A transformation of Markov jump processes and applications in genetic study
Xian Chen Zhi-Ming Ma
Discrete & Continuous Dynamical Systems - A 2014, 34(12): 5061-5084 doi: 10.3934/dcds.2014.34.5061
In this paper we provide a verifiable necessary and sufficient condition for a regular q-process to be again a q-process under a transformation of state space. The result as well as some other results on continuous states Markov jump processes is employed to investigate jump processes arising from the study in modeling genetic coalescent with recombination.
keywords: Markov jump process $\psi$-transform q-consistency q-process coalescent with recombination.
Chaos control in a pendulum system with excitations
Xianwei Chen Zhujun Jing Xiangling Fu
Discrete & Continuous Dynamical Systems - B 2015, 20(2): 373-383 doi: 10.3934/dcdsb.2015.20.373
This paper is devoted to investigate the problem of controlling chaos for a pendulum system with parametric and external excitations. By using Melnikov methods, the criteria of controlling chaos are obtained. Numerical simulations are given to illustrate the effect of the chaos control for this system, suppression of homoclinic chaos is more effective than suppression of heteroclinic chaos, and the chaotic motions can be suppressed to period-motions by adjusting parameters of chaos-suppressing excitation. Finally, we calculate the maximum Lyapunov exponents (LE) in parameter-plane and observe the frequency of chaos-suppressing excitation also play an important role in the process of chaos control.
keywords: bifurcation Melnikov methods. Pendulum equation chaos control
A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems
Xianjin Chen Jianxin Zhou
Conference Publications 2009, 2009(Special): 151-160 doi: 10.3934/proc.2009.2009.151
The aim of this paper is to numerically investigate multiple solutions of semilinear elliptic systems with zero Dirichlet boundary conditions

-$\Delta u=F_u(x;u,v),$   $x\in\Omega,
-$\Delta v=F_v(x;u,v),$   $x\in\Omega,

where $\Omega \subset \mathbb{R}^{N}$ ($N\ge 1$) is a bounded domain. A strongly coupled case where the potential $F(x;u,v)$ takes the form $|u|^{\alpha_1}|v|^{\alpha_2}$ with $\alpha_1, \alpha_2>1$ is specially studied. By using a local min-orthogonal method, both positive and sign-changing solutions are found and displayed.

keywords: min-orthogonal method Cooperative systems multiple solutions
How convolutional neural networks see the world --- A survey of convolutional neural network visualization methods
Zhuwei Qin Fuxun Yu Chenchen Liu Xiang Chen
Mathematical Foundations of Computing 2018, 1(2): 149-180 doi: 10.3934/mfc.2018008

Nowadays, the Convolutional Neural Networks (CNNs) have achieved impressive performance on many computer vision related tasks, such as object detection, image recognition, image retrieval, etc. These achievements benefit from the CNNs' outstanding capability to learn the input features with deep layers of neuron structures and iterative training process. However, these learned features are hard to identify and interpret from a human vision perspective, causing a lack of understanding of the CNNs' internal working mechanism. To improve the CNN interpretability, the CNN visualization is well utilized as a qualitative analysis method, which translates the internal features into visually perceptible patterns. And many CNN visualization works have been proposed in the literature to interpret the CNN in perspectives of network structure, operation, and semantic concept.

In this paper, we expect to provide a comprehensive survey of several representative CNN visualization methods, including Activation Maximization, Network Inversion, Deconvolutional Neural Networks (DeconvNet), and Network Dissection based visualization. These methods are presented in terms of motivations, algorithms, and experiment results. Based on these visualization methods, we also discuss their practical applications to demonstrate the significance of the CNN interpretability in areas of network design, optimization, security enhancement, etc.

keywords: Deep learning convolutional neural network CNN feature CNN visualization network interpretability

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