Search

Let ℳ (n , c) be the class of Moran sets with integer n ≥ 2 and real c satisfying n c < 1. It is well known that the Hausdorff dimension of any set in this class is s = − log c n. We show that for any E ∈ ℳ (n , c) , 2 s 2 (n − 1) c 1 − c s ≤ ℋ s (E) ≤ 1 , where ℋ s (E) denotes s -dimensional Hausdorff measure of E. For any a with 2 s 2 (n − 1) c 1 − c s ≤ a ≤ 1 , there exists a self-similar set E ∈ ℳ (n , c) such that ℋ s (E) = a. [ABSTRACT FROM AUTHOR]

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, -cubes, etc., as the subfamilies. [ABSTRACT FROM AUTHOR]

Given two metric spaces , it is well known that, where , denote, respectively, the Hausdorff and packing dimension of . In this paper, we show that, for any , there exist such that the following equalities hold simultaneously: This complete the related results of Wei et al. [C. Wei, S. Y. Wen and Z. X. Wen, Remarks on dimensions of Cartesian product sets, Fractals 24(3) (2016) 1650031]. [ABSTRACT FROM AUTHOR]

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast-slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast-slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov-Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov-Takens bifurcation point. [ABSTRACT FROM AUTHOR]

Electrical bursting is an activity which is universal in excitable cells such as neurons and various endocrine cells, and it encodes rich physiological information. As burst delay identifies that the signal integration has reached the threshold at which it can generate an action potential, the number of spikes in a burst may have essential physiological implications, and the transition of bursting in excitable cells is associated with the bifurcation phenomenon closely. In this paper, we focus on the transition of the spike count per burst of the pancreatic -cells within a mathematical model and bifurcation phenomenon in the Chay-Keizer model, which is utilized to simulate the pancreatic -cells. By the fast-slow dynamical bifurcation analysis and the bi-parameter bifurcation analysis, the local dynamics of the Chay-Keizer system around the Bogdanov-Takens bifurcation is illustrated. Then the variety of the number of spikes per burst is discussed by changing the settings of a single parameter and bi-parameter. Moreover, results on the number of spikes within a burst are summarized in ISIs (interspike intervals) sequence diagrams, maximum and minimum, and the number of spikes under bi-parameter value changes. [ABSTRACT FROM AUTHOR]