Facile preparation of a polydopamine‐based monolith for multiple monolithic fiber solid‐phase microextraction of triazine herbicides in environmental water samples

A new multiple monolithic fiber solid‐phase microextraction using a polydopamine‐based monolith as the extraction medium is proposed. The monolith was synthesized by facile in situ copolymerization of N‐methacryldopamine and dual cross‐linkers (divinylbenzene/ethylenedimethacrylate) in the presence of N ,N‐dimethylformamide. The effect of the contents of N‐methacryldopamine and porogen in the polymerization mixture on the extraction performance was investigated thoroughly. A series of characterization studies was performed to validate the structure and properties of the monolith. The prepared multiple monolithic fibers were used for the extraction of triazine herbicides in environmental water samples. After the optimization of the extraction parameters, a convenient, sensitive, cost‐effective, and environmentally friendly method for the determination of trace triazine herbicides in water samples was developed by coupling multiple monolithic fibers solid‐phase microextraction with high‐performance liquid chromatography and diode array detection. The results indicated that the limits of detection and quantification for the target compounds were 0.031–0.14 and 0.10–0.45 μg/L, respectively. Good precision and reproducibility were obtained with the relative standard deviations below 10%. The developed method was applied to the analysis of the triazine herbicides in different water samples (lake, river, and farmland waters). The recoveries of the method were in the range between 79.6 and 117%.     

The global weak solution for the shallow water wave model of moderate amplitude

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived.     

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.     

Efficient remainder rule

Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the given conditions, one may need to understand the concept of modulo arithmetic in number theory. The Chinese Remainder Theorem is a known method to solve these types of problems using modulo arithmetic. In this paper, an efficient remainder rule has been proposed based on basic mathematical concepts. These core concepts are as follows: basic remainder rules of divisions, linear equation in slope intercept form, arithmetic progression and the use of a graphing calculator. These are easily understood by students who have taken prealgebra or intermediate algebra.     

Rethinking pedagogy for second-order differential equations: a simplified approach to understanding well-posed problems

Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching ‘well posedness’ of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a first-order system of equations. We show that this excursion is unnecessary and present a direct approach regarding second- and higher-order problems that does not require an understanding of systems.     

Applications of ‐invariant distributions

In this sequel to the article 17 , a criterion for the regularity of fundamental solutions of differential operators with positive symbol is proved in analogy to Hörmander's criteria, see 9 , 12 . It implies that the temperate fundamental solution of the operator is not regular. Furthermore, a formula for the convolution of two ‐invariant distributions is presented, and, finally, L. Schwartz' question on the surjectivity of linear partial differential operators with constant coefficients on the space is completely answered in the case of ‐invariant differential operators.     

On the existence of positive least energy solutions for a coupled Schrödinger system with critical exponent

In this paper, we consider the following coupled Schrödinger system with critical exponent: where is a smooth bounded domain, λ > 0,μ≥0, and . Under certain conditions on λ and μ, we show that this problem has at least one positive least energy solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

This paper is concerned with the following systems of Kirchhoff‐type equations: Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.