“Mathematics and Symmetry/Asymmetry Section”—Editorial 2020–2021

As it is well known, the Mathematics and Symmetry/Asymmetry Section is one of the most active sections of the Symmetry journal. Along these lines, we want to state a brief summary of the activities done in 2022, and there is no better picture than by selecting 10 representative published works. These works properly cover the scope and type of papers that we have published this season.

In paper [1], a recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how to build such “dynamical polyhedra” for other icosahedral maps. First, algebra associated with the icosahedron determines a special family of maps with 60 periodic critical points. The topological behavior of each map is then worked out and results in a geometric algorithm, out of which emerges a system of edges—the dynamical polyhedron—in natural correspondence to a map’s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps’ geometric behavior fall into combinatorial classes, the presentation of which concludes the paper.

In paper [2], the effect of Magneto Hydro-Dynamics (MHD) on a polymer chain in the micro channel is studied by employing the Dissipative Particle Dynamics simulation (DPD) method.

In [3], a linear dynamical system under the action of potential and circulatory forces is considered. The matrix of potential forces is positive definite, and the main question is when the circulatory forces induce instability to the system. Different approaches to studying the problem are discussed and illustrated by examples. The case of multiple eigenvalues is also considered, and sufficient conditions of instability are obtained. Some issues of the dynamics of a nonlinear system with an unstable linear approximation are discussed. The behavior of trajectories in the case of unstable equilibrium is investigated, and an example of the chaotic behavior versus the case of bounded solutions is presented and discussed.

Paper [4] deals with the recent fact that stem cell transplantation therapy may inhibit inflammation during stroke and increase the presence of healthy cells in the brain. The novelty of this work, is to introduce a new mathematical model of stem cells transplanted to treat stroke.

Recent advances in experimental biology studies have produced large amounts of molecular activity data. In particular, individual patient data provide non-time series information for the molecular activities in disease conditions. The challenge is how to design effective algorithms to infer regulatory networks using the individual patient datasets and consequently address the issue of network symmetry. the objective of paper [5] is to develop an efficient pipeline to reverse-engineer regulatory networks based on the individual patient proteomic data.

Paper [6] presents a novel approach to improve the efficiency of resource allocation by means of different perspectives and ways of thinking. At the same time, paper [7] deals with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs).

The objective of paper [8] is to use a game-theoretic dynamic procedure to establish a mechanism that can be dynamically modified under relative symmetry at any time during operational processes. At paper [9], the axially symmetric propagation of bending waves in a thin Timoshenko-type cylindrical shell, interacting with a nonlinear elastic Winkler medium, is studied.

One fundamental step towards grasping the global dynamic structure of a population system involves characterizing the convergence behavior (specifically, how to characterize the convergence behavior). Finally, paper [10] focuses on the neutral functional differential equations arising from population dynamics.

We hope you enjoyed the topics in this section and that they motivate you to submit in the near future. You are very welcome!