounds improbable. Yet, here we report that brassinosteroid insensitive1 kinase inhibitor1 (BKI1)/membrane-associated kinase regulators (MAKRs) regulating a diverse function evolved into BKI1 and MAKR families from a common ancestor by respectively enhancing and losing ability to bind brassinosteroid receptor brassinosteroid insensitive1 (BRI1). The BKI1 family includes BKI1, MAKR1/BKI1-like (BKL) 1, and BKL2, while the MAKR family contains MAKR2-6. Seedless plants contain only BKL2. In seed plants, MAKR1/BKL1 and MAKR3, duplicates of BKL2, gained and lost the ability to bind BRI1, respectively. In angiosperms, BKL2 lost the ability to bind BRI1 to generate MAKR2, while BKI1 and MAKR6 were duplicates of MAKR1/BKL1 and MAKR3, respectively. In dicots, MAKR4 and MAKR5 were duplicates of MAKR3 and MAKR2, respectively. Importantly, BKI1 localized in the plasma membrane, but BKL2 localized to the nuclei while MAKR1/BKL1 localized throughout the whole cell. Importantly, BKI1 strongly and MAKR1/BKL1 weakly inhibited plant growth, but BKL2 and the MAKR family did not inhibit plant growth. Functional study of the chimeras of their N- and C-termini showed that only the BKI1 family was partially reconstructable, supporting stepwise evolution by a seesaw mechanism between their C- and N-termini to alternately gain an ability to bind and inhibit BRI1, respectively. Nevertheless, the C-terminal BRI1-interacting motif best defines the divergence of BKI1/MAKRs. Therefore, BKI1 and MAKR families evolved by gradually gaining and losing the same function, respectively, extremizing divergent evolution and adding insights into gene (BKI1/MAKR) duplication and divergence.

Shandong Peninsula is an important economic development area in China, and its superior geographical location and abundant resources provide a good foundation for rapid economic development. In order to further enhance the economic strength of Shandong Peninsula and the regional competitiveness of Shandong Peninsula city cluster, it is inevitable to strengthen the financial and fiscal support for the peninsula, integrate the financial structure and further expand the Shandong Peninsula Blue Economy Demonstration Zone within the national economic circle. influence. Starting from the significant problems in financial support on the Shandong Peninsula, this paper presents the problems of the peninsula city cluster in three aspects: financial resources, financial supervision, and urban-rural integration. The corresponding solution measures are also proposed to add another hand to the economic level of Shandong Peninsula.

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 second-order cone tensor eigenvalue complementarity problem. We reformulate second-order cone tensor eigenvalue complementarity problem as two constrained polynomial optimizations. For these two reformulated optimizations, Lasserre-type semidefinite relaxation methods are proposed to compute all second-order cone tensor complementarity eigenpairs. The proposed algorithms terminate when there are finitely many second-order cone complementarity eigenvalues. Numerical examples are reported to show the efficiency of the proposed algorithms.

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimzers and the same optimal value. In particular, we introduce a class of lifted stationary point of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that the whole sequence generated by SPG algorithm converges to a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm., 19 pages

In this paper, an SDP relaxation algorithm is proposed to test the copositivity of higher order tensors. By solving finitely many SDP relaxations, the proposed algorithm can determine the copositivity of higher order tensors. Furthermore, for any copositive but not strictly copositive tensor, the algorithm can also check it exactly. Some numerical results are reported to show the efficiency of the proposed algorithm.

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

Polynomial optimization problem with second-order cone complementarity constraints (SOCPOPCC) is a special case of mathematical program with second-order cone complementarity constraints (SOCMPCC). In this paper, we consider how to apply Lasserre's type of semidefinite relaxation method to solve SOCPOPCC. To this end, we first reformulate SOCPOPCC equivalently as a polynomial optimization and then solve the reformulated polynomial optimization with semidefinite relaxation method. For a special case of SOCPOPCC, we present another reformulation of polynomial optimization, which is of lower degree. SDP relaxation method is applied to solve the new polynomial optimization. Numerical examples are reported to show the efficiency of our proposed method.

Increasing building constructions have become one of the fastest-growing drivers of carbon emissions. Energy conservation and carbon reduction in buildings have become increasingly crucial in the context of global carbon neutrality. This paper assesses the annual total energy and carbon embodied in the ten most intensively used building materials in China, aiming to find potential CO2 reduction opportunities in the construction industry from a macroscopic perspective. The results show that: (1) the embodied energy and carbon of cement, steel, and brick account for more than 70% of the total embodied energy and carbon of all building materials; (2) differences in the embodied energy and carbon between steel-concrete buildings and brick-concrete buildings are not distinct; (3) disparities in the embodied energy and carbon of building materials between different regions are significant. The eastern and south-eastern regions consume excessive building materials and embody significantly higher energy and carbon than other regions. Several strategies are provided for China's building sector in energy and carbon reduction.

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

In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. For a third order tensor, its CP decomposition can be regarded as a special case of its triple decomposition. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank, and is strictly less than the middle value of the Tucker rank for an essential class of examples. These indicate that practical data can be approximated by low rank triple decomposition as long as it can be approximated by low rank CP or Tucker decomposition. This theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and video image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method.

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.

In this paper, we consider the positive semi-definite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semi-definite space tensor cone, give an error bound for the polyhedral outer approximation approach, and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem. We then study some other approaches for solving this structured convex program. These include the conic linear programming approach, the nonsmooth convex program approach and the bi-level program approach. Some numerical examples are presented.

A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, in order to get rid of the sign constraints. We study the first- and second-order optimality conditions of the original StBQP and the reformulated bi-quartic problem over the product of two Euclidean spheres. Furthermore, we discuss the one-to-one correspondence between the global/local solutions of StBQP and the global/local solutions of the reformulation. We introduce a continuously differentiable penalty function. Based upon this, the original problem is converted into the problem of locating an unconstrained global minimizer of a (specially structured) polynomial of degree eight.

This paper studies the relationship between the so-called bi-quadratic optimization problem and its semidefinite programming (SDP) relaxation. It is shown that each r-bound approximation solution of the relaxed bi-linear SDP can be used to generate in randomized polynomial time an $${\mathcal{O}(r)}$$ -approximation solution of the original bi-quadratic optimization problem, where the constant in $${\mathcal{O}(r)}$$ does not involve the dimension of variables and the data of problems. For special cases of maximization model, we provide an approximation algorithm for the considered problems.