First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.     

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold $\overline{M}$. We first assume the pull back Weitzenböck operator of $\overline{M}$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of $\overline{M}$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when $\overline{M}$ is the space form.     

Tailorable PC71BM Isomers: Using the Most Prevalent Electron Acceptor to Obtain High‐Performance Polymer Solar Cells

Despite being widely used as electron acceptor in polymer solar cells, commercially available PC₇₁BM (phenyl‐C₇₁‐butyric acid methyl ester) usually has a “random” composition of mixed regioisomers or stereoisomers. Here PC₇₁BM has been isolated into three typical isomers, α‐, β₁‐ and β₂‐PC₇₁BM, to establish the isomer‐dependent photovoltaic performance on changing the ternary composition of α‐, β₁‐ and β₂‐PC₇₁BM. Mixing the isomers in a ratio of α/β₁/β₂=8:1:1 resulted in the best power conversion efficiency (PCE) of 7.67 % for the polymer solar cells with PTB7:PC₇₁BM as photoactive layer (PTB7=poly[[4,8‐bis[(2‐ethylhexyl)oxy]benzo[1,2‐b:4,5‐b′]dithiophene‐2,6‐diyl][3‐fluoro‐2‐[(2‐ethylhexyl)carbonyl]thieno[3,4‐b]thiophenediyl]]). The three typical PC₇₁BM isomers, even though sharing similar LUMO energy levels and light absorption, render starkly different photovoltaic performances with average‐performing PCE of 1.28–7.44 % due to diverse self‐aggregation of individual or mixed PC₇₁BM isomers in the otherwise same polymer solar cells.     

EFFECTIVE SPECIFIC HEATS OF MULTI-PHASE THERMOELASTIC COMPOSITES

This paper studies the effective properties of multi-phase thermoelastic composites. Based on the Helmholtz free energy and the Gibbs free energy of individual phases, the effective elastic tensor, thermal-expansion tensor, and specific heats of the multi-phase composites are derived by means of the volume average of free-energies of these phases. Particular emphasis is placed on the derivation of new analytical expressions of effective specific heats at constant-strain and constant-stress situations, in which a modified Eshelby’s micromechanics theory is developed and the interaction between inclusions is considered. As an illustrative example, the analytical expression of the effective specific heat for a three-phase thermoelastic composite is presented.     

The enhanced biological activities of three Zn (II) complexes based on different 1,2,4‐triazole fungicides

In order to screen effective fungicides, three Zn (II) complexes, [Zn L ¹ ₄ (NO₃)₂]·2H₂O·2EtOH ( 1 ), [Zn L ² ₄ (NO₃)₂] ( 2 ), and [Zn L ³ ₄ (DMF)₂](NO₃)₂ ( 3 ), ( L ¹  = paclobutrazol, L ²  = diniconazole, and L ³  = hexaconazole), were synthesized and characterized by elemental analysis, FT‐IR spectroscopy, and single‐crystal XRD. The antifungal activities of these complexes were then evaluated against four selected fungi using the mycelial growth rate method. The resulting data indicate that all the complexes show the enhanced antifungal activities than the corresponding ligand and mixtures. And, the interactions between the metal salt and ligands in the three complexes seem to be synergistic. According to the study of the influence of the structures on the activity, complex 2 with C=C linkage and 2,4‐dichlorophenyl moieties enhances the bioactivity significantly, especially against Wheat gibberellic ( II ). Density functional theory (DFT) calculations were carried out to help explain the enhanced bioactivity of the Zn (II) complexes. Meanwhile, all complexes are excellent grow‐regulators, especially complex 3 . The resulting data show that the complexes based on triazole fungicides have the potential applications in agriculture.     

The Bishop-Phelps-Bollobás Theorem for operators from c 0 to uniformly convex spaces

We show that the pair of Banach spaces (c ₀, Y) has the Bishop-Phelps-Bollobás property when Y is uniformly convex. Further, when Y is strictly convex, if (c ₀, Y) has the Bishop-Phelps-Bollobás property then Y is uniformly convex for the case of real Banach spaces. As a corollary, we show that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on c ₀ × ? _p (1 < p < ∞).