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This paper introduces the basic knowledge of integral factors of first-order ordinary differential equations and Lie symmetry analysis. It then discusses the principle of constructing an integral factor of the first-order ordinary differential equation by the Lie symmetric method. Finally, it presents some examples to show the process of constructing the integral factor of the first-order ordinary differential equation and finding the general solutions.

This paper studies the supplier selection and order allocation (SS&OA) problem, where risks include a series of disruption scenarios with uncertain probability of occurrence. It is a challenge for industry decision-makers to balance the average cost and the level of risk under the ambiguity set for probabilities. To address this challenge, a two-stage distributionally robust (DR) Mean-CVaR model is presented for the SS&OA problem. A procedure is developed for constructing the ambiguity set, and Polyhedral and Box ambiguity sets are constructed to characterise the uncertain probabilities. The worst-case Mean-CVaR criterion is employed for the second-stage cost within the ambiguity set to trade off the expected cost and CVaR value. Three measures are incorporated to increase the resilience of the supply chain. The proposed robust model is reformulated into two mixed-integer linear programming models. A real case of the Huawei cell phone manufacturer is used to illustrate the validity of the proposed approach in numerical settings. Experimental results show that the new optimising approach can provide a robust SS&OA solution to immunise against the influence caused by uncertain probabilities. By comparative analyses, some management insights are obtained for industry decision-makers. [ABSTRACT FROM AUTHOR]

A ZrB2–copper–graphite composite was produced through powder metallurgy and was tested as a new electric brush material. The aim of this paper was to study the effect of ZrB2 addition on the composite's properties. Besides its physical properties such as density and resistivity, its mechanical properties, such as hardness, bending strength and wear resistance, were studied. A scanning electron microscope (SEM) was used to study the morphology of the wear surface, and a configured energy-dispersive spectrometer (EDS) was used to research the chemical composition of the samples. The results showed that, with the addition of ZrB2, the composite's properties such as density, resistivity, hardness, and bending strength improved significantly. Compared with samples without ZrB2, samples with the addition of 4% ZrB2 achieved a hardness of 87.5 HRA, which was improved by 45.8%, and a bending strength of 53.1 MPa, which was increased by nearly 50.0%. Composites with 1% content of ZrB2 showed the best wear resistance under non-conductive friction; however, under conductive friction, composites with 4% content of ZrB2 showed better wear resistance. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of bi-fractional Black–Scholes equation derived by the fractional G-Brownian motion. Then some exact solutions are obtained and the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, the new conservation theorem and the generalization of the Noether operators are developed to construct the conservation laws for bi-fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]