Existence and concentration of positive solutions for coupled Schrödinger equations

In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity.     

Chemometrics‐assisted chromatographic fingerprinting: An illicit methamphetamine case study

The volatile chemical constituents in complex mixtures can be analyzed using gas chromatography with mass spectrometry. This analysis allows the tentative identification of diverse impurities of an illicit methamphetamine sample. The acquired two‐dimensional data of liquid–liquid extraction was resolved by multivariate curve resolution alternating curve resolution to elucidate the embedded peaks effectively. This is the first report on the application of a curve resolution approach for chromatogram fingerprinting to identify particularly the embedded impurities of a drug of abuse. Indeed, the strong and broad peak of methamphetamine makes identifying the underlying peaks problematic and even impossible. Mathematical separation instead of conventional chromatographic approaches was performed in a way that trace components embedded in methamphetamine peak were successfully resolved. Comprehensive analysis of the chromatogram, using multivariate curve resolution, resulted in elution profiles and mass spectra for each pure compound. Impurities such as benzaldehyde, benzyl alcohol, benzene, propenyl methyl ketone, benzyl methyl ketone, amphetamine, N‐benzyl‐2‐methylaziridine, phenethylamine, N ,N ,α‐trimethylamine, phenethylamine, N ,α,α‐trimethylmethamphetamine, N‐acetylmethamphetamine, N‐formylmethamphetamine, and other chemicals were identified. A route‐specific impurity, N‐benzyl‐2‐methylaziridine, indicating a synthesis route based on ephedrine/pseudoephedrine was identified. Moreover, this is the first report on the detection of impurities such as phenethylamine, N ,α,α‐trimethylamine (a structurally related impurity), and clonitazene (as an adulterant) in an illicit methamphetamine sample.     

Positive ground state of coupled systems of Schrödinger equations in involving critical exponential growth

In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations: where the nonlinearities f₁(x,s) and f₂(x,s) are superlinear at infinity and have exponential critical growth of the Trudinger‐Moser type. The potentials V₁(x) and V₂(x) are nonnegative and satisfy a condition involving the coupling term λ(x), namely, λ(x)²<δ²V₁(x)V₂(x) for some 0<δ<1. For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and L^q‐estimates, we get regularity and asymptotic behavior.     

全空间上带Hardy-Sobolev项的拟线性椭圆方程解的存在性

讨论了全空间上一类带Hardy-Sobolev项的拟线性椭圆问题,利用集中紧原理和适当的实验函数,得到并验证了(PS)_c条件,从而证明了该问题非平凡解的存在性.     

带有时滞的非定常企业投资系统的Lyapunov稳定性

讨论了企业投资系统的Lyapunov稳定性,得到了企业投资系统渐进稳定的充分条件和稳定的必要条件,并给出了企业投资系统的临界积累率的表达式,这个问题的研究对于促进我国经济高速、稳定持续增长具有重要的理论意义和现实指导价值.     

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.