线性代数是蓝色的——大学非数学专业《线性代数》的课程设计

线性代数不仅是处理多元问题的有力工具,而且具有强烈的思辨性.大学本科教育应该是"泛专业的高等素质教育",作为公共基础课之一的《线性代数》,也应着眼于学生综合科学素质的培养,注意对学生进行思维训练.本文提出线性代数的教学理念:"提出处理多元问题的新要求,沿着多元整合的集成化思路推进,逐步把学生引上线性变换和线性空间的思维平台."为此,本文进行了"五模块、两阶段、三层次"的课程设计,希望对讲授该课程的教师有所裨益.     

A New Loop Algebra and Its Corresponding Multi-component Integrable Hierarchy

A type of new loop algebra $\tilde{G}_M$ is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy.     

New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy

A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.     

Matrix Lie Algebras and Integrable Couplings

Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.     

Comparative Effect of UV,UV/H2O2 and UV/H2O2/Fe on Terbuthylazine Degradation in Natural and Ultrapure Water

Different advanced oxidation processes (AOPs) (ultraviolet radiation, hydrogen peroxide photolysis and photo-Fenton) were applied to test the degradation of terbuthylazine in three types of water: (a) ultrapure water, (b) surface water from the Gaditana area (Los Hurones reservoir, Cádiz, Spain) and (c) groundwater from the Tempul spring in Jerez de la Frontera (Cádiz, Spain). The experiments were carried out on a laboratory scale, using two different types of reactors, batch and semi-continuous. In batch reactors, the most efficient process for the experiments carried out with both ultrapure water and underground groundwater was ultraviolet radiation, whereas for surface water from the Gaditana area, the process that obtained the best results was the photolysis of hydrogen peroxide with 2.5 mg L⁻¹ of H₂O₂. In semi-continuous reactors, the most efficient process was the photolysis of hydrogen peroxide with 2.5 mg L⁻¹ of H₂O₂ for all the matrices studied. In both types of reactors, terbuthylazine degradation percentages higher than 90% were achieved; the main difference was in the reaction time, which varied from minutes in the batch reactor to seconds in the semi-continuous reactor. In all the applied AOPs, N-terbutyl-6-hydroxy-N′ethyl-1,3,5-triazine-2,4-diamine (TBA-212) was generated as a reaction intermediate.     

Removal of Carbamazepine in Aqueous Solution by CoS2/Fe2+/PMS Process

Carbamazepine (CBZ), as a typical pharmaceutical and personal care product (PPCP), cannot be efficiently removed by the conventional drinking water and wastewater treatment process. In this work, the CoS₂/Fe²⁺/PMS process was applied for efficient elimination of CBZ. The CBZ removal efficiency of CoS₂/Fe²⁺/PMS was 2.5 times and 23 times higher than that of CoS₂/PMS and Fe²⁺/PMS, respectively. The intensity of DMPO-HO• and DMPO-SO4•− followed the order of Fe²⁺/PMS < CoS₂/PMS < CoS₂/Fe²⁺/PMS, also suggesting the CoS₂/Fe²⁺/PMS process has the highest oxidation activity. The effects of reaction conditions (e.g., CoS₂ dosage, Fe²⁺ concentration, PMS concentration, initial CBZ concentration, pH, temperature) and water quality parameters (e.g., SO42−, NO3−, H2PO4−, Cl−, NH4+, humic acid) on the degradation of CBZ were also studied. Response surface methodology analysis was carried out to obtain the best conditions for the removal of CBZ, which are: Fe²⁺ = 70 µmol/L, PMS = 240 µmol/L, CoS₂ = 0.59 g/L. The sustainability test demonstrated that the repeated use of CoS₂ for 8 successive cycles resulted in little function decrease (<10%). These findings suggest that CoS₂/Fe²⁺/PMS may be a promising method for advanced treatment of tailwater from sewage treatment plant.     

铁基金属-有机骨架及其复合物高级氧化降解水中新兴有机污染物

新兴有机污染物(Emerging organic contaminants,EOCs)是对人体健康及生态环境具有潜在或实质威胁的新型化学污染物.由于其被频繁使用且能在水生生态系统中持久性存在而对水生生物健康和安全造成严重威胁,故引起大众越来越多的关注.以活性污泥法为代表的传统水处理工艺通常不足以消除这些新兴有机污染物....     

Periodicities in Cluster Algebras and Cluster Automorphism Groups

We study the relations between two groups related to cluster automorphism groups which are defined by Assem,Schiffler and Shamchenko.We establish the relation-ships among (strict) direct cluster automorphism groups and those groups consisting of periodicities of labeled seeds and exchange matrices,respectively,in the language of short exact sequences.As an application,we characterize automorphism-finite cluster algebras in the cases of bipartite seeds or finite mutation type.Finally,we study the relation between the group Aut(A) for a cluster algebra A and the group AutMn(S) for a mutation group Mn and a labeled mutation class S,and we give a negative answer via counter-examples to King and Pressland's problem.     

Superposition rules, lie theorem, and partial differential equations

A rigorous geometric proof of the Lie theorem on nonlinear superposition rules for solutions of nonautonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of the Lie theorem for the case of systems of partial differential equations.     

Classification of extensions of torus algebra II

We use extension theory and algebraic methods to give a complete characterization of extensions of torus algebra by stable Cuntz algebras,and prove certain classification theorems of these extension algebras.