The recession cone and recession function are very important research objects in Convex Analysis. They have extensive applications in the optimization theory. Firstly, we study the properties of the recession cone and recession function. The positive homogeneity and subadditivity of recession function are mainly discussed. And the different methods are considered to prove these properties. Secondly, we discuss the unboundedness of the convex sets and convex functions by using recession cone and recession function.

This article explores the properties of Fibonacci sequences and their widespread applications.

Heat transfer fluids (HTFs) are critical in numerous industrial processes (e.g., the chemical industry, oil and gas, and renewable energy), enabling efficient heat exchange and precise temperature control. HTF degradation, primarily due to thermal cracking and oxidation, negatively impacts system performance, reduces fluid lifespan, and increases operational costs associated with correcting resulting issues. Regular monitoring and testing of fluid properties can help mitigate these effects and provide insights into the health of both the fluid and the system. To date, there is no extensive literature published on this topic, and the current narrative review was designed to address this gap. This review outlines the typical operating temperature ranges for industrial heat transfer fluids (i.e., steam, organic, synthetic, and molten salts) and then focuses specifically on organic and synthetic fluids used in industrial applications. It also outlines the mechanisms of fluid degradation and the impact of fluid type and condition. Other topics covered include the importance of fluid sampling and analysis, the parameters used to assess the extent of thermal degradation, and the management strategies that can be considered to help sustain fluid and system health. Operating temperature, system design, and fluid health play a significant role in the extent of thermal degradation, and regular monitoring of fluid properties, such as viscosity, acidity, and flash point, is crucial in detecting changes in condition (both early and ongoing) and providing a basis for decisions and interventions needed to mitigate or even reverse these effects. This includes, for example, selecting the right HTF for the specific application and operating temperature. This article concludes that by understanding the mechanisms of thermal degradation and implementing appropriate management strategies, it is possible to sustain the lifespan of thermal fluids and systems, ensure safe operation, and help minimise operational expenditure.

Based on the application of phase diagram calculation technique (CALPHAD), the Fe-Nd-B magnetic materials were investigated, and alloy design and microstructure evolution concerning. According to the thermodynamic database of Fe-Nd-B ternary system, the equilibrium solidification process of Fe78Nd15B7 alloy is simulated, and we explained well the reason of this experimental phenomenon by the metastable extension of the equilibrium phase diagram.

The Method of Discretization in Time (MDT) is a hybrid numerical technique intended to alleviate upfront the computational procedure of timedependent partial differential equations of parabolic type upfront. The MDT engenders a sequence of adjoint second order ordinary differential equations, wherein the space coordinate is the independent variable and time becomes an embedded parameter. Essentially, the adjoint second order ordinary differential equations are considered of “quasistationary” nature. In this work, the MDT is used for the analysis of unsteady heat conduction in regular bodies (large wall, long cylinder and sphere) accounting for nearly constant thermophysical properties, uniform initial temperature and surface heat flux. In engineering applications, the surface heat flux is customarily provided by electrical heating, radiative heating and pool fire heating. It is demonstrated that the approximate, semianalytical temperature solutions of the first adjoint “quasistationary” heat conduction equations using the first time jump are easily obtainable for each regular body. For enhanced acccuracy, regression analysis is applied to the deviations of the dimensionless surface temperature as a function of the dimensionless time for each regular body.

The effective drainage radius of coal seam is an important basis for the spacing of pre-drainage gas boreholes. To quickly and accurately determine the effective drainage radius, a new method was proposed. For the coal face where the desorbable gas content before mining has met the standard, the compliance of mine gas drainage rate was used as the basis to determine the effective drainage radius. The fluid-structure interaction model was constructed, numerical simulation of coal seam gas drainage was carried out by using COMSOL software, and the model was validated by combining the field test results. The results show that the new method has the advantage of short cycle. With the extension of drainage time, the increase of effective drainage radius gradually decreases, and finally reaches a relatively stable limit value, which conforms to the Langmuir function. The average error between numerical simulation and field test values of effective drainage radius is 4.9%, which proves that the model is reliable. This model can accurately predict the effective drainage radius under different coal seam gas contents and drainage times. The research results provide a new mean for determining the effective drainage radius of coal seam and the layout of gas drainage boreholes.