We study the exponential stability of traveling wave solutions of nonlinear systems of integral differential equations arising from nonlinear, nonlocal, synaptically coupled, excitatory-inhibitory neuronal networks. W...We study the exponential stability of traveling wave solutions of nonlinear systems of integral differential equations arising from nonlinear, nonlocal, synaptically coupled, excitatory-inhibitory neuronal networks. We have proved that exponential stability of traveling waves is equivalent to linear stability. Moreover, if the real parts of nonzero spectrum of an associated linear differential operator have a uniform negative upper bound, namely, max{Reλ: λ∈σ(L),λ≠ 0}≤-D, for some positive constant D, and λ = 0 is an algebraically simple eigenvalue of L, then the linear stability follows, where L is the linear differential operator obtained by linearizing the nonlinear system about its traveling wave and σ(L) denotes the spectrum of L. The main aim of this paper is to construct complex analytic functions (also called eigenvalue or Evans functions) for exploring eigenvalues of linear differential operators to study the exponential stability of traveling waves. The zeros of the eigenvalue functions coincide with the eigenvalues of L.展开更多
Surface-enhanced Raman spectroscopy(SERS) is an intense ongoing hot topic because it is an attractive tool for sensing or detecting molecules in trace amounts. Despite its high specificity and sensitivity, the SERS ...Surface-enhanced Raman spectroscopy(SERS) is an intense ongoing hot topic because it is an attractive tool for sensing or detecting molecules in trace amounts. Despite its high specificity and sensitivity, the SERS technique has not been established as a routine analytic method most likely due to the low reproducibility of the SERS signal. This review considers the influence factors to produce the poor reproducibility during the SERS measurement. This review starts with the discussion of calculation of surface-enhanced Raman intensity in order to explain the reason why it is so difficult to achieve a high reproducibility of SERS measurement from the origin of enhancement mechanism. Then we focus on the fabrication of SERS substrates generally including two types:① single particles and ② arrays on substrate that are directly used to detect molecules or other components.In addition, we discuss the molecule factors and optical system for the reproducibility for sample-to-sample or spot-to-spot on a substrate. In the final part of this review, some effects resulting in the irreproducibility of Raman bands' position from recent literatures are discussed.展开更多
文摘We study the exponential stability of traveling wave solutions of nonlinear systems of integral differential equations arising from nonlinear, nonlocal, synaptically coupled, excitatory-inhibitory neuronal networks. We have proved that exponential stability of traveling waves is equivalent to linear stability. Moreover, if the real parts of nonzero spectrum of an associated linear differential operator have a uniform negative upper bound, namely, max{Reλ: λ∈σ(L),λ≠ 0}≤-D, for some positive constant D, and λ = 0 is an algebraically simple eigenvalue of L, then the linear stability follows, where L is the linear differential operator obtained by linearizing the nonlinear system about its traveling wave and σ(L) denotes the spectrum of L. The main aim of this paper is to construct complex analytic functions (also called eigenvalue or Evans functions) for exploring eigenvalues of linear differential operators to study the exponential stability of traveling waves. The zeros of the eigenvalue functions coincide with the eigenvalues of L.
基金the National Natural Science Foundation of China(No.21375087)the Natural Science Foundation of Shanghai(No.13ZR1422100)
文摘Surface-enhanced Raman spectroscopy(SERS) is an intense ongoing hot topic because it is an attractive tool for sensing or detecting molecules in trace amounts. Despite its high specificity and sensitivity, the SERS technique has not been established as a routine analytic method most likely due to the low reproducibility of the SERS signal. This review considers the influence factors to produce the poor reproducibility during the SERS measurement. This review starts with the discussion of calculation of surface-enhanced Raman intensity in order to explain the reason why it is so difficult to achieve a high reproducibility of SERS measurement from the origin of enhancement mechanism. Then we focus on the fabrication of SERS substrates generally including two types:① single particles and ② arrays on substrate that are directly used to detect molecules or other components.In addition, we discuss the molecule factors and optical system for the reproducibility for sample-to-sample or spot-to-spot on a substrate. In the final part of this review, some effects resulting in the irreproducibility of Raman bands' position from recent literatures are discussed.