International Journal of Mathematics for Industry

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.

In this paper, we develop the Fudan-CCDC model by adding a source term to describe the imported infectors. The model is applied to analyze the situation of COVID-19 in East Asia, and then in Singapore. By data fitting, our model reveals that Singapore has a much higher isolation rate and earlier quarantine measures compared to other countries. We conclude that Singapore has been doing extraordinarily well on epidemic prevention and control. Finally we discuss the specific measures in Singapore’s success, and suggest other countries to learn from the Singapore’s style, so as to be well prepared in the future.

All rotating blades shed helical vortices which have a significant effect on the velocity over the blades and the forces acting on them. Nevertheless, knowledge of vortex behavior is not used in blade element theory (BET), the most common method to calculate the thrust produced by propellers and the power by wind turbines. Helical vortices of constant pitch and radius are also of fundamental interest as one of only three geometries that do not deform under their “self-induced” motion. This aspect of vortex theory is reviewed historically and the relationship with the forces acting on submerged bodies briefly reviewed. The development of helical vortex theory (HVT) in the 20th century is then described. It is shown that HVT allows BET to be used for a number of important problems that cannot be analyzed by current versions of the theory.

We consider the non-hermitian matrix-valued process of Elliptic Ginibre Ensemble. This model includes Dyson's Brownian motion model and the time evolution model of Ginibre ensemble by using hermiticity parameter. We show the complex eigenvalue processes satisfy the stochastic differential equations which are very similar to Dyson's model and give an explicit form of overlap correlations. As a corollary, in the case of 2-by-2 matrix, we also mention the relation between the diagonal overlap, which is the speed of eigenvalues, and the distance of the two eigenvalues.

Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space curve and the projected curves from several distinct directions, and (2) a surface and the apparent contours of projections from several distinct directions, in terms of differential geometry and singularity theory. In particular, formulae for recovering certain information on the original curves or surfaces from their projected images are given.

Since the global warming has recently become more severe causing many serious changes on the weather, economy, and society worldwide, lots of efforts have been put forward to prevent it. As one of the most important energy sectors, improvements in electric power grids are required to address the challenge of suppressing the carbon emission during electric generation especially when utilizing fossil-based fuels, while increasing the use of renewable and clean sources. This paper hence presents a novel optimization model for tackling the problems of optimal power scheduling and real-time pricing in the presence of a carbon constraint while taking into account a demand response possibility, which may provide a helpful method to limit the carbon emission from conventional generation while promoting renewable generation. The critical aspects include explicitly integrating the cost of emission with the total generation cost of conventional generation and combining it with the consumer satisfaction function. As such, conventional generation units must carefully schedule their power generation for their profits, while consumers, with the help from renewable energy sources, are willing to adjust their consumption to change the peak demand. Overall, a set of compromised solution called the Pareto front is derived upon which the conventional generating units choose their optimal generation profile to satisfy a given carbon constraint.

A graph is a mathematical model used to predict the topology of a given system. In chemical graph theory, a graph is designed by considering atoms as vertices and edges as bonds between atoms of a particular molecule. A topological index or molecular structure descriptor is a numeric quantity associated with the chemical constitution which correlated with various physiochemical properties of the chemical structure. In this paper, we study the $(a,b)$-Zagreb index of line graphs of the subdivision graphs of some chemical structures.

Motivated by an experimental result on the shape of liquid water supported by a small stainless helical wire, we find a class of stable helicoidal convex surfaces with constant mean curvature whose boundary consists of a single helix and two short arcs.

Media plays a vital role in disseminating any crucial information to the society. This helps the unaware population of the society to gain awareness regarding a particular information. But many times it is seen that, even though an information is being transmitted through the print or electronic media for a very long period of time, many youngsters although aware about the negative aspects of the information, act as unaware and sometimes face a fatal result as in the case of the Blue Whale Game.

In this paper, we have proposed a mathematical model depicting the effect of transmitting an information by media, particularly “Blue Whale Game”. We have concluded with the help of some simple calculations and figures, that because of the prolonged repeated transmission by the media regarding the blue whale game, curiousness got generated among the young generation and they tried to know more and more about the game, which in turn made them follow the game as designed by the game developer and in many cases turned out to be lethal.