On the Tutte–Krushkal–Renardy polynomial for cell complexes

Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte–Krushkal–Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte–Krushkal–Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte–Krushkal–Renardy polynomial.     

Emergence of density dynamics by surface interpolation in elementary cellular automata

A classic problem in elementary cellular automata (ECAs) is the specification of numerical tools to represent and study their dynamical behaviour. Mean field theory and basins of attraction have been commonly used; however, although the first case gives the long term estimation of density, frequently it does not show an adequate approximation for the step-by-step temporal behaviour; mainly for non-trivial behaviour. In the second case, basins of attraction display a complete representation of the evolution of an ECA, but they are limited up to configurations of 32 cells; and for the same ECA, one can obtain tens of basins to analyse. This paper is devoted to represent the dynamics of density in ECAs for hundreds of cells using only two surfaces calculated by the nearest-neighbour interpolation. A diversity of surfaces emerges in this analysis. Consequently, we propose a surface and histogram based classification for periodic, chaotic and complex ECA.     

有超车规则的交通流的元胞自动机模型

分析和评论了2014年美国大学生数学建模竞赛（MCM）A题,以及获得outstanding winner奖的6篇论文。首先对试题进行分析,并结合已有文献指出A题的主要解题思路;然后,通过评述获奖论文,指出学生论文的优点及存在的问题;最后,对今年赛题的某些问题进行讨论。     

细胞神经网络的指数稳定性

该文研究了具有可变时滞的随机细胞神经网络的指数 稳定性,应用Razumikhin定理与Lyapunov函数,建立了这种细胞神经网络均方指数稳定与几乎必然指数稳定的两类判据,一类是时滞无关而另一类是时滞相关.     

Lattice games without rational strategies

We show that the lattice games of Guo and Miller support universal computation, disproving their conjecture that all lattice games have rational strategies. We also state an explicit counterexample to that conjecture: a three dimensional lattice game whose set of winning positions does not have a rational generating function.     

CO2-switchable vesicles-network structure transition and drug release property

A series of amphiphilic triblock polymers based on poly(ethylene glycol) (PEG) and two symmetrical poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) blocks was synthesized via the Atom Transfer Radical Polymerization (ATRP) method. Conductivity, pH, and viscosity tests demonstrated the CO₂-switchability jointly; Cryogenic transmission electron microscopy (Cryo-TEM), Dynamic light scattering (DLS) revealed the self-assembly morphology transformation from unilamellar vesicle to network structure when bubbling CO₂. These changes were all attributed to the protonation of tertiary amine groups in PDMAEMA blocks and the mechanism was proved by ^?H NMR. The vesicles have a relatively low release rate of drug; once stimulated by CO₂, the release rate will be accelerated. The polymeric vesicle has the possibility to find potential applications in drug delivery and release domains.     

Evolution of groups with a hierarchical structure

The universal occurrence of a hierarchical structure and its dynamic behavior in various types of group, living or abstract, are discussed. Here the word “group” refers not only to tangible aggregation but also to invisible aggregation of social psychological and of geopolitical meaning. The evolution of these groups is simulated using a model of agents distributed on the lattices of cellular grids. It is assumed that agents, fearing isolation, interact asymmetrically with each other with regard to exchange of “power”. As an indicator of hierarchy, the Gini coefficient is introduced. Example calculations are made for the aggregation, fusion and fission of animal groups, and for the appearance of a powerful empire and the rise and fall of supremacy. It is shown that such abstract objects evolve with time in accordance with the universal rules of groups common to birds and fish.     

Optimal Conditions for Existence of Vesicles Under Electric Field

Giant vesicles have been of intense interest as the model system for cell membranes. To bring numerous applications of the vesicles into full play, the existence time of the vesicles becomes particularly important. In this work, we explored the existence time of the vesicles in distilled water under four kinds of representative alternative current (AC) electric fields with different energizing time. When the application time of AC electric field reached 1 hour, the existence time of the vesicles reached the maximum value. As the applied electric field time increased, the existence time of the vesicles decreased. Thus, after obtaining the best vesicles on the carbon fiber electrode, disconnecting the external electric field is very necessary. Based on the in situ observation using inverted microscope, the disappearing phenomena of the vesicles were presented. In addition, it is found that vesicles in phosphate buffer solutions (PBSs) can last about 2–3 days when the electric field was applied for 1 hour, which was much longer than that in distilled water.