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A new reliable algorithm for computing the H2-norm of linear time-varying periodic (LTP) systems via the periodic Lyapunov differential equation (PLDE) is proposed. By taking full advantage of the periodicity, the transition matrix of the underlying LTP system associated with the PLDE is effectively computed by developing a novel extended precise integration method based on Fourier series expansion, where the time-consuming work for the computation of the matrix exponential and its related integrals in every sub-interval is avoided. Then, a highly accurate and efficient algorithm for the PLDE is derived using the block form of the transition matrix. Thus, the H2-norm is evaluated by solving a simple first-order ordinary differential equation. Finally, two numerical examples are presented and compared with other algorithms to verify the numerical accuracy and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The linear nonhomogeneous ordinary differential equations with two-point boundary value problems (TPBVP) are solved by using homogeneous equations precise integration method and nonhomogeneous term precise integration method. The matrix exponent method and interval mixed energy method (Riccati method) are used to transform the two-point boundary value problems into initial value problems, and then the general initial value problems can be solved by using precise integration method of recursive schemes. The method presented can avoid solving inversion of system matrix, and with small computational cost the numerical computation precision of nonhomogeneous term can attain to the numerical computation precision of homogeneous general solution by using precise integration method. The validity of the method presented is proved by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: (Nd0.4Pr0.6)9Fe76−x Nb x B15 (x=0–5) flakes were fabricated by slow quenching from master alloys. Effects of Nb on crystallization behavior and magnetic properties of flakes have been systematically investigated. For the sample with x=0, the metastable compound (Nd,Pr)2Fe23B3 crystallized as a precursor from amorphous state and then transformated to the (Nd,Pr)2Fe14B and α-Fe phases, and the (Nd,Pr)1.1Fe4B4 phase was formed on further increasing annealing temperature to 700°C. So the crystallization process of the (Nd0.4Pr0.6)9Fe76B15 alloy was as follows: amorphous phase (AP)→AP′+(Nd,Pr)2Fe23B3→(Nd,Pr)2Fe14B+α-Fe→(Nd,Pr)2Fe14B+α-Fe+(Nd,Pr)1.1Fe4B4. For the sample with x=4, the α-Fe phase was first formed from amorphous matrix, and then the (Nd,Pr)2Fe14B and Fe3B phases subsequently crystallized from residual amorphous phase. So the crystallization process of the (Nd0.4Pr0.6)9Fe72Nb4B15 alloy was as follows: amorphous phase (AP)→α-Fe+AP′ →α-Fe+Fe3B+(Nd,Pr)2Fe14B. The addition of Nb to (Nd0.4Pr0.6)9Fe76B15 alloy could suppress the formation of the metastable (Nd,Pr)2Fe23B3 phase and nonmagnetic (Nd,Pr)1.1Fe4B4 phase. Meanwhile, microstructure of (Nd0.4Pr0.6)9Fe76B15 flakes was significantly refined by introducing Nb. After optimal annealing treatment, the intrinsic coercivity increased from 397.3kA/m for the Nb-free sample to 1297kA/m for the 5at% Nb-doped sample. [Copyright &y& Elsevier]

Metal-organic frameworks (MOFs) are very promising self-sacrificing templates for the large-scale fabrication of new functional materials owing to their versatile functionalities and tunable porosities. Most conventional metal oxide electrodes derived from MOFs are limited by the low abundance of incorporated metal elements. This study reports a new strategy for the synthesis of multicomponent active metal oxides by the pyrolysis of polymetallic MOF precursors. A hollow N-doped carbon-coated ZnO/ZnCo 2 O 4 /CuCo 2 O 4 nanohybrid is prepared by the thermal annealing of a polymetallic MOF with ammonium bicarbonate as a pore-forming agent. This is the first report on the rational design and preparation of a hybrid composed of three active metal oxide components originating from MOF precursors. Interestingly, as a lithium-ion battery anode, the developed electrode delivers a reversible capacity of 1742 mAh g -1 after 500 cycles at a current density of 0.3 mA g -1 . Furthermore, the material shows large storage capacities (1009 and 667 mAh g -1 ), even at high current flow (3 and 10 A g -1 ). The remarkable high-rate capability and outstanding long-life cycling stability of the multidoped metal oxide benefits from the carbon-coated integrated nanostructure with a hollow interior and the three active metal oxide components., (© 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)