The Lattice of（∈,∈∨qk）-fuzzy Filters in a Given R0-algebra

The purpose of this paper is to further study the(∈,∈∨q_k)-fuzzy filter theory in R_0-algebras. Some new properties of(∈, ∈∨ q_k)-fuzzy filters are given. Representation theorem of(∈,∈∨q_k)-fuzzy filter which is generated by a fuzzy set is established. It is proved that the set consisting of all(∈, ∈∨q_k)-fuzzy filters on a given R_0-algebra, under the partial order, forms a complete distributive lattice.

Generalized （∈, ∈ ∨ q）-fuzzy Ideals in BCK-algebras

In this paper, as a generalization of the notion of(∈, ∈∨ q)-fuzzy ideals, we introduced the notion of(∈, ∈∨ qδ)-fuzzy ideals and investigated their properties in BCKalgebras. Several equivalent characterizations of(∈, ∈∨ qδ)-fuzzy ideals are obtained and relations between(∈, ∈∨ qδ)-fuzzy ideals and ideals are discussed in BCK-algebras.

N(2，2，0)代数的(∈,∈∨q_（λ，μ）)-模糊理想

对N(2,2,0)代数(∈,∈∨q_(λ,μ))-模糊理想进行了详细的研究。给出了N(2,2,0)代数广义模糊理想和点态化(∈,∈∨q_(λ,μ))-模糊理想的概念,讨论了两者之间的等价关系。给出了(∈,∈∨q_(λ,μ))-模糊理想的一些等价刻画,并研究了其相关性质。讨论了(∈,∈∨q_(λ,μ))-模糊理想的同态像与同态原像的关系。最后研究了(∈,∈∨q_(λ,μ))-模糊理想的直积。

格蕴涵代数的（∈,∈∨q~[k]）-fuzzy LI-理想

运用fuzzy集的思想和方法研究格蕴涵代数的LI-理想.引入了(∈,∈∨q[k])-fuzzy LI-理想和(∈,∈∨q[k])-fuzzy格理想的概念,讨论了它们的性质及相互关系.获得了(∈,∈∨q[k])-fuzzy LI-理想的若干等价刻画;证明了在格H蕴涵代数中(∈,∈∨q[k])-fuzzy LI-理想和(∈,∈∨q[k])-fuzzy格理想相互等价的结论.

A New Rational Algebraic Approach to Find Exact Analytical Solutions to a （2＋1）-Dimensional System

In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions. The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions, and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on （2＋1）-dimensional dispersive long-wave equation.

A Generalized Variable-Coefficient Algebraic Method Exactly Solving （3＋1）-Dimensional Kadomtsev-Petviashvilli Equation

A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.

Comment on “Computer Algebra and Solutions to the Karamoto-Sivashinsky Equation” [Commun. Theor. Phys. （Beijing, China） 43 （2005） 39]

In a recent article [Commun. Theor. Phys. (Beijing, China) 43 (2005) 39], Xie et al. improved the extended tanh function method by introducing a generalized Riccati equation and its new solutions. Then they choose the Karamoto-Sivashinsky (KS) equation to illustrate their approach and obtain many exact solutions of the KS equation.So they claim that, by using their method, one not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear evolution equations. In this comment, we will show that the claim is incorrect.

BCS Ground State and XXZ Antiferromagnetic Model as SU（2）,SU（1,1） Coherent States：AN Algebraic Diagonalization Method

An algebraic diagonalization method is proposed. As two examples, the Hamiltonians of BCS ground stateunder mean-field approximation and XXZ antiferromagnetic model in linear spin-wave frame have been diagonalized byusing SU(2), SU(1,1) Lie algebraic method, respectively. Meanwhile, the eigenstates of the above two models are revealedto be SU(2), SU(1,1) coherent states, respectively. The relation between the usual Bogoliubov Valatin transformationand the algebraic method in a special case is also discussed.

Y（sl（2）） Algebra Application in Extended Hydrogen Atom and Monopole Models

We present the extended hydrogen atom and monopole-hydrogen atom theory through generalizing the usual hydrogen atom model and with a monopole model respectively, in which Y (sl(2) ) algebras are realized. We derive the Hamiltonians of the two models based on the Y(sl(2) ) and the generalized Pauli equation. The energy spectra of the systems are also given in terms of Yangian algebra and quantum mechanics.

Exact Solutions of Two Coupled Harmonic Oscillators Related to the Sp（4,R） Lie Algebra

Exact solutions of the eigenvalue problem of two coupled harmonic oscillators related to the Sp(4, R) Lie algebra are derived by using an algebraic method. It is found that the energy spectrum of the system is determined by one-boson excitation energies built on a vector coherent state of Sp(4, R) U(2).

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.

泛逻辑学中UB代数系统的(∈,∈∨q)-fuzzy滤子

何华灿教授给出了理想状态下的泛逻辑学的形式演绎系统β,并证明了该系统的可靠性。并且提出了理想状态下的泛逻辑学对应的代数系统-UB代数,并讨论了它们的性质。在以上这些结果的基础上,引入UB代数的(∈,∈∨q)-fuzzy滤子和(∈,∈∨q)-fuzzy关联滤子的概念,获得了它们的若干等价刻画,证明了(∈,∈∨q)-fuzzy关联滤子的扩张定理。

N(2,2,0)代数的(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊理想

将模糊点和模糊集间的"∈(属于)"和"q_(λ,μ)(广义重于)"关系推广为"∈~δ(Ω-属于)"和"q_(λ,μ)~δ(Ω-重于)"关系,提出了(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊代数。将(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊代数与N(2,2,0)代数相结合,给出了点态化(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊理想和Ω_(λ,μ)-模糊理想的概念,证明了两者之间的等价关系,研究了它们的一些基本性质;最后提出了点态化(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊子代数和Ω_(λ,μ)-模糊子代数的定义,研究了(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊理想和(∈~δ,∈~δ∨q~δ_(λ,μ))-模糊子代数的相互关系。

BCK/BCI-代数的(∈,∈∨q)-模糊软理想

在BCK/BCI-代数中引入(∈,∈∨q)-模糊软理想的概念,研究并获得了若干代数性质.定义了BCK/BCI-代数的模糊软同态和模糊软同构的概念,证明了(∈,∈∨q)-模糊软理想的模糊软同构(态)像(原像)仍为(∈,∈∨q)-模糊软理想这一结论.

BCK-代数的广义(∈,∈∨q)-模糊蕴涵理想

为推广BCK-代数的(∈,∈∨q)-模糊蕴涵理想,引入了(∈,∈∨q_k)-模糊蕴涵理想的概念,研究了相关性质;指出了模糊蕴涵理想和(∈,∈∨q)-模糊蕴涵理想都是(∈,∈∨q_k)-模糊蕴涵理想,反之不然;给出了BCK-代数的模糊子集成为(∈,∈∨q_k)-模糊蕴涵理想的条件;证明了(∈,∈∨q_k)-模糊蕴涵理想都是(∈,∈∨q_k)-模糊理想,用实例说明其逆命题不真;给出了(∈,∈∨q_k)-模糊蕴涵理想的若干等价刻画,讨论了BCK-代数的基于蕴涵的模糊蕴涵理想。

布尔代数的(∈,∈∨q_(λ,μ))-模糊理想

文中给出了布尔代数的(∈,∈∨q(λ,μ))-模糊理想及广义模糊理想的新概念,得到了(∈,∈∨q(λ,μ))-模糊理想的一些等价刻画及基本性质。

格蕴涵代数的(∈,∈∨q_(λ,μ))-模糊滤子

给出了点态化(∈,∈νq_(λ,μ))-模糊滤子和广义模糊滤子的概念,讨论了两者之间的等价关系;得到了(∈,∈νq_(λ,μ))-模糊滤子的若干等价刻画,并研究了其相关性质;讨论了(∈,∈νq_(λ,μ))-模糊滤子的同态像与同态原像的基本性质;最后研究了(∈,∈vq_(λ,μ))-模糊滤子的直积。

格蕴涵代数的(∈，∈∨q(λ，μ))-模糊素滤子

将((∈,∈∨q_((λ,μ))))-模糊代数应用于格蕴涵代数,提出了点态化((∈,∈∨q_((λ,μ))))-模糊素滤子和(λμ)-模糊素滤子的概念,并从以下几个方面对((∈,∈∨q_((λ,μ))))-模糊素滤子进行了详细的研究:讨论了((∈,∈∨q_((λ,μ))))-模糊素滤子和(λμ)-模糊素滤子的等价关系;研究了((∈,∈∨q_((λ,μ))))-模糊素滤子的相关性质;得到了特定条件下((∈,∈∨q_((λ,μ))))-模糊素滤子的若干等价刻画,建立了((∈,∈∨q_((λ,μ))))-模糊素滤子的扩张定理;探讨了((∈,∈∨q_((λ,μ))))-模糊素滤子的同态像与同态原像。

N(2,2,0)代数的(∈,∈∨q_(λ,μ))-模糊结合理想

引入N(2,2,0)代数结合理想的概念。利用(∈,∈∨q(λ,μ))-模糊代数,给出了N(2,2,0)代数广义(λ,μ)-模糊结合理想和点态化模糊结合理想的概念,讨论了两者之间的等价关系;给出了(∈,∈∨q(λ,μ))-模糊结合理想的一些等价刻画,研究了其相关性质;讨论了(∈,∈∨q(λ,μ))-模糊结合理想的同态像与同态原像的关系;还研究了(∈,∈∨q(λ,μ))-模糊结合理想的直积。

Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L∞(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L2(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .