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In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

The brown rat (Rattus norvegicus) is found wherever humans live and transmits many diseases, and its breeding produced the laboratory rat used widely in medical research. Here, we sequenced whole genomes from 118 rats to explore the origin and dispersal routes of the brown rat and the domestication of the laboratory rat. We showed that brown rats migrated about 3600 years ago from southern East Asia, rather than Northern Asia as formerly suggested, to the Middle East and then to Europe and Africa. Many genes involved in the immune system experienced positive selection in the wild brown rat, while genes involved in the nervous system and energy metabolism showed evidence of artificial selection during the domestication of laboratory strains. Our findings demystify the puzzling origin and migration of brown rats and reveal the impact of evolution and domestication on this animal.

A nonlinear finite element model of fixed ended cold-formed steel stiffened lipped channel short columns with and without perforations was developed and verified against experiment results. Material and geometrical non-linearities were included in the finite element model. It was proved by comparison analysis that using weighted average tensile yield stresses of the flat and corner areas and the mean geometrical imperfections along the length of the specimens was suitable for predicting the failure mode, ultimate load and load versus displacement curves of the channel short column. The results got by FEA coincided with the experimental results.

Plate-like Bi2.5Na3.5Nb5O18 particles were used as templates to fabricate grain-oriented Na0.5Bi0.5TiO3—BaTiO3 (NBTBT) ceramics by reactive-templated grain growth. The effects of sintering conditions on the grain orientation and microstructure of the textured NBTBT ceramics were investigated, and the kinetic mechanism of grain growth is discussed. The results show that textured ceramics were successfully obtained with orientation factor more than 0.6. NBTBT specimens are composed of strip-like grains and equiaxed shaped grains. The textured ceramics have a microstructure with strip-like grains aligning in the direction parallel to the casting plane and exhibit an {h00} preferred orientation. The degree of grain orientation increases initially, then decreases with increasing sintering temperature and soaking time. The maximum texture fraction is 0.69 when sintered at 1185 °C for 6 h. The kinetic exponent n and activation energy Q of the two types of grain in textured NBTBT ceramics were calculated. The results show that the grain growth mechanism of oriented grains is controlled not only by grain lattice diffusion, but also by grain boundary diffusion. The grain growth mechanism of matrix grains is mainly controlled by the grain boundary curvature.

The textured (BiNa)0.5TiO3–BaTiO3 ceramic with a preferred 〈h00〉 orientation was fabricated by tape casting with the plate-like Bi2.5Na3.5Nb5O18 (BINN5) particles as template. Effects of processing parameters and the content of template on the microstructure and the texture fraction of NBT-based ceramics were investigated. The results show that the texture fraction increases with increasing sintering temperature and the increasing content of BINN5. The increased sintering temperature and BINN5 content result in the grain size and the grain shape changing from irregular to plate like. These factors were responsible for the increased degree of orientation. The texture fraction had a maximum value when the sintering temperature is 1225 °C and the optimal content of template is 20 wt.%.

Grain-oriented Na0.5Bi0.5TiO3-BaTiO3 (NBTBT) ceramics were fabricated by reactive-templated grain growth using plate-like Bi2.5Na3.5Nb5O18 (BNN) as templates. The specimens are composed of NBTBT perovskite phase and BNN lay-structured phase. Textured ceramics have a brick-wall microstructure with strip-like grains aligning in the direction parallel to the casting plane and exhibit an {h00} preferred orientation. The texture fraction increases initially, and then decreases with increasing sintering temperature. The optimal sintering temperature is 1,185 °C where the texture fraction has a maximum value of 0.58 and d33 is 98 pC/N. The textured NBTBT ceramics show evidence of relaxor ferroelectrics with diffuse phase transition and frequency dispersion because of composite biphasic structure.

One-dimensional needle-like KSr 2 Nb 5 O 15 (KSN) and two-dimensional plate-like Bi 2.5 Na 3.5 Nb 5 O 18 (BNN) particles were used as templates to fabricate grain-oriented Na 0.5 Bi 0.5 TiO 3 –BaTiO 3 (NBTBT) ceramics by reactive-templated grain growth. The effects of the template concentration on the microstructure and orientation of NBTBT ceramics were investigated, and the mechanism of grain growth was discussed. The results show that NBTBT textured ceramics cannot be obtained with KSN template, since the needle-like KSN particles were aligned randomly along the tape casting direction. Ceramics contain perovskite NBTBT and tungsten–bronze-type KSN phase. Some cucurbit-like KSN grains resulted from the defects of needle-like KSN templates were detected. Textured ceramics with orientation factor more than 60% were obtained successfully when the plate-like BNN was used as templates. The results show that the texture fraction of NBTBT textured ceramics increase with increasing the content of BNN. The textured ceramics exhibited { h 0 0} preferred orientation have brick wall microstructure with strip-like grains aligning in the same plane as the casting plane. The reaction and crystal formation of textured NBTBT ceramics with plate-like BNN templates are by diffusion–recrystallization process. Finally the selecting rules of suitable templates are proposed.